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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
Yuri Kondratiev (Bielefeld) Spectral Analysis 2 / 27
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.
Yuri Kondratiev (Bielefeld) Spectral Analysis 3 / 27
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, ....
Yuri Kondratiev (Bielefeld) Spectral Analysis 4 / 27
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)
Yuri Kondratiev (Bielefeld) Spectral Analysis 5 / 27
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.
Yuri Kondratiev (Bielefeld) Spectral Analysis 6 / 27
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.
Yuri Kondratiev (Bielefeld) Spectral Analysis 7 / 27
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).
Yuri Kondratiev (Bielefeld) Spectral Analysis 8 / 27
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.
Yuri Kondratiev (Bielefeld) Spectral Analysis 9 / 27
Theorem (Low dimensions)
Let d = 1, 2 and ∫Rd|x|2a(x)dx <∞.
Then for all V 6= 0 the ground state exists.
Yuri Kondratiev (Bielefeld) Spectral Analysis 10 / 27
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.
Yuri Kondratiev (Bielefeld) Spectral Analysis 11 / 27
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→∞.
Yuri Kondratiev (Bielefeld) Spectral Analysis 12 / 27
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.
Yuri Kondratiev (Bielefeld) Spectral Analysis 13 / 27
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.
Yuri Kondratiev (Bielefeld) Spectral Analysis 14 / 27
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|.
Yuri Kondratiev (Bielefeld) Spectral Analysis 15 / 27
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→∞.
Yuri Kondratiev (Bielefeld) Spectral Analysis 16 / 27
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→∞.
Yuri Kondratiev (Bielefeld) Spectral Analysis 17 / 27
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?
Yuri Kondratiev (Bielefeld) Spectral Analysis 18 / 27
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?
Yuri Kondratiev (Bielefeld) Spectral Analysis 19 / 27
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→∞.
Yuri Kondratiev (Bielefeld) Spectral Analysis 20 / 27
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.
Yuri Kondratiev (Bielefeld) Spectral Analysis 21 / 27
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.
Yuri Kondratiev (Bielefeld) Spectral Analysis 22 / 27
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 .
Yuri Kondratiev (Bielefeld) Spectral Analysis 23 / 27
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.
Yuri Kondratiev (Bielefeld) Spectral Analysis 24 / 27
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.
Yuri Kondratiev (Bielefeld) Spectral Analysis 25 / 27
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.
Yuri Kondratiev (Bielefeld) Spectral Analysis 26 / 27
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.
Yuri Kondratiev (Bielefeld) Spectral Analysis 27 / 27