Spectral analysis of non-local Markov generatorskondrat/Presentations/BI_SEOUL.… · Spectral analysis of non-local Markov generators Yuri Kondratiev Bielefeld University, Germany

Spectral analysis of non-local Markovgenerators

Yuri Kondratiev

Bielefeld University, GermanyNational Dragomanov University, Kyiv

Yuri Kondratiev (Bielefeld) Spectral Analysis 1 / 27

Introduction

Schrodinger operators in physics:

H = −∆ + V = H0 + V

In MathSciNet: 17547 references!

Stochastics:

Markov generator ∆→Heat semigroup Tt = et∆ →Heat kernel pt(x, y)→Transition probability Pt(x, dy) = pt(x, y)dy →Brownian motion Bt


Random walks and compound Poisson

For a given 0 ≤ a ∈ L1(Rd) (normalized) consider an operator

Laf(x) =

∫Rda(x− y)[f(y)− f(x)]dy

in spaces Cb(Rd) or L2(Rd). This operator is a Markov generator andcorresponding Markov process Xt is a random walk in Rd. This process has usefulrepresentation as a compound Poisson process:

Xt =

Nt∑k=1

ξk,

where ξk are iid random vectors with the distribution

P (dx) = a(x)dx

and Nt is homogeneous Poisson process.


Singular kernels

There is a special case when

a(x) =b(x)

|x|β

with a regular b(x). This situation is essentially different:

La = L1 + L2,

with unbounded L1 (a la fractional Laplacian) and bdd L2. Then spectralproperties are defined by L1.This case is widely studied:

Chen, Grigoryan, Kassmann, P.Kim, Kumagai ...

For the case of non-singular kernel we have much less informations. Particularresults by Berestitsky, Coville, Rossi, ....


Basic references

Grigor’yan, Alexander; Kondratiev, Yuri; Piatnitski, Andrey; Zhizhina, ElenaPointwise estimates for heat kernels of convolution-type operators. Proc. Lond.Math. Soc. (3) 117 (2018), no. 4, 849 – 880.

Kondratiev, Yuri; Molchanov, Stanislav; Piatnitski, Andrey; Zhizhina, ElenaResolvent bounds for jump generators. Appl. Anal. 97 (2018), no. 3, 323 – 336.

Kondratiev, Yu.; Molchanov, S.; Vainberg, B. Spectral analysis of non-localSchrdinger operators. J. Funct. Anal. 273 (2017), no. 3, 1020 – 1048.

Kondratiev, Yuri; Molchanov, Stanislav; Pirogov, Sergey; Zhizhina, Elena Onground state of some non local Schrdinger operators. Appl. Anal. 96 (2017), no.8, 1390 –1400.

Anatoly N. Kochubei, Yuri Kondratiev, Ground States for Nonlocal SchrdingerType Operators on Locally Compact Abelian Groups, arXiv:1807.09491, to appearin J.Spectral Theory (2019)


Schrodinger type operators in biology

Contact model in the continuum: K/Skorokhod 2006.Density evolution in a population

∂u

∂t= Lu, u = u(t, x), x ∈ Rd, t ≥ 0.

Lu(x) = −m(x)u(x) +

∫Rda(x− y)u(y)dy.

Assumptions:

0 ≤ a ∈ L1(Rd) ∩ Cb(Rd),∫Rda(x)dx = 1.

m ∈ Cb(Rd), 0 ≤ m(x) ≤ 1, m(x)→ 1, x→∞.

For m = 1 (critical value) we have a stationary regime.


Another formL = L0 + V,

L0f(x) =

∫Rda(x− y)[f(y)− f(x)]dy, V (x) = 1−m(x),

0 ≤ V (x) ≤ 1.

Biological meaning of V :

local fluctuation below critical value of mortality.


Ground state problem

We are searching for the maximal eigenvalue λ > 0 s.t.

Lψλ = λψλ.

ψλ is called the ground state. In the case of L2(Rd) we have the uniqueness ofψλ > 0 (positivity improving semigroup).

Lemma

1) If ψλ ∈ Cb(Rd), then ψλ(x)→ 0, x→∞.2) If ψλ ∈ L2(Rd), then ψλ ∈ Cb(Rd) and ψλ → 0, x→∞.3) If ψλ ∈ Cb(Rd) and V ∈ L2(Rd), then ψλ ∈ L2(Rd).


Existence of ground states in Cb(Rd)

Theorem (local paradise)

Assume that there exists δ > 0 s.t.

∀x ∈ Bδ(x0) ⊂ Rd V (x) = 1(i.e.,m(x) = 0).

Then the ground state of L0 + V exists.

Theorem (happy island )

Assume that for certain β ∈ (0, 1) there exists R > 0 s.t.

∀x ∈ BR(x0) V (x) ≥ β.

Then there exists R(β) s.t. for all R ≥ R(β) the ground state exists.


Theorem (Low dimensions)

Let d = 1, 2 and ∫Rd|x|2a(x)dx <∞.

Then for all V 6= 0 the ground state exists.


Ground states in L2(Rd)

This case is even simpler due to the s.-a. of L. In particular, all statements ofprevious theorems are valid in L2(Rd).We can consider also the subcritical regime m > 1. For this value the populationwill be degenerated:

0 < u(t, x) ≤ Ce−αt.

We rewriteL = L0 + (1−m(x)) = L0 + V (x)− h,

with V (x) = m−m(x), h = m− 1. We assume a bdd support for V .

TheoremAssume there exists δ > 0 s.t.

∀x ∈ Bδ(x0) V (x) = m.

Then the ground state exists.


Density asymptotic

Theorem

Assume there exists a unique ground state ψ > 0 of L = L0 + V in L2(Rd) andλ > 0 be the maximal eigenvalue. Then in L2(Rd)

u(t, x) = Ceλtψ(x)(1 + o(1)), t→∞.

In Cb(Rd) for any bdd domain D ⊂ Rd holds∫D

u(t, x)dx→∞, t→∞.


Front propagation

Under previous assumptions about ground states, the density

∂u

∂t= (L0 + V )u, u = u(t, x), x ∈ Rd, t ≥ 0.

will grow point-wisely. Define the propagation front

F (t) = {x ∈ Rd | u(t, x) = 1}.

To analyze F (t) we need an additional assumption about the jump kernel:

a(x) ≤ Ce−|x|α

, α > 1 ultra light tails.

Theorem

The front F (t) has the form

F (t) = {|x| = C(λ)(λt+1− d

2log t) +O(1)}, t→∞.

The density u(t, x) grows (decays) exponentially in time uniformly in x in anyregion inside (outside) the front whose distance from the front exceeds γt withsome γ > 0.


Resolvent bounds

Denote Rλ(L0) = Rλ the resolvent operator. For the kernel Rλ(x, y) of theresolvent we have

Rλ(x, y) = (1 + λ)−1(δ(x− y) +Gλ(x− y)).

We will study asymptotic Gλ(x) which essentially depends on the behavior of thejump kernel a(x) in the operator L0.


Polynomial tails

Assume

c−(1 + |x|)−(d+α) ≤ a(x) ≤ c+(1 + |x|)−(d+α)

with some α > 0.

Theorem

There exist constant 0 < C−(λ) ≤ C+(λ) s.t.

C−(1 + |x|)−(d+α) ≤ Gλ(x) ≤ C+(1 + |x|)−(d+α).

Further,C+(λ) = O(λ−(d+α+2)), λ→ 0

and

Gλ(x) ≥ C0

λ(1 + |x|)−(d+α)

for large enough |x|.


Light tails

Consider the casea(x) ≤ Ce−δ|x|

for some C, δ > 0.

Theorem

There exist positive constants k(λ),m(λ) s.t.

Gλ(x) ≤ k(λ)e−m(λ)|x|

andk(λ)→∞, m(λ) = O(λ) λ→∞.


Applications I

We come back to the Schrodinger operator L0 + V assuming previous conditionson V for the existence of the g.s. ψλ(x).

Theorem

Let V ∈ C0(Rd) and V 6= 0. If

c−(1 + |x|)−(d+α) ≤ a(x) ≤ c+(1 + |x|)−(d+α)

thenc−(λ)(1 + |x|)−(d+α) ≤ ψλ(x) ≤ c+(λ)(1 + |x|)−(d+α).

For the case of light tails

ψλ(x) ≤ k(λ)e−m(λ)|x|.

For d = 1ψλ(x) = e−q(λ)|x|(c(λ) + o(1)), x→∞.


Application II

Let us consider an infection spreading model in which the density of infectedpopulation is described by the equation (coming from a microscopic IPS)

∂u

∂t(t, x) = L0u(t, x)−mu(t, x) + f(x).

Here m > 0, 0 ≤ f ∈ C0(Rd) is a source function.For f = 0 we have a model in which u(t, x)→ 0, t→∞ exponentially fast.

How the source of the infection may affect the density of infected population?


Let w(x) be the stationary solution. Then in both spaces L2(Rd) and Cb(Rd)holds

‖u(t, ·)− w(·)‖ → 0, t→∞.

For the case of polynomially decaying kernel we have

c−(1 + |x|)−(d+α) ≤ w(x) ≤ c+(1 + |x|)−(d+α)

That means that even a very local source may produce a long range spreading ofthe infection. In such a case it is hopeless to grow the recovering intensity m (thatmeans essentially a very expensive medical care). We shall control the source.

In the case of light tails w(x) will decay exponentially fast. Then we have anotherway to control infection spreading by essential restrictions on the contacts in thesociety. Is this way acceptable politically?


NLSO on locally compact Abelian groups

Let G be a second countable locally compact non-compact Abelian group, µdenotes the Haar measure on G.As above we consider an operator

Lu = L0u+ V (x)u,

where

L0u(x) =

∫Ga(x− y)[u(y)− u(x)]dµ(y).

We assume V ∈ Cb(G), 0 ≤ V (x) ≤ 1 and V (x)→ 0, x→∞.


Let us consider the probability measure µa(dx) = a(x)µ(dx) on G. Ifξ1, . . . , ξn, . . . are independent G-valued random variables with their probabilitylaws µa, then the random walk with initial point S0 is the Markov chain

Sn = S0 + ξ1 + · · ·+ ξn.

A RW is said to be recurrent if for some compact neighborhood M of 0

∞∑k=1

P{Sk ∈M | S0 = 0} =∞.

Introduce the following assumption

(G0) : The minimal subgroup generated by supp(µa) coincides with G.

Theorem

Let condition (G0) satisfied and the corresponding RW is recurrent. For anyV 6= 0 satisfying conditions above the ground state of the operator L0 + V inCb(G) exists.


The p-adic case

Motivations: protein dynamics, Avetisov et. al., 2002-2014

Consider the case G = Qnp . For N ∈ Z denote

BN = {x ∈ Qnp | |x| ≤ pN}.

Theorem

Suppose V (x) = 1 for x ∈ BN . Then the ground state of L exists.

Theorem

Assume that for some β ∈ (0, 1) there exists N ∈ Z s.t.

β ≤ V (x) ≤ 1, x ∈ BN .

Then the ground state exists if N = N(β) is sufficiently large.


Heat kernels

In the case of singular kernels there are known global estimations for the relatedheat kernels, e.g., [Grigoryan, Kumagai], [Chen,Kim,Kumagai]. For the regularkernels the situation is completely different.For the semigroup etL0 the (heat) kernel has form

Gt(x) = e−tδ(x) + v(t, x).

Our aim is to study v(t, x).Consider first the particular case of a Gaussian jump kernel

a(x) =1

(4π)d/2e−

|x|24 .


TheoremFor the Gaussian jump kernel the following asymptotic for t→∞ holds:1) For any r > 0 and |x| ≤ rt1/2

v(t, x) =1

(4π)d/2e−

|x|24t (1 + o(t−1/4)).

2) For any r > 0 and |x| = rt1+δ2 with 0 < δ < 1

log v(t, x)|x|24t

→ −1.


Theorem (continuation)

3) For any r > 0 and |x| = rt

log v(t, x)

t→ −Φ(r),

where0 ≤ Φ(r) ≤ r2/4,

Φ(r) = r2/4(1 + o(1)), r →) + 0,

Φ(r) = r(log r)1/2(1 + o(1)), r →∞.

4) If |x| > t1+δ2 with δ > 1, then

log v(t, x)

|x|(log |x|t )1/2→ −1.


Concluding remarks

These results about heat kernels are extended (with proper modifications) to thecase of general light tails kernels.

The case of heavy tails is an open problem. In general, spectral properties ofNLSO may depend essentially on properties of jump kernels. That is importantdifference comparing with the case of diffusion (uniformly elliptic) generators.Possible analogies can be related with diffusion generators withdegenerated/abnormal diffusion coefficients.

Considered CTRW in Rd have σ-finite invariant measures. Ergodicity of suchMarkov processes, LLN, CLT etc. are essentially (surpizingly?) unknown.


There are non-linear non-local evolution equations appearing in Markov dynamicsof birth-and-death IPS in the continuum, e.g, spatial logistic models. Theseequation do appear as the result of the mesoscopic scaling on the initialmicroscopic models.We refer to recent works

Dmitri Finkelshtein, Yuri Kondratiev, Pasha Tkachov:

————————-Existence and properties of traveling waves for doubly nonlocal Fisher-KPPequationsElectron. J. Differential Equations, Vol. 2019 (2019), No. 10, pp. 1-27.————————-Doubly nonlocal Fisher-KPP equation: Speeds and uniqueness of traveling waves,to appear in Journal of Mathematical Analysis and Applications (2019)————————Finkelshtein, Dmitri; Kondratiev, Yuri; Molchanov, Stanislav; Tkachov, PashaGlobal stability in a nonlocal reaction-diffusion equation. Stoch. Dyn. 18 (2018),no. 5, 1850037, 15 pp.————————+ several papers in ArXiv.


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Spectral analysis of non-local Markov generatorskondrat/Presentations/BI_SEOUL.… · Spectral analysis of non-local Markov generators Yuri Kondratiev Bielefeld University, Germany