Paul Lescot

Perturbations of Generalized Mehler Semigroups and Applications to Stochastic Heat Equations with Levy Noise and Singular Drift

In this paper we solve the Kolmogorov equation and, as a consequence, the martingale problem corresponding to a stochastic differential equation of type dXt=AXtdt+b(Xt)dt+dYt, on a Hilbert space E, where (Yt)t≥0 is a Levy process on E,A generates a C0-semigroup on E and b:E→E. Our main point is to allow unbounded A and also singular (in particular, non-continuous) b. Our approach is based on perturbation theory of C0-semigroups, which we apply to generalized Mehler semigroups considered on L2(μ), where μ is their respective invariant measure. We apply our results, in particular, to stochastic heat equations with Levy noise and singular drift.

The Global Random Attractor for a Class of Stochastic Porous Media Equations

We prove new L 2-estimates and regularity results for generalized porous media equations “shifted by” a function-valued Wiener path. To include Wiener paths with merely first spatial (weak) derivates we introduce the notion of “ζ-monotonicity” for the non-linear function in the equation. As a consequence we prove that stochastic porous media equations have global random attractors. In addition, we show that (in particular for the classical stochastic porous media equation) this attractor consists of a random point.

GENERATORS OF MEHLER-TYPE SEMIGROUPS AS PSEUDO-DIFFERENTIAL OPERATORS

We study semigroups (Pt)t ≥ 0 on a Hilbert space E, given by a Mehler-type formula: Under reasonable assumptions, the Lp(E,μ)-generator of (Pt)t ≥ 0 turns out to be expressible as a pseudo-differential operator, provided μ is an invariant measure for (Pt)t ≥ 0. The question of Lp-uniqueness is also answered positively.

The martingale problem for pseudo-differential operators on infinite-dimensional spaces

A martingale problem for pseudo-differential operators on infinite dimensional spaces is formulated and the existence of a solution is proved. Applications to infinite dimensional “stable-like” processes are presented. §0. Introduction The purpose of this paper is to formulate and solve a martingale problem for pseudo-differential operators on infinite dimensional state space. We thus provide a routine machinery to construct (non-trivial) infinite dimensional processes which are merely càdlàg and have not been obtained before by other means. We emphasize, however, that this work is only on existence of solutions to these martingale problems, not on uniqueness. The uniqueness, which is known to be already extremely difficult in infinite dimensions if we are merely dealing with “differential” (i.e., local) operators, will be studied in a forthcoming paper. The main motivation of the present work (see [FuR 97] and also [BRS 96]) is to make a contribution to the development of a theory of pseudo-differential operators in infinite dimensions, since this theory has proved to be so powerful in finite dimensions, e.g. in proving index theorems. The state spaces E treated here are duals to countably nuclear spaces and the definition of a pseudo-differential operator p( · ,D) on E is taken from [Hr82, Hr87]. There are several possibilities to introduce pseudodifferential operators in infinite dimensions. One of them is based on the Received November 10, 1995. Revised September 8, 1997. Financial support of the Sonderforschungsbereich 343 (Bielefeld), EC-Human-Capital and Mobility (Contract No. ERBCHBGCT 920016), the International Science Foundation (Grant No. M38000), and the Russian Foundation of Fundamental Research is gratefully acknowledged. 101 102 V. BOGACHEV, P. LESCOT AND M. RÖCKNER idea to use Fourier transforms of measures as test functions. The approach to Feynman integrals developed in [AH-K76, AH-K77] was based on this idea. Earlier in [Fom68] infinite-dimensional differential operators were considered in the framework of the duality between spaces of functions and spaces of measures (replacing the finite-dimensional duality between the Schwartz test function space S and its dual S′). This latter approach was then further developed by several authors. The most convenient version for us is that due to A. Yu. Hrennikov (cf. [Hr82] and also [Hr87] and the references therein). In particular, the domain of p( · ,D) is hence a space of Fourier transforms of measures. Thus one can avoid the use of a reference measure, as Lebesgue measure is in finite dimensions (which does not exist on our state space E if dim E = ∞). We refer to Sections 1 and 2 for precise details, but emphasize here that it definitely is a natural and direct way to extend the definition of a pseudo-differential operator to infinite dimensions. A further justification for this from a probabilistic point of view is given by the results of this paper. After discussing examples (cf. Examples 2.2 below) and giving the necessary probabilistic definitions at the end of Section 2, we then prove the existence of the solution to the martingale problem for p( · ,D) on E in Section 3. We thus extend fundamental work of D.W. Stroock (cf. [St75]) and, in particular, the more recent work by W. Hoh (cf. [H92, H94], see also [H95a, H95b]). The problems in the proof arise in part from the nonmetrizability of E and hence of DE (i.e., the space of càdlàg paths from R+ to E equipped with the Skorohod topology). But here we benefit a lot from the tightness results in [M83] resp. [J86], i.e., the characterization of tightness of laws of càdlàg processes on E in terms of tightness of the laws of their “one-dimensional component processes.” By these results the approximation method in [H92, H94] can be shown to extend to the infinite dimensional case. We particularly concentrate on the problems which arise from the measure theory on DE which a-priori is much more difficult than that on D n (but was well-analyzed in [J86]). Another substantial difficulty we had to face in our proof is the lack of “localizing” functions on E with uniformly bounded images under p( · ,D). The existence of such functions is crucial in the proof of Hoh (cf. [H92, Lemma 2.10]) in the case E = R. We overcome this difficulty by proving the existence of suitable localizing functions for “one-dimensional components of the processes” (cf. Claim 1 in Section 3) which turns out to be sufficient. Finally, it should be emphasized that the reduction to one-dimensional components in order to prove tightMARTINGALE PROBLEMS IN INFINITE DIMENSIONS 103 ness according to [M83], [J86], unfortunately does not take us back to the finite (or even one-) dimensional situation which was solved in [H92, H94]. The reason is that these components do not solve a one-dimensional martingale problem, since the operator p( · ,D) obviously introduces interactions between the different components and since the underlying filtration (Ft)t≥0 has to be the one generated by the full process and not the one generated by the respective component. The results of this paper have been announced at conferences in Warwick and Bielefeld in Summer 1994, as well as in several invited talks e.g. at the University of California, San Diego, October 1994, and the MittagLeffler-Institute, Stockholm, March 1995. §1. Definitions, notation and preliminary results In this paper E will denote the topological dual F ′ of a real countably nuclear Fréchet space F endowed with the strong topology. We equip the topological dual E′ of E with the topology of uniform convergence on bounded sets. Then E′ = F (as topological vector spaces; cf. e.g. [GV 64, p. 61] or [S71, Chap. III, §7 and Chap. IV, §5]). Let B(E), B(E′) denote the Borel σ-algebras of E, E′ respectively. Remark 1.1. Since E′ (= F ) is separable (see e.g. [GV 64, p. 73]), there exist ξn ∈ E ′, n ∈ N, separating the points of E. Since E is a Lusin space (i.e., the continuous one-to-one image of a Polish space, cf. [Sch73]), by [Sch73, Lemma 18, p. 108] we have that B(E) = σ(ξn|n ∈ N). Let M := M b (E ′) be the space of complex-valued measures on B(E′) with bounded total variation, and 〈 , 〉 the dualization between E and E′, i.e., 〈x, ξ〉 = ξ(x), x ∈ E, ξ ∈ E′. We recall that the Fourier transform of μ ∈ M is defined by F(μ)(x) := ∫ E′ ei〈x,ξ〉μ(dξ), x ∈ E . We set W := F(M). W will play the role of a “test-function” space (cf. [AHK76, AH-K77] and also [Hr82, p. 779]). Since F is one-to-one F−1:W → M is well-defined. Let Cb(E; C) denote the space of all bounded continuous complex valued functions on E. The following is standard for the type of state spaces considered here. 104 V. BOGACHEV, P. LESCOT AND M. RÖCKNER Lemma 1.2. W ⊂ Cb(E; C). Proof. see [Sch73, Theorem 3, p. 239 or Theorem 1, p. 193]. We define the set of cylinder functions FS(E; C) with base functions in S(R; C) (:= the space of complex-valued Schwartz test functions) for some m ∈ N by FS(E; C) := {f(ξ1, . . . , ξm) | m ∈ N, f ∈ S(R ; C), ξ1, . . . , ξm ∈ E ′}. Lemma 1.3. FS(E; C) ⊂ W . Proof. Let φ ∈ FS(E; C), φ = f(ξ1, . . . , ξm), m ∈ N, f ∈ S(R ; C) and ξ1, . . . , ξm ∈ E ′. Then there exists g ∈ S(R, C) such that