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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
Analysis of SPDEs arising in path sampling, part II: The nonlinear case
 In preparation
"... In many applications it is important to be able to sample paths of SDEs conditional on observations of various kinds. This paper studies SPDEs which solve such sampling problems. The SPDE may be viewed as an infinite dimensional analogue of the Langevin SDE used in finite dimensional sampling. In th ..."
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Cited by 7 (6 self)
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In many applications it is important to be able to sample paths of SDEs conditional on observations of various kinds. This paper studies SPDEs which solve such sampling problems. The SPDE may be viewed as an infinite dimensional analogue of the Langevin SDE used in finite dimensional sampling. In this paper nonlinear SDEs, leading to nonlinear SPDEs for the sampling, are studied. In addition, a class of preconditioned SPDEs is studied, found by applying a Green’s operator to the SPDE in such a way that the invariant measure remains unchanged; such infinite dimensional evolution equations are important for the development of practical algorithms for sampling infinite dimensional problems. The resulting SPDEs provide several significant challenges in the theory of SPDEs. The two primary ones are the presence of nonlinear boundary conditions, involving first order derivatives, and a loss of the smoothing property in the case of the preconditioned SPDEs. These challenges are overcome and a theory of existence, uniqueness and ergodicity developed in sufficient generality to subsume the sampling problems of interest to us. The Gaussian theory developed in Part I of this paper considers Gaussian SDEs, leading to linear Gaussian SPDEs for sampling. This Gaussian theory is used as the basis for deriving nonlinear SPDEs which effect the desired sampling in the nonlinear case, via a change of measure. 1
Ergodic theory for SDEs with extrinsic memory
, 2008
"... We develop a theory of ergodicity for a class of random dynamical systems where the driving noise is not white. The two main tools of our analysis are the strong Feller property and topological irreducibility, introduced in this work for a class of nonMarkovian systems. They allow us to obtain a cr ..."
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Cited by 7 (3 self)
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We develop a theory of ergodicity for a class of random dynamical systems where the driving noise is not white. The two main tools of our analysis are the strong Feller property and topological irreducibility, introduced in this work for a class of nonMarkovian systems. They allow us to obtain a criteria for ergodicity which is similar in nature to the DoobKhas’minskii theorem. The second part of this article shows how it is possible to apply these results to the case of stochastic differential equations driven by fractional Brownian motion. It follows that under a nondegeneracy condition on the noise, such equations admit a unique adapted stationary solution. 1
Maslowski B.: Lower estimates of transition densities and bounds on exponential ergodicity for stochastic PDE’s
"... A formula for the transition density of a Markov process defined by an infinitedimensional stochastic equation is given in terms of the Ornstein–Uhlenbeck bridge and a useful lower estimate on the density is provided. As a consequence, uniform exponential ergodicity and Vergodicity are proved for ..."
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Cited by 5 (2 self)
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A formula for the transition density of a Markov process defined by an infinitedimensional stochastic equation is given in terms of the Ornstein–Uhlenbeck bridge and a useful lower estimate on the density is provided. As a consequence, uniform exponential ergodicity and Vergodicity are proved for a large class of equations. We also provide computable bounds on the convergence rates and the spectral gap for the Markov semigroups defined by the equations. The bounds turn out to be uniform with respect to a large family of nonlinear drift coefficients. Examples of finitedimensional stochastic equations and semilinear parabolic equations are given. 1. Introduction. The
Nonsymmetric OrnsteinUhlenbeck Generators
, 1999
"... We review some properties of non symmetric OrnsteinUhlenbeck generators in L p (), where is an invariant measure. We provide a necessary and sufficient condition for the Poincare Inequality and the Logarithmic Sobolev Inequality, thus extending the result of Rothaus and Simon to the nonsymmetric ..."
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Cited by 2 (1 self)
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We review some properties of non symmetric OrnsteinUhlenbeck generators in L p (), where is an invariant measure. We provide a necessary and sufficient condition for the Poincare Inequality and the Logarithmic Sobolev Inequality, thus extending the result of Rothaus and Simon to the nonsymmetric Gaussian case. Next, we show that the same condition is necessary and sufficient for the compact embedding of the Sobolev space W 1,p Q () into L p (). We provide also necessary and sufficient condition for the OrnsteinUhlenbeck operator to be selfadjoint and in this case its domain in L p () is completely characterized.
ASYMPTOTICS OF SOLUTIONS TO SEMILINEAR STOCHASTIC WAVE EQUATIONS
, 2006
"... Largetime asymptotic properties of solutions to a class of semilinear stochastic wave equations with damping in a bounded domain are considered. First an energy inequality and the exponential bound for a linear stochastic equation are established. Under appropriate conditions, the existence theorem ..."
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Cited by 2 (0 self)
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Largetime asymptotic properties of solutions to a class of semilinear stochastic wave equations with damping in a bounded domain are considered. First an energy inequality and the exponential bound for a linear stochastic equation are established. Under appropriate conditions, the existence theorem for a unique global solution is given. Next the questions of bounded solutions and the exponential stability of an equilibrium solution, in meansquare and the almost sure sense, are studied. Then, under some sufficient conditions, the existence of a unique invariant measure is proved. Two examples are presented to illustrate some applications of the theorems. 1. Introduction. Semilinear
INVARIANT MANIFOLDS FOR STOCHASTIC MODELS IN FLUID DYNAMICS
, 2010
"... paper is dedicated to Peter Imkeller on his Sixtieth Birthday celebration This paper is a survey of recent results on the dynamics of Stochastic Burgers equation (SBE) and twodimensional Stochastic Navier–Stokes Equations (SNSE) driven by affine linear noise. Both classes of stochastic partial diff ..."
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paper is dedicated to Peter Imkeller on his Sixtieth Birthday celebration This paper is a survey of recent results on the dynamics of Stochastic Burgers equation (SBE) and twodimensional Stochastic Navier–Stokes Equations (SNSE) driven by affine linear noise. Both classes of stochastic partial differential equations are commonly used in modeling fluid dynamics phenomena. For both the SBE and the SNSE, we establish the local stable manifold theorem for hyperbolic stationary solutions, the local invariant manifold theorem and the global invariant flag theorem for ergodic stationary solutions. The analysis is based on infinitedimensional multiplicative ergodic theory techniques developed by D. Ruelle [22] (cf.[20, 21]). The results in this paper are based on joint work of the author with T. S. Zhang and H. Zhao ([17–19]). Keywords: Stochastic Burgers equation; stochastic Navier–Stokes equation; cocycle;
DISSIPATIVE STOCHASTIC EVOLUTION EQUATIONS DRIVEN BY GENERAL GAUSSIAN AND NONGAUSSIAN NOISE
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