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112
Stochastic calculus with anticipative integrands
 PROBAB. THEORY RELATED FIELDS 78
, 1988
"... 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 ..."
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Cited by 106 (12 self)
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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.
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 97 (13 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...
Uniqueness of the Invariant Measure for a Stochastic PDE Driven by Degenerate Noise
, 2001
"... We consider the stochastic GinzburgLandau equation in a bounded domain. We assume the stochastic forcing acts only on high spatial frequencies. The lowlying frequencies are then only connected to this forcing through the nonlinear (cubic) term of the GinzburgLandau equation. Under these assumpti ..."
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Cited by 67 (16 self)
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We consider the stochastic GinzburgLandau equation in a bounded domain. We assume the stochastic forcing acts only on high spatial frequencies. The lowlying frequencies are then only connected to this forcing through the nonlinear (cubic) term of the GinzburgLandau equation. Under these assumptions, we show that the stochastic PDE has a unique invariant measure. The techniques of proof combine a controllability argument for the lowlying frequencies with an infinite dimensional version of the Malliavin calculus to show positivity and regularity of the invariant measure. This then implies the uniqueness of that measure. Contents 1 Introduction 2 2 Some Preliminaries on the Dynamics 5 3 Controllability 6 4 Strong Feller Property and Proof of Theorem 1.1 9 5 Regularity of the Cutoff Process 11 5.1 Splitting and Interpolation Spaces . . . . . . . . . . . . . . . . . . . 12 5.2 Proof of Theorem 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 12 5.3 Smoothing Properties of the...
A version of Hörmander’s theorem for the fractional Brownian motion. Probab. Theory Related Fields 139
, 2007
"... It is shown that the law of an SDE driven by fractional Brownian motion with Hurst parameter greater than 1/2 has a smooth density with respect to Lebesgue measure, provided that the driving vector fields satisfy Hörmander’s condition. The main new ingredient of the proof is an extension of Norris ..."
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Cited by 42 (13 self)
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It is shown that the law of an SDE driven by fractional Brownian motion with Hurst parameter greater than 1/2 has a smooth density with respect to Lebesgue measure, provided that the driving vector fields satisfy Hörmander’s condition. The main new ingredient of the proof is an extension of Norris ’ lemma to this situation. Contents 1
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 40 (15 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 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 32 (14 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
On the existence of smooth densities for jump processes
 Probab. Theory Related Fields
, 1996
"... Abstract. We consider a Lévy process Xt and the solution Yt of a stochastic differential equation driven by Xt; we suppose that Xt has infinitely many small jumps, but its Lévy measure may be very singular (for instance it may have a countable support). We obtain sufficient conditions ensuring the ..."
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Cited by 31 (1 self)
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Abstract. We consider a Lévy process Xt and the solution Yt of a stochastic differential equation driven by Xt; we suppose that Xt has infinitely many small jumps, but its Lévy measure may be very singular (for instance it may have a countable support). We obtain sufficient conditions ensuring the existence of a smooth density for Yt; these conditions are similar to those of the classical Malliavin calculus for continuous diffusions. More generally, we study the smoothness of the law of variables F defined on a Poisson probability space; the basic tool is a duality formula from which we estimate the characteristic function of F. Mathematics Subject Classification (1991). 60H07 60J75 60J30 Suppose that we are given a continuous diffusion process Yt; it can be represented as a functional of a Wiener process. The aim of Malliavin’s calculus introduced in [8] is to prove, by means of probabilistic methods, the existence of a smooth density for Yt. The basic tool is an integration by parts formula on the Wiener space which enables to prove,
Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths
, 2011
"... We consider differential equations driven by rough paths and study the regularity of the laws and their long time behaviour. In particular, we focus on the case when the driving noise is a rough path valued fractional Brownian motion with Hurst parameter ..."
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Cited by 27 (6 self)
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We consider differential equations driven by rough paths and study the regularity of the laws and their long time behaviour. In particular, we focus on the case when the driving noise is a rough path valued fractional Brownian motion with Hurst parameter
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 23 (6 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.
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 21 (9 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.