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LOCAL ORGANIZER
Moritz Kaßmann
SCIENTIFIC COMMITTEE
Rodrigo BañuelosCyril Imbert
Grzegorz KarchTakashi Kumagai
József LörincziRené Schilling
NONLOCALAnalysis, Probability,
Geometry and Applicati ons
OPERATORSJuly, 9th - 14th, 2012
Nonlocal OperatorsAnalysis, Probability, Geometry and Applications
Center for Interdisciplinary Research (ZiF), Bielefeld
July 9–14, 2012
Supported by the Collaborative Research Centre SFB 701
Local Organizer:
Moritz Kaßmann
Scientific Committee:
Rodrigo BañuelosCyril ImbertGrzegorz KarchTakashi KumagaiJózsef LörincziRené Schilling
Cover design: Stefan Adamick, ZiFLATEX typesetting: Matthieu Felsinger, SFB 701
Contents
1 Summer School on Nonlocal Operators 7
2 Program 9
3 Abstracts of the talks 19
4 Poster Presentations 59
5 Practical Information 61
6 List of Participants 67
Conference Poster 71
Summer Schoolon Nonlocal OperatorsJuly 4–6 2012, BielefeldPreceding this conference the Department of Mathematics and the CollaborativeResearch Centre SFB 701 hosted a summer school on nonlocal operators.
Zhen-Qing Chen
University of Washingtonon
Potential theory for jumpprocesses
Luis Silvestre
University of Chicagoon
Regularity theory for nonlocaloperators
The purpose of this intense 3-day school on nonlocal operators was to teach recentresults and techniques in the area of jump processes and integro-differential operatorsof fractional order.
Six lectures given by Prof. Z.-Q. Chen addressed methods from probability and stochas-tic analysis in the study of nonlocal operators. The focus was on the potential theory ofLévy-type processes and more general jump processes generated by nonlocal Dirichletforms, including fine estimates on heat kernels.
Six lectures given by Prof. L. Silvestre concentrated on methods from analysis andpartial differential operators. Here the focus was on regularity estimates for solutionsto fully nonlinear equations driven by nonlocal operators.
Further information can be found here:http://www.math.uni-bielefeld.de/nloc-school
Bielefeld, 2012 7
Program
Monday, July 9
08:30 – 09:00 Registration
09:00 – 10:00 Opening
10:00 – 10:45 Zhen-Qing ChenPerturbation by non-local Operators
10:45 – 11:20 Ante MimicaRegularity estimates of harmonic functions for jumpprocesses
11:20 – 11:50 Coffee Break
11:50 – 12:25 Christine GeorgelinOn Neumann and oblique derivatives boundary conditionsfor non-local equations
12:25 – 13:00 Mateusz KwaśnickiEstimates of harmonic functions for non-local operators
13:00 – 14:30 Lunch
14:30 – 15:15 Peter ImkellerConstruction of stochastic processes with singular jumpcharacteristics as solutions of martingale problems
15:15 – 15:50 Ilya PavlyukevichSmall noise asymptotics of integrated Ornstein-UhlenbeckProcesses driven by α-stable Lévy processes
15:50 – 16:20 Coffee Break
16:20 – 16:55 Rupert FrankUniqueness and nondegeneracy of ground states fornon-local equations in 1D
16:55 – 17:40 Yuri KondratievNon-local evolutions as kinetic equations for Markovdynamics
17:45 – 18:45 Poster Session
19:00 – Dinner at ZiF
Bielefeld, 2012 9
Tuesday, July 10
09:00 – 09:45 Mark MeerschaertThe Inverse Stable Subordinator
09:45 – 10:20 Hans-Peter SchefflerFractional governing equations for coupled continuous timerandom walks
10:20 – 10:50 Coffee Break
10:50 – 11:35 Renming SongStability of Dirichlet heat kernel estimates of non-localoperators under perturbations
11:35 – 12:10 Bartłomiej DydaComparability and regularity estimates for symmetricnon-local Dirichlet forms
12:10 – 12:45 Matthieu FelsingerLocal regularity for parabolic nonlocal operators
12:45 – 14:15 Lunch
14:15 – 15:00 María del Mar GonzálezFractional order operators in conformal geometry
15:00 – 15:35 Giampiero PalatucciAsymptotics of the s-perimeter as s 0
15:35 – 16:05 Coffee Break
16:05 – 16:50 Francesca Da LioAnalysis of fractional harmonic maps
16:50 – 17:25 Armin SchikorraKnot-energies and fractional harmonic maps
17:25 – 18:10 Julio RossiA Monge-Kantorovich mass transport problem for a discretedistance
10 Nonlocal Operators
Wednesday, July 11
09:00 – 09:45 David ApplebaumMartingale transforms and Lévy processes on Lie groups
09:45 – 10:20 Victoria KnopovaParametrix construction for the transition probabilitydensity of some Lévy-type processes
10:20 – 10:50 Coffee Break
10:50 – 11:35 Niels JacobNon-locality, non-isotropy, and geometry
11:35 – 12:20 Yannick SireSmall energy regularity for a fractional Ginzburg-Landausystem
12:30 – 13:45 Lunch
13:45 – Excursion
Bielefeld, 2012 11
Thursday, July 12
09:00 – 09:45 Michael RöcknerSub- and supercritical stochastic quasi-geostrophic equation
09:45 – 10:20 Naotaka KajinoNon-regularly varying and non-periodic oscillation of theon-diagonal heat kernels on self-similar fractals
10:20 – 10:50 Coffee Break
10:50 – 11:35 Alexander Grigor’yanEstimates of heat kernels of Dirichlet forms
11:35 – 12:10 Tomasz GrzywnyEstimates of the Poisson kernel of a half-line forsubordinate Brownian motions
12:10 – 12:45 Nathaël AlibaudContinuous dependence estimates for fractal/fractionaldegenerate parabolic equations
12:45 – 14:15 Lunch
14:15 – 15:00 Richard LehoucqPeridynamic non-local mechanics
15:00 – 15:35 Etienne EmmrichAnalysis of the peridynamic model in non-local elasticity
15:35 – 16:05 Coffee Break
16:05 – 16:50 Ralf MetzlerAgeing and ergodicity breaking in anomalous diffusion
16:50 – 17:25 Björn BöttcherConstructive approximation of Feller processes withunbounded coefficients
17:25 – 18:10 Alexander BendikovMarkov semigroups on totally disconnected sets
12 Nonlocal Operators
Friday, July 13
09:00 – 09:45 Panki KimAn Lp-theory of stochastic PDEs with random fractionalLaplacian operator
09:45 – 10:20 Mohammud FoondunStochastic heat equation with spatially coloured randomforcing
10:20 – 10:50 Coffee Break
10:50 – 11:35 Alexis VasseurIntegral variational problems
11:35 – 12:10 Enrico ValdinociNon-local non-linear problems
12:10 – 12:45 Russell SchwabOn Alexandrov-Bakelman-Pucci type estimates forintegro-differential equations
12:45 – 14:15 Lunch
14:15 – 15:00 Krzysztof Bogdan3G, 4G and perturbations
15:00 – 15:35 Yuishi ShiozawaConservation property of symmetric jump-diffusionprocesses
15:35 – 16:05 Coffee Break
16:05 – 16:50 Enrico PriolaUniqueness for singular SDEs driven by stable processes
16:50 – 17:25 Suleyman UlusoyNon-local conservation laws and related Keller-Segel typesystems
17:25 – 18:10 Piotr BilerBlowup of solutions to generalized Keller-Segel model
Bielefeld, 2012 13
Saturday, July 14
09:00 – 09:45 Igor SokolovFractional subdiffusion and subdiffusion-reaction equations:Physical Motivation and Properties
09:45 – 10:20 Enrico ScalasCharacterization of the fractional Poisson process
10:20 – 10:55 Piotr GarbaczewskiLévy flights, Lévy semigroups and fractional quantummechanics
10:55 – 11:25 Coffee Break
11:25 – 12:00 Paweł SztonykUpper estimates of transition densities for stable-dominatedsemigroups
12:00 – 12:45 Zoran VondračekPotential theory of subordinate Brownian motions withGaussian components
12:45 – 14:15 Lunch
14 Nonlocal Operators
Bielefeld, 2012 15
8:30
– 9
:00
Reg
istr
atio
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9:00
– 1
0:00
Ope
ning
9:00
– 9
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45 –
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209:
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10:
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pro
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Re
gula
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est
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of h
arm
on
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nct
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for
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p p
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tab
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of
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t h
eat
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est
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Jaco
bN
on-
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non
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org
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On
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und
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all
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Est
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arm
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or n
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lsin
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r p
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Imke
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Co
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toch
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roce
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aps
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Th
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day
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day
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ay
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Rö
ckne
rS
ub-
an
d su
per
criti
cal s
toch
astic
qu
asi-
geo
stro
ph
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qu
atio
n
Kim
An
L^p
the
ory
of
stoc
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tic P
DE
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an
dom
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l La
plac
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ope
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kolo
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ract
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bdiff
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bdi
ffu
sio
n-r
ea
ctio
n
equ
atio
ns:
Ph
ysic
al M
otiv
atio
n a
nd
Pro
per
ties
Ka
jino
No
n-re
gula
rly v
ary
ing
and
no
n-p
erio
dic
osc
illa
tion
o
f the
on
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gon
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ea
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elf-
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ond
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chas
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eat
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pat
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forc
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Sca
las
Ch
ara
cter
izat
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of t
he f
ract
iona
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isso
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oce
ss
Ga
rbac
zew
ski
Le
vy fl
igh
ts,
Levy
se
mig
rou
ps
an
d fr
act
iona
l qu
ant
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me
cha
nic
s
Grig
or'y
an
Est
ima
tes
of h
eat
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els
of
Diri
chle
t fo
rms
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sse
urIn
teg
ral v
aria
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l pro
ble
ms
Grz
ywny
Est
ima
tes
of t
he P
ois
son
ke
rne
l of
a h
alf-
line
for
subo
rdin
ate
Bro
wn
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s
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ldin
oci
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n-lo
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on
-line
ar
pro
ble
ms
Szt
on
ykU
ppe
r e
stim
ate
s of
tra
nsi
tion
de
nsi
ties
for
stab
le-d
omin
ate
d s
em
igro
ups
Alib
au
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on
tinu
ous
dep
end
enc
e e
stim
ate
s fo
r fr
act
al/f
ract
ion
al d
ege
ne
rate
pa
rab
olic
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atio
ns
Sch
wa
bO
n A
lexa
ndro
v-B
ake
lma
n-P
ucci
type
est
ima
tes
for
inte
gro
-diff
ere
ntia
l equ
atio
ns
Von
dra
ček
Po
ten
tial t
heo
ry o
f su
bo
rdin
ate
Bro
wn
ian
m
otio
ns
with
Ga
uss
ian
co
mp
one
nts
Leh
ouc
qP
erid
yna
mic
no
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ech
an
ics
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gda
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4G
an
d p
ert
urb
atio
ns
Em
mric
hA
na
lysi
s o
f th
e pe
ridyn
am
ic m
ode
l in
non
-loca
l e
last
icity
Sh
ioza
wa
Co
nser
vatio
n p
rop
erty
of
sym
me
tric
jum
p-d
iffu
sio
n
pro
cess
es
Me
tzle
rA
gei
ng
an
d e
rgo
dici
ty b
reak
ing
in a
no
mal
ous
diff
usio
n
Prio
laU
niq
ue
ness
for
sin
gula
r S
DE
s d
rive
n b
y st
able
p
roce
sses
Bö
ttch
er
Co
nstr
uct
ive
ap
pro
xim
atio
n o
f Fe
ller
pro
cess
es
with
un
bou
nd
ed c
oeff
icie
nts
Ulu
soy
No
n-lo
cal c
on
serv
atio
n la
ws
an
d re
late
d K
elle
r-S
ege
l typ
e s
yste
ms
Be
ndik
ov
Mar
kov
sem
igro
ups
on
tota
lly d
isco
nne
cte
d se
ts
Bile
rB
low
up
of s
olut
ions
to g
en
era
lized
Kel
ler-
Se
gel
m
od
el
Abstracts of the talksNathaël Alibaud – Continuous dependence estimates for fractal/fractional
degenerate parabolic equations . . . . . . . . . . . . . . . . . . . . . . . . 21David Applebaum – Martingale transforms and Lévy processes on Lie groups 21Alexander Bendikov – Markov semigroups on totally disconnected sets . . . . 22Piotr Biler – Blowup of solutions to generalized Keller–Segel model . . . . . . 22Krzysztof Bogdan – 3G, 4G and perturbations . . . . . . . . . . . . . . . . . . 23Björn Böttcher – Constructive Approximation of Feller Processes with
unbounded coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Zhen-Qing Chen – Perturbation by Non-Local Operators . . . . . . . . . . . . 25Francesca Da Lio – Analysis of Fractional Harmonic Maps . . . . . . . . . . . 26Bartłomiej Dyda – Comparability and regularity estimates for symmetric
nonlocal Dirichlet forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Etienne Emmrich – Analysis of the peridynamic model in nonlocal elasticity . 28Matthieu Felsinger – Local Regularity for Parabolic Nonlocal Operators . . . . 29Mohammud Foondun – Stochastic heat equation with spatially coloured
random forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Rupert Frank – Uniqueness and nondegeneracy of ground states for non-local
equations in 1D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Piotr Garbaczewski – Lévy flights, Lévy semigroups and fractional quantum
mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Christine Georgelin – On Neumann and oblique derivatives boundary
conditions for non-local equations . . . . . . . . . . . . . . . . . . . . . . 31María del Mar González – Fractional order operators in conformal geometry . 32Alexander Grigor’yan – Estimates of heat kernel of Dirichlet forms . . . . . . 33Tomasz Grzywny – Estimates on the Poisson kernel of a half-line for
subordinate Brownian motions . . . . . . . . . . . . . . . . . . . . . . . . 34Peter Imkeller – Construction of stochastic processes with singular jump
characteristics as solutions of martingale problems . . . . . . . . . . . . 35Niels Jacob – Non-locality, Non-Isotropy, and Geometry . . . . . . . . . . . . 36Naotaka Kajino – Non-regularly varying and non-periodic oscillation of the
on-diagonal heat kernels on self-similar fractals . . . . . . . . . . . . . . 36Panki Kim – An Lp-theory of stochastic PDEs with random fractional
Laplacian operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37Victoria Knopova – Parametrix construction for the transition probability
density of some Lévy-type processes . . . . . . . . . . . . . . . . . . . . . 38Yuri Kondratiev – Non-local evolutions as kinetic equations for Markov
dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Mateusz Kwaśnicki – Estimates of harmonic functions for non-local operators 39Richard Lehoucq – Peridynamic nonlocal mechanics . . . . . . . . . . . . . . 40
Bielefeld, 2012 19
Mark Meerschaert – Solutions to a nonlocal fractional wave equation . . . . . 41Ralf Metzler – Ageing and Ergodicity Breaking in Anomalous Diffusion . . . . 42Ante Mimica – Regularity estimates of harmonic functions for jump processes 42Giampiero Palatucci – Asymptotics of the s-perimeter as s 0 . . . . . . . . 44Ilya Pavlyukevich – Small noise asymptotics of integrated Ornstein–Uhlenbeck
processes driven by α-stable Lévy processes . . . . . . . . . . . . . . . . . 46Enrico Priola – Uniqueness for singular SDEs driven by stable processes . . . 46Michael Röckner – Sub- and supercritical stochastic quasi-geostrophic equation 47Julio Rossi – A Monge-Kantorovich mass transport problem for a discrete
distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Enrico Scalas – Characterization of the fractional Poisson process . . . . . . . 48Hans-Peter Scheffler – Fractional governing equations for coupled continuous
time random walks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Armin Schikorra – Knot-energies and Fractional Harmonic Maps . . . . . . . 49Russell Schwab – On Aleksandrov-Bakelman-Pucci type estimates for integro
differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50Yuichi Shiozawa – Conservation property of symmetric jump-diffusion processes 50Yannick Sire – Small energy regularity for a fractional Ginzburg-Landau system 52Igor Sokolov – Fractional Subdiffusion and Subdiffusion-Reaction equations:
Physical Motivation and Properties . . . . . . . . . . . . . . . . . . . . . 52Renming Song – Stability of Dirichlet heat kernel estimates of non-local
operators under perturbations . . . . . . . . . . . . . . . . . . . . . . . . 53Paweł Sztonyk – Upper estimates of transition densities for stable-dominated
semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Suleyman Ulusoy – Non-local Conservation laws and related Keller-Segel type
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55Enrico Valdinoci – Nonlocal nonlinear problems . . . . . . . . . . . . . . . . . 56Alexis Vasseur – Integral variational problems . . . . . . . . . . . . . . . . . . 56Zoran Vondraček – Potential theory of subordinate Brownian motions with
Gaussian components . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
20 Nonlocal Operators
Nathaël Alibaud
Continuous dependence estimates for fractal/fractional degenerate parabolicequations
This talk will be concerned with the Cauchy problem
ut + div f(u) + (−4)α2 ϕ(u) = 0, u(0) = u0,
where α ∈ (0, 2) and ϕ is a nondecreasing nonlinearity. It will focus on continuousdependence estimates on the data; i.e. given another solution
vt + div g(v) + (−4)β2ψ(u) = 0, v(0) = v0,
we will see how u−v can be bounded by differences between (f, α, ϕ, u0) and (g, β, ψ, v0).For instance, if α = 2 and if u and v have the same data, excepted ψ 6= ϕ, B. Cockburnand G. Gripenberg have shown in 1999 that
‖u(·, t)− v(·, t)‖L1 = O(‖√ϕ′ −
√ψ′‖∞
). (3.1)
In the fractional case, we shall see that
‖u(·, t)− v(·, t)‖L1 =
O(‖(ϕ′) 1
α − (ψ′)1α ‖∞
), α > 1,
O (‖ϕ′ lnϕ′ − ψ′ lnψ′‖∞) , α = 1,O (‖ϕ′ − ψ′‖∞) , α < 1,
giving in particular a new proof of (3.1) as α→ 2. In the case where ψ = ϕ but β 6= α,we shall see that
‖u(·, t)− v(·, t)‖L1 = O(|α− β|).All these results are optimal.
Joint work with Simone Cifani and Espen R. Jakobsen (Norwegian University of Sci-ence and Technology, Trondheim, Norway).
David Applebaum
Martingale transforms and Lévy processes on Lie groups
We obtain martingale transforms that are built in a natural way from a generic Lévyprocess in a Lie group G. Using sharp inequalities due to Burkholder we then constructa class of linear operators that are bounded on Lp(G,m) (where m is a Haar measure)
Bielefeld, 2012 21
for all 1 < p < ∞. When the group is compact, we use Peter-Weyl theory to exhibitspecific examples of these operators as Fourier multipliers. These include second orderRiesz transforms, imaginary powers of the Laplacian and operators associated withsubordinated Brownian motion.Talk based on joint work with Rodrigo Bañuelos (Purdue).
The paper is available at: http://arxiv.org/abs/1206.1560
Alexander Bendikov
Markov semigroups on totally disconnected sets
This is joint work with A. Grigor’yan and Ch. Pittet. Let (X, d) be a metric space.We assume that it is totally disconnected and locally compact. Then there exists anultra-metric d′ which generates the same topology on X. Given a measure m on Xwe construct a symmetric Markov semigroup (Pt) on X. We give upper- and lower-bounds of its transition function and its Green function, give a criterion of transience,estimate moments of the corresponding Markov process and describe the spectrum ofits Markov generator. In a particular case when X is the field of p-adic numbers, ourconstruction recovers The Vladimirov operator. Even in this well-established settingsome of our results are new.
Piotr Biler
Blowup of solutions to generalized Keller–Segel model
The existence and nonexistence of global in time solutions is studied for a class ofequations generalizing the chemotaxis model of Keller and Segel. These equationsinvolve Lévy diffusion operators and general potential type nonlinear terms.
We will consider the following nonlinear nonlocal evolution equation
∂tu+ (−∆)α/2u+∇ · (uB(u)) = 0,
for (x, t) ∈ Rd × R+, where the anomalous diffusion is modeled by a fractional powerof the Laplacian, α ∈ (1, 2), and the linear (vector) operator B is defined as
B(u) = ∇((−∆)−β/2u)
for β ∈ (1, d], and d ≥ 2.
22 Nonlocal Operators
Krzysztof Bogdan
3G, 4G and perturbations
Schrödinger perturbations of transition densities p by functions q are widely studied,see, e.g., [3, 4]. Local integral smallness of q with respect to p plays a role in theseconsiderations. In a series of recent papers (see, e.g., [1]) we propose the followingassumption,
t∫s
∫X
ps,u(x, z)q(u, z)pu,t(z, y)dzdu ≤ [η +Q(s, t)]ps,t(x, y), (3.2)
where q ≥ 0, 0 ≤ η < 1 and 0 ≤ Q(s, u) +Q(u, t) ≤ Q(s, t) for s < u < t. Under (3.2),the perturbation, p, of p by q enjoys the following upper bound,
ps,t(x, y) ≤ ps,t(x, y)
(1
1− η
)1+Q(s,t)/η
.
The assumption (3.2) is quite friendly if p allows for the so-called 3P inequality:
ps,u(x, z) ∧ pu,t(z, y) ≤ Cps,t(x, y), s < u < t.
The Gaussian transition density, however, fails to satisfy such 3P.
I will report a joint work in progress with Karol Szczypkowski on a new bound inspiredby [2] and called 4P, which applies to the Gaussian kernel.
[1] K. Bogdan, T. Jakubowski & S. Sydor: Estimates of perturbation series for kernels.http://arxiv.org/abs/1201.4538
[2] T. Jakubowski & K. Szczypkowski: Time-dependent gradient perturbations of frac-tional Laplacian. J. Evol. Equ. 10(2):319–339 (2010)
[3] V. Liskevich & Y. Semenov: Two-sided estimates of the heat kernel of the Schrödingeroperator. Bull. London Math. Soc. 30(6):596–602 (1998)
[4] Q. S. Zhang: A sharp comparison result concerning Schrödinger heat kernels. Bull.London Math. Soc. 35(4):461–472 (2003)
Bielefeld, 2012 23
Björn Böttcher
Constructive Approximation of Feller Processes with unbounded coeffi-cients
Feller semigroups and Feller processes are well studied objects. But their constructionis in general difficult. In particular, most of the existing construction methods only al-low the construction of Feller processes with bounded coefficients, i.e., they require thatthe symbol of the generator is bounded uniformly with respect to the space variable.We will show how this boundedness assumption can be relaxed.
Let (Xt)t≥0 be an Rd-valued Feller process. In general its generator A is a nonlocaloperator which has, given that the test functions are in the domain, a representationas a pseudo-differential operator with a symbol q. Moreover, the symbol q has a Lévy-Khinchin representation:
q(x, ξ) = c(x)− il(x)ξ + ξQ(x)ξ +
∫y 6=0
(1− eiyξ +
iyξ
1 + |y|2)N(x, dy), (3.3)
where c(x) ≥ 0, l(x) ∈ Rd, Q(x) = (qjk(x))j,k=1,...,d ∈ Rd×d is a positive semidefinitematrix and N(x, .) is a measure which integrates |y|2
1+|y|2 on Rd\0.Conversely, it is a natural to ask:
When is an operator A, with symbol q given by (3.3), the generator of a Fellerprocess?
Thus one wants to construct a Feller process starting with a given symbol q. A commonassumption in such constructions is: limr→∞ supx sup|ξ|≤ 1
r|q(x, ξ)| = 0. Our main
result shows that this condition can be relaxed.
Theorem. Let q(., .) : Rd × Rd → C be a function such that
limr→∞
sup|y|≤r
sup|ξ|≤ 1
r
|q(y, ξ)| = 0.
For each k ∈ N let (Xkt )t≥0 be a Feller process with semigroup (T kt )t≥0, such that its
generator Ak satisfies C∞c ⊂ D(Ak) and the symbol qk(x, ξ) of Ak∣∣C∞c
satisfies
|qk(x, ξ)| ≤ |q(x, ξ)| for all x, ξ ∈ Rd,qk(x, ξ) = q(x, ξ) for all |x| ≤ k, ξ ∈ Rd
andXk.∧τk
Bk(0)
d= X l
.∧τ lBk(0)
for l ≥ k.
24 Nonlocal Operators
Then the operator (−q(x,D), C∞c ) has an extension which generates a Feller processwith semigroup
Ttu = limk→∞
T kt u
for u ∈ C∞.
[1] B. Böttcher: On the construction of Feller processes with unbounded coefficients. Elec-tron. Commun. Probab. 16:545–555 (2011)
Zhen-Qing Chen
Perturbation by Non-Local Operators
Let d ≥ 1 and 0 < β < α < 2. In this talk, we consider fractional Laplacian ∆α/2 :=−(−∆)α/2 on Rd perturbed by non-local operator
Sbf(x) := A(d,−β)
∫Rd
(f(x+ z)− f(x)− 〈∇f(x), z1|z|≤1〉
) b(x, z)|z|d+β
dz,
where b(x, z) is a bounded measurable function on Rd×Rd with b(x, z) = b(x,−z) forx, z ∈ Rd, and A(d,−β) is a normalizing constant so that Sb = ∆β/2 when b(x, z) ≡ 1.We address the existence and uniqueness of the fundamental solution qb(t, x, y) to thenon-local operator Lb = ∆α/2 + Sb. We show that if b(x, z) is continuous in x, thenqb(t, x, y) ≥ 0 if and only if b(x, z) ≥ 0.
The following is one of the main results we will present during the talk. For a ≥0, denote by pa(t, x, y) the fundamental function of ∆α/2 + a∆β/2 (or equivalently,the transition density function of the Lévy process Yt + a1/βZt, where Y and Z are(rotationally) symmetric α-stable process and symmetric β-stable processes on Rd thatare independent to each other). It is known that
pa(t, x, y) (t−d/α ∧ (at)−d/β
)∧(
t
|x− y|d+α+
at
|x− y|d+β
).
Denote by S(Rd) the space of tempered functions on Rd.
Theorem. Let b(x, z) be a bounded non-negative function on Rd × Rd satisfyingb(x, z) = b(x,−z). For each x ∈ Rd, the martingale problem for (Lb,S(Rd)) withinitial value x is well-posed. These martingale problem solutions Px, x ∈ Rd forma strong Markov process Xb with infinite lifetime, which possesses a jointly continu-ous transition density function qb(t, x, y) with respect to the Lebesgue measure on Rd.
Bielefeld, 2012 25
Moreover, the following holds.
(i) The transition density function qb(t, x, y) can be explicitly constructed as follows.Define qb0(t, x, y) := p0(t, x, y) and
qbn(t, x, y) :=
t∫0
∫Rd
qbn−1(t− s, x, z)Sbzp0(s, z, y)dzds for n ≥ 1.
There is ε > 0 so that∑∞
n=0 qbn(t, x, y) converges absolutely on (0, ε] × Rd × Rd
and qb(t, x, y) =∑∞
n=0 qbn(t, x, y).
(ii) qb(t, x, y) = p0(t, x, y) +
t∫0
∫Rd
qb(t− s, x, z)Sbzp0(s, z, y)dzds.
(iii) For every A > 0, there are positive constants ck = ck(d, α, β,A), k = 1, · · · , 4,such that for any non-negative function b(x.z) on Rd×Rd with b(x, z) = b(x,−z)and ‖b‖∞ ≤ A, we have
c1e−c2t p inf b(t, x, y) ≤ qb(t, x, y) ≤ c3e
c4t p‖b‖∞(t, x, y) on (0,∞)× Rd × Rd.
Our study is partially motivated by the consideration of the following stochastic dif-ferential equation dXt = dYt + c(Xt−)dZt. Its solution has infinitesimal generator Lbwith b(x, z) = |c(x)|β .This talk is based on a joint work with Jieming Wang.
Francesca Da Lio
Analysis of Fractional Harmonic Maps
In this talk we present regularity and compactness results for fractional weak harmonicmaps into manifolds. These maps are critical points of nonlocal functionals of the form
L(u) =
∫Rn
|∆α/2u(x)|pdx , (3.4)
where αp = n, u : Rn → N , N is a smooth k-dimensional sub-manifold of Rm whichis compact and without boundary. This kind of variational problems appears as sim-plified models for renormalized energy in general relativity, (see [1]) . For simplicity weconsider here the case of weak 1/2−harmonic maps from the real line into a sphere
26 Nonlocal Operators
(n = 1, α = 1/2, p = 2) . In this particular case the Lagrangian (3.4) is also in-variant under the trace of conformal maps that keep invariant the half space R2
+ (thewell-known Möbius group) .
The key point in our results is first a formulation of the Euler-Lagrange equationfor (3.4) (the so-called 1/2−harmonic map equation) in the form of a nonlocal linearSchrödinger type equation with a 3-terms commutators in the right-hand-side. Wethen establish a sharp estimate for these commutators by using the Littlewood-Paleydecomposition and the theory of para-products.
[1] S. Alexakis & R. Mazzeo: The Willmore functional on complete minimal surfaces inH3: boundary regularity and bubbling.http://arxiv.org/abs/1204.4955
[2] F. Da Lio: Compactness and Bubbles Analysis for Half-Harmonic Maps into Spheres.Preprint
[3] F. Da Lio: Fractional Harmonic Maps into Manifolds in odd dimension n>1.http://arxiv.org/abs/1012.2741
[4] F. Da Lio & A. Schikorra: n/p-Harmonic maps: regularity for the sphere case.http://arxiv.org/abs/1202.1151
[5] F. Da Lio & T. Rivière: Three-term commutator estimates and the regularity of 12 -
harmonic maps into spheres. Anal. PDE 4(1):149–190 (2011)
[6] F. Da Lio & T. Rivière: Sub-criticality of non-local Schrödinger systems with antisym-metric potentials and applications to half-harmonic maps. Adv. Math. 227(3):1300–1348(2011)
[7] A. Schikorra: Regularity of n/2-harmonic maps into spheres. J. Differential Equations252(2):1862–1911 (2012)
Bartłomiej Dyda
Comparability and regularity estimates for symmetric nonlocal Dirichletforms
We will discuss sufficient conditions on the kernel k such that the following twoquadratic forms
EkB(u, u) =
∫∫BB
(u(y)− u(x)
)2k(x, y) dy dx,
E(α)B (u, u) =
∫∫BB
(u(y)− u(x)
)2|x− y|−d−α dy dx,
Bielefeld, 2012 27
are comparable. Namely, the sufficient conditions will be:
[(K)] for almost every x, y ∈ Rd
L(x− y) ≤ k(x, y) ≤ U(x− y) ,
for some functions L,U : Rd → [0,∞) satisfying L(x) = L(−x), U(x) = U(−x)for almost every x ∈ Rd, L 6= 0 on a set of positive measure
[(U)] There exists C1 > 0 such that for every r ∈ (0, 1]∫Rd
(r2 ∧ |z|2
)U(z) dz ≤ C1r
2−α.
[(L)] There exist a > 1 and C2, C3 such that every annulus Ba−n+1 \ Ba−n(n = 0, 1, . . .) contains a ball Bn with radius C2a
−n, such that
L(z) ≥ C3(2− α)|z|−d−α, z ∈ Bn.
We will discuss the independence of the constant on α when α→ 2−.
As an application, we will provide a weak Harnack inequality and a-priori estimates inHölder spaces for solutions u ∈ L∞(Rd) ∩Hα/2
loc (B) to integrodifferential equations ofthe following form
EkB(u, φ) = 0 for every φ ∈ C∞c (B).
The talk is based on a joint work with Moritz Kassmann (Universität Bielefeld).
[1] B. Dyda: On comparability of integral forms. J. Math. Anal. Appl. 318(2):564–577 (2006)
[2] B. Dyda & M. Kassmann: Comparability and regularity estimates for symmetric nonlocalDirichlet forms.http://arxiv.org/abs/1109.6812
[3] M. Kassmann: A new formulation of Harnack’s inequality for nonlocal operators. C. R.Math. Acad. Sci. Paris 349(11-12):637–640 (2011)
Etienne Emmrich
Analysis of the peridynamic model in nonlocal elasticity
Peridynamics is a rather new nonlocal continuum theory that avoids spatial deriva-tives. It is believed to be suited for the description of fracture and other materialfailure. From the mathematical point of view, the peridynamic model exhibits several
28 Nonlocal Operators
difficulties: nonlocality, nonlinearity, time delay, and multiscale behaviour. The anal-ysis and numerical analysis of the peridynamic model is, therefore, still at the verybeginning.
In this talk, we give a survey of results known so far. In particular, we focus on thequestion of existence of solutions and the question of the limit of vanishing nonlocality.
Matthieu Felsinger
Local Regularity for Parabolic Nonlocal Operators
This talk will be concerned with regularity results for weak solutions to parabolicequations
∂tu− Lu = f,
where L is a nonlocal integro-differential operator of differentiability order α ∈ (α0, 2).L can be considered as a generalization of the fractional Laplacian −(−∆)α/2 in thesense that generalized integral kernels kt(x, y) |x− y|−d−α are allowed.
Local a priori estimates of Hölder norms and a weak Harnack inequality are proved.These results are robust with respect to α 2, i.e. all constants appearing in the proofscan be chosen independently of α. In this sense, the presentation is a generalization ofMoser’s result from 1971.
We employ localization techniques and the Lemma of Bombieri-Giusti.
The talk is based on a joint preprint with M. Kassmann, seehttp://arxiv.org/abs/1203.2126.
Mohammud Foondun
Stochastic heat equation with spatially coloured random forcing
The aim of this talk is to present some results concerning the long term behaviour ofa class of stochastic heat equations. More precisely, we look at the following class ofequations.
∂tu(t, x) = Lu(t, x) + σ(u(t, x))W (t, x),
Bielefeld, 2012 29
under suitable conditions. Here L is the generator of a Lévy process and is hence anintegro-differential operator. W is the random term and is assumed to be Gaussian.After explaining how to make sense of these types of equations, we will give conditionsunder which the second-moment of the solution grows exponentially. If time permits,we will specialise to the case where L is a fractional Laplacian and discuss finite timeblow-up of the second moment of the solution under some suitable conditions.
Rupert Frank
Uniqueness and nondegeneracy of ground states for non-local equations in1D
We prove uniqueness of energy minimizing solutions Q for the nonlinear equation(−∆)sQ + Q − Qα+1 = 0 in 1D, where 0 < s < 1 and 0 < α < 4s
1−2s for s < 1/2and 0 < α < ∞ for s ≥ 1/2. Here (−∆)s is the fractional Laplacian. As a techni-cal key result, we show that the associated linearized operator is nondegenerate, in thesense that its kernel is spanned by Q′. This solves an open problem posed by Weinsteinand by Kenig, Martel and Robbiano.
The talk is based on joint work with E. Lenzmann.
Piotr Garbaczewski
Lévy flights, Lévy semigroups and fractional quantum mechanics
Lévy-Schrödinger semigroups derive from additive perturbations of symmetric stablegenerators, [1]-[6]. Their integral kernels can be Doob-transformed into transition prob-ability density functions of the jump-type process which, under confining conditions,admit non-Gaussian invariant (asymptotic) probability density functions (pdfs). Therelated dynamical pattern of behavior is inequivalent to that obtained via gradientperturbations of fractional (e.g. symmetric stable) noise generators, [2, 3]. Undersuitable restrictions, Lévy semigroups admit an analytic continuation in time to theunitary dynamics, viewed as a signature of the quantum behavior, [1, 4]. The spectralproperties of the involved non-local Hamiltonian-type operators (negative semigroupgenerators) are of utmost importance, [4, 6]. Their ground states φ determine invari-ant pdfs ρ = |Φ|2 of jump-type processes, identical with those Doob-deduced fromthe corresponding semigroup. An issue of the abnormal (heavy tailed) asymptotic of
30 Nonlocal Operators
diffusion-type processes is to be mentioned, [7].The talk is supposed to give a brief outline of the present author’s research on Lévystable and quasi-relativistic processes, motivated by physicist’s intuitions (specificallyso-called Schrödinger boundary data problem and the non-Langevin modeling of Lévyflights) and struggles with the understanding and proper usage of the grand mathe-matical formalism for that subject matter.
[1] P. Garbaczewski, J. R. Klauder & R. Olkiewicz: Schrödinger problem, Lévy pro-cesses, and noise in relativistic quantum mechanics. Phys. Rev. E (3) 51(5, part A):4114–4131 (1995)
[2] P. Garbaczewski & R. Olkiewicz: Cauchy noise and affiliated stochastic processes.J. Math. Phys. 40(2):1057–1073 (1999)
[3] P. Garbaczewski & R. Olkiewicz: Ornstein-Uhlenbeck-Cauchy process. J. Math.Phys. 41(10):6843–6860 (2000)
[4] P. Garbaczewski & V. Stephanovich: Lévy flights in confining potentials. Phys. Rev.E 80:031113 (2009)
[5] P. Garbaczewski & V. Stephanovich: Lévy flights in inhomogeneous environments.Phys. A 389(21):4419–4435 (2010)
[6] P. Garbaczewski & V. Stephanovich: Lévy targeting and the principle of detailedbalance. Phys. Rev. E 84:011142 (2011)
[7] P. Garbaczewski, V. Stephanovich & D. Kedzierski: Heavy-tailed targets and(ab)normal asymptotics in diffusive motion. Physica A: Statistical Mechanics and itsApplications 390(6):990–1008 (2011)
Christine Georgelin
On Neumann and oblique derivatives boundary conditions for non-localequations
On this talk we will present different Neumann type boundary value problems for non-local equations related to Lévy processes. Since these equations are nonlocal, Neumanntype problems can be obtained in many ways, depending on the kind of “reflection”we impose on the outside jumps. We will focus on rather simple linear equations setin half-space domains and consider different models of reflection and rather generalnon-symmetric Lévy measures. We derive the Neumann/reflection problems through atruncation procedure on the Lévy measure, and we will present comparison, existence,and some regularity results using a viscosity solution theory. The reflection modelsthat we consider include cases where the underlying Lévy processes are reflected, pro-jected, and/or censored upon exiting the domain.If there is enough time, we will say few words about the approach of Lions & Sznitman
Bielefeld, 2012 31
in order to give a sense to Neumann and oblique derivatives boundary conditions forpartial integro-differential equations set on more general domains Ω.
These results are joint works with G. Barles, E. Chasseigne and E. Jakobsen.
María del Mar González
Fractional order operators in conformal geometry
In the talk we will have a closer look at the relations between scattering operatorsof asymptotically hyperbolic metrics [8] and Dirichlet-to-Neumann operators for uni-formly degenerate elliptic boundary value problems [2], as described in the joint workwith A. Chang [3]. This allows to define the fractional conformal Laplacian opera-tor, which is a nonlocal operator, but still satisfies the geometric conformal covarianceproperty and it coincides with the classical construction of [7] at the integer powers. Inaddition, one may also define the fractional Q-curvature of a manifold. Although theprecise meaning of this new curvature is not clear yet, it may be understood througha variation formula for weighted volume [4].
There are still many open problems: on one hand, on a compact manifold one mayformulate the fractional Yamabe problem (joint with J. Qing [6]). On the other hand,the noncompact case is much less well understood. Some cases of singular metricson spheres were considered in [5], while some other constructions on hyperbolic-likemanifolds are described in [1].
[1] V. Banica, M. d. M. González & M. Saez: Some constructions for the fractionalLaplacian on noncompact manifolds. In preparation
[2] L. Caffarelli & L. Silvestre: An extension problem related to the fractional Lapla-cian. Comm. Partial Differential Equations 32(7-9):1245–1260 (2007)
[3] S.-Y. A. Chang & M. d. M. González: Fractional Laplacian in conformal geometry.Adv. Math. 226(2):1410–1432 (2011)
[4] M. d. M. González: A weighted notion of renormalized volume related to the fractionalLaplacian. To appear in Pacific. J. Math.
[5] M. d. M. González, R. Mazzeo & Y. Sire: Singular solutions of fractional orderconformal Laplacians. To appear in Journal of Geometric Analysis
[6] M. d. M. González & J. Qing: Fractional conformal Laplacians and fractional Yamabeproblems. Preprint.
[7] C. R. Graham, R. Jenne, L. J. Mason & G. A. J. Sparling: Conformally invariantpowers of the Laplacian. I. Existence. J. London Math. Soc. (2), 46(3):557–565, 1992.
[8] C. R. Graham & M. Zworski: Scattering matrix in conformal geometry. Invent. Math.,152(1):89–118, 2003
32 Nonlocal Operators
Alexander Grigor’yan
Estimates of heat kernel of Dirichlet forms
Let (M,d, µ) be a metric measure space and (E ,F) be a regular Dirichlet form inL2 (M,µ). It is well known, any regular Dirichlet form has the generator L that is anon-positive definite self-adjoint operator in L2 (M,µ), which in turn gives rise to theheat semigroup Pt = etL and an associated Hunt process Xtt≥0 onM. If the operatorPt in L2 (M,µ) has an integral kernel, then it is called the heat kernel and is denotedby pt (x, y).
We discuss in the talk two questions:
Q1. What are “nice” estimates of the heat kernels that one can expect?
Q2. How to prove such estimates?
To Q1 the following dichotomy was proved by Grigor’yan and Kumagai in 2008. As-sume that the heat kernel in the general setting satisfies the following estimate
pt (x, y) C
tα/βΦ
(cd(x, y)
t1/β
), (3.5)
with some monotone decreasing function Φ. Then α = dimHM and Φ can be only oftwo types:
1. Φ (s) exp(−cs
ββ−1
)(sub-Gaussian function)
2. Φ (s) (1 + s)−(α+β) (stable-like function)
To Q2. Denote by B (x, r) metric balls in (M,d). It was proved by Grigor’yan andTelcs 2010, that sub-Gaussian estimate (3.5) is equivalent to the conjunction of thefollowing conditions:
(i) the locality, that is, the Hunt process Xt is a diffusion;
(ii) the volume regularity: µ (B (x, r)) Crα;(iii) the mean exit time regularity: ExτB(x,r) Crβ where τΩ is the first exit time ofXt from the set Ω;
(iv) the uniform elliptic Harnack inequality for positive harmonic functions.
Analogous result for stable-like estimate (that corresponds to a jump process) is stillmissing.
Let us discuss upper bound in (3.5). The necessary and sufficient conditions for theon-diagonal upper bound
pt (x, x) ≤ Ct−α/β (3.6)
(that is independent of Φ) are well-known. We present the equivalent conditions forthe upper bounds in (3.5) in terms of (3.6), volume growth, certain tail estimate of the
Bielefeld, 2012 33
heat kernel, and upper bound of the jump kernel J (x, y) of (E ,F). Namely, if J = 0that is, if (E ,F) is local, then the upper bound holds with sub-Gaussian function, andif J (x, y) ≤ C
d(x,y)α+β, then with the stable-like function. These results were proved
by Grigor’yan and Hu 2008 for the local case and by Grigor’yan, Hu, Lau 2011 fornon-local case.
Tomasz Grzywny
Estimates on the Poisson kernel of a half-line for subordinate Brownianmotions
Let Tt be a subordinator i.e. an increasing Lévy process starting from 0. The Laplacetransform of a subordinator is of the form Ee−λTt = e−tψ(λ), λ ≥ 0, where ψ is called theLaplace exponent of T . ψ is a Bernstein function and has the following representation:
ψ(λ) = a+ bλ+
∫(0,∞)
(1− e−λu)µ(du),
where a, b ≥ 0 and µ is a Lévy measure on (0,∞). Let Bt be a Brownian motion in R.
Definition 1. We say that a function f satisfies USC(α) if there exist constants β < αand σ > 0 such that, for all x, λ ≥ 1:
|f(λx)| ≤ σλβ|f(x)|.
We say that a function f satisfies LSC(α) if there exist constants β > α and σ > 0such that, for all x, λ ≥ 1:
|f(λx)| ≥ σλβ|f(x)|.
The purpose of this talk is to present some recent results about subordinate Brownianmotions BTt on R. We give new forms of estimates for the Lévy and potential densityof the subordinator near zero. For example:
Theorem 1. Let µ(du) = µ(u)du, where µ(u) is a non-increasing function. Then thefollowing conditions are equivalent.(i) There exist constants c, t0 such that µ(t) ≤ cµ(2t), for t ≤ t0.(ii) ψ(n) satisfies USC(0), for some n ∈ N.(iii) µ(t) ≈ |ψ(n)|(t−1)
tn+1 , for t ≤ t1 and some n ∈ N.
These results provide us to find estimates for the Lévy and potential density of the
34 Nonlocal Operators
subordinate Brownian motion BTt near origin. Next we show the asymptotic behaviourof the derivative of the renewal function V ′ of the ascending ladder-height process.Using these results we find estimates for the Poisson kernel of a half-line. In particularwe get the following estimates.
Theorem 2. Let ψ be a complete Bernstein function and ψ(∞) = ∞. If ψ′ satisfiesUSC(0) and ψ′
ψ1/2 satisfies LSC(−3/2), then for x, |z| < 1 we have
P(0,∞)(x, z) ≈V (x)
V (|z|)V ′(x− z)V (x− z) ≈
ψ1/2(|z|−2)
ψ1/2(x−2)
ψ′((x− z)−2)
(x− z)3ψ((x− z)−2).
[1] T. Grzywny & M. Ryznar: Potential theory of one-dimensional subordinate Brownianmotions. Preprint (2012)
[2] A. Mimica & P. Kim: Harnack inequalities for subordinate Brownian motions. Electron.J. Probab. 17:no. 37, 1–23 (2012)
[3] P. Kim, R. Song & Z. Vondraček: Potential theory of subordinate Brownian mo-tions revisited. Interdisciplinary Mathematical Sciences - Vol. 13, Stochastic Analysis andApplications to Finance (2012)
Peter Imkeller
Construction of stochastic processes with singular jump characteristics assolutions of martingale problems
We construct Lévy processes with discontinuous jump characteristics in form of weaksolutions of appropriate stochastic differential equations, or related martingale prob-lems with non-local operators. For this purpose we prove a general existence theoremfor martingale problems in which a sequence of operators generating Feller processesapproximates an operator with a range containing discontinuous functions. The ap-proach crucially depends on uniform estimates for the probability densities of the ap-proximating processes derived from properties of the associated symbols. The theoremis applicable to stable like processes with discontinuous stability index.
This talk is based on work with N. Willrich (WIAS Berlin).
Bielefeld, 2012 35
Niels Jacob
Non-locality, Non-Isotropy, and Geometry
Many non-local operators, for example many generators of Lévy or Lévy-type processes,are also non-isotropic, i.e. they have a symbol which grows in the co-variable differentlyin different directions, and in addition, in general no reasonable notion of a principalsymbol exist. This rules out standard geometric considerations when dealing with theseoperators. However, these symbols often define their own natural geometry which issuitable to study these operators and their off-springs such as associated semi-groupsor stochastic processes.
Naotaka Kajino
Non-regularly varying and non-periodic oscillation of the on-diagonal heatkernels on self-similar fractals
The purpose of this talk is to present the author’s result in a recent preprint [2] onon-diagonal oscillatory behavior of the canonical heat kernels on self-similar fractals.Note that this talk is on diffusions on fractals and has nothing to do with non-localoperators.
Let K be either a nested fractal or a generalized Sierpiński carpet, which is a compactsubset of the Euclidean space, and let pt(x, y) be the transition density of the Brownianmotion on K. Then it is well-known that there exist c1, c2 ∈ (0,∞) and ds ∈ [1,∞)such that for any x ∈ K,
c1 ≤ tds/2pt(x, x) ≤ c2, t ∈ (0, 1].
The exponent ds is called the spectral dimension of K. Then it is natural to ask howtds/2pt(x, x) behaves as t ↓ 0 and in particular whether the limit
limt↓0
tds/2pt(x, x) (3.7)
exists. WhenK is a nested fractal, the author has proved in [1] that the limit (3.7) doesnot exist for “generic" (hence almost every) x ∈ K under very weak assumptions on K.The proof of this fact, however, heavily relied on the two important features of nestedfractals — they are finitely ramified (i.e. can be made disconnected by removing finitelymany points) and highly symmetric. In particular, the result of [1] is not applicable togeneralized Sierpiński carpets, which are infinitely ramified.
The main results of [2] have overcome this difficulty by a completely different method,
36 Nonlocal Operators
thereby establishing the non-existence of the limit (3.7) for “generic" x ∈ K when Kis an arbitrary generalized Sierpiński carpet. More strongly, we have the followingassertion under a quite general setting of a self-similar Dirichlet form on a self-similarset K.
[ (NRV)]p(·)(x, x) does not vary regularly at 0 for “generic" x ∈ K, if
lim supt↓0
pt(y, y)
pt(z, z)> 1 for some y, z ∈ K \ V0. (3.8)
[ (NP)]“Generic" x ∈ K does not admit a periodic function G : R → R suchthat
pt(x, x) = t−ds/2G(− log t) + o(t−ds/2) as t ↓ 0, if (3.9)
lim inft↓0
pt(y, y)
pt(z, z)> 1 for some y, z ∈ K \ V0. (3.10)
We will also see that the conditions (3.8) and (3.10) can be easily verified for most(though not all) typical self-similar fractals.
[1] N. Kajino: On-diagonal oscillation of the heat kernels on post-critically finite self-similarfractals. Probability Theory and Related Fields, in press (2012)http://dx.doi.org/10.1007/s00440-012-0420-9
[2] N. Kajino: Non-regularly varying and non-periodic oscillation of the on-diagonal heatkernels on self-similar fractals. Preprint (2012)
Panki Kim
An Lp-theory of stochastic PDEs with random fractional Laplacian operator
In this talk, we introduce an Lp-theory of a class of parabolic stochastic equations withrandom fractional Laplacian operator. The driving noises of the equations are generalLevy processes. Uniqueness and existence results in Sobolev spaces will be introduced.
This is a joint work with Kyeong-Hun Kim.
Bielefeld, 2012 37
Victoria Knopova
Parametrix construction for the transition probability density of some Lévy-type processes
The talk is devoted to the parametrix construction of the fundamental solution to theequation
∂
∂tu(t, x) = L(x,D)u(t, x), t > 0, x ∈ R, (3.11)
where for v ∈ C∞0 (R)
L(x,D)v(x) :=
∫R
(u(x+ y)− u(x))m(x, y)µ(dy),
µ is a Lévy measure, and m(x, y) satisfies some mild regularity assumptions. Weshow that the fundamental solution to (3.11) can be constructed by Levi’s parametrixmethod, and derive the estimates for this solution.
The talk is based on the on-going research work with Alexei Kulik.
Yuri Kondratiev
Non-local evolutions as kinetic equations for Markov dynamics
Non-linear PDEs are widely used in phenomenological models of complex systems. Asan important example we may mention reaction-diffusion equations (RDE)
∂u
∂t= 4u+ f(u), u = u(t, x)
which we meet in combustion theory, bacterial growth, nerve propagation, epidemiol-ogy, genetics etc. A particular case is the celebrated Fisher equation corresponding tof(s) = s(1− s).We observe growing attention to non-local versions of RDE:
∂u
∂t= J ∗ u− u+ f(u), u = u(t, x).
u(0, x) = φ(x), x ∈ Rd,where
0 ≤ J ∈ L1, ‖J‖1 = 1
38 Nonlocal Operators
is a jump kernel and non-linearity f may be local or non-local as well. Here are fewreferences: Coville, Dupaigne (2008), Ignat, Rossi (2010), Berestycki, Nadin, Ryzhik(2009), Pan, Li, Lin (2009), Zhang, Li, Sun (2010). Especially, I would like to mentionrecent monograph "Nonlocal Diffusion Problems", F. Andreu-Vaillo et. al., AMS ,MSM v. 165 (2010). Actually, such kind of equations was introduced by Kolmogorov,Petrovsky and Piskunov in 1937 as a way to derive Fisher equation.
The aim of the presented talk is to describe how non-linear non-local evolutional equa-tions appear as kinetic equations for interacting particle systems. More precisely, wewill show that several models of stochastic dynamics for continuous systems (micro-scopic level) in a scaling limit (Vlasov, Lebowitz-Penrose, etc.) may be described bymeans of so-called kinetic equations (mesoscopic level) for the particle density. Theseequations contain (as a rule) certain convolutional terms which may enter as non-localdiffusion generators as well as in other quite different forms.
Mateusz Kwaśnicki
Estimates of harmonic functions for non-local operators
I will present the results of my recent work with Krzysztof Bogdan and Takashi Ku-magai, contained in [1]. We consider a non-local operator A, which generates a Fellersemigroup on a metric measure space X . For example, A can be a fractional powerof the Laplace operator (on a Riemannian manifold or on a fractal set), or a pseudo-differential operator with sufficiently smooth coefficients. We prove local supremumestimate for nonnegative subharmonic functions in a ball. Next, we show the boundaryHarnack inequality (BHI).
More precisely, let∫
(f(y)− f(x))ν(x, y)m(dy) be the non-local part of A. We assumethat ν(x, y) = ν(y, x) is bounded above and below by a positive constant on everycompact subset of X × X \ diagX . The same condition is imposed on the kernel ofthe resolvent operator (1 − A)−1. Furthermore, we require that A generates a Fellerand strong-Feller semigroup, and that the C0 domain of A contains bump functions:functions equal to 1 on a given compact set and vanishing outside a given (larger) openset. Under these assumptions,
supB′
f ≤ c∫
(B′)c
f(z)ν(x0, z)m(dz) (3.12)
for all globally nonnegative functions f subharmonic (with respect to A) in the ballB = B(x0, R). Here B′ = B(x0, r), 0 < r < R and c = c(x0, r, R). The supremum
Bielefeld, 2012 39
estimate (3.12) is used to prove BHI: for any open D,
f(x)
g(x)≤ c f(y)
g(y)(x, y ∈ B′ ∩D) (3.13)
holds for all globally nonnegative functions f , g harmonic (with respect to A) in B∩D,vanishing in B\D and continuous on B∩∂D. Here again B = B(x0, R), B′ = B(x0, r),0 < r < R and c = c(x0, r, R).
Our results cover also the non-symmetric case under an appropriate duality assump-tion. They extend previous works on BHI for non-local operators in several directions,and are new even for many pseudo-differential operators with constant coefficients.Probabilistic methods are used in the proof of (3.12).
[1] K. Bogdan, T. Kumagai & M. Kwaśnicki: Boundary Harnack inequality for Markovprocesses with jumps. In preparation
Richard Lehoucq
Peridynamic nonlocal mechanics
The peridynamic balance of linear momentum was postulated by Silling [1] to allow theconsideration of discontinuous motion, or deformation. The nonlocal force and energydensities are given by ∫
Ω
(t(x,x′)− t(x′,x)
)dx′,
∫Ω
(t(x,x′) · v(x, t)− t(x,x′) · v(x′, t)
)dx′ −
∫Ω
(h(x,x′)− h(x′,x)
)dx′,
and are the nonlocal analogues for the classical force and energy densities
∇ · σ, ∇ · (σv)−∇ · q.
The integral operators sum forces and power expenditures among volumes separated bya finite distance and so represent nonlocal interaction. This is in contrast to the classicaldensities in which interaction only occurs between volumes in direct contact—the in-teraction is therefore deemed local. The integral operators obviate special treatment atpoints of discontinuity conventionally employed because spatial derivatives are avoided.The resulting balance laws extend the classical theory of continuum mechanics to allowfor jumps.
My talk first introduces the nonlocal balances of linear momentum and energy [2]. Acrucial aspect is that the integrands are antisymmetric with respect to an interchange
40 Nonlocal Operators
of x and x′. This is a necessary and sufficient condition for the balance laws to be ad-ditive, an action-reaction principle to hold, and that the nonlocal flux is an alternatingform [3]. The nonlocal balance laws may be derived using the principles of statisticalmechanics [4].
Nonlocal constitutive relations naturally arise when the gradient operator is replacedwith its nonlocal analogue. This generalized notion of kinematics is what enables dis-continuous motion to be modeled, leads to well-posed balance laws and thermodynamicrestriction so that the second law of thermodynamics is not violated. My presentationends with a review of an emerging mathematical theory [5] for nonlocal diffusion [6]and the peridynamic Navier equation. Critical to this analysis is the introduction ofvolume constraints, the nonlocal analogue of boundary conditions. These constraintsenable the use of function spaces where a trace operator may not be defined. The math-ematical analysis is also facilitated by the introduction of a nonlocal vector calculus.
[1] S. A. Silling: Reformulation of elasticity theory for discontinuities and long-range forces.J. Mech. Phys. Solids 48(1):175–209 (2000)
[2] S. A. Silling & R. B. Lehoucq: Peridynamic Theory of Solid Mechanics. Advances inApplied Mechanics 44 (2010)
[3] Q. Du, M. Gunzburger, R. B. Lehoucq & K. Zhou: A nonlocal vector calcu-lus, nonlocal volume-constrained problems, and nonlocal balance laws. To appear in inM3AS:Mathematical Models and Methods in Applied Sciences
[4] R. B. Lehoucq & M. P. Sears: Statistical mechanical foundation of the peridynamicnonlocal continuum theory: Energy and momentum conservation laws. Phys. Rev. E84:031112 (2011)
[5] E. Emmrich & R. B. Lehoucq: Peridynamics: a nonlocal continuum theory. To appearin the proceedings of the Sixth International Workshop Meshfree Methods for PartialDifferential Equations
[6] Q. Du, M. Gunzburger, R. B. Lehoucq & K. Zhou: Analysis and approximation ofnonlocal diffusion problems with volume constraints. To appear in the SIAM review
Mark Meerschaert
Solutions to a nonlocal fractional wave equation
A nonlocal fractional wave equation with attenuation has been proposed to modelsound wave conduction in heterogeneous media. This equation has an exact analyticalsolution written in terms of stable densities. The solution is causal only when the stableindex is less than one. However, applications to medical ultrasound require a stableindex between one and two. In this talk, a stochastic model will be presented to explainthe appearance of the stable density in the solution, and an alternative causal solution
Bielefeld, 2012 41
will be developed, based on the hitting time of a positively skewed stable Lévy motionwith drift. Finally, some open problems will be discussed, including alternatives to thestable density, anisotropic media, and model coefficients that vary in space.
[1] M. Caputo: Linear models of dissipation whose Q is almost frequency independent. II.Fract. Calc. Appl. Anal. 11(1):4–14 (2008), reprinted from Geophys. J. R. Astr. Soc. 13(1967), no. 5, 529–539
[2] W. Chen & S. Holm: Fractional Laplacian time-space models for linear and nonlinearlossy media exhibiting arbitrary frequency power-law dependency. J. Acoust. Soc. Am.(2004)
[3] J. F. Kelly, R. J. McGough & M. M. Meerschaert: Analytical time-domain Green’sfunctions for power-law media. J. Acoust. Soc. Am. 124(5):2861–2872 (2008)
[4] M. Meerschaert, P. Straka, Y. Zhou & R. McGough: Stochastic solution to atime-fractional attenuated wave equation. Submitted.http://www.stt.msu.edu/users/mcubed/StochWave.pdf
[5] T. Szabo: Causal theories and data for acoustic attenuation obeying a frequency power-law. J. Acoust. Soc. Am. 97:14–24 (1995)
Ralf Metzler
Ageing and Ergodicity Breaking in Anomalous Diffusion
In 1905 Einstein formulated the laws of diffusion, and in 1908 Perrin published hisNobel-prize winning studies determining Avogadro’s number from diffusion measure-ments. With similar, more refined techniques the diffusion behaviour in complex sys-tems such as the motion of tracer particles in living biological cells or the tracking ofanimals and humans is nowadays measured with high precision. Often the diffusionturns out to deviate from Einstein’s laws. This talk will discuss the basic mechanismsleading to such anomalous diffusion as well as point out its consequences. In partic-ular the unconventional behaviour of non-ergodic, ageing systems will be discussedwithin the framework of continuous time random walks. Indeed, non-ergodic diffusionin the cytoplasm of living cells as well as in membranes has recently been demonstratedexperimentally.
Ante Mimica
42 Nonlocal Operators
Regularity estimates of harmonic functions for jump processes
In this talk we discuss probabilistic methods of proving regularity of harmonic functionsfor a class of non-local operators.
Consider a kernel n : Rd × (Rd \ 0) → [0,∞) that is symmetric and such that thereis a cone V ⊂ Rd whose apex is at origin such that for any x ∈ Rd
n(x, h) ≥ |h|−d−α`(|h|), h ∈ V .
Here α ∈ (0, 2) and ` : (0,∞) → (0,∞) is bounded away from zero and varies slowlyat zero, i. e.
limr→0+
`(λr)
`(r)= 1 for all λ > 0 .
Furthermore, we assume that for some κ > 1,
n(x, h) ≤ κ|h|−d−α`(|h|) for all h ∈ Rd, h 6= 0 .
We can associate a non-local operator L to the kernel n:
Lf(x) =
∫Rd\0
(f(x+ h)− f(x)− 〈∇f(x), h〉1|h|≤1
)n(x, h) dh, f ∈ C2
b (Rd).
Assume that there exists a strong Markov process X = Xt,Pxx∈Rd,t≥0 with pathsthat are right-continuous with left limits that is associated to L in the following sense:for any f ∈ C2
b (Rd) and x ∈ Rd, the stochastic processf(Xt)− f(x)−t∫
0
Lf(Xs) ds
t≥0
is a Px-martingale .
A bounded function u : Rd → R is said to be harmonic in an open set D ⊂ Rd if forany open set B ⊂ Rd such that B ⊂ D and any x ∈ B, the process f(Xt)t≥0 is aPx-martingale.
In this case we have the following a-priori regularity estimates of harmonic functions:there exist constants c > 0 and γ ∈ (0, 1) such that for all r ∈ (0, 1), x0 ∈ Rd and allfunctions u that are harmonic in the ball Br(x0), the following holds
|u(x)− u(y)| ≤ c‖u‖∞( |x− y|
r
)γfor all x, y ∈ Br/2(x0).
We use method of Bass and Levin (see [1]) which is based on an estimate of Krylov-Safonov-type. We show how to prove such an estimate in our case.
Bielefeld, 2012 43
The case α = 0 is also discussed. In this case it is not clear whether Hölder continuityestimates holds. Nevertheless, something can be said about the modulus of continuityof harmonic functions for some classes of non-local operators/stochastic processes.
The talk is based on [2, 3, 4].
[1] R. F. Bass & D. A. Levin: Harnack inequalities for jump processes. Potential Anal.17(4):375–388 (2002)
[2] A. Mimica & M. Kassmann: Analysis of jump processes with nondegenerate jumpingkernels.http://arxiv.org/abs/1109.3678
[3] A. Mimica & P. Kim: Harnack inequalities for subordinate Brownian motions. Electron.J. Probab. 17:no. 37, 1–23 (2012)
[4] A. Mimica: On harmonic functions of symmetric Lévy processes. To appear in Ann. Inst.H. Poincaré Probab. Statist. (2012)http://arxiv.org/abs/1109.3676
Giampiero Palatucci
Asymptotics of the s-perimeter as s 0
Given s ∈ (0, 1) and a bounded open set Ω ⊂ Rn, the s-perimeter of a (measurable)set E ⊆ Rn in Ω is defined as
Pers(E; Ω) := L(E ∩ Ω, (CE) ∩ Ω)
+ L(E ∩ Ω, (CE) ∩ (CΩ)) + L(E ∩ (CΩ), (CE) ∩ Ω),(3.14)
where CE = Rn \ E denotes the complement of E, and L(A,B) denotes the followingnonlocal interaction term
L(A,B) :=
∫A
∫B
1
|x− y|n+sdx dy ∀A,B ⊆ Rn.
This notion of s-perimeter and the corresponding minimization problem were intro-duced in [4] (see also [14, 15], where some functionals related to the one in (3.14) havebeen analyzed in connection with fractal dimensions).
Recently, the s-perimeter has inspired a variety of literature in different directions,both in the pure mathematical settings (for instance, as regards the regularity of sur-faces with minimal s-perimeter, see [7, 1, 3, 13]) and in view of concrete applications(such as phase transition problems with long range interactions, see [5, 11, 12]). Ingeneral, the nonlocal behavior of the functional is the source of major difficulties, con-
44 Nonlocal Operators
ceptual differences, and challenging technical complications. We refer to [9, 8] for anintroductory review on this subject.
The limits as s 0 and s 1 are somehow the critical cases for the s-perimeter, sincethe functional in (3.14) diverges as it is. Nevertheless, when appropriately rescaled,these limits seem to give meaningful information on the problem. In particular, it wasshown in [6, 2] that (1− s)Pers approaches the classical perimeter functional as s 1(up to normalizing multiplicative constants), and this implies that surfaces of minimals-perimeter inherit the regularity properties of the classical minimal surfaces for ssufficiently close to 1 (see [7]).
As far as we know, the asymptotic as s 0 of sPers was not studied yet (see how-ever [10] for some results in this direction), and this is the question that we would liketo address in this talk. That is, we are interested in the quantity
µ(E) := lims0
sPers(E; Ω)
whenever the limit exists.
We will prove necessary and sufficient conditions for the existence of such limit, byalso providing an explicit formulation in terms of the Lebesgue measure of E and Ω.Moreover, we will construct examples of sets for which the limit does not exist.
Work in collaboration with S. Dipierro, A. Figalli and E. Valdinoci. Available athttp://arxiv.org/abs/1204.0750
[1] M. C. Caputo & N. Guillen: Regularity for non-local almost minimal boundaries andapplications.http://arxiv.org/abs/1003.2470
[2] L. Ambrosio, G. De Philippis & L. Martinazzi: Gamma-convergence of nonlocalperimeter functionals. Manuscripta Math. 134(3-4):377–403 (2011)
[3] B. Barros Barrera, A. Figalli & E. Valdinoci: Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces.http://arxiv.org/abs/1202.4606v1
[4] L. Caffarelli, J.-M. Roquejoffre & O. Savin: Nonlocal minimal surfaces. Comm.Pure Appl. Math. 63(9):1111–1144 (2010)
[5] L. A. Caffarelli & P. E. Souganidis: Convergence of nonlocal threshold dynamicsapproximations to front propagation. Arch. Ration. Mech. Anal. 195(1):1–23 (2010)
[6] L. Caffarelli & E. Valdinoci: Uniform estimates and limiting arguments for nonlocalminimal surfaces. Calc. Var. Partial Differential Equations 41(1-2):203–240 (2011)
[7] L. Caffarelli & E. Valdinoci: Regularity properties of nonlocal minimal surfaces vialimiting arguments. Preprinthttp://www.ma.utexas.edu/mp_arc-bin/mpa?yn=11-69
[8] E. D. Nezza, G. Palatucci & E. Valdinoci: Hitchhiker’s guide to the fractionalSobolev spaces. Bull. Sci. Math.
Bielefeld, 2012 45
http://dx.doi.org/10.1016/j.bulsci.2011.12.004
[9] G. Franzina & E. Valdinoci: Geometric analysis of fractional phase transition inter-faces. Preprinthttp://cvgmt.sns.it/paper/1782/
[10] V. Maz’ya & T. Shaposhnikova: On the Bourgain, Brezis, and Mironescu theoremconcerning limiting embeddings of fractional Sobolev spaces. J. Funct. Anal. 195(2):230–238 (2002)
[11] O. Savin & E. Valdinoci: Density estimates for a variational model driven by theGagliardo norm.http://arxiv.org/abs/1007.2114
[12] O. Savin & E. Valdinoci: Γ-convergence for nonlocal phase transitions.http://arxiv.org/abs/1007.1725
[13] O. Savin & E. Valdinoci: Regularity of nonlocal minimal cones in dimension 2.http://arxiv.org/abs/1202.0973
[14] A. Visintin: Nonconvex functionals related to multiphase systems. SIAM J. Math. Anal.21(5):1281–1304 (1990)
[15] A. Visintin: Generalized coarea formula and fractal sets. Japan J. Indust. Appl. Math.8(2):175–201 (1991)
Ilya Pavlyukevich
Small noise asymptotics of integrated Ornstein–Uhlenbeck processes drivenby α-stable Lévy processes
We study the behaviour of a one-dimensional integrated Ornstein–Uhlenbeck processdriven by an α-stable Lévy process of small amplitude. We show that the continuousintegrated Ornstein–Uhlenbeck process converges to the driving càdlàg α-stable Lévyprocess in the Skorokhod M1-topology. In particular this allows us to determine thelimiting distribution of its first passage times.
This is a joint work with R. Hintze (FSU Jena).
Enrico Priola
Uniqueness for singular SDEs driven by stable processes
46 Nonlocal Operators
In [2] the authors have established pathwise uniqueness for the following one dimen-sional SDE
dXt = b(Xt)dt+ dLt, X0 = x ∈ R, (3.15)
where b : R → R is continuous and bounded and L = (Lt) is a standard α-stableprocess with α ∈ [1, 2). They have also shown an example of non-uniqueness when0 < α < 1 and b is β-Hölder continuous and bounded with α+ β < 1.
Simultaneously with [2], the limit case α = 2 (i.e., when L is a Wiener process) hasbeen considered in [4] where uniqueness was proved even if b : R → R is only Boreland bounded (see also [3], [1] and the references therein for uniqueness results in moredimensions).
In this talk we show a multidimensional extension of [2] when b : Rn → Rn, n ≥ 1,is β-Hölder continuous and bounded and L is a non-degenerate symmetric α-stableprocess. The proof uses analytic regularity results at the level of the non-local Kol-mogorov equation plus an Itô-Tanaka trick which is related to the Zvonkin method. Wealso prove the stochastic flow property of the solutions and their differentiability withrespect to the initial data. In the final part possible generalizations of the previousresults will be considered as well.
[1] N. V. Krylov & M. Röckner: Strong solutions of stochastic equations with singulartime dependent drift. Probab. Theory Related Fields 131(2):154–196 (2005)
[2] H. Tanaka, M. Tsuchiya & S. Watanabe: Perturbation of drift-type for Lévy pro-cesses. J. Math. Kyoto Univ. 14:73–92 (1974)
[3] A. J. Veretennikov: Strong solutions and explicit formulas for solutions of stochasticintegral equations. Mat. Sb. (N.S.) 111(153)(3):434–452, 480 (1980)
[4] A. K. Zvonkin: A transformation of the phase space of a diffusion process that willremove the drift. Mat. Sb. (N.S.) 93(135):129–149, 152 (1974)
Michael Röckner
Sub- and supercritical stochastic quasi-geostrophic equation
Consider the 2D stochastic quasi-geostrophic equation on the torus T2 for generalparameter α ∈ (0, 1)
dΘ(t) + (−∆)αΘ(t)dt+ u(t) · ∇Θ(t)dt = G(t,Θ(t))dW (t),
u(t) = −∇⊥((−∆)12 Θ(t)),
G(0) = Θ0
Bielefeld, 2012 47
on the Hilbert space
H :=
Θ ∈ L2(T2) :
∫T2
Θ(ξ)dξ = 0.
The talk will give a survey on recent results on this equation. These include existenceof weak solutions for additive noise, existence of mar- tingale solutions and Markovselections for multiplicative noise and under some condition pathwise uniqueness for allα ∈ (0, 1). Further- more, in the subcritical case α > 1
2 , we prove existence and unique-ness of (probabilistically) strong solutions. In addition, we prove er- godicity for α > 2
3 ,provided the noise is non-degenerate. In this case, the convergence to the (unique)invariant measure is exponen- tially fast. We establish the large deviation principle forthe stochastic quasi-geostrophic equations for α > 1
2 with small multiplicative noise.An analogous result is also obtained for the small time asymptotics. (These resultsare joint with Wei Liu.) The existence of a random at- tractor for the solutions ofthe stochastic quasi-geostrophic equation for α > 1
2 driven by real multiplicative noiseand additive noise is also established. Time permitting, we shall also report on a veryrecent result about ergodicity in the general subcritical case α > 1
2 and for degeneratenoise.
Joint work with Rongchan Zhu and Xiangchan Zhu.
Julio Rossi
A Monge-Kantorovich mass transport problem for a discrete distance
This talk is concerned with a Monge-Kantorovich mass transport problem in which inthe transport cost we replace the Euclidean distance with a discrete distance. We fixthe length of a step and the distance that measures the cost of the transport dependsof the number of steps that is needed to transport the involved mass from its originto its destination. For this problem we construct special Kantorovich potentials, andoptimal transport plans via a nonlocal version of the PDE-formulation given by Evansand Gangbo for the classical case with the Euclidean distance. We also study howthis problems, when rescaling the step distance, approximate the classical problem.In particular we obtain, taking limits in the reescaled nonlocal formulation, the PDE-formulation given by Evans-Gangbo for the classical problem.
Joint work with N. Igbida, J. M. Mazon and J. Toledo.
48 Nonlocal Operators
Enrico Scalas
Characterization of the fractional Poisson process
The fractional Poisson process (FPP) is a counting process with independent andidentically distributed inter-event times following the Mittag-Leffler distribution. Thisprocess is very useful in several fields of applied and theoretical physics including modelsfor anomalous diffusion. Contrary to the well-known Poisson process, the fractionalPoisson process does not have stationary and independent increments. It is not a Levyprocess and it is not a Markov process. I present formulas for its finite-dimensionaldistribution functions, fully characterizing the process. Some recent applications tofinance of these results are briefly discussed.
[1] M. Politi, T. Kaizoji & E. Scalas: Full characterization of the fractional Poissonprocess.http://arxiv.org/abs/1104.4234
[2] E. Scalas & M. Politi: A parsimonious model for intraday European option pric-ing. Economics Discussion Papers, No. 2012-14, Kiel Institute for the World Econ-omy. Discussion paper available at: http://www.economics-ejournal.org/economics/discussionpapers/2012-14
Hans-Peter Scheffler
Fractional governing equations for coupled continuous time random walks
In a continuous time random walk (CTRW), a random waiting time precedes eachrandom jump. The CTRW is coupled if the waiting time and the subsequent jumpare dependent random variables. The CTRW is used in physics to model diffusingparticles. Its scaling limit is governed by an anomalous diffusion equation. Someapplications require an overshoot continuous time random walk (OCTRW), where thewaiting time is coupled to the previous jump. This talk develops stochastic limit theoryand governing equations for CTRW and OCTRW. The governing equations involvecoupled space-time fractional derivatives. In the case of infinite mean waiting times,the solutions to the CTRW and OCTRW governing equations can be quite different.
Armin Schikorra
Bielefeld, 2012 49
Knot-energies and Fractional Harmonic Maps
We will present a proof that curves (knots) which are stationary for the Moebius energyare smooth in the critical dimension. This energy was introduced by O’Hara (Topology,30(2):241–247, 1991), and it was shown by Freedman, He and Wang (Ann. of Math.(2), 139(1):1–50, 1994), and He (Comm. Pure Appl. Math., 53(4):399–431, 2000) thatminimizers of this energy are smooth, using a geometric argument employing cruciallythe Moebius invariance of this energy.
Our approach, however, which shows regularity even for critical points, does notrely on this geometric invariance, but rather uses analytic methods from HarmonicAnalysis and Potential Theory inspired by the regularity arguments of fractional har-monic maps by Da Lio-Riviere (APDE, 4(1):149–190, 2011; Advances in Mathematics,227:1300–1348, 2011), and S. (J. Differential Equations, 252:1862–1911, 2012; Preprint,arXiv:1103.5203, 2011).
Joint work with S. Blatt and P. Reiter.
Russell Schwab
On Aleksandrov-Bakelman-Pucci type estimates for integro differential equa-tions
Despite much recent (and not so recent) attention to solutions of integro-differentialequations of elliptic type, it is surprising that a result such as a comparison theoremwhich can deal with only measure theoretic norms (e.g. L-n and L-infinity) of the righthand side of the equation has gone unexplored. For the case of second order equationsthis result is known as the Aleksandrov-Bakelman-Pucci estimate (and dates back tocirca 1960s), which says that for supersolutions of uniformly elliptic equation Lu=f,the supremum of u is controlled by the L-n norm of f (n being the underlying dimen-sion of the domain). We will discuss this estimate in the context of fully nonlinearintegro-differential equations and present a recent result in this direction.
Joint work with Nestor Guillen, see http://arxiv.org/abs/1101.0279.
Yuichi Shiozawa
Conservation property of symmetric jump-diffusion processes
50 Nonlocal Operators
We say that a Markov process is conservative if the associated particle stays at thestate space forever. This property is one of important global path properties of Markovprocesses. In particular, there are many results on the conservativeness criterion ofsymmetric diffusion processes, in terms of the volume growth of the underlying measureand the growth of the “coefficient”, established by Grigor’yan, Davies, Ichihara, Takeda,Oshima, Sturm,....
Motivated by the recent progress on the analysis of jump processes, there also havebeen results on the conservativeness criterion of symmetric jump(-diffusion) processesgenerated by regular Dirichlet forms ([1, 2, 3, 4]). In [1, 2, 3], the volume of theunderlying measure is allowed to grow exponentially, but the coefficients are assumedto be bounded. In contrast with this, we allow in [4] the coefficients to be unbounded;however, since the explicit form of the L2-generator is needed for the proof, we assumethat the state space is Rd and the underlying measure is the Lebesgue measure on Rd.Furthermore, we also need the assumption on the “drift” parts which may entail thecontinuity on the coefficients.
A purpose in this talk is to establish a conservativeness criterion for symmetric jump-diffusion processes generated by regular Dirichlet forms, in terms of the volume growthof the underlying measure and the growth of the coefficients. Moreover, by using thiscriterion, we remove the conditions in [4] as we mentioned before. We also generalizethe results in [1, 2, 3, 4] so that we allow the volume of the underlying measure to growexponentially and the coefficients to be unbounded at the same time. We do not knowabout the sharpness of our criterion in general; however, we can show the sharpnessfor a class of time changed Dirichlet forms.
We finally give examples related to symmetric stable-like processes and censored stable-like processes.
Example(cf. [4, Example 3.1]) Fix α ∈ (0, 2). Let c(x, y) be a nonnegative Borelfunction on Rd × Rd \ diag such that c(x, y) = c(y, x) for any (x, y) ∈ Rd × Rd \ diag.Let m be a positive Radon measure on Rd such that, for some positive constants β > 0and C1, C2 with C2 > C1 > 0,
C1rβ ≤ m(Bx(r)) ≤ C2r
β for any x ∈ Rd and r > 0. (3.16)
We define
E(u, v) =1
2
∫∫Rd×Rd\diag
(u(x)− u(y))(v(x)− v(y))c(x, y)
|x− y|α+βm(dx)m(dy). (3.17)
We now assume the following on c(x, y).
For 0 < |x− y| < 1,
c(x, y) (1 + |x|2) log(2 + |x|) + (1 + |y|2) log(2 + |y|);
Bielefeld, 2012 51
For |x− y| ≥ 1,c(x, y) (1 + |x|2)p + (1 + |y|2)p
for some p ∈ [0, α/2).
By our result, (E ,F) is conservative.
[1] A. Grigor’yan, X. Huang & J. Masamune: On stochastic completeness of jumpprocesses. To appear in Math. Z.http://dx.doi.org/10.1007/s00209-011-0911-x
[2] J. Masamune & T. Uemura: Conservation property of symmetric jump processes. Ann.Inst. Henri Poincaré Probab. Stat. 47(3):650–662 (2011)
[3] J. Masamune, T. Uemura & J. Wang: On the conservativeness and the recurrence ofsymmetric jump-diffusion. Preprint
[4] YY. Shiozawa & T. Uemura: Explosion of jump-type symmetric dirichlet forms onRd;. To appear in J. Theoret. Probab.http://dx.doi.org/10.1007/s10959-012-0424-5
Yannick Sire
Small energy regularity for a fractional Ginzburg-Landau system
I will describe an epsilon-regularity result for a non local Ginzburg-Landau equationinvolving the fractional laplacian.
Joint work with V. Millot.
Igor Sokolov
Fractional Subdiffusion and Subdiffusion-Reaction equations: Physical Mo-tivation and Properties
The main topic of the talk is: How do kinetic equations with nonlocal operators (e.g.with time-fractional derivatives describing slowly fading memory of the system) emergewhen describing physical models. This understanding is important both in order toget an intuition about possible types of the corresponding equations and in order tosee strengths and limitations of the corresponding approaches. For this we first discussphysical models leading to subdiffusion, and consider the whole chain of reasoning and
52 Nonlocal Operators
approximations leading from the physical model of a particle (say, a charge carrier)in a random potential field via continuous-time random walks and generalized mas-ter equation to the time-fractional subdiffusion equation. We moreover discuss howour intuition based on the time-local (in this case – Markovian) behavior gets uselesswhen passing to systems with long-time memory. We do this considering an exampleof derivation of the subdiffusive-reaction equations, which are found to contain non-linearity in the memory kernel depending on reaction, a property which is absent inmemoryless variants of the theory. The properties of the corresponding equations andtheir solutions are discussed using the simplest examples of the isomerization (A⇒ B)reaction and of the autocatalytic conversion (A + B ⇒ 2B) reaction (a nonlocal vari-ant of the FKPP-equation), the last one leading to a very peculiar front propagationproperties.
Renming Song
Stability of Dirichlet heat kernel estimates of non-local operators underperturbations
Recently, sharp two-sided Dirichlet heat kernel estimates have been obtained for severalclasses of discontinuous processes (or non-local operators), including symmetric stableprocesses, censored stable processes, relativistic stable processes and mixtures of stableprocesses. In this talk I will present some results on the stablility of Dirichlet heatkernel estimates of non-local operators under gradient perturbations and Feynman-Kac perturbations.
I will first present stability results for the Dirichlet heat kernel estimates of symmetricα-stable processes, α ∈ (1, 2), under gradient perturbations. A Radon measure µ onRd is said to be in the Kato class Kd,α−1 if
limr→0
supx∈Rd
∫B(x,r)
|ν|(dy)
|x− y|d+1−α = 0.
For µ = (µ1, · · · , µd) with µj ∈ Kd,α−1, we define an α-stable process with drift µ as aweak solution of an SDE. We can show that this SDE has a unique solution and, whenD is a C1,1 open set, the Dirichlet heat kernel of the α-stable process with drift µ inD is comparable that of the α-stable process in D.
Then I will present stability results for the Dirichlet heat kernels of (not necessarilysymmetric) Hunt processes under nonlocal Feynman-Kac perturbations. Our assump-tions on the nonlocal Feynman-Kac perturbation are also Kato class type conditions.
The results presented are contained in the 4 references.
[1] Z.-Q. Chen, P. Kim & R. Song: Dirichlet heat kernel estimates for fractional Laplacian
Bielefeld, 2012 53
with gradient perturbation. To appear
[2] Z.-Q. Chen, P. Kim & R. Song: Stability of Dirichlet heat kernel estimates for non-localoperators under Feynman-Kac perturbation. Preprint
[3] P. Kim & R. Song: Stable process with singular drift. Preprint
[4] P. Kim & R. Song: Dirichlet heat kernel estimates for stable processes with singulardrifts in unbounded C1,1 open sets. Preprint
54 Nonlocal Operators
Paweł Sztonyk
Upper estimates of transition densities for stable-dominated semigroups
We consider Feller semigroups with symmetric jump intensity f(x, y) dominated bythat of the rotation invariant stable Lévy process, i.e.,
f(x, y) ≤M φ(|y − x|)|y − x|α+d
, x, y ∈ Rd, y 6= x,
for a positive constant M and a bounded function φ(s) satisfying some additionalassumptions. The assumptions are satisfied for example by φ(s) =
(em(1−sγ) ∧ 1
)(1 ∨
s)β , where γ ∈ (0, 1] and β ∈ (−∞, d/2 + α− 1/2), m > 0.
For the semigroup of operators Pt, t ≥ 0 with the generator
Aϕ(x) = limε↓0
∫|y−x|>ε
(ϕ(y)− ϕ(x)) f(x, y) dy,
we obtain the following estimate.
There exists p : (0,∞)× Rd × Rd → [0,∞) such that
Ptϕ(x) =
∫Rd
ϕ(y)p(t, x, y) dy, x ∈ Rd, t > 0, ϕ ∈ C∞(Rd),
andp(t, x, y) ≤ C1e
C2t min
(t−d/α,
tφ(|y − x|)|y − x|α+d
), x, y ∈ Rd, t > 0.
Suleyman Ulusoy
Non-local Conservation laws and related Keller-Segel type Systems
Non-local conservation laws have gained a lot of attention recently. In this talk, wewill briefly discuss our proposed Levy mixed hyperbolic-parabolic equations. If timepermits, our recent findings on a related Keller-Segel type system will also be intro-duced.
Joint work with Kenneth H. Karlsen.
Bielefeld, 2012 55
Enrico Valdinoci
Nonlocal nonlinear problems
I would like to discuss some questions related to some semilinear equations driven by anonlocal elliptic operator. As a specific example, we consider the fractional Allen-Cahnequation
(−∆)su = u− u3, s ∈ (0, 1),
in which the classical Laplace operator is replaced by a fractional Laplacian.
Particular interest will be devoted to the qualitative properties of the solutions, suchas:
symmetry (e.g., whether or not a global, bounded, monotone solution dependsonly on one Euclidean variable),
density estimates of the level sets (e.g., what measure of space is occupied by|u| < 1/2),asymptotic behaviors (e.g., the limit properties of the blow-up uε(x) = u(x/ε)and the corresponding Γ-convergence issues).
The limit interfaces of these problems are related to both the local and the nonlocalperimeter functionals, depending on whether s ∈ [1/2, 1) or s ∈ (0, 1/2). On this topic,I would like to discuss some rigidity and regularity results (for instance, regularity ofs-minimizers for any s ∈ (0, 1/2) in dimension n = 2, and for any s ∈ ((1/2)− ε0, 1/2)in dimension n ≤ 7). Some open problems will be presented as well.
Alexis Vasseur
Integral variational problems
We will present, in this talk, the proof of the existence of classical solutions for a classof non-linear integral variational problems. Those types of equations are typically usedin nonlocal image and signal processing. They involve nonlinear versions of fractionaldiffusion operators. The method is based on De Giorgi-Nash-Moser techniques. Itextends to fully nonlinear settings a previous work on the regularity of solutions to theSurface Quasi-Geostrophic equations (SQG).
Joint work with L. Caffarelli and Ch.-H. Chan.
56 Nonlocal Operators
Zoran Vondraček
Potential theory of subordinate Brownian motions with Gaussian compo-nents
In this talk I will look at a subordinate Brownian motion with a Gaussian componentand a rather general discontinuous part. The assumption on the subordinator is thatits Laplace exponent is a complete Bernstein function with a Levy density satisfying acertain growth condition near zero. The main result that I will present is a boundaryHarnack principle with explicit boundary decay rate for non-negative harmonic func-tions of the process in C1,1 open sets. I will also discuss an example showing that theboundary Harnack principle fails for processes with finite range jumps. As a conse-quence of the boundary Harnack principle, one can establish sharp two-sided estimateson the Green function of the subordinate Brownian motion in any bounded C1,1 openset D and identify the Martin boundary of D with respect to the subordinate Brownianmotion with the Euclidean boundary.
Joint work with Panki Kim and Renming Song.
Bielefeld, 2012 57
Poster Presentations
Ishak Derrardjia
Fixed point techniques and stability for neutral nonlinear differential equa-tions with unbounded delay
We use the contraction mapping theorem to obtain stability results of the scalar non-linear neutral differential equation with functional delay
x′(t) = −ax(t) + b(t)x2(t− r(t)) + c(t)x(t− r(t))x′(t− r(t)).
[1] T. A. Burton: Liapunov functionals, fixed points, and stability by Krasnoselskii’s theo-rem. Nonlinear Stud. 9(2):181–190 (2002)
[2] T. A. Burton: Stability and periodic solutions of ordinary and functional-differentialequations, volume 178 of Mathematics in Science and Engineering. Academic Press Inc.,Orlando, FL (1985)
[3] T. A. Burton: Stability by fixed point theory or Liapunov theory: a comparison. FixedPoint Theory 4(1):15–32 (2003)
[4] T. A. Burton & T. Furumochi: Fixed points and problems in stability theory forordinary and functional differential equations. Dynam. Systems Appl. 10(1):89–116 (2001)
Luz Roncal
Fractional Laplacian on the torus
We consider the fractional powers of the Laplacian (−∆T)σ/2, 0 < σ < 2, on thetorus T. This operator can be defined in the obvious way by using Fourier series.We are interested in obtaining pointwise formulas for (−∆T)σ/2f(x) as well as Hölderestimates and interior and boundary Harnack’s inequalities.
To deal with these problems we use a novel approach to fractional operators based onthe application of the language of semigroups, as introduced by Stinga and Torrea.Indeed, by using the Poisson semigroup on the torus, we derive integro-differentialpointwise formulas (avoiding the computation of inverse Fourier series) and regularityproperties on Hölder spaces. Clearly, this semigroup method permits us to see thatthe fractional Laplacian on the torus is a nonlocal operator.
Bielefeld, 2012 59
On the other hand, we generalize the Caffarelli-Silvestre extension problem to fractionalpowers of second order differential operators Lγ , for any noninteger positive γ. Againthe semigroup language allows us to write an explicit formula for the solution of theextension which involves the heatdiffusion semigroup e−tL. Our result extends theone obtained by Stinga and Torrea for 0 < γ < 1. With this we prove interior andboundary Harnack’s inequalities for (−∆T)σ/2 .
Joint work with P. R. Stinga.
Pablo Raúl Stinga
Harnack’s inequality for fractional nonlocal equations
We show Harnack’s inequalities for solutions to fractional nonlocal equations of theform
Lσu = 0, u ≥ 0, in Ω ⊆ Rn,
where Lσ, 0 < σ < 1, is the fractional power of a Laplacian L. Our examples includesecond order divergence form elliptic operators with potentials; the radial Laplacianor, more generally, Bessel operators; the Laplacian on bounded domains and also oper-ators arising in classical orthogonal expansions (Hermite, Laguerre and ultrasphericaloperators).
The main idea is to apply a novel point of view of the Caffarelli–Silvestre extensionproblem of [1] for the fractional Laplacian. This new viewpoint was introduced byStinga and Torrea in [3]. It consists in the use of the language of semigroups, theprincipal tool being the heat-diffusion semigroup e−tL. Such a general method appliesto any Laplacian L. In this way we can take advantage of local methods, like Gutiérrez’sHarnack inequality for degenerate Schrödinger equations of [2], to obtain our results.
[1] L. Caffarelli & L. Silvestre: An extension problem related to the fractional Lapla-cian. Comm. Partial Differential Equations 32(7-9):1245–1260 (2007)
[2] C. E. Gutierrez: Harnack’s inequality for degenerate Schrödinger operators. Trans. Am.Math. Soc. 312(1):403–419 (1989)
[3] P. R. Stinga & J. L. Torrea: Extension problem and Harnack’s inequality for somefractional operators. Comm. Partial Differential Equations 35(11):2092–2122 (2010)
60 Nonlocal Operators
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Bielefeld, 2012 65
Here is a list of pubs that we recommended to the participants of the summer school. For specificrecommendations of restaurants please contact your colleagues from Bielefeld or the ZiF-staff.
Accumulation Points
• Corner Arndstr./Große-Kurfürsten-Str.5mins walking distance from the tram station “Siegfriedplatz” and the central station “Haupt-bahnhof”
Wunderbar, Arndtstraße 21Bar & Restaurant
Café Berlin, Große-Kurfürsten-Str. 65Café & Restaurant
Desperado, Arndtstr. 20Bar
Westside Lounge, Arndtstr. 18(Cocktail-)Bar & Restaurant
• Neues BahnhofsviertelWithin spitting distance on the North side of the central station “Hauptbahnhof”. Beside thebars, clubs and restaurants listed below there are also a cinema and a bowling center located inthis area.
Edelweiß, Boulevard 1(Cocktail-)Bar & Restaurant
Mexim’s, Ostwestfalenplatz 2(Cocktail-)Bar & Mexican Restaurant
Stereo, Boulevard 1Club
La Pampa, Boulevard 4Steakhouse
Puccini’s, Boulevard 4Italian Restaurant
Wok and Roll, Boulevard 5Asian Restaurant
• Arndtstr. (Downtown)5mins walking distance from the tram station “Jahnplatz” and the central station “Hauptbahnhof”
Mokkaklatsch, Arndtstr. 11(Cocktail-)Bar
Nichtschwimmer, Arndtstr. 6(Cocktail-)Bar & Restaurant
Las Tapas, Arndtstr. 7Spanish Restaurant
Mellow Gold, Karl-Eilers-Str. 22Bar
• Old Town / Klosterplatz5-10mins walking distance from the tram stations “Jahnplatz” or “Rathaus”
Brauhaus Joh. Albrecht, Hagenbruchstr. 8German Restaurant
Irish Pub, Mauerstr. 38
Rockcafé, Neustädter Str. 25Bar
3eck, Ritterstraße 21Bar & Restaurant
Atoms
Plan B, Friedrichstr. 65Tram stations nearby: Hauptbf., SiegfriedplatzBar
Bernstein, Niederwall 2Tram stations nearby: Jahnplatz(Cocktail-)Bar
List of ParticipantsLuis Guillermo Alcuna ValverdePurdue University, USA
Nathaël AlibaudUniversité de Besançon, France
David ApplebaumUniversity of Sheffield, UK
Jong-Chun BaeSeoul National University, Republic of Ko-rea
Rodrigo BañuelosPurdue University, USA
Ali Ben AmorUniversität Bielefeld, Germany
Mahmoud Ben FredjFaculté des Sciences de Monastir, Tunisia
Alexander BendikovUniversity of Wroclaw, Poland
Piotr BilerUniversity of Wroclaw, Poland
Krzysztof BogdanWroclaw University of Technology, Poland
Björn BöttcherTechnische Universität Dresden, Germany
Toralf BurghoffFriedrich-Schiller-Universität Jena, Ger-many
Zhen-Qing ChenUniversity of Washington, USA
Francesca Da LioUniversita di Padova, Italy
Latifa DebbiUniversität Bielefeld, Germany
Zhang DengUniversität Bielefeld, Germany
Ishak DerrardjiaEl Tarf, Algeria
Bartłomiej DydaUniversität Bielefeld, Germany
Khalifa El MabroukEcole Supérieure des Sciences et Technolo-gie, Tunisia
Etienne EmmrichTechnische Universität Berlin, Germany
Mouhamed Moustapha FallGoethe-Universität Frankfurt, Germany
Matthieu FelsingerUniversität Bielefeld, Germany
Mohammud FoondunLoughborough University, UK
Rupert FrankUniversity of Princeton, USA
Uta FreibergUniversität Siegen, Germany
Piotr GarbaczewskiUniversity of Opole, Poland
Bielefeld, 2012 67
Christine GeorgelinUniversité François Rabelais, France
María del Mar GonzálezUniversitat Politècnica de Catalunya,Spain
Alexander Grigor’yanUniversität Bielefeld, Germany
Tomasz GrzywnyTechnische Universität Dresden, Germany
Walter HohUniversität Bielefeld, Germany
Julian HollenderTechnische Universität Dresden, Germany
Jiaxin HuUniversität Bielefeld, Germany
Cyril ImbertUniversité Paris Est Créteil CNRS, France
Peter ImkellerHumboldt-Universität Berlin, Germany
Niels JacobSwansea University, UK
Sven JarohsGoethe-Universität, Frankfurt am Main,Germany
Naotaka KajinoUniversität Bielefeld, Germany
Diana KämpfeUniversität Bielefeld, Germany
Grzegorz KarchUniversity of Wroclaw, Poland
Moritz KaßmannUniversität Bielefeld, Germany
Panki KimSeoul National University, Republic of Ko-rea
Victoria KnopovaTaras Shevchenko National University ofKyiv, Ukraine
Yuri KondratievUniversität Bielefeld, Germany
Takahashi KumagaiKyoto University, Japan
Mateusz KwaśnickiWroclaw University of Technology, Poland
Richard LehoucqSandia National Laboratories, USA
Xiaohua LiUniversität Bielefeld, Germany
Luis LópezMarseille, France
József LörincziLoughborough University, UK
Antonios ManoussosUniversität Bielefeld, Germany
Mark MeerschaertMichigan State University, USA
Ralf MetzlerUniversität Potsdam, Germany
Ante MimicaUniversität Bielefeld, Germany
Giampiero PalatucciUniversità degli Studi di Parma, Italy
68 Nonlocal Operators
Ilya PavlyukevichFriedrich-Schiller-Universität Jena, Ger-many
Ling PeiUniversität Bielefeld, Germany
Xuhui PengUniversität Bielefeld, Germany
Xue PengUniversität Bielefeld, Germany
Enrico PriolaUniversità degli Studi di Torino, Italy
Diana Camelia PutanUniversität Bielefeld, Germany
Marcus RangUniversität Bielefeld, Germany
Michael RöcknerUniversität Bielefeld, Germany
Luz RoncalUniversidad de La Rioja, Spain
Julio RossiUniverisad de Alicante, Spain
Nikola SandrićUniversity of Zagreb, Croatia
Enrico ScalasUniversità del Piemonte Orientale, Italy
Hans-Peter SchefflerUniversität Siegen, Germany
Armin SchikorraMax-Planck-Institut Leipzig, Germany
René SchillingTechnische Universität Dresden, Germany
Russell SchwabCarnegy Mellon University, USA
Marina SertićUniversität Bielefeld, Germany
Yuichi ShiozawaOkayama University, Japan
Yannick SireUniversité Aix-Marseille III, France
Igor SokolovHumboldt-Universität Berlin, Germany
Renming SongUniversity of Illinois, USA
Pablo Raul StingaUniversidad de La Rioja, Spain
Kohei SuzukiUniversity Kyoto, Japan
Paweł SztonykWroclaw University of Technology, Poland
Felix ThielHumboldt-Universität Berlin, Germany
Erwin ToppUniversidad de Chile, Chile
Suleyman UlusoyZirve University, Turkey
Enrico ValdinociUniversità di Roma Tor Vergata, Italy
Alexis VasseurUniversity of Texas, USA
Paul VoigtUniversität Bielefeld, Germany
Bielefeld, 2012 69
Zoran VondračekUniversity of Zagreb, Croatia
Vanja WagnerUniversity of Zagreb, Croatia
Sven WiesingerUniversität Bielefeld, Germany
Zhuo-Ran XiaoUniversität Bielefeld, Germany
70 Nonlocal Operators
Analysis, Probability, Geometry and Applications
SCIENTIFIC COMMITTEERODRIGO BAÑUELOS CYRIL IMBERT GRZEGORZ KARCH
Nonlocal OperatorsJuly 9th - 14th, 2012 at ZiF Bielefeld
TAKASHI KUMAGAI JÓZSEF LÖRINCZI RENÉ SCHILLING
LOCAL ORGANIZER - MORITZ KASSMANN
Zentrum für interdisziplinäre Forschung (ZiF)Universität BielefeldWellenberg 1D-33615 Bielefeld
Sonderforschungsbereich (SFB) 701Universität BielefeldPostfach 10 01 31D-33501 Bielefeld