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10
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...
On Regularity Of Transition Probabilities And Invariant Measures Of Singular Diffusions Under Minimal Conditions
, 1999
"... Let A = (a ij ) be a matrixvalued Borel mapping on a domain# # R d , let b = (b i ) be a vector field on # and let L A,b # = a ij # x i # x j # + b i # x i #. We study Borel measures on# that satisfy the elliptic equation L # A,b = 0 in the weak sense: # L A,b # d = 0 for all # ..."
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Cited by 8 (3 self)
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Let A = (a ij ) be a matrixvalued Borel mapping on a domain# # R d , let b = (b i ) be a vector field on # and let L A,b # = a ij # x i # x j # + b i # x i #. We study Borel measures on# that satisfy the elliptic equation L # A,b = 0 in the weak sense: # L A,b # d = 0 for all # # C # 0(#2 We prove that, under mild conditions, has a density. If A is locally uniformly nondegenerate, A # H p,1 loc and b # L p loc for some p > d, then this density belongs to H p,1 loc . Actually, we prove Sobolev regularity for solutions of certain generalized nonlinear elliptic inequalities. Analogous results are obtained in the parabolic case. These results are applied to transition probabilities and invariant measures of diffusion processes.
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
A generalization of Hasminskii's theorem on existence of invariant measures for locally integrable drifts
, 1998
"... We prove existence of invariant measures for diffusions on IR n whose drift are merely locally in L p with p ? n. Our only global assumption is that a Lyapunovtype function exists. Subsequently, we generalize this result to the case where IR n is replaced by a Riemannian manifold. AMS 1991 Su ..."
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Cited by 5 (4 self)
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We prove existence of invariant measures for diffusions on IR n whose drift are merely locally in L p with p ? n. Our only global assumption is that a Lyapunovtype function exists. Subsequently, we generalize this result to the case where IR n is replaced by a Riemannian manifold. AMS 1991 Subject classification: Primary: 60H10, 47D07 Secondary: 47D06, 60J60 Keywords and phrases: invariant measures, elliptic equations, diffusion processes. Running head: Existence of invariant measures 0 Introduction A wellknown theorem due to Hasminskii [1], [2] states that a diffusion process in IR n (or, more generally, in a smooth manifold) with locally Lipschitzian coefficients possesses an invariant probability provided there exists a certain Lyapunovtype function for the generator of this diffusion. In applications one of the best known cases is where the drift term is coercive. In recent years there has been a growing interest in diffusion processes with more general, possibly singul...
Elliptic Equations for Measures on Infinite Dimensional Spaces and Applications
, 1999
"... We introduce and study a new concept of a weak elliptic equation for measures on infinite dimensional spaces. This concept allows one to consider equations whose coefficients are not globally integrable. By using a suitably extended Lyapunov function technique, we derive a priori estimates for the s ..."
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Cited by 4 (3 self)
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We introduce and study a new concept of a weak elliptic equation for measures on infinite dimensional spaces. This concept allows one to consider equations whose coefficients are not globally integrable. By using a suitably extended Lyapunov function technique, we derive a priori estimates for the solutions of such equations and prove new existence results. As an application, we consider stochastic Burgers, reactiondiffusion, and NavierStokes equations and investigate the elliptic equations for the corresponding invariant measures. Our general theorems yield a priori estimates and existence results for such elliptic equations. We also obtain moment estimates for Gibbs distributions and prove an existence result applicable to a wide class of models.
L_pAnalysis Of Finite And Infinite Dimensional Diffusion Operators
"... This paper consists of lectures given at the C.I.M.E. summer school on Kolmogorov equations held at Cetraro in 1998. The purpose of these lectures was to present an approach to Kolmogorov equations in infinite dimensions which is based on an L p ()analysis of the corresponding diffusion operator ..."
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Cited by 4 (1 self)
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This paper consists of lectures given at the C.I.M.E. summer school on Kolmogorov equations held at Cetraro in 1998. The purpose of these lectures was to present an approach to Kolmogorov equations in infinite dimensions which is based on an L p ()analysis of the corresponding diffusion operators w.r.t. suitably chosen measures. The main ideas and aims are explained, and an as complete as possible presentation is given of what has been achieved in this respect over the last few years.
Strong uniqueness for a class of infinite dimensional Dirichlet operators and applications to stochastic quantization
, 1997
"... Strong and Markov uniqueness problems in L 2 for Dirichlet operators on rigged Hilbert spaces are studied. An analytic approach based on apriori estimates is used. The extension of the problem to the L p setting is discussed. As a direct application essential self adjointness and strong uni ..."
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Cited by 3 (1 self)
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Strong and Markov uniqueness problems in L 2 for Dirichlet operators on rigged Hilbert spaces are studied. An analytic approach based on apriori estimates is used. The extension of the problem to the L p setting is discussed. As a direct application essential self adjointness and strong uniqueness in L p is proved for the generator (with initial domain the bounded smooth cylinder functions) of the stochastic quantization process for Euclidean quantum field theory in finite volume ae R 2 . AMS Subject Classification Primary: 47 B 25, 81 S 20 Secondary: 31 C 25, 60 H 15, 81 Q 10 Key words and phrases: Dirichlet operators, essential selfadjointness, C 0  semigroups, generators, stochastic quantization, Markov uniqueness, apriori estimates Running head: Strong uniqueness for Dirichlet operators 1 Introduction The theory of Dirichlet forms is a rapidly developing field of modern analysis which has intimate relationships with potential theory, probability theory, diffe...
Uniqueness of invariant measures and maximal dissipativity of diffusion operators on L¹
, 1999
"... It is proved that there exists at most one probability measure on R d , so that L = 0, where L = a ij @ i @ j + b i @ i , provided L; C 1 0 (R) is maximally dissipative on L 1 (R d ; ) for at least one , so that L = 0. Here it is assumed that (a ij ) is nondegenerate, a ..."
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Cited by 2 (1 self)
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It is proved that there exists at most one probability measure on R d , so that L = 0, where L = a ij @ i @ j + b i @ i , provided L; C 1 0 (R) is maximally dissipative on L 1 (R d ; ) for at least one , so that L = 0. Here it is assumed that (a ij ) is nondegenerate, a ij 2 H p;1 loc , and b i 2 L p loc . We also present a whole class of examples (even for a ij = ij ), where L = 0 has more than one solution. Furthermore, recent related results are reviewed.
Strong Uniqueness For Certain Infinite Dimensional Dirichlet Operators And Applications To Stochastic Quantization
, 1998
"... . Strong and Markov uniqueness problems in L 2 for Dirichlet operators on rigged Hilbert spaces are studied. An analytic approach based on apriori estimates is used. The extension of the problem to the L p setting is discussed. As a direct application essential selfadjointness and strong un ..."
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. Strong and Markov uniqueness problems in L 2 for Dirichlet operators on rigged Hilbert spaces are studied. An analytic approach based on apriori estimates is used. The extension of the problem to the L p setting is discussed. As a direct application essential selfadjointness and strong uniqueness in L p is proved for the generator (with initial domain the bounded smooth cylinder functions) of the stochastic quantization process for Euclidean quantum field theory in finite volume ae R 2 . 1. Introduction The theory of Dirichlet forms is a rapidly developing field of modern analysis which has intimate relationships with potential theory, probability theory, differential equations and quantum physics. We refer to the monographs [16], [17], [20] and [32] where the theory of Dirichlet forms with applications to different branches of analysis and probability theory is presented. Though the abstract general theory is well developed, specific analytic questions remain open whe...
L^p uniqueness of nonsymmetric diffusion operators with singular drift coefficients: I. The finitedimensional case
"... Two uniqueness results for C 0 semigroups on weighted L p spaces over R n generated by operators of type \Delta + fi \Delta r with singular drift fi are proven. A key ingredient in the proofs is the verification of some kind of "weak Kato inequality" which seems to break down exactly for thos ..."
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Two uniqueness results for C 0 semigroups on weighted L p spaces over R n generated by operators of type \Delta + fi \Delta r with singular drift fi are proven. A key ingredient in the proofs is the verification of some kind of "weak Kato inequality" which seems to break down exactly for those drift singularities where L p uniqueness breaks down as well.