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On the long-time behaviour of a class of parabolic SPDE’s : monotonicity methods and exchange of stability

Benjamin Bergé, Bruno Saussereau (2005)

ESAIM: Probability and Statistics

In this article we prove new results concerning the structure and the stability properties of the global attractor associated with a class of nonlinear parabolic stochastic partial differential equations driven by a standard multidimensional brownian motion. We first use monotonicity methods to prove that the random fields either stabilize exponentially rapidly with probability one around one of the two equilibrium states, or that they set out to oscillate between them. In the first case we can...

On the long-time behaviour of a class of parabolic SPDE's: monotonicity methods and exchange of stability

Benjamin Bergé, Bruno Saussereau (2010)

ESAIM: Probability and Statistics

In this article we prove new results concerning the structure and the stability properties of the global attractor associated with a class of nonlinear parabolic stochastic partial differential equations driven by a standard multidimensional Brownian motion. We first use monotonicity methods to prove that the random fields either stabilize exponentially rapidly with probability one around one of the two equilibrium states, or that they set out to oscillate between them. In the first case we can...

On the short time asymptotic of the stochastic Allen–Cahn equation

Hendrik Weber (2010)

Annales de l'I.H.P. Probabilités et statistiques

A description of the short time behavior of solutions of the Allen–Cahn equation with a smoothened additive noise is presented. The key result is that in the sharp interface limit solutions move according to motion by mean curvature with an additional stochastic forcing. This extends a similar result of Funaki [Acta Math. Sin (Engl. Ser.)15 (1999) 407–438] in spatial dimension n=2 to arbitrary dimensions.

On the small time asymptotics of the two-dimensional stochastic Navier–Stokes equations

Tiange Xu, Tusheng Zhang (2009)

Annales de l'I.H.P. Probabilités et statistiques

In this paper, we establish a small time large deviation principle (small time asymptotics) for the two-dimensional stochastic Navier–Stokes equations driven by multiplicative noise, which not only involves the study of the small noise, but also the investigation of the effect of the small, but highly nonlinear, unbounded drifts.

Pathwise uniqueness for stochastic PDEs

Giuseppe Da Prato (2015)

Banach Center Publications

We consider a stochastic evolution equation in a separable Hilbert spaces H or in a separable Banach space E with a Hölder continuous perturbation on the drift. We review some recent result about pathwise uniqueness for this equation.

Positivity of the density for the stochastic wave equation in two spatial dimensions

Mireille Chaleyat-Maurel, Marta Sanz-Solé (2003)

ESAIM: Probability and Statistics

We consider the random vector u ( t , x ̲ ) = ( u ( t , x 1 ) , ⋯ , u ( t , x d ) ) , where t > 0 , x 1 , ⋯ , x d are distinct points of ℝ 2 and u denotes the stochastic process solution to a stochastic wave equation driven by a noise white in time and correlated in space. In a recent paper by Millet and Sanz–Solé [10], sufficient conditions are given ensuring existence and smoothness of density for u ( t , x ̲ ) . We study here the positivity of such density. Using techniques developped in [1] (see also [9]) based on Analysis on an abstract Wiener space, we characterize the set of...

Positivity of the density for the stochastic wave equation in two spatial dimensions

Mireille Chaleyat–Maurel, Marta Sanz–Solé (2010)

ESAIM: Probability and Statistics

We consider the random vector u ( t , x ̲ ) = ( u ( t , x 1 ) , ⋯ , u ( t , x d ) ) , where t > 0, x1,...,xd are distinct points of ℝ 2 and u denotes the stochastic process solution to a stochastic wave equation driven by a noise white in time and correlated in space. In a recent paper by Millet and Sanz–Solé [10], sufficient conditions are given ensuring existence and smoothness of density for u ( t , x ̲ ) . We study here the positivity of such density. Using techniques developped in [1] (see also [9]) based on Analysis on an abstract Wiener space, we characterize...

Probabilistic analysis of singularities for the 3D Navier-Stokes equations

Franco Flandoli, Marco Romito (2002)

Mathematica Bohemica

The classical result on singularities for the 3D Navier-Stokes equations says that the 1 -dimensional Hausdorff measure of the set of singular points is zero. For a stochastic version of the equation, new results are proved. For statistically stationary solutions, at any given time t , with probability one the set of singular points is empty. The same result is true for a.e. initial condition with respect to a measure related to the stationary solution, and if the noise is sufficiently non degenerate...

Probability density for a hyperbolic SPDE with time dependent coefficients

Marta Sanz-Solé, Iván Torrecilla-Tarantino (2007)

ESAIM: Probability and Statistics

We prove the existence and smoothness of density for the solution of a hyperbolic SPDE with free term coefficients depending on time, under hypoelliptic non degeneracy conditions. The result extends those proved in Cattiaux and Mesnager, PTRF123 (2002) 453-483 to an infinite dimensional setting.

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