Application of homogenization and large deviations to a nonlocal parabolic semi-linear equation

. We study the behavior of the solution for a class of nonlocal partial differential equation of parabolic-type with non-constant coefficients varying over length scale δ and nonlinear reaction term of scale 1 /ε , related to stochastic differential equations driven by multiplicative isotropic α -stable L´evy noise ( 1 < α < 2 ). The behavior is required as ε tends to 0 with δ small compared to ε . Our homogenization method is probabilistic. Since δ decreases faster than ε , we may apply the large deviations principle with homogenized coefficients.


Introduction
Let ε, δ > 0 small enough.Our aim in this article is to study the behavior of u ε,δ : R d −→ R of the following nonlocal partial differential equation (PDE) with parabolic-type : where the linear operator L α ε,δ is a nonlocal integro-differential operator of Lévy-type given by

Application of homogenization and LDP to a nonlocal parabolic PDE
Here B is the unit open ball in R d centering at the origin, and ν α,ε −1 (dy) := 1 ε ν α (dy) = ε −1 dy |y| d+α is the isotropic α-stable Lévy measure.In this paper, we use Einstein's convention that the repeated indices in a product will be summed automatically.
The combinatorial effects of homogenization and large deviation principle (LDP) is a classical problem which goes back to P. Baldi [1] at the end of 20'th century .Such a problem has been most extensively investigated by Freidlin and Sowers [7] in stochastic differential equations (SDE) and linear parabolic PDE on the whole of R d .Huang et al. [9] recently studied a nonlocal problem from the mathematical point of view of homogenization theory.They considered the nonlocal parabolic linear equation without the viscosity (large deviations principle) parameter ε, with linear reaction term of scale 1  δ α−1 .Inspired by [1,7], the work in this paper is highly motivated by the consideration to combine the two principles in a compatible way, for a class of semilinear parabolic PDE.The present paper will only focus on the subcritical case 1 < α < 2. There are both probabilistic and analytical difficulties for the supercritical case 0 < α ⩽ 1.All things considered, the nonlocal part has lower order than the drift part, so that one cannot regard the drift as a perturbation of the nonlocal operator.We first give the rate function S 0,t of the large deviations, in fact since δ tends faster to zero than ε this function is expressed by the homogenized coefficients of the PDE (1.1), next we express the solution of PDE (1.1) by the use of Backward stochastic differential equations (BSDE) in [2] and the Feynman-Kac formula, then we consider an auxiliary equation solved by ε log u ε,δ .The limit of this auxiliary equation helps us to find the limit of u ε,δ when both ε, δ tend to zero.We show in the end that there exists a function V * (which depends on S 0,t ) such that u ε,δ tends to zero if (t, x) ∈ {V * < 0} and tends to 1 in the interior of {V * = 0}.
We organize the paper as follows.In Section (2), we present some general assumptions and definitions.Section (3) contains the results of large deviations principle.In Section (4), we study the behavior of the solution of the PDE (1.1).

Preliminaries
By B r we means the open ball in R d centering at the origin with radius r > 0, we shall omit the subscript when the radius is one.We denote by C k C k b with integer k ⩾ 0 the space of (bounded) continuous functions possessing (bounded) derivatives of orders not greater than k.We shall explicitly write out the domain if necessary.Denote by (this seminorm can also be defined for the case γ ′ = 1, which is exactly the Lipschitz seminorm).In the sequel, the torus T d := R d /Z d will be used frequently.Denote by D := D R + ; T d the space of all T d -valued càdlàg functions on R + , equipped with the Skorokhod topology.We shall always identify the periodic function on R d of period 1 with its restriction on the torus T d .
Alioune COULIBALY (H.1)We assume that lim Let Ω, F, P, {F t } t⩾0 be a filtered probability space endowed with a Poisson random measure N α,ε −1 on be a d-dimensional isotropic α-stable Lévy process given by Given ε > 0, x ∈ R d , consider the following: or more precisely, Before continuing, we list some general assumptions for the PDE (1.1) and the nonlocal the SDE (2.1).We consider u 0 ∈ C b R d , R + and we set sup We assume that f : R d × R → R is periodic in each direction with respect to the first argument, and it verifies : And we assume that ii) For every y ∈ R d , the function x → (σ(x, y), c(x, y)) is periodic of period 1 in each component.
iii iv) The initial functions u 0 is continuous.

Application of homogenization and LDP to a nonlocal parabolic PDE
The function σ : R d × R d −→ R d satisfies the following conditions (for some comments see, [9]). (H.3) There exists a constant C > 0, such that for any ii) The oddness condition : for all x, y ∈ R d , σ(x, −y) = −σ(x, y).
iii) The Jacobian matrix with respect to the second variable and there exists a constant C > 0 such that (∇ y σ(x, y) Let us introduce the linear operator A σ,ν α defined as By virtue of the oddness condition and the symmetry of the jump intensity measure ν α , we can rewrite the operator A σ,ν α as : (see, [9]) where the kernel ν σ,α is given by Next, to move the SDE (2.1) to the torus T d , we define Xε,δε where and with ε δ L α,ε −1 by virtue of the self-similarity.We shall also consider the limit SDE (2.5), namely where, heuristically by the L'Hôpital's rule, σ(x, y) = ∇ y σ(x, 0)y is the point-wise limit of ε δε σ •, δε ε • as ε ↓ 0. We need a stronger convergence as follows: (H.4)For every y ∈ R d , 1 η σ(x, ηy) −→ (∇ y σ(x, 0))y uniformly in x ∈ R d , as η → 0.
Let us set L α be the linear integro-partial differential operator given by By requirement there exists a L α -Feller process on R d and by periodicity assumption on the coefficients such a process induces a process X which is a strong Markov process on the torus T d , moreover the L α -process is ergodic (see, [9]).We denote by µ its unique invariant measure on T d , B T d .In order to do the homogenization for the SDE X ε,δε (2.1), we need the following be in force ( [3,8,10]): (H.5)The centering condition : Thanks to [9, Proposition 4.11], there is a unique periodic solution b ∈ C α+β of the Poisson equation which satisfies the estimate ∥ b∥ α+β ⩽ C ∥ b∥ 0 + ∥b∥ β . (2.9) Now we set

Large deviation principle
The theory of large deviations is concerned with events A for which probability P X ε,δε ∈ A converges to zero exponentially fast as ε → 0 (see, [4]).The exponential decay rate of such probabilities is typically expressed in terms of a rate function J mapping R d into [0, +∞].Our method allows us to characterize the LDP by analysing the logarithmic moment generating function [4,Chap. 2.3].Initially the corresponding rate function is identified as the Legendre transform of the limit (when it exists) of the logarithmic moment generating function defined as:

If we set
Xε,δε then we have by Itô's formula where

Before proceeding, let us define for all z ∈ T d and for all
Now, by Girsanov's formula, we have where Ẽ is the expectation operator with respect to the probability P defined as e ⟨θ,σ( Xε,δε s ,y)⟩ − 1 − θ, σ Xε,δε s , y 1 B (y) ν α (dy)ds .
Let us set, for all z ∈ T d , for all θ ∈ R d : and let us set Ψ ε θ ∈ C α+β T d be the unique solution of Such a solution Ψ ε θ must exist again by the assumptions on the coefficients and the Fredholm alternative.So applying Itô's formula to Alioune COULIBALY Then putting (3.5) into the equation (3.3), we obtain where Ê is the expectation operator with respect to the probability P defined as Since the coefficients are bounded, we first notice that Recall an elementary result.
We let r = (δ ε /ε) γ for some γ ∈ R that will be chosen for B r .It follows from Lemma 3.1 that for all φ ∈ C α+β T d (see [9,Appendix]) : On the other hand, using a similar estimate once again, it follows Let J denote the Fenchel-Legendre transform of J .Then we have where ϱ(r) := r log r − r + 1, r ∈ R * + .Now, we state our main result.Theorem 3.2.Fix T > 0 and assume (H.1) -(H.5) hold true.Then for every x ∈ R d , the family X ε,δε : ε > 0 of R d -valued random variables has a large deviations principle with good rate function Next, let us consider Since the function J is convex we can show that