- Open Access
Fokker-Planck equation for Kolmogorov operators associated to stochastic PDE with multiplicative noise
Advances in Difference Equations volume 2014, Article number: 222 (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.
Let H be a separable Hilbert space (with norm and inner product ), and be the Banach space of all bounded linear operators from H to H. We consider the stochastic differential equation in H
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.
It is well known that under Hypothesis 2.1, (1.1) has a unique mild solution (see, for instance, [1, 2]), that is, for any the process is adapted to the filtration , and it is continuous in mean square and fulfills the integral equation
for ℙ-a.e. and any . Moreover, a straightforward computation shows that for any there exists such that
where the expectation is taken with respect to ℙ. As we shall see in Lemma 2.4, estimates (1.3) and (1.4) allow us to define the transition semigroup associated to (1.2) in the space (see below for a precise definition), by the setting
It is not hard to prove that maps into and satisfies the semigroup property, see , but it is not a strongly continuous semigroup. However, we can define the infinitesimal generator of in as
Naturally, we are interested in the connections between and the Kolmogorov operator
where is a suitable function and Tr means trace.
In this paper, we will study the relationship between Kolmogorov operator and the infinitesimal generator of the transition semigroup and solve the Fokker-Planck equation corresponding to (i.e. the dual of the Kolmogorov equation). As the first main result (see Theorem 3.1 below), we show that extends the Kolmogorov operator defined on the domain , which consists of the linear span of the real and imaginary parts of the functions
and the set is a π-core (cf. Section 2) for .
Let be the space of all finite Borel measures on H, be the set of all such that , where is the total variation of μ, the second main result is that for any , there exist a family of measures fulfilling
and the Fokker-Planck equation
Moreover, this solution is given by , where is a semigroup defined by
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.
We list some notation which are applied in this paper. Let H be a separable real Hilbert space (norm , inner product ), and represent its topological dual space. Let be the space of all finite Borel measures on H, and for any , denote the set of all such that by , where is the total variation of ν. We denote the space of all linear bounded operators from H into H by , endowed with the norm
and let be the space of all Hilbert-Schmidt operators , endowed with the Hilbert-Schmidt norm. If E is a Banach space (norm ), we denote as the linear space of all continuous and bounded mappings , endowed with the norm
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 .
If and , we shall identify with the unique element h of H such that
If and , we shall identify with the unique linear operator such that
(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 .
(H3) There exists such that
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 .
Proposition 2.2 Under Hypothesis 2.1, for any initial condition , problem (1.1) has a unique mild solution , that is, for any , the process is adapted to the filtration , and it is continuous in mean square and fulfills the integral equation
for ℙ-a.e. and all , . Moreover, for any , , there exists such that
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 ).
Definition 2.3 (i) A sequence is said to be π-convergent to a function if for any we have
Similarly, the m-indexed sequence is said to be π-convergent to if for any there exists an i-indexed sequence , such that
We shall write
or as , when the sequence has one index.
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.
Lemma 2.4 Formula (1.5) defines a semigroup of operators , in , and there exist a family of probability measures and two constants , such that
, for any , , ;
for any , , the function , is continuous;
for any and ;
for any and any sequence such that as , we have as , for any .
Lemma 2.5 Let be the mild solution of problem (1.1) and let , be the associated transition semigroups in the space defined by (1.5). Let also be the associated infinitesimal generators, defined by (1.6). Then
for any , we have and , ;
for any , , the map , is continuously differentiable and ;
given and as in Lemma 2.4, for any the linear operator on done by
satisfies, for any
We call the resolvent of K at λ.
Definition 2.6 We say that is a π-core for the operator , if D is π-dense in and for any there exist and an m-indexed sequence such that
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].
Proposition 3.2 For any , there exist and an m-indexed sequence such that
Moreover, if , we can choose the sequence in such a way that (3.1) holds and
3.1 The case
If , we consider the transition semigroup associated to the stochastic differential equation in H
We recall that the mild solution is given by the process
where , and the semigroup is defined by setting
We define the infinitesimal generator of the semigroup in as in (1.6), with L replacing by K and replacing .
Proposition 3.3 Assuming that the conditions of Theorem 3.1 hold, for any we have and
The set is a π-core for , and for any there exist and an m-indexed sequence such that
Moreover, if we can choose the sequence in such a way that (3.6), (3.7) hold and
for any .
Proof By the mean value theorem, for any
where is a Borel function from H into . Therefore
We first show that
Again, by the mean value theorem,
where . However, for arbitrary
Since , as , the relation (3.11) holds.
Assertion. For any , let denote the right-hand side of (3.10), and set(3.12)
For any , set , then
We shall use the following Burkholder estimate, see ; for a constant , we have
By the Minkowski inequality, on account of hypothesis (H3) and (2.5), it renders
and due to , for any , there exists a constant such that
Moreover, , as , therefore, taking into account (3.15), , as uniformly with respect to .
On the other side, from the Fubini theorem and the properties of the stochastic integral for cylindrical Winner processes (see, for instance, ), it follows that
where the sequence is an orthonormal basis in H, by hypotheses (H2), (H3), and (3.3), for any it follows that
Moreover, in view of (3.10), we have
considering hypotheses (H0), (H3), and (2.5), it yields
On account of (3.15) and , this implies
thus, and (3.5) holds.
Similar to the proof given in , now we prove that the set is a π-core for . Let , for any , set . Clearly, and as . By Proposition 3.2, for any we fix a sequence (for the sake of simplicity we assume that the sequence has only one index) such that as . Set
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 .
And likewise, for any we have
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. □
Theorem 3.4 Assume that the conditions of Theorem 3.1 hold, let be the semigroup (1.5) and be the semigroup (3.4), we denote by , the corresponding infinitesimal generators in . Then we have and
for any .
Proof Let be the mild solution of (1.1) and be the mild solution of (3.2), for any , taking into account that
by the Taylor formula we have ℙ-a.s.
Then we get
for any , with the help of Luigi Manca’s result (see , Theorem 4.1), we have
As is easily seen, . Moreover,
taking into account hypothesis (H3) and (2.5), it yields
and due to , for any , there exists a constant such that
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].
Lemma 3.6 Let us assume Hypothesis 2.1 and that , . Then the semigroup defined in (1.5) maps into , and for any , we have
where is the mild solution of the equation in H
Proposition 3.7 Under the conditions of Lemma 3.6, we assume that the function κ in hypothesis (H3) satisfies , , for two constants and , let be the infinitesimal generator of . Then there exist two constants and such that for any , the resolvent of K at λ maps into and we have
Proof Let , for any , and for any we have
where is the mild solution of (3.19), that is,
applying the Minkowski inequality and the generalization of maximal inequality of martingales to stochastic convolution, see [, Lemma 7.7 and Proposition 7.8], it follows that
where c is a given positive constant. By Hypothesis 2.1 and , , we have
for any , considering that , we obtain
By the Hölder inequality, one can obtain
Let , multiplying (3.21) by and taking into account , for , yields
Now from the Gronwall inequality it follows that
where the constants , , thus
By (iii) of Lemma 2.5, we have
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 ).
We take a sequence of nonnegative twice differentiable functions such that
Let be the orthonormal projection of H onto , be the orthonormal basis in H. We will identify with , the mappings and are defined by
It is easy to check that , for any . Moreover, , as for all , and , , satisfy (H1)-(H3) in Hypothesis 2.1.
Let be the semigroup
where is the mild solution of (1.1) with , replacing F, G. Clearly, we have
and for any , there exists a constant such that
for any , . We denote the infinitesimal generator of the semigroup in , defined as in (1.6) with , replacing K, . Also all the statements of Lemmas 2.4 and 2.5 hold for and . Combining (3.33), it is straightforward to see that the resolvent of satisfy
for any , .
Proposition 3.8 The set is a π-core for , and for any there exist and an m-indexed sequence such that
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
This section is devoted to studying the following Fokker-Planck equation for the 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).
Definition 4.1 Given , we say that a family of measures is a solution of the Fokker-Planck equation (4.1) if the following are fulfilled:
the total variation of the measure satisfies(4.2)
for any and any , it holds(4.3)
From Theorem 3.1 we know the relationships between the Kolmogorov operator and the infinitesimal generator of the transition semigroup , defined by (1.6). Then we firstly study the measure value equation
Theorem 4.2 Assume that the conditions of Theorem 3.1 hold, let be the semigroup associated to the SPDE (1.1) defined by (1.5) and be its infinitesimal generator defined by (1.6). Then the formula
defines a semigroup of linear and continuous operators on that maps into . Moreover, for any there exist a unique family of measures such that
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 .
To prove the uniqueness of the solution, we assume that is a solution of (4.1). For any and , there exists a sequence (for simplicity we assume that this sequence has only one index) such that
and we have
Now observe that for some and for any , so
Taking into account (4.2) and applying the dominated convergence theorem, it yields
this implies that is a solution of the measure equation (4.4). On the other hand such a solution is unique and is given by by Theorem 4.2, that is, for any we have
By Proposition 3.2, (4.11) still holds for any , this implies , . This concludes the proof. □
Cerrai S: Stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term. Probab. Theory Relat. Fields 2003, 125: 271-304. 10.1007/s00440-002-0230-6
Stannat W: Stochastic partial differential equations: Kolmogorov operators and invariant measures. Jahresber. Dtsch. Math.-Ver. 2011, 113: 81-109. 10.1365/s13291-011-0016-9
Manca L: Fokker-Planck equation for Kolmogorov operators with unbounded coefficients. Stoch. Anal. Appl. 2009, 27: 747-769. 10.1080/07362990902976579
Goldys B, Kocan M: Diffusion semigroups in spaces of continuous functions with mixed topology. J. Differ. Equ. 2001, 173: 17-39. 10.1006/jdeq.2000.3918
Manca L: Kolmogorov equations for measures. J. Evol. Equ. 2008, 8: 231-262. 10.1007/s00028-008-0335-1
Da Prato G, Zabczyk J Encyclopedia of Mathematics and Its Applications 44. In Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge; 1992.
Da Prato G Advanced Courses in Mathematics - CRM Barcelona. In Kolmogorov Equations for Stochastic PDEs. Birkhäuser, Basel; 2004.
Da Prato G, Debussche A: m -Dissipativity of Kolmogorov operators corresponding to Burgers equations with space-time white noise. Potential Anal. 2007, 26: 31-55. 10.1007/s11118-006-9021-5
Da Prato G, Zabczyk J London Mathematical Society Lecture Notes 293. In Ergodicity for Infinite Dimensional Systems. Cambridge University Press, Cambridge; 1996.
Da Prato G, Zabczyk J London Mathematical Society Lecture Notes 283. In Second Order Partial Differential Equations in Hilbert Space. Cambridge University Press, Cambridge; 2002.
Manca L: On a class of stochastic semilinear PDEs. Stoch. Anal. Appl. 2006, 24: 399-426. 10.1080/07362990500522452
Bogachev VI, Da Prato G, Röckner M: Fokker-Planck equations and maximal dissipativity for Kolmogorov operators with time dependent singular drifts in Hilbert spaces. J. Funct. Anal. 2009, 256: 1269-1298. 10.1016/j.jfa.2008.05.005
Bogachev VI, Da Prato G, Röckner M: Existence results for Fokker-Planck equations in Hilbert spaces. Progress in Probability 63. In Seminar on Stochastic Analysis, Random Fields and Applications VI. Edited by: Dalang R, Dozzi M, Russo F. Springer, Basel; 2011:23-35. (Centro Stefano Franscini, Ascona, May 2008)
Bogachev VI, Röckner M: Elliptic equations for measures on infinite dimensional spaces and applications. Probab. Theory Relat. Fields 2001, 120: 445-496. 10.1007/PL00008789
Priola E: On a class of Markov type semigroups in spaces of uniformly continuous and bounded functions. Stud. Math. 1999, 136: 271-295.
Peszat S, Zabczyk J: Strong Feller property and irreducibility for diffusions on Hilbert spaces. Ann. Probab. 1995, 23: 157-172. 10.1214/aop/1176988381
This work was partially supported by NNSF of China (Grant No. 11171122).
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
About this article
Cite this article
Shi, Y., Liu, B. Fokker-Planck equation for Kolmogorov operators associated to stochastic PDE with multiplicative noise. Adv Differ Equ 2014, 222 (2014). https://doi.org/10.1186/1687-1847-2014-222
- Kolmogorov operator
- stochastic PDE
- transition semigroup
- infinitesimal generator
- Fokker-Planck equation