On Jumps Stochastic Evolution Equations With Application of Homogenization and Large Deviations

Abstract We consider a class of jumps and diffusion stochastic differential equations which are perturbed by two parameters: ε (viscosity parameter) and δ (homogenization parameter) both tending to zero. We analyse the problem taking into account the combinatorial effects of the two parameters ε and δ. We prove a Large Deviations Principle estimate for jumps stochastic evolution equation in case that homogenization dominates.


Introduction
Let ε, δ > 0, we consider the following stochastic differential equations (SDE) : where {W t : t 0} is a standard Brownian motion and L ε,δ := { L ε,δ t : t 0 } is a Poisson point process with continuous compensator, independent of W, both defined on a given filtered probability space (Ω, F , P, F) with F := {F t : t 0} being the P-completion of the filtration F .More precisely, we assume that L ε,δ takes the form: where k is a given measurable function, and N is a Poisson random counting measure on R d with mean Lèvy measure (or intensity measure) ν.The coefficients b, c, σ are subject to suitable conditions.
The main idea of this paper is to get a variational representation formulas for the logarithmic moment generating function associated with the process X x,ε,δ (1), and to deduct the Large Deviations principle (LDP).For Brownian functional with combinatorial effect of LDP and Homogenization theory, Freidlin and Sowers, 1999, have shown three classical regimes depending on the relative rate at which the small viscosity parameter ε and the homogenization parameter δ tend to zero.They have proved some Large Deviations estimates which are very effective for SDE.Up to our knowledge, there are no results on the combination of Homogenization and LDP for Poisson functional.However, there are still few results on the large deviation for stochastic evolution equations with Poisson jumps (see for example, Röckner & Zang, 2007).At the same time, Large deviations for stochastic evolution equations driven by Brownian motions have been investigated in various literatures.Motivated by the above works, we would like to prove some large deviations properties for such Markov processes with diffusion and jumps, in the sense that the homogenization parameter tends to zero faster than the viscosity parameter.
The paper is organized as follows.In Section (2) we set up some notation, make precise our hypothesis and give the outline of the main technique for showing LDP.In Section (3) we state the main result and give its proof.

Preliminaries
In this section, we closely follow the notations in (Freidlin & Sowers, 1999).Before continuing, we recall with no loss of generality δ := δ ε and we assume that

Notations and Formulation
Denote expectation with respect to P by E and the gradient operator by ∇.We have already defined ⟨. ) , and we also assume that We assume that b, c in (1) are in We now turn our attention to the Poisson part.We first consider a Poisson random measure N ε −1 (., .) on [0, T } × R d defined on the space probability (Ω, F , P), with Lèvy measure ε −1 ν such that the standard integrability condition holds: ∫ The compensator of εN ε −1 is thus the deterministic measure ε Nε −1 (dtdy) := dtν(dy) on [0, T } × R d .In this paper we shall be interested in Poisson point process of class (QL), namely a point process whose counting measure has continuous compensator (see, Ikeda & Watanabe, 1981).More precisely, in light of the representation theorem of the Poisson point process (Ikeda & Watanabe, 1981, Chap. II, Theorem 7.4), we shall assume that L ε,δ ε is a pure jump process of the following form : where with respect to first variable, integrable with respect to dtdy, so that the counting measure of L ε,δ ε , denoted by N L ε,δε (dtdy) takes the form : and its compensator is therefore NL ε,δε (dtdy ) (s)ν(dy)dt, and hence continuous, i.e.L ε,δ ε has continuous statistic.
The Markov processes X ε,δ ε that we consider include jump processes and diffusion.Next let's write down its generator on twice continuously differentiable functions with compact support by where the matrix a := ) is factored as a := σσ * , and * denotes the tanspose.We set and require the following : (Growth) there exists constant C 2 such that for any By requirement there exists a L-diffusion with jumps on R d and by periodicity assumption on the coefficients such a process induces process X which has both a diffusion component and a jump component on the d-dimensional torus T d , moreover the L-diffusion-part process is ergodic.We denote by m its unique invariant measure.In order for the process with generator L ε,δ ε to have a limit in law as ε → 0, we need the following condition to be in force.
To move the SDE (1) to the torus T d , we define the pull-back which satisfies the SDE : where The infinitesimal generator L ε,δ ε of the diffusion component associated to the process X ε,δ ε is This generator tends to L defined in (7), as ε → 0. .Finally, we give some definitions.

Usual Formulas
The proof of Theorem 3.1, in Section 3, relies on explicit calculation of the logarithm moments of X ε,δ ε and the followings Girsanov's formula and Itô's formula.Before proceeding, let us introduce some space.
For E locally compact, let H 2 (T, µ) be the linear space of all equivalence classes of mappings F : [0, T ] × E × Ω −→ R which coincide almost everywhere with respect to dt ⊗ dµ ⊗ dP and which satisfy the following conditions : • F is predictable; We endow H 2 (T, µ) with the inner product ⟨F, G⟩ T,µ := ) is a real separable Hilbert space.
Let N p be a Poisson random measure on R + × (E\{0}) with intensity measure µ, according to a given F T -adapted, σ-finite point process p which is independent of the Brownian motion W. Let N p be the associated compensated Poisson random measure.Now we have (see, Applebaum, 2009, Chapter 5, Section 2) Lemma 2.1 (Girsanov's formula).
Let X be a Lèvy process such that e X is a martingale, i.e: We suppose that there exists C > 0 such that For L ∈ H 2 (T, µ) we define Finally, we define Let Q be the probability measure on (Ω, F T ) defined as: Then under Q, B t is a Brownian motion and N t is a Q-martingale.
Next define a d-dimensional semi-martingale Y t := (Y 1 , . . ., Y d ) by where • M t is locally continuous square integrable (F t )-martingale and M 0 := 0; • A t is a continuous (F t )-adapted process whose almost all sample functions are of bounded variation on each finite interval and A 0 := 0; • g is (F t )-predictable and for t > 0, • f is (F t )-predictable and for t > 0, We have (see, for example Ikeda & Watanabe, 1981, Theorem 5.1) Lemma 2.2 (Itô's formula).
Let F be a function of class C 2 on R d and Y t a d-dimensional semi-martingale given in (11).Then the stochastic process F (Y t ) is also a (F t )-semi-martingale and the following formula holds :

Outline of the LDP Characterization
The theory of large deviations is concerned with events A for which probability P ( X x,ε,δ ε ∈ A ) converges to zero exponentially fast as ε → 0. The exponential decay rate of such probabilities is typically expressed in terms of a rate function The following standard result is the main technique for showing that X x,ε,δ ε has LDP.For T > 0 and x ∈ R d , define Now define g T,x (θ) := lim When this limit exists, we then have (see, for example Dembo & Zeitouni, 1993, Chap. 2.3) ii.) the origin is in the interior of the set Then the random variables have a Large Deviations Principle with rate function I T,x defined by

Large Deviations Principle
Before proceeding, let us set By assumption on a, the matrix a 0 is strictly positive-definite.Then letting a −1 0 be its inverse matrix, we define the norm θ where S ν denotes the support of ν.Let con ( Q a 0 ,ν ) be the smallest convex cone that contains Q a 0 ,ν and define Let ri(A) denotes the relative interior of a set A. In light of the characterization lemma (see, Ikeda & Watanabe, 1981, Lemma 10.2.3) to the effective domain of J(θ) := inf ) .

valued random variables has a large deviations principle with good rate function
Proof.
We set Xε,δ , by the Itô's formula, we have ) So, we are going to consider the logarithm cumulant generating function of X x,ε,δ ε , g ε T,x (12).By Girsanov's formula, we have where E is the expectation operator with respect to the probability P defined as and let us set Such a solution Ψ must exist again by the assumptions on the coefficients (see, for instance, Pardoux & Veretennikov, 2001).So applying Itô formula to )] N (δ ε /ε) 2 (dyds).
Then, for all θ ∈ R d , we have the following asymptotic.
, .⟩ as the standard Euclidean inner product on R d , and ∥.∥ as the associated norm.Let C p ( R d , R d ) be the collection of continuous mapping from R d into R d which are periodic of period 1 in each coordinate of the argument and let ∥.∥ C p (R d ,R d ) be the associated sup norm.Let T d be the d-dimensional torus of size 1, and let ∥.∥ C(R d ,R d ) be the standard sup norm on C