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36
Regularity of Invariant Measures on Finite and Infinite Dimensional Spaces and Applications
 J. Funct. Anal
, 1994
"... In this paper we prove new results on the regularity (i.e., smoothness) of measures ¯ solving the equation L ¯ = 0 for operators of type L = \Delta +B \Delta r on finite and infinite dimensional state spaces E. In particular, we settle a conjecture of I. Shigekawa in the situation where \Delta = ..."
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Cited by 23 (12 self)
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In this paper we prove new results on the regularity (i.e., smoothness) of measures ¯ solving the equation L ¯ = 0 for operators of type L = \Delta +B \Delta r on finite and infinite dimensional state spaces E. In particular, we settle a conjecture of I. Shigekawa in the situation where \Delta = \Delta H is the GrossLaplacian, (E; H; fl) is an abstract Wiener space and B = \Gammaid E +v where v takes values in the CameronMartin space H . Using Gross' logarithmic Sobolevinequality in an essential way we show that ¯ is always absolutely continuous w.r.t. the Gaussian measure fl and that the square root of the density is in the Malliavin test function space of order 1 in L 2 (fl). Furthermore, we discuss applications to infinite dimensional stochastic differential equations and prove some new existence results for L ¯ = 0. These include results on the "inverse problem", i.e., we give conditions ensuring that B is the (vector) logarithmic derivative of a measure. We also prove ...
Stochastic Burgers and KPZ equations from particle systems
 Comm. Math. Phys
, 1997
"... We consider the weakly asymmetric exclusion process on the one dimensional lattice. It has been proven that, in the diffusive scaling limit, the density field evolves according to the Burgers equation [8, 19, 14] and the fluctuation field converges to a generalized OrnsteinUhlenbeck process [8, 10] ..."
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Cited by 18 (2 self)
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We consider the weakly asymmetric exclusion process on the one dimensional lattice. It has been proven that, in the diffusive scaling limit, the density field evolves according to the Burgers equation [8, 19, 14] and the fluctuation field converges to a generalized OrnsteinUhlenbeck process [8, 10]. We analyze instead the density fluctuations beyond the hydrodynamical scale and prove that their limiting distribution solves the (non linear) Burgers equation with a random noise on the density current. We also study an interface growth model, for which the microscopic dynamics is a SolidOnSolid type deposition process. We prove that the fluctuation field, if suitably rescaled, converges to the KardarParisiZhang equation. This provides a microscopic justification of the so called kinetical roughening, i.e. the non Gaussian fluctuations in some nonequilibrium processes. Our main tool is the ColeHopf transformation and its microscopic version. We also exploit the (known) connection bet...
The Theory Of Generalized Dirichlet Forms And Its Applications In Analysis And Stochastics
, 1996
"... We present an introduction (also for nonexperts) to a new framework for the analysis of (up to) second order differential operators (with merely measurable coefficients and in possibly infinitely many variables) on L²spaces via associated bilinear forms. This new framework, in particular, covers b ..."
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Cited by 18 (1 self)
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We present an introduction (also for nonexperts) to a new framework for the analysis of (up to) second order differential operators (with merely measurable coefficients and in possibly infinitely many variables) on L²spaces via associated bilinear forms. This new framework, in particular, covers both the elliptic and the parabolic case within one approach. To this end we introduce a new class of bilinear forms, socalled generalized Dirichlet forms, which are in general neither symmetric nor coercive, but still generate associated C0 semigroups. Particular examples of generalized Dirichlet forms are symmetric and coercive Dirichlet forms (cf. [FOT], [MR1]) as well as time dependent Dirichlet forms (cf. [O1]). We discuss many applications to differential operators that can be treated within the new framework only, e.g. parabolic differential operators with unbounded drifts satisfying no L p conditions, singular and fractional diffusion operators. Subsequently, we analyz...
Quasiregular Dirichlet forms: Examples and counterexamples
, 1993
"... We prove some new results on quasiregular Dirichlet forms. These include results on perturbations of Dirichlet forms, change of speed measure, and tightness. The tightness implies the existence of an associated right continuous strong Markov process. We also discuss applications to a number of exam ..."
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Cited by 14 (7 self)
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We prove some new results on quasiregular Dirichlet forms. These include results on perturbations of Dirichlet forms, change of speed measure, and tightness. The tightness implies the existence of an associated right continuous strong Markov process. We also discuss applications to a number of examples including cases with possibly degenerate (sub)elliptic part, diffusions on loops spaces, and certain FlemingViot processes.
Generalized Mehler semigroups and applications
, 1994
"... We construct and study generalized Mehler semigroups (p t ) t#0 and their associated Markov processes M. The construction methods for (p t ) t#0 are based on some new purely functional analytic results implying, in particular, that any strongly continuous semigroup on a Hilbert space H can be extend ..."
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Cited by 10 (4 self)
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We construct and study generalized Mehler semigroups (p t ) t#0 and their associated Markov processes M. The construction methods for (p t ) t#0 are based on some new purely functional analytic results implying, in particular, that any strongly continuous semigroup on a Hilbert space H can be extended to some larger Hilbert space E, with the embedding H # E being HilbertSchmidt. The same analytic extension results are applied to construct strong solutions to stochastic differential equations of type dX t = CdW t + AX t dt (with possibly unbounded linear operators A and C on H) on a suitably chosen larger space E. For Gaussian generalized Mehler semigroups (p t ) t#0 with corresponding Markov process M, the associated (nonsymmetric) Dirichlet forms (E , D(E)) are explicitly calculated and a necessary and sufficient condition for path regularity of M in terms of (E , D(E)) is proved. Then, using Dirichlet form methods it is shown that M weakly solves the above stochastic differential ...
Ergodicity for the stochastic dynamics of quasiinvariant measures with applications to Gibbs states
, 1997
"... The convex set M a of quasiinvariant measures on a locally convex space E with given "shift"RadonNikodym derivatives (i.e., cocycles) a = (a tk ) k2K 0 ; t2R is analyzed. The extreme points of M a are characterized and proved to be nonempty. A specification (of lattice type) is constructed ..."
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Cited by 10 (2 self)
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The convex set M a of quasiinvariant measures on a locally convex space E with given "shift"RadonNikodym derivatives (i.e., cocycles) a = (a tk ) k2K 0 ; t2R is analyzed. The extreme points of M a are characterized and proved to be nonempty. A specification (of lattice type) is constructed so that M a coincides with the set of the corresponding Gibbs states. As a consequence, via a wellknown method due to DynkinFollmer a unique representation of an arbitrary element in M a in terms of extreme ones is derived. Furthermore, the corresponding classical Dirichlet forms (E ; D(E )) and their associated semigroups (T t ) t?0 on L 2 (E; ) are discussed. Under a mild positivity condition it is shown that 2 M a is extreme if and only if (E ; D(E )) is irreducible or equivalently, (T t ) t?0 is ergodic. This implies timeergodicity of associated diffusions. Applications to Gibbs states of classical and quantum lattice models as well as those occuring in Euclidean...
Regularity of Invariant Measures: The Case of NonConstant Diffusion Part
 J. Funct. Anal
, 1994
"... We prove regularity (i.e., smoothness) of measures on R d satisfying the equation L = 0 where L is an operator of type Lu = tr(Au 00 ) +B \Delta ru. Here A is a Lipschitz continuous, uniformly elliptic matrixvalued map and B is merely square integrable. We also treat a class of correspon ..."
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Cited by 7 (4 self)
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We prove regularity (i.e., smoothness) of measures on R d satisfying the equation L = 0 where L is an operator of type Lu = tr(Au 00 ) +B \Delta ru. Here A is a Lipschitz continuous, uniformly elliptic matrixvalued map and B is merely square integrable. We also treat a class of corresponding infinite dimensional cases where IR d is replaced by a locally convex topological vector space X . In this cases is absolutely continuous w.r.t. a Gaussian measure on X and the square root of the RadonNikodym density belongs to the Malliavin test function space ID 2;1 . AMS Subject Classification (1991) Primary: 35B65, 60H10 Secondary: 35R15, 60H15, 47D06, 28C20 1 Department of Mechanics and Mathematics, Moscow State University, 119899 Moscow, Russia 2 School of Mathematics, Minneapolis, MN 55455, USA 3 Fakultat fur Mathematik, Universitat Bielefeld, Universitatsstr. 25, D33615 Bielefeld, Germany 1 Introduction and the main result in finite dimensions The purpose of this...
Stochastic calculus of generalized Dirichlet forms and applications to stochastic differential equations in infinite dimensions
"... this paper we systematically develop a stochastic calculus for generalized Dirichlet forms (cf. [St1]). In particular, we show Fukushima's decomposition of additive functionals and an Itotype formula in this framework. The class of generalized Dirichlet forms is much larger than the wellstudied cl ..."
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Cited by 7 (3 self)
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this paper we systematically develop a stochastic calculus for generalized Dirichlet forms (cf. [St1]). In particular, we show Fukushima's decomposition of additive functionals and an Itotype formula in this framework. The class of generalized Dirichlet forms is much larger than the wellstudied class of symmetric and coercive Dirichlet forms as in [FOT 94] resp. [MR 92] and time dependent Dirichlet forms as in [O 92]. It contains examples of an entirely new kind (cf. Section 6, [St1]). Therefore, the results obtained in this paper lead to extensions of the corresponding results in the "classical" theories. In particular the proofs are "locally" completely different (cf. e.g. Theorem 2.3 and Theorem 2.5; though for the reader's convenience we tried to follow the line of argument in [FOT 94] as closely as possible). This difference has several reasons: First of all we do not assume any sector condition; in certain cases we have to handle Equasilowersemicontinuous
First Order Perturbations Of Dirichlet Operators: Existence And Uniqueness
, 1996
"... We study perturbations of type B \Delta r of Dirichlet operators (L 0 ; D(L 0 )) associated with Dirichlet forms of type E 0 (u; v) = 1=2 R hru; rviH d¯ on L 2 (E; ¯) where E is a finite or infinite dimensional Banach space E. Here H denotes a Hilbert space densely and continuously embed ..."
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Cited by 5 (1 self)
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We study perturbations of type B \Delta r of Dirichlet operators (L 0 ; D(L 0 )) associated with Dirichlet forms of type E 0 (u; v) = 1=2 R hru; rviH d¯ on L 2 (E; ¯) where E is a finite or infinite dimensional Banach space E. Here H denotes a Hilbert space densely and continuously embedded in E. Assuming quasiregularity of (E 0 ; D(E 0 )) we show that there always exists a closed extension of Lu := L 0 u + hB; ruiH that generates a subMarkovian C 0 semigroup of contractions on L 2 (E; ¯) (resp. L 1 (E; ¯)), if B 2 L 2 (E; H;¯) and R hB; ruiH d¯ 0; u 0. If D is an appropriate core for (L 0 ; D(L 0 )) we show that there is only one closed extension of (L; D) in L 1 (E; ¯) generating a strongly continuous semigroup. In particular we apply our results to operators of type \Delta H +B \Delta r, where \Delta H denotes the GrossLaplacian on an abstract Wiener space (E; H; fl) and B = \Gammaid E + v, where v takes values in the CameronMartin s...