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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 ...
On Ito's formula for multidimensional Brownian motion
 and Related Fields
, 2000
"... . Consider a ddimensional Brownian motion X = (X 1 ; : : : ; X d ) and a function F which belongs locally to the Sobolev space W 1;2 . We prove an extension of Ito's formula where the usual second order terms are replaced by the quadratic covariations [f k (X); X k ] involving the weak fir ..."
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Cited by 9 (0 self)
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. Consider a ddimensional Brownian motion X = (X 1 ; : : : ; X d ) and a function F which belongs locally to the Sobolev space W 1;2 . We prove an extension of Ito's formula where the usual second order terms are replaced by the quadratic covariations [f k (X); X k ] involving the weak first partial derivatives f k of F . In particular we show that for any locally squareintegrable function f the quadratic covariations [f(X); X k ] exist as limits in probability for any starting point, except for some polar set. The proof is based on new approximation results for forward and backward stochastic integrals. Key words: Ito's formula, Brownian motion, stochastic integrals, quadratic covariation, Dirichlet spaces, polar sets. Supported in part by ONR grant # N000149610262 and NSF grant # 9401109INT 1. Introduction The behavior of a smooth function F on R d along the paths of ddimensional Brownian motion is described as follows by Ito's formula. Let P x be the distribu...
Finite Dimensional Approximation of Diffusion Processes on Infinite Dimensional Spaces
, 1996
"... We prove that the laws of diffusion processes M on E associated with Dirichlet forms of type E(u; v) = R E hA(z)ru(z); rv(z)i H¯(dz), where H , E are separable Hilbert spaces, are the weak limits of laws of finite dimensional diffusions. These are associated with the image Dirichlet forms obtaine ..."
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Cited by 4 (3 self)
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We prove that the laws of diffusion processes M on E associated with Dirichlet forms of type E(u; v) = R E hA(z)ru(z); rv(z)i H¯(dz), where H , E are separable Hilbert spaces, are the weak limits of laws of finite dimensional diffusions. These are associated with the image Dirichlet forms obtained from E under projections from E onto finite dimensional subspaces in H. As a byproduct we obtain Hoelder continuity of the sample paths as well as a new existence proof for the infinite dimensional diffusion M.
Vector fields on mapping spaces and related Dirichlet forms and diffusions
 OSAKA J. MATH
, 1996
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Martingale Decomposition of Dirichlet Processes on the Banach space C 0 [0,1]
"... We prove that for a given symmetric Dirichlet form of type E(u; v) = R E hA(z)ru(z); rv(z)i H ¯(dz) with E = C 0 [0; 1] and H = classical CameronMartin space the corresponding diffusion process (under P ¯ ) can be decomposed into a forward and a backward E valued martingale. Applications to p ..."
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Cited by 1 (0 self)
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We prove that for a given symmetric Dirichlet form of type E(u; v) = R E hA(z)ru(z); rv(z)i H ¯(dz) with E = C 0 [0; 1] and H = classical CameronMartin space the corresponding diffusion process (under P ¯ ) can be decomposed into a forward and a backward E valued martingale. Applications to prove tightness of laws of diffusions of the above kind are given. AMS Subject Classification Primary: 60J60 Secondary: 60G44, 31C25, 60H10, 60B11 Key words: diffusions on Banach spaces, Dirichlet forms, Dirichlet processes, martingale decomposition Running head: Infinite dimensional Dirichlet processes 1 Department of Mathematics, Imperial College of Science, Technology and Medicine, Huxley Building, 180 Queen's Gate, London, U.K. 2 Fakultat fur Mathematik, Universitat Bielefeld, Postfach 100131, D33501 Bielefeld, Germany on leave from HSH, Haugesund, Norway 1 Introduction and framework Let E = C 0 [0; 1] be the Banach space of all continuous functions on [0; 1] with initial value...
Probabilistic Representations and Hyperbound Estimates for Semigroups
, 1998
"... In this paper, we study lower order perturbations of a symmetric second order differential operator generating a hypercontractive semigroup. We give a probabilistic representation of the in general not subMarkovian semigroup associated with the perturbed operator and prove that the perturbed semigr ..."
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In this paper, we study lower order perturbations of a symmetric second order differential operator generating a hypercontractive semigroup. We give a probabilistic representation of the in general not subMarkovian semigroup associated with the perturbed operator and prove that the perturbed semigroup is also hypercontractive under some exponential integrability conditions on the coefficients. 1 Introduction Let X be a locally convex Hausdorff topological vector space which is Souslinean and let (H; !; ?H ) be a Hilbert space continuously and densely embedded into X . Let ¯ be a probability measure on (X; B(X)), where B(X) stands for the Borel oealgebra. The starting point of this paper is a second order symmetric differential operator Lu(x) = 1 2 div(A(x)ru(x)) (1) on L 2 (X; ¯) determined by a quadratic form (E ; D(E)), which satisfies a LogSobolev inequality. This is equivalent to the associated semigroup being hypercontractive (see e.g. [8]). Consider now a lower order pertu...