SPECIAL SESSION 45 191

Special Session 45: Stochastic and Deterministic Dynamical Systems,and Applications

Tomas Caraballo, University of Sevilla, SpainJose Valero Cuadra, University Miguel Hernandez (Elche), Spain

Maria Garrido-Atienza, University of Sevilla, Spain

The aim of this session is to o↵er an overview on recent results concerning the asymptotic behaviour ofsolutions of stochastic and deterministic partial and ordinary di↵erential equations. The main topics of thesession are: existence and properties of pullback attractors for stochastic and non-autonomous equations,stability, stabilization, attractors for equations without uniqueness, dynamics of equations with delay andfinite-dimensional dynamics for dynamical systems.

Lyapunov exponents for autonomous and non-autonomous systems

Francisco BalibreaUniversity of Murcia, Spainbalibrea@um.esM.Victoria Caballero

It is an extended idea to associate positive Lyapunovexponents to instability of orbits in both autonomousand non-autonomous systems and stability to neg-ative values of such exponents. But such issue isnot true in general. We will consider separatelythe two cases stating under what conditions the for-mer statement is true. We will construct one andtwo dimensional systems to show the di↵erencesbetween the autonomous and non-autonomous situ-ations. Obviously the non-autonomous case is toomuch more complicated. We will extend the resultsto n-dimensional systems of both types.

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Analytical and numerical results on escape ofbrownian particles

Carey CaginalpBrown University, USAcarey caginalp@brown.edu

A particle moves with Brownian motion in a material,e.g. a unit disc, with reflection from the boundariesexcept for a portion (called a ”window” or ”gate”)in which it is absorbed. A key question involves thedi↵erences between the small, finite step size and thelimiting Brownian motion. The stochastic problemof Brownian motion can be transformed into an el-liptic PDE with mixed boundary conditions. Ourwork confirms an asymptotic formula that had beenobtained by Chen and Friedman. Furthermore, weobtain an exact solution for a gate of any size.

The main problems are to determine the firsthitting time and spatial distribution. Also given isthe probability density of the location where a par-ticle hits if initially the particle is at the center oruniformly distributed. Numerical simulations of thestochastic process with finite step size and su�cientamount of sample paths are compared with the ex-act solution to the Brownian motion (the limit of zero

step size), providing an empirical formula for the di-vergence. Histograms of first hitting times are alsogenerated. These problems have applications to cellbiology and materials science.

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Asymptotic behavior of linear elliptic prob-lems with Dirichlet conditions on randomvarying domains

Carmen Calvo-JuradoUniversidad de Extremadura, Spainccalvo@unex.esJuan Casado-Dıaz, Manuel Luna-Laynez

Given (⌦,F , P ) a probability space, we study theasymptotic behavior of the solutions of linear ellipticproblems posed in a fixed domain O ⇢ RN , N � 3,satisfying Dirichlet boundary conditions on a randomsequence of “holes” �n(!) ⇢ O, P -a.e. ! in ⌦. Un-der assumptions about the size and the distributionof the holes, analogously to the classical Cioranescu-Murat “strange term”, we show the existence of anew term of order zero in the limit equation. Theproof of this result is based on the ergodic theoryand Nguetseng and Allaire’s two scale convergencemethod.

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Asymptotic behaviour of lattice systems per-turbed by additive noise

Tomas CaraballoUniversidad de Sevilla, Spaincaraball@us.es

In this talk we study the asymptotic behaviour ofsolutions of a first-order stochastic lattice dynamicalsystem perturbed by noise. We do not assume anyLipschitz condition on the nonlinear term, just acontinuity assumption together with growth and dis-sipative conditions, so that uniqueness of the Cauchyproblem fails to be true. Using the theory of multi-valued random dynamical systems we prove the ex-istence of a random compact global attractor.

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192 9th AIMS CONFERENCE – ABSTRACTS

Fractional stochastic porous media equations

Maria Garrido-AtienzaUniversity of Seville, Spainmgarrido@us.es

The aim of this talk is to study the existence anduniqueness of solutions of stochastic porous mediaequations driven by fractional Brownian motion,when the Hurst parameter H > 1/2. Using the the-ory of random dynamical systems we also discusssome results on the asymptotic behavior of thesesolutions.

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Cahn-Hilliard equations with memory e↵ects

Maurizio GrasselliPolitecnico di Milano, Italymaurizio.grasselli@polimi.itC.G. Gal, C. Cavaterra

We introduce a modified Cahn-Hilliard equation pro-posed by P. Galenko et al. in order to account forrapid spinodal decomposition in deep supercoolingglasses or in binary alloys. In this equation thedynamics of the order parameter depends on pasthistory of the chemical potential. Such dependenceis expressed through a time convolution integral char-acterized by a smooth nonnegative exponentially de-creasing memory kernel. We intend to present someresults on this non-standard equation with specialregard to dynamic boundary conditions and long-time behavior of solutions. In particular, we discussthe convergence of solutions when the memory kernelapproaches the Dirac mass.

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A fractional stochastic Schrodinger equation

Wilfried GreckschMartin-Luther-University, Germanywilfried.grecksch@mathematik.uni-halle.de

We consider (⌦,F , (Ft)t�0, P ) to be a filtered com-plete probability space.Let (V, (·, ·)V ) and (H, (·, ·))be separable complex Hilbert spaces,such that(V,H, V ⇤) forms a triplet of rigged Hilbert spaces.Let K be a separable real Hilbert space.We assume⇣W (t)

⌘

t�0to be a K-valued cylindrical Wiener pro-

cessadapted to the filtration (Ft)t�0 and (Bh(t))t�0

to be a K-valuedcylindrical fractional Brownian mo-tion with Hurst index h 2 ( 1

2, 1) adapted to the

filtration (Ft)t�0.We study the properties of the variational solutionX of the following stochastic nonlinear evolution

equation of Schrodinger type

(X(t), v) = (X0, v)

� i

Z t

0

hAX(s), vids

+ i

Z t

0

(f(s,X(s)), v)ds

+ i(

Z t

0

g(s,X(s))dW (s), v)

+ i(

Z t

0

b(s)dBh(s), v)

for a.e. ! 2 ⌦ and all t 2 [0, T ], v 2 V .Especially, conditions are given such that the Malli-avin derivative of the solution process exists.

References[1] Grecksch, W.; Lisei, H. Stochastic NonlinearEquations of Schrodinger Type.Stoch. Anal.Appl.29(2011), No 4, 631-653.

[2] Grecksch, W.; Lisei, H. Stochastic SchrodingerEquation Driven by Cylindrical Wiener Process andFractional Brownian Motion.Stud. Univ. Babes-Bolyai Math. 56(2011), No.2, 381-391.

�!1 ⇧1 �On the global attractor for the Kazhikhov-Smagulov equations

Juan Gutierrez SantacreuUniversidad de Sevilla, Spainjuanvi@us.es

In this talk we will study the existence of theglobal attractor for the Kazhikhov-Smagulov equa-tion which govern the dynamics of the two fluidshaving di↵erent densities subject to Fick’s law:8><

>:

@t⇢ + u · r⇢ � ��⇢ = 0 in ⌦ ⇥ (0,1),⇢@tu + (⇢u · r)u � µ�u + rp

��(r⇢ · r)u � �(u · r)r⇢ = ⇢f in ⌦ ⇥ (0,1),r·u = 0 in ⌦ ⇥ (0,1),

for a bounded two-dimensional domain ⌦, withinitial conditions

u(x, t) = 0, @n⇢(x, t) = 0 (x, t) 2 @⌦⇥ (0,1),

and the boundary conditions

⇢(x, 0) = ⇢0(x), u(x, 0) = u0(x) x 2 ⌦.

Here u : ⌦ ⇥ (0,1) ! R2 is the velocity vectorfield, p : ⌦ ⇥ (0,1) ! R is the pressure scalarfield, and ⇢ : ⌦⇥ (0,1) ! R is the density scalarfield. The parameters µ and � are assumed tobe constant and represent the kinematic viscos-ity and a mass di↵usion coe�cient, respectively.

To simplify the discussion, we will only focuson the evolution of the velocity. For this, we will

SPECIAL SESSION 45 193

take a fixed initial density data ⇢0. Then we willprove that the velocity trajectories of the weaksolutions of the Khazhikov-Smagulov equationsare absorbed by a connected, compact invariantset A in the natural phase space for the Navier-Stokes problem.

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Features of fast living: on the weak se-lection for longevity in degenerate birth-death processes.

Hyejin KimUniversity of Michigan, Ann Arbor, USAkhyejin@umich.eduYen Ting Lin, Charles Doering

Deterministic descriptions of dynamics of com-peting species with identical carrying capaci-ties but distinct birth, death, and reproductionrates predict steady state coexistence with pop-ulation ratios depending on initial conditions.Demographic fluctuations described by a Marko-vian birth-death model break this degeneracy. Anovel large carrying capacity asymptotic theoryconfirmed by conventional analysis and simula-tions reveals a weak preference for longevity inthe deterministic limit with finite-time extinc-tion of one of the competitors on a time scaleproportional to the total carrying capacity.

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Some generalizations of the Cahn-Hilliardequation

Alain MiranvilleUniversite de Poitiers, Francemiranv@math.univ-poitiers.fr

Our aim in this talk is to discuss a Cahn-Hilliardmodel with a proliferation term which has, e.g.,applications in biology. In particular, we willdiscuss the asymptotic behavior of the system interms of finite-dimensional attractors.

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SDEs driven by fractional Brownian mo-tion: continuous dependence on the Hurstparameter

Andreas NeuenkirchUniversity of Mannheim, Germanyneuenkirch@kiwi.math.uni-mannheim.deM.J. Garrido-Atienza

In this talk, we study the stochastic di↵erentialequation

dXHt = a(XH

t )dt+ �(XHt )dBH

t ,

t 2 [0, T ], XH0 = x0 2 Rd, where a : Rd ! Rd

and � : Rd ! Rd,m satisfy standard smooth-ness assumptions and BH = (BH

t )t2[0,T ] is anm-dimensional fractional Brownian motion withHurst parameter H 2 (0, 1) defined on a prob-ability space (⌦,A,P). Using tools from roughpath theory we show that the law of the solu-tion XH = (XH

t )t2[0,T ] depends continuously onH 2 [1/2, 1), i.e. we have

PXH

�! PXH0for H ! H0

with H,H0 2 [1/2, 1). Moreover, in the case ofadditive noise we give a stronger pathwise con-tinuous dependence result for H 2 (0, 1) and dis-cuss applications to random attractors of suchequations.

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Localization of solutions to stochasticporous media equations: finite speed ofpropagation

Michael RoecknerUniversity of Bielefeld, Germanyroeckner@math.uni-bielefeld.deViorel Barbu

We present a localization result for stochasticporous media equations with linear multiplica-tive noise. More precisely, we prove that thesolution process P-a.s. has the property of ”fi-nite speed propagation of disturbances” in thesense of [Antontsev/Shmarev, Nonlinear Analy-sis 2005].

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194 9th AIMS CONFERENCE – ABSTRACTS

The e↵ects of noise on sliding motion

David SimpsonThe University of British Columbia, Canadadsimpson@math.ubc.caRachel Kuske

Vector fields that are discontinuous alongcodimension-one surfaces are used as mathe-matical models of a wide range of physical sys-tems involving a discontinuity or switch such asrelay control systems and vibro-impacting sys-tems. Trajectories evolve on discontinuity sur-faces whenever the vector field on both sides ofthe surface points towards the surface. This isknown as sliding motion and is formulated bythe method of Filippov. To investigate the pos-sible role of background vibrations, parametricuncertainty and model error, we analyze the ef-fect of small, additive, white Gaussian noise onsliding trajectories in a simple two-dimensional,discontinuous vector field. We find that the noisepushes orbits slightly o↵ the discontinuity surfaceand that this may induce a significant change inthe observed dynamics. In addition, for thissystem we show that the mean of the stochas-tic solution limits on Filippov’s definition as thenoise amplitude is taken to zero.

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Su�cient and necessary criteria for ex-istence of pullback attractors for non-compact random dynamical systems

Bixiang WangNew Mexico Tech, USAbwang@nmt.edu

We study pullback attractors of non-autonomousnon-compact dynamical systems generated bydi↵erential equations with non-autonomous de-terministic as well as stochastic forcing terms.We first introduce the concepts of pullback at-tractors and asymptotic compactness for suchsystems. We then prove a su�cient and neces-sary condition for existence of pullback attrac-tors. We also introduce the concept of completeorbits for this sort of systems and use these spe-cial solutions to characterize the structures ofpullback attractors. For random systems con-taining periodic deterministic forcing terms, weshow the pullback attractors are also periodic.As an application, we prove the existence of aunique pullback attractor for Reaction-Di↵usionequations on unbounded domains with both de-terministic and random external terms.

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