Soft and hard wall in a stochastic reaction diffusion equation

We consider a stochastically perturbed reaction diffusion equation in a bounded interval, with boundary conditions imposing the two stable phases at the endpoints. We investigate the asymptotic behavior of the front separating the two stable phases, as the intensity of the noise vanishes and the size of the interval diverges. In particular, we prove that, in a suitable scaling limit, the front evolves according to a one-dimensional diffusion process with a non-linear drift accounting for a "soft" repulsion from the boundary. We finally show how a "hard" repulsion can be obtained by an extra diffusive scaling.Comment: 33 page

Dobrushin states in the \phi^4_1 model

We consider the van der Waals free energy functional in a bounded interval with inhomogeneous Dirichlet boundary conditions imposing the two stable phases at the endpoints. We compute the asymptotic free energy cost, as the length of the interval diverges, of shifting the interface from the midpoint. We then discuss the effect of thermal fluctuations by analyzing the \phi^4_1-measure with Dobrushin boundary conditions. In particular, we obtain a nontrivial limit in a suitable scaling in which the length of the interval diverges and the temperature vanishes. The limiting state is not translation invariant and describes a localized interface. This result can be seen as the probabilistic counterpart of the variational convergence of the associated excess free energy.Comment: 34 page

Stability of the instanton under small random perturbations

We consider the process obtained as solution of the Allen-Cahn equation perturbed by white noise with initial condition an instanton. (That is, the stationary solution of the Allen-Cahn equation which interpolates between the two constant stationary solutions.) We show that, as the intensity of the noise goes to zero, the process remains close to the initial instanton for times that go to infinity conveniently. At the scale we consider, one sees an infinitesimal shift of the instanton, given by a Brownian motion.Stochastic partial differential equations Stability of interfaces

Some results on small random perturbations of an infinite dimensional dynamical system

We consider a small random perturbation of a non-linear heat equation with Dirichlet boundary conditions on an interval. The equation can be thought of as a gradient type dynamical system in the space of continuous functions of the interval. It has two stable equilibrium configurations, and several saddle points. We prove that, with probability growing to one in the limit as the strength of the noise goes to zero, the tunnelling between the two stable configurations occurs close to the saddle points with lowest potential. This was suggested by Faris and Jona-Lasinio (1982), who introduced the model. We also prove stability of time averages along a path of the process, in the sense introduced by Cassandro, Galves, Olivieri and Vares (1984), as part of their characterization of metastability for stochastic systems.random perturbations infinite dimensional dynamical systems metastability large deviations

2

Stochastic phase field equations: existence and uniquenes

Soft and hard wall in a stochastic reaction diffusion equation

We consider a stochastically perturbed reaction diffusion equation in a bounded interval, with boundary conditions imposing the two stable phases at the endpoints. We investigate the asymptotic behavior of the front separating the two stable phases, as the intensity of the noise vanishes and the size of the interval diverges. In particular, we prove that, in a suitable scaling limit, the front evolves according to a one-dimensional diffusion process with a non-linear drift accounting for a "soft" repulsion from the boundary. We finally show how a "hard" repulsion can be obtained by an extra diffusive scaling. © 2008 Springer-Verlag

Stochastic Phase Field Equations: Existence and Uniqueness

We consider a conservative system of stochastic PDE's, namely a one dimensional phase field model perturbed by an additive space-time white noise. We prove a global existence and uniqueness result in a space of continuous functions on ℝ+ × ℝ. This result is obtained by extending previous results of Doering [3] on the stochastic Allen-Cahn equation