Fokker-Planck equation for Kolmogorov operators associated to stochastic PDE with multiplicative noise
© Shi and Liu; licensee Springer 2014
Received: 1 March 2014
Accepted: 28 July 2014
Published: 15 August 2014
In this paper, we show that the Kolmogorov operator associated to a stochastic PDE with multiplicative noise can be extended to the infinitesimal generator of the corresponding transition semigroup in a proper weighted space. Then we apply the result to obtain the existence and uniqueness of a solution for the Fokker-Planck equation involving the Kolmogorov operator .
MSC:60H15, 35R15, 47D07.
where is the infinitesimal generator of a strongly continuous semigroup in H, , and are measurable mappings, and is a cylindrical Wiener process on H, defined on a filtered probability space , and adapted to some filtration that is assumed to be right-continuous and complete in the sense that contains all P-null sets. The precise assumptions on F and G are given in Hypothesis 2.1 below.
where is a suitable function and Tr means trace.
and the set is a π-core (cf. Section 2) for .
where is the topological dual space of ; see Theorem 4.3.
For the problems with additive noise case (that is, G is a constant in (1.1)), it is worth mentioning, the papers [3–5] have studied the problems by means of the Ornstein-Uhlenbeck semigroup method. However, for the multiplicative noise case, the stochastic convolution in (1.2) is no longer a martingale, so this method and the Itô formula can not be used to a mild solution, thus new technology is needed. In particular, Da Prato and Zabczyk  studied the problem (1.1) by the semigroup method, based on the classical fixed point theorem, and used the factorization method to get an estimation of the stochastic convolution, which is a generalization of maximal inequality of martingales to stochastic convolution, and plays an important role in the following sections.
On the other hand, we remark that a great deal of research has been devoted to the extension of a differential operator like (1.6) to the infinitesimal generator of a diffusion semigroup in the space , , where ν is an invariant measure for the semigroup (see, for example, [7–11] and references therein). In fact, if ν is an invariant measure for the semigroup (1.5), then the semigroup (1.5) can be extended to a strongly continuous contraction semigroup in , .
Kolmogorov equations for measures in infinite dimensional space have been the object of many authors (see, e.g., [12–14] and references therein). For example, Bogachev and Röckner  considered the existence of measure valued solutions for the equation involving second order partial differential operators in infinite dimensional spaces. However, as an extension of the existing theory in , in this paper we pay attention to the existence and uniqueness of the solution of the Fokker-Planck equation for Kolmogorov operators associated to the SPDEs with multiplicative noise case which is important.
We organize the rest of this paper as follows. Some notation and preliminary results are in Section 2. In Section 3, we prove that extends the Kolmogorov operator , and the set is a π-core for . Finally, the existence and uniqueness of the solution for the Fokker-Planck equation (1.9) are given in Section 4; see Theorem 4.3.
is a Banach space. Moreover, represents the subspace of of all functions which are Fréchet differentiable on H with a continuous and bounded derivative , and the space for all can be defined analogously. We shall write , for short. For any , let be the space of all functions such that the function , belongs to . The space is a Banach space, endowed with the norm . In the following, we shall denote by the topological dual space of .
(H0) is the infinitesimal generator of a strongly continuous semigroup of type , i.e. there exist and such that , .
(H1) Let be a continuous vector-field.
(H2) Let be strongly continuous, and for all .
for all ,
for all .
The following result is about the existence and uniqueness of a mild solution for (1.1), the proof essentially follows by Theorem 1.7 in .
As the semigroups of operators which we will deal with are not strongly continuous, we introduce the notion of π-convergence in the space (see ).
- (ii)For any subset we say that φ belongs to the π-closure of D, and we denote it by , if there exist and an m-indexed sequence such that
Finally, we shall say that a subset is π-dense in if .
Notice that since the convergence is pointwise we cannot take a diagonal sequence. However, in order to avoid heavy notations, we shall often assume that the sequence has one index.
As an extension of the π-convergent to the space (), a sequence is said to be π-convergent to a function if we have as in . Similarly, we can define π-dense in the space ().
Here we introduce some properties for the transition semigroup in , which can be proved by a similar argument to , and they play an important role in the proof of the results in the following sections.
, for any , , ;
for any , , the function , is continuous;
for any and ;
for any and any sequence such that as , we have as , for any .
for any , we have and , ;
for any , , the map , is continuously differentiable and ;
- (iii)given and as in Lemma 2.4, for any the linear operator on done by
We call the resolvent of K at λ.
3 A core for operator K
In this section, we give the first main result which is a better understanding of the relationships between the infinitesimal generator K and the Kolmogorov differential operator defined by (1.7).
Theorem 3.1 Assume, besides Hypothesis 2.1, that there exist two constants and such that the function κ in hypothesis (H3) satisfies , , then the operator is an extension of , that is, for any , we have and . Moreover, is a π-core for .
We split the proof in several steps. In Proposition 3.3 we will prove Theorem 3.1 in the case , then Corollary 3.5 will show that is an extension of and for any . Finally, Proposition 3.8 will complete the proof by the proper approximation sequence of F and G.
Firstly, we need the following approximation result, proved in [, Proposition 2.7].
3.1 The case
We define the infinitesimal generator of the semigroup in as in (1.6), with L replacing by K and replacing .
for any .
Since , as , the relation (3.11) holds.
Assertion. For any , let denote the right-hand side of (3.10), and set(3.12)
Moreover, , as , therefore, taking into account (3.15), , as uniformly with respect to .
thus, and (3.5) holds.
for any , so we can show that the sequence fulfills (3.6) by a straightforward computation because of the continuity of the function , , for any .
where we have used the continuity of and the fact that (cf. Lemmas 2.4 and 2.5). In fact that any limit above is equibounded in with respect to the corresponding index by the construction of , thus (3.7) holds.
If , (3.8) can be proved by Proposition 3.2. □
for any .
hence, and .
In the same way, for any , we can prove that , so this completes the proof. □
By Proposition 3.3 and Theorem 3.4 we have the following.
Corollary 3.5 is an extension of and for any we have and .
3.2 The case
The following lemma is proved in [, Chapter 7].
for any . Therefore, (3.16) follows. □
3.3 The general case and the proof of Theorem 3.1
From the above argument it remains to prove that is a π-core for for the proof of Theorem 3.1. For this we introduce the following approximation result (see ).
It is easy to check that , for any . Moreover, , as for all , and , , satisfy (H1)-(H3) in Hypothesis 2.1.
for any , .
Proof The proof goes along the same lines as that of Lemma 4.6 in , with some important changes. Namely, instead of Propositions 4.3 and 4.5 in , we have to use the modifications which correspond to Propositions 3.3 and 3.7 above. □
4 Fokker-Planck equation for Kolmogorov operator
where the Kolmogorov operator is defined by (1.7).
To give a precise meaning of this problem, we introduce the notion of solution of (4.1).
- (i)the total variation of the measure satisfies(4.2)
- (ii)for any and any , it holds(4.3)
for any , . Finally, the solution of (4.4) is given by , .
Proof In order to proof our result, we can follow almost the same arguments as in the proof of Theorem 1.2 in  by Luigi Manca. We omit the details here. □
As a consequence we get the second main result.
Theorem 4.3 Assume that the conditions of Theorem 3.1 hold, for any there exists a unique solution of measures of (4.1), and this solution is given by .
Proof Let be the infinitesimal generator defined by (1.6), by Theorem 3.1 we find that is a π-core for , and that , for any . So combining Theorem 4.2 it is easy to show that is a solution of the Fokker-Planck equation (4.1) for any .
By Proposition 3.2, (4.11) still holds for any , this implies , . This concludes the proof. □
This work was partially supported by NNSF of China (Grant No. 11171122).
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