Stabilization of partial differential equations by noise

We provide an example of a class of partial differential equations being stabilized (in terms of Lyapunov exponents) by noise. In particular, we show that the stability of the heat equation can be improved by adding a stochastic term to the equation. We also give an example of an unstable PDE made stable by noise.Lyapunov exponents Stochastic partial differential equations Partial differential equations Heat equation

Stabilization of Evolution Equations By Noise With Application to Partial Differential Equations of Parabolic Type.

We consider a deterministic equation of evolution X 0 (t) = AX(t)dt, in a separable Hilbert space. We prove that if A generates a C 0 - semigroup, then this equation can be stabilized, in terms of Lyapunov exponents, by noise. Then we apply this abstract result to partial differential equations of parabolic type. We also compute the Lyapunov exponents of these PDEs, both deterministic and stochastic, as functions of the eigenvalues of the operator A. 1. Introduction. The present paper is a development of [10], where we have constructed an example of a class of partial differential equations being stabilized (in 1 Research partially supported by KBN grant 2 P03A 016 16 A.A. Kwieci'nska Stabilization of evolution equations 2 terms of Lyapunov exponents) by noise. In this paper we provide sufficient conditions for exponential stabilization of abstract evolution equations. We apply these results to parabolic partial differential equations extending thus the example from [10]. First..

Random dynamical systems arising from iterated function systems with place-dependent probabilities

We show that if an iterated function system with place-dependent probabilities admits an invariant and attractive measure, then it has the structure of a random dynamical system (in the sense of Ludwig Arnold).Random dynamical system Iterated function system Markov operator Invariant measure Attractive measure