On well-posedness of semilinear stochastic evolution equations on spaces
We establish well-posedness in the mild sense for a class of stochastic semilinear evolution equations on spaces, driven by multiplicative Wiener noise, with a drift term given by an evaluation operator that is assumed to be quasi-monotone and polynomially growing, but not necessarily continuous. In particular, we consider a notion of mild solution ensuring that the evaluation operator applied to the solution is still function-valued, but satisfies only minimal integrability conditions. The proofs rely on stochastic calculus in Banach spaces, monotonicity and convexity techniques, and weak compactness in spaces.
The purpose of this work is to prove well-posedness (existence, uniqueness and continuous dependence of solutions on the initial datum) to stochastic evolution equations (SEEs) of the type
where , is a linear -accretive operator on , with a bounded domain in and , is an increasing function of polynomial growth (without any continuity assumption), is a cylindrical Wiener noise on a separable Hilbert space , and is a (random) map from to satisfying suitable Lipschitz continuity conditions. Precise assumptions on the notion of solution and on the data of the problem are given in Section 3. In particular, we adopt three notions of solution, that depend on the integrability properties of : strict mild and mild solution are defined to be such that almost surely and that , respectively (here stands for the underlying probability space); on the other hand, generalized solutions are defined as limits of strict mild solutions, so that, in general, may not have any integrability. The first notion of solution is the simplest but also the most restrictive in terms of assumptions on the data of the problem. The second notion is the most natural if one wants to be function-valued, while satisfying minimal integrability conditions. The last notion, motivated by analogous constructions in the deterministic setting, apart of being the least demanding, is useful in several contexts, for instance in the study of Kolmogorov operators and Markovian semigroups associated to SPDEs (cf. e.g. ).
Our approach to the well-posedness problem is based, on the probabilistic side, on stochastic calculus for processes with values in Banach spaces (of which we use only the “simpler” version on spaces with type 2), and, on the analytic side, on methods from the theory of (nonlinear) -accretive operators and convex analysis. Some ideas developed here, concerning strict mild and generalized solutions, already appeared, in a more primitive form, in [21, 23] and, in a slightly different context, in .
There is a rather large literature on semilinear dissipative SEEs, up-to-date references to which can be found, e.g., in . Here we shall only discuss how our results compare to other recent ones that are most closely related. A widely used technique to study the well-posedness of (1) consists in the reduction of the equation to a deterministic evolution equation with random coefficients, roughly speaking “by subtracting the stochastic convolution”. To the best of our knowledge, the sharpest result obtained through this reduction is due to Barbu , who proved existence and uniqueness of mild solutions to (1) assuming that , is the negative Laplacian, does not depend on (i.e. the noise is additive), and, most importantly, the stochastic convolution
where denotes the semigroup generated by , is continuous in time and space, and satisfies , where is a primitive of . On the other hand, no polynomial bound on is assumed. Our setting allows much more flexibility and no assumption is made on the stochastic convolution, but we need an extra polynomial growth assumption on . A (partial) extension of our results to the case of general (i.e. removing the growth assumption) and is provided in a forthcoming joint work with L. Scarpa , thus considerably improving on the result of . In another vein, global well-posedness in the mild sense of (1) is obtained in  assuming that is an analytic semigroup and that is polynomially bounded and locally Lipschitz continuous on (not as function of !). The approach is through approximation of the coefficients and extension of local solutions. Even though the condition on is very restrictive, adapting ideas from , and considerably improving results thereof, well-posedness in spaces of continuous functions is obtained, allowing to be monotone and locally Lipschitz, now only as a function of . Incidentally, in , hence also in , the above-mentioned reduction to a PDE with random coefficients is again used, although in a more sophisticated way. While the reasoning in  relies on stochastic calculus in Hilbert spaces and ad hoc arguments, the improvements in  depend in an essential way on stochastic calculus in Banach spaces. We also use techniques from this calculus (although in a less sophisticated way), but we do not need any local Lipschitzianity assumption, although we obviously cannot consider solvability in spaces of continuous functions.
Our proofs do not employ at any stage the reduction to a deterministic equation with random coefficients. In fact, following the classical approach of constructing solutions to regularized equations and then passing to the limit in an appropriate topology (cf. e.g. [1, 7] for the deterministic theory), all the necessary estimates are obtained by stochastic calculus arguments, rather than by classical calculus. Namely, the essential tool is Itô’s formula for -valued processes (even though, as explained in Remark 4 below, the classical formula for real processes would suffice). Using techniques from convex analysis and the theory of nonlinear -accretive operators, we then show that, thanks to the above-mentioned estimates, solutions to regularized equations converge to a process that solves the original equation.
The rest of the text is organized as follows: in Section 2 we collect several tools used in the proof of the main results. Everything except the content of the last subsection is known and is included here for the readers’ convenience. Our main results are stated in Section 3. In Sections 4, 5, and 6 we prove well-posedness in the strict mild, generalized, and mild sense, respectively.
Acknowledgments. A large part of the work for this paper was done while the author was visiting the Interdisziplinäres Zentrum für Komplexe Systeme (IZKS) at the University of Bonn. The author is very grateful to Sergio Albeverio, his host, for the kind hospitality and the excellent working conditions.
In this section we introduce notation and recall some facts that will be used in the rest of the text.
Let , with fixed, be a filtered probability space satisfying the “usual” conditions (see e.g. ), and let denote expectation with respect to . All stochastic elements will be defined on this stochastic basis, and any expression involving random quantities will be meant to hold -almost surely, unless otherwise stated. Throughout the paper, stands for a cylindrical Wiener process on a (fixed) separable Hilbert space .
Given and a Banach space , we shall denote by the set of -valued random variables such that
and by the set of measurable111Since we never need weak measurability, measurable will always mean strongly measurable., adapted -valued processes such that
Both spaces are Banach spaces for , and quasi-Banach spaces for . The space , when endowed with the equivalent (quasi-)norm
will be denoted by .
The domain and range of a map will be denoted by and , respectively. The standard notation will be used for the space of linear bounded operators between two Banach spaces and . If and are metric spaces, stands for the set of Lipschitz maps such that
We shall omit the indication of the spaces and when it is clear what they are.
Throughout this section we shall simply write , , to mean the usual Lebesgue spaces over a generic -finite measure space .
Finally, we shall use the notation to mean that is less than or equal to modulo a constant, with subscripts to emphasize its dependence on specific quantities. Completely analogous meaning have the symbols and .
2.1 Convex functions and subdifferentials
Let be a convex function. Then, for any , ,
where denotes the subdifferential of at . The above inequality defines , which is a subset of , in the sense that , by definition, if it satisfies (2) for all . If is differentiable at , then reduces to a singleton and coincides with . The following mean-value theorem holds (cf. [16, Theorem 2.3.4, p. 179]): if is finite-valued, one has, for any , ,
where is any selection of the subdifferential .
Given a maximal monotone graph (see §2.3 below), there exists a convex function , called the potential of , such that . The converse is also true, i.e. the map defines a maximal monotone graph of for any convex function .
The (Legendre-Fenchel) conjugate of the convex (proper, lower semicontinuous) function is defined as
is itself a convex (proper, lower semicontinuous) function. The definition obviously implies for all , , with equality if and only if , which in turn is equivalent to . Moreover, if is everywhere finite on , then is superlinear at infinity, i.e.
In particular, if for all , or, equivalently, the domain of is , then is finite-valued on and is superlinear at infinity (see, e.g., [16, Chapter E] for all these facts).
2.2 Duality mapping and differentiability of the norm
Let be a Banach space with (topological) dual . The duality mapping of is the map
If is strictly convex, then is single-valued and continuous from , endowed with the strong topology, to , endowed with the weak topology (i.e. is demicontinuous). Moreover, if is uniformly convex, then is uniformly continuous on bounded subsets of . For instance, all Hilbert spaces and all spaces with are uniformly convex (hence also strictly convex), and their duality mappings are single-valued and demicontinuous. In particular, if , , one has
On the other hand, the duality mapping of is multivalued: in fact, if , one has
Moreover, one has , where and stands for the subdifferential in the sense of convex analysis. The (-th power of the) norm of spaces with is in fact very regular: setting , one has , with
where stands for the space of bilinear forms on . In particular, for any ,
and, by Hölder’s inequality,
2.3 -accretive operators
A subset of is called accretive if, for every , , there exists such that . An accretive set is called -accretive if . One often says that is a multivalued (nonlinear) mapping on , rather than a subset of . Through the rest of this subsection, we shall assume that is an -accretive subset of .
The Yosida approximation (or regularization) of is the family of (single-valued) operators on defined by
The following properties will be extensively used:
with Lipschitz constant bounded by ;
for all ;
for all ;
if is single-valued and , are uniformly convex, then as for all .
with Lipschitz constant bounded by ;
as for all .
If is uniformly convex, then the -accretive set is demiclosed, i.e. it is closed in , where stands for endowed with its weak topology. More precisely, if strongly in and weakly in as , then .
Let , . If is a maximal monotone graph in , then the (multivalued) evaluation operator associated to is an -accretive subset of . The operator is defined on as
Note that the graph of a (discontinuous) increasing function is a monotone subset of , but it is not maximal monotone. However, the graph defined by
where is the jump set of , is maximal monotone and (clearly) extends . We shall not explicitly distinguish below among an increasing function , its maximal monotone extension , and the associated evaluation operator .
The proofs of the above facts (and much more) can be found, for instance, in [1, §2.3].222Formula (3.12) in op. cit. contains a misprint: should be replaced by .
2.4 -Radonifying operators
We shall use only basic facts from the rich and powerful theory of -Radonifying operators. For more information we refer to, e.g., the survey .
Let , be real separable Hilbert spaces and , Banach spaces. An operator is said to be -Radonifying if there exists an orthonormal basis of such that
where is a sequence of independent identically distributed standard Gaussian random variable on a probability space . One can shows that does not depend on the choice of the orthonormal basis . The set of all such that is finite is itself a Banach space with norm , and is a two-sided ideal of , i.e. and imply
The following convergence result is a simple corollary of the ideal property: if strongly in as , i.e. in for all , then
If , , by a simple application of the Khinchin-Kahane inequalities it follows that if and only if
for all orthonormal bases of , and . Moreover, the mapping
is an isomorphism of Banach spaces, where one can take
2.5 Stochastic calculus in Banach spaces
Let be a UMD Banach space. For any and progressively measurable process , the stochastic integral of with respect to is a well-defined -valued local martingale that satisfies Burkholder inequality
If has type 2 (this is the case if , ), one has the continuous embedding333If , , the only case of interest for us, the embedding is just an obvious consequence of Minkowski’s inequality: .
for all (the case follows by Lenglart’s domination inequality, see ). Note that, if , in view of the isomorphism mentioned at the end of last subsection, the above inequalities can be equivalently written only in terms of norms. In other words, for our purposes the use of -Radonifying norms amounts only to adopting a convenient language. For further details we refer to  and references therein.
We shall also need Itô’s formula for -valued processes, and we use the version of , which is valid for UMD-valued processes. For our purposes, however, previous less general versions (cited in ) would also do, as well as the very specific one of , where only the -th power of the norm is considered. Let us first introduce some notation: if is a bilinear form on and , we set
for which it is easily seen that
Let be a UMD Banach space, and consider the -valued process
is measurable, adapted and such that ;
is -measurable, adapted, stochastically integrable with respect to , and such that .
For any , one has
2.6 Estimates for linear equations
Given a Banach space and a linear -accretive operator on , for any -valued or -mapping , we shall write, for any , .
We first prove an estimate that will be used repeatedly in the following.
Let be a linear -accretive operator on , and consider the unique mild solution to the equation
where , , and satisfy the assumptions of Theorem 1 (with . If , then
It is not difficult to verify that is the unique strong solution to
(cf. e.g. [22, Lemma 6]). Itô’s formula then yields444From now on we shall occasionally omit the indication of the time parameter, if no confusion may arise, for notational compactness.
where by accretivity of on , and by contractivity of on . We are thus left with
We are now going to pass to the limit as in this inequality. One clearly has as because converges strongly to the identity in as . By the triangle inequality,
The following reasoning is to be understood to hold for each fixed in a subset of of full -measure. Since and in , hence also in measure, for all , and is continuous, it follows that
in measure for all . Moreover,
and , which imply, by the dominated convergence theorem,
By a completely analogous argument one shows that , hence also that
Let us now show that
in probability. Recall that, for the sequence of continuous local martingales , one has in probability if and only if in probability (see e.g. [17, Proposition 17.6]). We have
and, by the triangle inequality,
pointwise in the time variable, and because converges strongly to the identity in . The above also yields
where, since and ,
Therefore, by the dominated convergence theorem,
We now establish a maximal inequality for stochastic convolutions that might be interesting in its own right (see Remark 4 below). We shall use the following notation, already used in the Introduction:
Let and . If satisfies the hypothesis of Theorem 1, then the stochastic convolution has (a modification with) continuous paths and
We proceed in two steps, first assuming that takes values in , then removing this assumption.
Step 1. Let us assume for the moment that . As in the proof of Proposition 2, it is easy to see that is the unique strong solution to
Then Itô’s formula yields
we can write
Let be arbitrary but fixed. Then with
Therefore, by Itô’s formula for real processes,
where we have applied Young’s inequality555From now on, whenever we apply Young’s inequality, we shall mostly state only the exponents used. in the form