Results 1  10
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54
Stochastic calculus with anticipative integrands, Probab. Theory Related Fields 78
, 1988
"... Summary. We study the stochastic integral defined by Skorohod in [24] of a possibly anticipating integrand, as a function of its upper limit, and establish an extended It6 formula. We also introduce an extension of Stratonovich's integral, and establish the associated chain rule. In all the results, ..."
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Cited by 53 (8 self)
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Summary. We study the stochastic integral defined by Skorohod in [24] of a possibly anticipating integrand, as a function of its upper limit, and establish an extended It6 formula. We also introduce an extension of Stratonovich's integral, and establish the associated chain rule. In all the results, the adaptedness of the integrand is replaced by a certain smoothness requirement. 1.
Analysis and Geometry on Configuration Spaces
, 1997
"... In this paper foundations are presented to a new systematic approach to analysis and geometry for an important class of infinite dimensional manifolds, namely, configuration spaces. More precisely, a differential geometry is introduced on the configuration space \Gamma X over a Riemannian manifold X ..."
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Cited by 50 (8 self)
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In this paper foundations are presented to a new systematic approach to analysis and geometry for an important class of infinite dimensional manifolds, namely, configuration spaces. More precisely, a differential geometry is introduced on the configuration space \Gamma X over a Riemannian manifold X. This geometry is "nonflat" even if X = IR d . It is obtained as a natural lifting of the Riemannian structure on X. In particular, a corresponding gradient r \Gamma , divergence div \Gamma , and LaplaceBeltrami operator H \Gamma = \Gammadiv \Gamma r \Gamma are constructed. The associated volume elements, i.e., all measures ¯ on \Gamma X w.r.t. which r \Gamma and div \Gamma become dual operators on L 2 (\Gamma X ; ¯), are identified as exactly the mixed Poisson measures with mean measure equal to a multiple of the volume element dx on X. In particular, all these measures obey an integration by parts formula w.r.t. vector fields on \Gamma X . The corresponding Dirichlet...
Spectral gaps in Wasserstein distances and the 2D stochastic NavierStokes equations
, 2006
"... We develop a general method that allows to show the existence of spectral gaps for Markov semigroups on Banach spaces. Unlike most previous work, the type of norm we consider for this analysis is neither a weighted supremum norm nor an Ł ptype norm, but involves the derivative of the observable as ..."
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Cited by 15 (7 self)
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We develop a general method that allows to show the existence of spectral gaps for Markov semigroups on Banach spaces. Unlike most previous work, the type of norm we consider for this analysis is neither a weighted supremum norm nor an Ł ptype norm, but involves the derivative of the observable as well and hence can be seen as a type of 1–Wasserstein distance. This turns out to be a suitable approach for infinitedimensional spaces where the usual Harris or Doeblin conditions, which are geared to total variation convergence, regularly fail to hold. In the first part of this paper, we consider semigroups that have uniform behaviour which one can view as an extension of Doeblin’s condition. We then proceed to study situations where the behaviour is not so uniform, but the system has a suitable Lyapunov structure, leading to a type of Harris condition. We finally show that the latter condition is satisfied by the twodimensional stochastic NavierStokers equations, even in situations where the forcing is extremely degenerate. Using the convergence result, we show shat the stochastic NavierStokes equations ’ invariant measures depend continuously on the viscosity and the structure of the forcing. 1
A Quantum Nonadaptive Itô Formula and Stochastic Analysis in Fock scale
 Journal of Funct. Analysis
, 1991
"... Abstract. A generalized definition of quantum stochastic (QS) integrals and differentials is given in the free of adaptiveness and basis form in terms of Malliavin derivative on a projective Fock scale, and their uniform continuity and QS differentiability with respect to the inductive limit converg ..."
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Cited by 14 (8 self)
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Abstract. A generalized definition of quantum stochastic (QS) integrals and differentials is given in the free of adaptiveness and basis form in terms of Malliavin derivative on a projective Fock scale, and their uniform continuity and QS differentiability with respect to the inductive limit convergence is proved. A new form of QS calculus based on an inductive ⋆–algebraic structure in an indefinite space is developed and a nonadaptive generalization of the QS Itô formula for its representation in Fock space is derived. The problem of solution of general QS evolution equations in a Hilbert space is solved in terms of the constructed operator representation of chronological products, defined in the indefinite space, and the unitary and *–homomorphism property respectively for operators and maps of these solutions, corresponding to the pseudounitary and ⋆–homomorphism property of the QS integrable generators, is proved. 1.
An Extension Of Hörmander's Theorem For Infinitely Degenerate SecondOrder Operators
, 1995
"... . We establish the hypoellipticity of a large class of highly degenerate second order differential operators of Hormander type. The hypotheses of our theorem allow Hormander's general Lie algebra condition to fail on a collection of hypersurfaces. The proof of the theorem is probabilistic in nature. ..."
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Cited by 11 (4 self)
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. We establish the hypoellipticity of a large class of highly degenerate second order differential operators of Hormander type. The hypotheses of our theorem allow Hormander's general Lie algebra condition to fail on a collection of hypersurfaces. The proof of the theorem is probabilistic in nature. It is based on the Malliavin calculus and requires new sharp estimates for diffusion processes in Euclidean space. 1. Introduction. Let X 0 ; : : : ; Xn denote a collection of smooth vector fields defined on an open subset D of R d , and c : D ! R a smooth function. Consider the second order differential operator L := 1 2 n X i=1 X 2 i +X 0 + c: (1:1) Let Lie(X 0 ; : : : ; Xn ) be the Lie algebra generated by the vector fields X 0 ; : : : ; Xn . According to the theorem of Hormander ([H], Theorem 1.1), L is hypoelliptic on D if the vector space Lie(X 0 ; : : : ; Xn )(x) has dimension d at every x 2 D. Hormander's condition characterizes hypoellipticity for operators of the form (1...
Degenerate Stochastic Differential Equations, Flows And Hypoellipticity
 Pitman Monographs and Surveys in Pure and Applied Mathematics
, 1995
"... this article we shall study stochastic hereditary systems on R ..."
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Cited by 10 (0 self)
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this article we shall study stochastic hereditary systems on R
Statistical Aspects of the fractional stochastic calculus
 ANN. STAT
, 2007
"... We apply the techniques of stochastic integration with respect to the fractional Brownian motion and the theory of regularity and supremum estimation for stochastic processes to study the maximum likelihood estimator (MLE) for the drift parameter of stochastic processes satisfying stochastic equati ..."
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Cited by 10 (5 self)
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We apply the techniques of stochastic integration with respect to the fractional Brownian motion and the theory of regularity and supremum estimation for stochastic processes to study the maximum likelihood estimator (MLE) for the drift parameter of stochastic processes satisfying stochastic equations driven by fractional Brownian motion with any level of Holderregularity (any Hurst parameter). We prove existence and strong consistency of the MLE for linear and nonlinear equations. We also prove that a version of the MLE using only discrete observations is still a strongly consistent estimator.
Differential geometry of Poisson spaces
, 1996
"... We identify the differential geometry of the Poisson measure over R^d. We prove an integration by parts formula and determine explicitly the corresponding gradient, divergence, Dirichlet form, generalized Laplacian and heat semigroup. We also briefly describe immediate generalizations, for example ..."
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Cited by 9 (0 self)
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We identify the differential geometry of the Poisson measure over R^d. We prove an integration by parts formula and determine explicitly the corresponding gradient, divergence, Dirichlet form, generalized Laplacian and heat semigroup. We also briefly describe immediate generalizations, for example to Poisson measures over (infinite dimensional) manifolds or other random fields (resp. Gibbs measures).
Brownian sheet and capacity
, 1999
"... Summary. The main goal of this paper is to present an explicit capacity estimate for hitting probabilities of the Brownian sheet. As applications, we determine the escape rates of the Brownian sheet, and also obtain a local intersection equivalence between the Brownian sheet and the additive Brownia ..."
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Cited by 8 (4 self)
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Summary. The main goal of this paper is to present an explicit capacity estimate for hitting probabilities of the Brownian sheet. As applications, we determine the escape rates of the Brownian sheet, and also obtain a local intersection equivalence between the Brownian sheet and the additive Brownian motion. Other applications concern quasi–sure properties in Wiener space.
A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs
"... We present a theory of hypoellipticity and unique ergodicity for semilinear parabolic stochastic PDEs with “polynomial ” nonlinearities and additive noise, considered as abstract evolution equations in some Hilbert space. It is shown that if Hörmander’s bracket condition holds at every point of this ..."
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Cited by 8 (7 self)
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We present a theory of hypoellipticity and unique ergodicity for semilinear parabolic stochastic PDEs with “polynomial ” nonlinearities and additive noise, considered as abstract evolution equations in some Hilbert space. It is shown that if Hörmander’s bracket condition holds at every point of this Hilbert space, then a lower bound on the Malliavin covariance operator Mt can be obtained. Informally, this bound can be read as “Fix any finitedimensional projection Π on a subspace of sufficiently regular functions. Then the eigenfunctions of Mt with small eigenvalues have only a very small component in the image of Π.” We also show how to use a priori bounds on the solutions to the equation to obtain good control on the dependency of the bounds on the Malliavin matrix on the initial condition. These bounds are sufficient in many cases to obtain the asymptotic strong Feller property introduced in [HM06]. One of the main novel technical tools is an almost sure bound from below on the size of “Wiener polynomials, ” where the coefficients are possibly nonadapted stochastic processes satisfying a Lipschitz condition. By exploiting the polynomial structure of the equations, this result can be used to replace Norris ’ lemma, which is unavailable in the present context. We conclude by showing that the twodimensional stochastic NavierStokes equations and a large class of reactiondiffusion equations fit the framework of our theory. Contents 1