We prove that the stochastic Nonlinear Schrödinger (NLS) equation is the limit of NLS equation with random potential with vanishing correlation length. We generalize the perturbed test function method to the context of dispersive equations. Apart from the difficulty of working in infinite dimension, we treat the case of random perturbations which are not assumed uniformly bounded.
We study in this work the limit of Non-Linear Schrödinger Equation with randomness. More precisely, we consider the following problem
on the domain , with regular initial data. Such an equation occurs in many situations, for instance in optical fibers dynamics (see [1,25]). More generally, the Nonlinear Schrödinger equation is an equation describing wave propagation in a nonhomogeneous dispersive medium and random effects often enter the description via a potential. Here we consider such a random potential which depends on time and space with a scaling of the form .
Under such a scaling, we are in the situation of approximation-diffusion. The random term formally converges to a spatially dependent white noise in time and we expect to obtain a white noise driven stochastic partial differential equation at the limit. Such stochastic non-linear Schrödinger equations are used in the physics literature and have been mathematically studied, for example in a conservative version by Debussche and de Bouard ([9–11]…). Barbu, Röckner and Zhang proved in [2] well posedness results in both conservative and nonconservative cases thanks to rescaling transformations, while Brzezniak and Millet studied in [5] the stochastic Nonlinear Schrödinger equation on a two-dimensional manifold. We can also cite [21] and [22] where the authors study the one dimensional -critical and supercritical cases for Nonlinear Schrödinger equation with spatially correlated noise and space time white noise. We believe that it is important to prove that these equations are limits of equations with realistic noises such as (1.1).
Our basic tool is the Perturbed Test Function method. This method provides an elegant way for approximation-diffusion problems; it was first introduced by Papanicolaou, Stroock and Varadhan [23] in a finite dimensional case, and one can find many applications in the book of Fouque, Garnier, Papanicolaou and Solna [18]. Generalizations in infinite dimension have been recently developped, for instance in [12,13] or [16]. Our aim is to develop this method in the context of the Nonlinear Schrödinger equation with a random potential. The difficulty is that the fundamental object for this method is the infinitesimal generator of the Markov process associated to (1.1), which is a complicated object since we work in infinite dimension. Besides the Perturbed Test Function method is based on compactness arguments which are not trivial in the case of (1.1) whose space variable lives in the full space . We overcome this latter problem by working in a weighted space.
In [2], the main tool to study the stochastic Nonlinear Schrödinger equation is the rescaling transform. This changes the stochastic equation containing a spatially dependent white noise into a Nonlinear Schrödinger with continuous in time random terms. The counterpart is that the Laplace operator is replaced by a linear differential operator with varying coefficient. It is not clear how to use the scaling transform for our study since these linear operators would depend on ε. Moreover, the rescaling transform is useful only for purely multiplicative noise. Our arguments immediately extend to more general equations, for instance if is replaced by for a Nonlinear operator of Nemitsky type Λ with sublinear growth such that is real valued. For the sake of clarity we decided to restrict to the case .
The driving noise m is an ergodic Markov process with values in a Sobolev space to be described below. In all the articles mentioned above on approximation-diffusion problem for partial differential equations, this noise is assumed to be uniformly bounded. This is important to get a priori estimates. In this article, we introduce a new argument which allows to replace this assumption by a much more satisfactory one: we only assume that m has sufficiently many finite moments. This introduces several difficulties. In particular the correctors cannot be bounded uniformly in ε. We use a stopping time argument which, together with a control of the growth of stationary process, allows to obtain a bound on the correctors. Another difficulty is to obtain bounds on the solutions of (1.1) uniform in ε. In particular, the control of the energy is very delicate. Our idea has been used and improved in a recent work on approximation diffusion for kinetic equations (see [24]).
We work with solutions in the Sobolev space and assume that the non linear term is subcritical:
Moreover, we work with global solutions and, when , we need a further assumption. Namely, we assume
Our main result can be stated informally as follows. Precise assumptions are given below.
For alland, assume thatconverges in law toin the space, see (
2.1
), then for allthe-valued process, solution of (
1.1
) with initial dataconverges in law into X solution ofwith initial data, where F, Q are defined respectively by (
2.26
) and (
2.36
) below, and W is a cylindrical Wiener process on.
The last two terms in (1.4) actually correspond to the Stratonovitch noise . The covariance operator Q is described below and is explicit in terms of the correlation of m, see (2.24) and (2.36) below. As mentioned above, we could consider a more general noise. Let be a map with sublinear growth such that for all . If in (1.1), the random term is replaced by , the limit equation would be the stochastic Nonlinear Schrödinger with the Stratonovitch noise .
The article is organized as follows. We first introduce the notations and state preliminary results on the Schrödinger equation and on the driving process m. In Section 3, we adapt Kato’s method (see [6]) to get global existence of the process . The estimates are obtained by classical manipulations of the equation and blow up when . To avoid this we adapt the perturbed test function method to our problem in Section 4 and 6.1. This enables us to get both tightness of the process and the expression of the infinitesimal generator of the limit. In Section 5 we use all the results proved in Section 3 and Skorohod Theorem (see [3]) to prove the weak convergence of to X. Finally, the Appendix is devoted to details about an example of process m that can be considered in (1.1) and to proofs of technical estimates.
In this article , C or c denote constants whose value may change from one line to the other and which unless explicitely stated are independent of ε or of the smoothing parameter δ introduced below. They may eventually depend on other parameters such as T, d, σ, R or η and if needed we may precise the dependences by denoting for example .
Preliminaries and main result
Notations
Throughout this paper, for , we denote by the Lebesgue space of p integrable -valued functions on , endowed with the usual norm. For , is the inner product of given by
For , we use the usual Sobolev space of tempered distributions such that , where , endowed with the usual norm
where denotes the Fourier transform.
For and , we also use the standard Sobolev spaces consisting of functions which are in as well as their derivatives up to order m. It is classical that .
We also consider the same spaces for -valued functions, they are denoted by the same symbols with replaced by . When there is no ambiguity, we omit and or . For instance, we simply write for .
In this article, the following weighted Sobolev spaces are particularly useful. We define for ,
endowed with the norm . Such weighted spaces are commonly used in the theory of the deterministic Schrödinger equation. The group introduced below has some smoothing effect in these spaces for instance ([6], Section 2.5). They are also useful to study finite time blow-up ([6], Section 6.5).
Given a Banach space E, (resp. ) denotes the space of real valued continuous (resp. ) functions on E. And (resp. ) is the space of bounded continuous functions (resp. bounded as well as their derivatives up to order k). Finally is the subset of of functions with polynomial growth:
When H, K are Hilbert spaces, we denote by the space of linear operators from H to K. If , we simply write . We also denote by the space of Hilbert–Schmidt operators from H to K.
Given a Hilbert space H endowed with a scalar product and a Banach space E continuously embedded in H, we use the common abuse of notations for the duality between E and :
We denote by the group associated to the linear homogeneous equation and defined by . The solution of (1.1) is taken in the mild sense.
As already mentioned we assume that (1.2), (1.3) hold so that we are able to prove global well posedness of (1.1) in (see [6] and Proposition 3.1 below).
The energy of is denoted by and is given by
By Sobolev embeddings, we know that for when , when and when . Thus, by (1.2), this is a well defined quantity for .
In order to justify the computations when getting energy estimates, we may need a regularization procedure, so we choose ρ a mollifier, namely a function which satisfies
Finally for , we define and stands for the convolution of u and v, when it makes sense.
The random process m
We assume that m is a centered, càdlàg, stochastically continuous, stationary -valued process, for so that and E is an algebra, on a probability space adapted to a filtration (see [7] for the basic theory of Hilbert space valued Markov processes). In particular that is real valued. We also define the rescaled process
which is centered stationary -valued process -adapted, where .
Note that, as it is often the case in the study of partial differential equations, functions depending on space and time are seen as functions depending on time with values in a space of spatially dependent functions. That is, in the case of the process m, we use the identification . With this in mind, the rescaling above may be writen:
The process m is supposed to be an homogeneous Markov process. We denote by the transition semigroup associated to m, its infinitesimal generator. For simplicity, we assume that there exists a Markov process on adapted to the filtration such that and, for bounded borelian function φ on E, , .1
We do not really need this and could require the existence of such that only in distribution. We think that this slightly stronger assumption allows to lighten the definitions and proofs below. Note that, with this assumption, we have a process for all , even if m does not visit the whole space E.
Recall that a borelian function φ on E is in the domain of the infinitesimal generator if for
is an integrable martingale. It is in general difficult to describe completely the domain of an infinitesimal generator. Here we only require that sufficiently many function are in the range of the generator. More precisely, we assume that there exist sets and included in such that included in the domain of and is included in the range of . For , we assume that there exists such that for
is an integrable martingale. In other words, is in the domain of and . This is an ergodicity assumption on m. Below we require that some specific functions are in
We assume that has a unique invariant measure ν, which is the invariant law of m. Clearly, our setting requires that
We need that contains sufficiently many functions. Let us define for :
We assume that for each , and there exist continuous such that
Informally, this says:
Then we define for :
and assume that for each , , and there exist and continuous such that
Again, informally this may be written as:
and
We need to invert a further function of n. For , we define
and assume that for each , and there exists continuous such that
Informally:
In previous articles on approximation-diffusion for PDEs, the process is assumed to be almost surely bounded: for all , a.s. for some deterministic constant K. This boundedness assumption is replaced here by the weaker assumption that there exist and a constant such that the following estimate holds
By stationarity of m, this implies that for all :
Note that this implies:
and
for any .
We need some control on the growth of the functions introduced above. We assume that there exist and such that:
In many situations, is given by:
and, in particular, for each , is an integrable function: . Here we do not need this but we simply use the assumption that there exists such that:
and that is given by
We also use the following object in next section. Note that assumptions (2.19) and (2.22) imply that, for , is integrable over , so that we can define:
It is not difficult to check that . We assume that
where denotes the Laplace operator with respect to the first variable of k.
We use the notation
Thanks to (2.22), it can be seen that .
We end this section with the following Lemma. It is similar to [8, Lemma 15.4.4] and is fundamental to avoid the uniform boundedness assumption on the process m.
For all, and, andwhen ε goes to 0.
For , denote by the random variable
then by the Markov inequality and (2.20) we have for all δ and k
and choosing δ such that , we have
so we get by the Borel–Cantelli lemma that for almost every , there exists such that for . It follows that, for , and
with the random variable defined by . Finally, since
and the probability of the right-hand side event goes to 0, when , under the condition , this ends the proof. □
We naturally define the -stopping time by
with the convention that when this set is empty.
Note it is possible that is equal to 0, this is the case when . Otherwise, when , we have .
We will use many times that , which together with Lemma 2.1 yields
An example
An example of assumptions on the process m that can be checked in practice and are sufficient to satisfy all the above hypotheses is the following:
For every , is a stochastically continuous Markov process associated to the semigroup .
is Feller.
For every and , is finite and there exist , such that for every
There exist , , such that for any we can construct a coupling of such that
By coupling, we mean that the law of (resp. is the same as (resp. ). It follows classically from this last assumption that has a unique invariant measure ν which is exponentially mixing. We finally need
ν is centered: , and for any
We then take m as a stationary process with law ν such that .
Define
For , we may define
Our assumptions imply that and that (2.6) is an integrable martingale. We define
Clearly for any , we may take and can be constructed and follow our assumptions.
Let us now construct a process satisfying 1 to 5. We consider the following stochastic equation in a Hilbert space H: for ,
where A is an unbounded operator on dense in H, invertible with a compact inverse, which generates an analytic semigroup and such that
and
for some , , . Assume moreover that, for all , is Hilbert–Schmidt on H and
with , .
Remark that if , where is the Laplace operator, then A satisfies our assumptions provided that .
The nonlinear term G can be chosen in various ways, for simplicity we assume that it is Lipschitz bounded. The covariance of the noise σ is an invertible operator on H. It has been proved in [14] that (2.33) defines a Markov process in H. We consider a continuous linear invertible operator and take
where , ν being the invariant measure of , and, for , being the unique solution of (2.33).
When and , one may consider . Then, Λ maps H into E and (2.25) is satisfied for . At the end of Section 2.4, we introduce a further assumption which is satisfied for . More generally, Λ can be the solution map associated to an elliptic equation of sufficiently high order and containing a confining potential.
The conditions (1), (2) and (4) follow from corresponding properties on proved in [14]. Concerning (3) and (5), they may be proved by classical computations based on the change of unknown
in (2.33) and energy estimates. In the Appendix we give details and prove that the process satisfies conditions 1 to 5 and that these conditions are indeed sufficient to construct as above.
We could also build an example based on a Markov chain in E as in [15].
The covariance operator
Let Q be the linear operator defined by
Since , Q maps into itself.
The following lemma, whose proof can be found in [20], is useful to prove that Q is non-negative.
(Wiener–Kintchine).
Letbe a real-valued stationary and centered process, we setand assume C is integrable on. Definingwe have
We now use the last assumptions on m and to show the following properties:
The operator Q has finite trace on, is self-adjoint and non-negative:for all. Moreover the following identities holdwhere ν is the law of.
We have assumed that so , and k is obviously symmetric. Moreover the stationarity of m yields
and we easily get (2.37). Let us now prove the positivity of Q. We define , which is centered stationary process, and denote by its correlation function. Then definitions (2.36) and (2.37) of k and Q and Lemma 2.2 yield
Finally, we write thanks to (2.23) with :
this proves (2.38) □
Thanks to Proposition 2.1, we may define the operator which is Hilbert–Schmidt on . Let us denote by q its kernel. Then, we have:
We need a little more smoothness on this operator. We have seen that , therefore
with F defined in (2.26). We also know that is Hilbert–Schmidt from to since:
We assume that it is also γ-radonifying (see [4] or [9] for the definition) from to for any . It is shown in [4] that this amounts to assume that . This is true for instance if for as can be seen from Sobolev embeddings since is γ-radonifying from to for any when it is Hilbert–Schmidt from to .
Under these properties on and F, it is shown in [10] that (1.4) has a unique global solution in .
Main result
We state here our main result, where we recall all the assumptions needed on the process m.
Let,,, and letsuch thatand
Consider the randomly perturbed Nonlinear Schrödinger equationwhere m is a centered-adapted, stationary-valued homogeneous Markov process for which there existandsuch thatAssume furthermore that (
2.21
), (
2.22
), (
2.23
) and (
2.25
) are satisfied. Finally, consider F and Q respectively defined in (
2.26
), (
2.36
), with k defined in (
2.24
) which can be written aswhere q is the kernel of the operator. Assume thatfor any. Then for alland, assuming thatconverges in law toindefined in (
2.1
), the-valued processsolution of the above Schrödinger equation with initial dataconverges in law into X solution ofwith initial conditionand whereis a cylindrical Wiener process on.
The proof of this theorem is detailed in Section 5. We first study equation (1.1) and the generator of the couple . This allows to introduce the perturbed test function method which is the key tool to obtain tightness and prove the main result.
The equation
The Cauchy problem
This subsection is devoted to prove the following proposition. It states the existence of and provides a bound in . This bound is obtained by standard arguments but is not uniform in ε. A uniform bound is obtained with more sophisticated tools below.
Given, then almost surely, there exists a unique mild solutionof (
1.1
) with initial datawhich lies in,, for any T. Moreover, we have,such that. for:withandis the negative part of λ:ifandfor. Finally ifis-measurable then the processis-adapted.
We solve (1.1) pathwise and consider the equation
We use here Kato’s method (see [6, Theorem 4.4.1]) to get existence and uniqueness of for a small enough . We recall that is the fixed-point of ϕ defined for by:
in the space
endowed with the distance
Kato’s method proves that ϕ is a contraction for well chosen M, , and allows to get the continuity in time of the solution.
Since ϕ given by (3.2) maps -adapted processes onto -adapted processes, and that is obtained by iterating ϕ, this gives that is -adapted.
Moreover, it is easy to verify that . To prove the bound on the gradient, we use the energy defined in (2.3) and a regularization is necessary to justify the computations. We denote now by for all . And we compute :
Since we get that in and similarly converges to in as δ goes to 0. The other terms are treated similarly and after integration in time, and we obtain
which leads to, using the Cauchy–Schwarz and Young inequalities
This implies (3.1) when . For , we use the Gagliardo–Nirenberg and Young inequalities and have for and
Thus condition (1.3) implies that the energy provides a control on the norm of the gradient. In fact, the energy bound (3.3), the Gronwall lemma and the conservation of the norm yield the result. Global existence follows from (3.1) and the conservation of the norm. □
Persistence in the spaces
The result given in this subsection is a modification of [6, Theorem 6.5.1].
Assumefor somethenalmost surely for any. More precisely there existsa deterministic constant such that
Since the constant does not depend on ε, this result shows that if we are able to prove a bound on the norm of the solution uniform in ε, a uniform bound in ε on the weighted norm follows immediately.
Recall that we denote by . Given , denote by the function
Note that:
and similarly, since is real valued, . It follows:
Note that the last term vanishes. Since is sufficiently smooth, no regularization argument in the computation above is needed.
We have and since :
It follows:
Since for , , , and we deduce
which, thanks to the Young inequality and the Gronwall lemma, leads to
Finally, by Fatou’s lemma, letting :
which is exactly (3.5). Then by (3.1), we see that the right-hand side of (3.5) is bounded, which implies . Since we already know that it is continuous with values in , we deduce , where the subscript w indicates weak continuity in time. Letting in the first part of (3.6) we see that
and since the right-hand side is continuous, we get the strong continuity: □
Generator of
In order to determine the law of the limiting process X, we need to identify the generator of . Clearly, is not a Markov process, because its increments depend on , but the couple is a Markov process, since m is.
We compute now the generator of the process . We are able to compute this generator acting on functions such as in the next definition. There are many such functions. In particular, we can choose functions independent on n: for , which allow to characterize the dynamic or a diffusion process. When we apply the generator to these functions, other functions of the form , are needed, the correctors, and thanks to the assumptions on these also are good test function in the sense of the definition below. We also use the energy and associated correctors to obtain bounds independent on ε.
We consider functions depending on , for such function φ we denote by its differential with respect to the variable u.
(Good test function).
We say that , is a good test function if the following holds:
Ψ is continuously differentiable with respect to u, the differential is denoted by .
Ψ is subpolynomial in u and n; , , ,
is continuous from to .
is continuous with respect to the norm and satisfies the following subpolynomial bound in u: , , ,
,
is continuous and subpolynomial in u and n; , , ,
With these good test functions, we may identify the generator of the Markov process . For the definition of the predictable quadratic variation of a martingale, we refer to [19] and recall that it coincides with the quadratic variation when the martingale is continuous.
For a good test function φ, the infinitesimal generatorofis given by the formula:for,. More precisely, iffor all p:is a càdlàg and integrablezero-mean martingale. If furthermoreandis also a good test function, its continuous quadratic variation is given by:
Let and be a subdivision of with . Given g a -measurable and bounded function, we have
where
with
and
To treat , we write and since is regularizing and subpolynomial, we have
Also, since , we know that is continuously embedded in and by duality is continuously embedded in . We deduce:
which gives, thanks to (3.1) and (2.19), the uniform (in ) integrability of Moreover, since is almost surely continuous and is stochastically continous, converges to 0 when in probability, so I converges to 0, as tends to 0.
We claim that
Indeed, for , (see Definition 3.1) and
is a -martingale. As and g are -measurable, we get
Since satisfies the polynomial bound in Definition 3.1 we get the uniform integrability of . The convergence in probability to 0 comes from the continuity of .
The quadratic variation is characterized by the property that is a martingale. Let and be a subdivision of with . Recall the following sequence of identity:
where the last equality follows from
and
Hence, it suffices to show that the right hand side of (3.9) goes to zero as .
Let us write:
Applying the above to implies that the process
is a martingale for the filtration generated by . We can now write
and
Again, the inequalities (2.19) and (3.1) and uniform integrability can be used to prove that the three terms of the right hand side go to zero under the extra assumption that . □
This proof is not completely rigourous. Indeed, we have differentiated with respect to t but we do not know whether is with values in . This is easily overcome by a regularization argument as in the proof of Proposition 3.1: we replace φ by and let at the end of the proof.
The perturbed test function method
Correctors
From the expression of , we see that negative powers of ε are present. The term of order −2 cancels if φ does not depend on n. Since we are interested only in the behaviour of when , it is natural to consider such functions. To treat the −1 order term we need to add correctors to φ. Assuming , are good test functions we have
Recall that the notation denotes the differential of φ with respect to u. Let us compute formally the correctors. As already mentionned, the order term vanishes because we chose a function φ depending only on u. The first corrector is chosen so that the order term cancels. It is formally given by:
with defined in (2.8), (2.9), (2.10).
The second corrector enables to identify the limit generator. The average with respect ot ν of the third line is given by
and we choose such that:
In this way, we formally get and we indeed identify the limit generator.
We do not need to justify rigorously the above computation for many functions. As we shall see below, for our purpose, it is sufficient to consider test functions of the form: for , . It is clearly a good test function and satisfies all assumptions of Proposition 3.3.
(First corrector).
Letwith,. Then there existsa good test function such that:Moreover,and satisfies all assumptions of Proposition
3.3
.
For any , . Therefore:
where ν is the law of , which is centered. Thanks to our assumptions, is given by
By (2.21), we easily see that this is a good test function and Proposition 3.3 applies. □
Note that thanks to (2.21), we have:
Moreover, for ,
We now compute the second corrector. For the test function , we have
The equation for then writes:
The following proposition is again a straigthforward application of our assumptions.
(Second corrector).
Letwith,. Then there existsa good test function such that:Moreover,is also a good test function.
Let , then the right hand side of (4.2) is of the form (2.11). It follows that exists and by (2.13), (2.14) is given by:
Similarly, for , the right hand side of (4.2) is of the form (2.16) and by (2.17), (2.18) is given by:
Thanks to (2.21), we have in both cases:
It follows that and are good test functions. □
(Perturbed test-function method).
Let, where,, and,given by Propostions
4.1
and
4.2
. For, we define. Thenverifies for,:
The processis a càdlàg and integrablezero-mean martingale.
We treat the case . The case is similar but lengthier. The first assertion clearly follows from the bound we have written above on and .
By definition of and :
Recalling the expressions of , , we have
and
We have seen in the proof of Proposition 3.3 that
The estimate of the second assertion follows easily. By Proposition 4.1, Proposition 4.2 we know that is a good test function. Therefore Proposition 3.3 applies and the third point is also clear. □
Tightness of the process
In this subsection, we aim to obtain tightness of the family of stopped processes where . The definition (2.27) of depends on α. We choose such that .
The crucial ingredient used in previous works on diffusion-approximation in infinite dimension is an assumption on uniform boundedness of the driving process in the adequate functional space, which would be in our case, w.r.t. ε (see [16] and [12]). Under our weaker assumptions, the result remains true provided we use the stopping time . We will see that this is sufficient to conclude.
Assuming. in the spacefor somethen the family of processis tight infor.
This result strongly relies on the following a priori estimate.
Let. Assume that, and letbe the solution of (
1.1
) with initial dataandthe stopping time introduced in (
2.27
). Then for any stopping time, there exists a constantdepending on T and p but not on ε such that for:
The proofs of Lemma 4.1 and Proposition 4.4 are technical and are postponed to Section 6.
We remark that this lemma, together with (3.5) and assuming show that the following inequality holds
The proof of Theorem 1.1 is divided into 3 steps. First we identify the SPDE associated to , then we prove the weak convergence of to X solution of (1.4), linked to , and finally we conclude using the uniqueness of the solution.
Step 1: Identification of the limiting generator
For with , we use the functionals and which are clearly good test functions. We now compute and ,
where F is given by (2.26) (see (2.38)). In the case of , we again have
but now the term does not vanish,
Let us denote now by q the kernel of where Q is given by (2.36), we have
and
Finally, we have
and
Step 2: Convergence
Given , by Propostion 4.4, we have a subsequence of , still denoted by , of law and a probability measure P on such that
Since is compact and is separable, is also separable, and by Skohorod theorem (see [3]) there exist a probability space and random variables , on with values in such that
and and . For , we use the test function and , the correctors given by Propositions 4.1 and 4.2. We define , then by Proposition 4.3, the process
is a martingale for the filtration , and the stopped process is also a martingale, that is for all and
Moreover, we easily have
Then we get:
We now use Proposition 4.3 and Lemma 4.1 and notice that when to get the bound:
Similarly:
By the embedding (see (1.2)) and the Hölder inequality, we have
Lemma 4.1 and Hölder inequality yield
with when by Lemma 2.1 and the uniform integrability. Similarly we have
with when .
Finally, we obtain
where C does not depend on ε. Moreover, as and have the same law, then (5.3) is also true by replacing by , and by . Since φ, and g are continuous from to (the continuity of requires the continuity of the nonlinearity which is given by the embedding ), taking the limit , we get
Then the process
is a martingale with respect to the filtration generated by . Note that this martingale is continuous.
Similarly, we can pass to the limit in the definition of the quadratic variation and obtain that the quadratic variation of is given by:
Note that this step requires the use of the perturbed test function method applied to .
From (5.1) and (5.2), we deduce
The continuous -valued martingale
has the quadratic variation
then, using the martingale representation theorem (see [8]) and up to enlarging the probability space, there exists a cylindrical Wiener process W such that:
Step 3: Uniqueness of the limit Note that also satisifies (4.4). Letting , we deduce that for any is a martingale solution of (1.4). Using the integral form of (1.4), we see that it has paths in . We know from [2] that under our assumptions that (1.4) has a unique solution with paths in . This implies uniqueness in law for martingale solutions.
As P is the law of , we deduce this is the law of the solution of (1.4). By uniqueness of the limit, we conclude that the whole sequence converges in law to X, in the space of probability measure of .
Finally, we obviously have for ,
and together with Lemma 2.1 yields the convergence in probability of to 0.
Using finally [3, Theorem 4.1], we obtain the weak convergence of to X in . □
As seen in Section 3, a straight application of standard energy arguments gives a very bad dependance on ε. The idea is to use the perturbed test function. This mimics Itô formula which is used to get a priori estimates for the limit equation with white noise (see [10]).
If one tries to use similar arguments to those of Proposition 4.3 with the functional , defined by (2.3), in place of linear or quadratic functional, this requires a lot of smoothness on and the useless assumption (). We proceed slightly differently. We first smooth the functional and take advantage of the various cancelations before constructing the correctors.
We give the proof only for . The general case is not more complicated but is lengthier.
We consider the functional , where is the mollifier introduced at the end of Section 2.1. We claim that it is a good test function. Indeed, we have for
and there is no difficulty to verify that satisfies the condition in Definition 3.1. By Proposition 3.3, we know that the process
is a martingale. It can be seen that when and . Moreover, we have
and we proved in Proposition 3.1 that the second term of the right-hand side converges to 0, when . Similarly we have when δ tends to 0
so that
since m is real valued. In the same way, we have
Finally, as δ converges to 0, converges almost surely to
and since is a -martingale, from (3.1) and the dominated convergence theorem, we deduce immediately that is also a -martingale.
We define then the first corrector
This is not a good test function. For , we set . There is no difficulty to see that is a good test function and then, we obtain by Proposition 3.3 that the process
is a martingale. After computations, we have
after integration by parts, we get
and
Finally, taking the limit and using
we get that the process given by
is a martingale.
We finally consider the test function
where was defined in (2.12), (2.13), (2.15). Again,thanks to (2.21) and the assumptions on , it is not difficult to check that is a good test function.
Using Proposition 3.3, we know that
is a -martingale. After computation we obtain :
where we strongly used that n is real-valued.
Consequently, we know that the following process is a martingale, with respect to the filtration .
Hence, since , given by (2.27), is a bounded -stopping time, the process is a martingale. From the identity above, the conservation, (6.2) and (6.4) we have
We recall that and that, when , according to Lemma 2.1. Hence using (2.21), (3.4),
then for ε small enough we get
The Gronwall Lemma and the conservation of the norm give us that for any
Then, taking τ a stopping time such that , we can do the same computations as before and get a similar bound as (6.5):
and using the bound (6.6) we are able to conclude. □
To prove the tightness of the sequence in the space , we use Aldous criterion, which can be found in [3, Theorem 16.10] in the finite dimensional case. In the case of an infinite dimensional separable space H, the hypothesis (16.22) in [3] has to be replaced. Let us state the criterion we use.
Let H be a complete separable space, andbe a sequence of process onsuch that, for any,is H valued. Assume that
For every, for everythere exists a compact set, such that
For every, there existsuch that for, and, if τ is a stopping time then
Then the sequenceis tight in.
We can prove this result using the same proof as for [3, Theorem 16.10], but instead of hypothesis (16.22) we assume (a) in [17, Theorem 7.9].
Lemma 4.1 and Proposition 3.2 ensure that satisfies (6.7) for , . Indeed, according to these two lemmas, we have
with independent on ε.
Thus for any and :
for R large enough independent on ε. Thus lives in a bounded set of with probability larger than . It remains to prove that the embedding is compact. Let be a bounded sequence in . Then by compact embedding and diagonal extraction, there exists a subsequence which converges to for every . Now we compute:
for any , thus converges to u in , and is bounded in , so that by interpolation the subsequence also converges to u in .
We also need to prove that satisfies the second condition of Proposition 6.1. We first prove that (6.8) holds with H replaced by . For this we use the Perturbed Test Function method, and the equality:
First we apply the Perturbed Test Function method to . We compute :
We choose the first corrector in order to cancel the term of negative order in ε, and recalling that we get
on which we have a bound using hypothesis (2.21):
Now we apply the infinitesimal generator to :
Thus, treating the operator as a kernel operator with kernel , we set
where we recall that formally . Finally, defining , we obtain that:
First we get an estimate of in (6.11). We have
and for the other term in the right-hand side of (6.11) we use the assumptions made on to get the bound:
Finally we obtain the following bound on :
with and defined in (2.11), (2.13), (2.14), (2.16), (2.17), (2.18).
Now we work on the different terms of the expression of in (6.12). First we deal with the terms of order 0 in ε. Using integrations by parts and (3.4) we get
Then we have
where is defined in (2.24). We get:
We get the same estimate for the last term of order 0:
Now we focus on the terms of order 1 in ε in (6.12). We recall the expression of the first corrector . Thus
We use integrations by parts to deal with the terms involving and we get that
For the other terms, we use (3.4) to get
Finally
It remains to control the terms coming from the introduction of the second corrector written in (6.11). We have
where denotes the kernel of the operator and is defined in (2.18). We start by the estimate of . We can easily bound the terms involving :
For the other term, compute:
We can use this bound to control the terms which do not involve in :
Finally we have
It remains to control . We use the same computations as for and the bounds (3.4) and (6.19) to get:
Finally, estimates (6.13), (6.17), (6.20), (6.21) coupled with Lemma 2.1, conservation of the -norm and Lemma 4.1 give us for the stopping time defined in (2.27):
where , denote the stopped process , .
We use this last estimate and the fact that
is a martingale to compute for any stopping time τ:
Finally, for δ and ε small enough we get:
Then we apply the Pertubed Test Function method on for a fixed function . Computations lead us to choose two correctors:
and the infinitesimal generator applied to is
Here again there are several quantities which need to be bounded. We use similar computations as previously done for and we obtain:
where also depends on the -norm of h. Now taking τ a stopping time and , knowing that conditioning by
is a martingale, we obtain
Thus for ε, δ small enough, we have
Finally, gathering (6.22), (6.24) and using the Markov inequality, we get that for any stopping time τ and , for ε, δ small enough:
We then use an interpolation inequality to write for :
It follows for any :
Thanks to Lemma 4.1 we can choose M such that the last term is smaller than . Then, using (6.25), we can choose R such that the first term of the right hand is less than . Thus Aldous criterion is satisfied, the process is tight in . □
Footnotes
Acknowledgement
This work was conducted within the France 2030 framework porgramme, Centre Henri Lebesgue ANR-11-LABX-0020-01.
Details about the example in Section 2.3
Consider solution of (2.33):
where , and with H a Hilbert space. Then we define
with a continuous invertible operator and where ν is the invariant measure of . In order to show that m satisfies the assumptions in E, it is sufficient to show that these assumptions are satisfied by X in H. It is straightforward that is stochastically continuous, and it is proved in [14] that satisfies the coupling assumption 4. We first prove that the moment of order 2 of is bounded. Let us define
Thanks to the assumptions made on A and σ we know that there exists such that for any ,
where is the domain of the operator . Besides is solution of
Now we can start to prove that satisfies the different assumptions 1 to 5. The following lemma ensures that verifies 2.
Now we are interested in the existence and uniqueness of an invariant measure for the process .
It remains to prove that satisfies assumptions 3 and 5. In this aim we decompose and, using (A.6), we have for any
which gives us that assumption (3) is satisfied. Indeed, all the moments of are bounded.
Given , this inequality generalizes into:
Choosing such that , we obtain, for ,
Under our assumptions, we know that the left hand side converges to . We deduce:
Finally we proved that satisfies assumptions (1) to (5) (except the zero-mean property), and so does according to its definition in (2.35).
Now we can work on the construction of the functionals . We state the following result, which ensures the existence of .
So far we have proved that it is possible to construct which satisfies our assumptions. Let us now construct the other functionals.
We have the same result for .
Finally, we need to construct a last functional , which is slightly different to the previous ones.
We have constructed the last functional in a slightly different way than in (2.18), but taking we recover the same expression for .
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