The α-dependence of the invariant measure of stochastic real Ginzburg-Landau equation driven by α-stable Lévy processes

Abstract

This paper is intended to establish the connection between stochastic dynamical systems with non-Gaussian noises and stochastic dynamical systems with Gaussian noises, by considering the convergence behavior of invariant probability measures for stochastic real Ginzburg-Landau equation driven by cylindrical α-stable Lévy process, in the limit α→2. Indeed, we prove that the invariant measure of stochastic real Ginzburg-Landau equation on torus and driven by cylindrical α-stable Lévy processes converges to the invariant measure of stochastic real Ginzburg-Landau equation forced by cylindrical Brownian motions under the Wasserstein distance, as α tends to 2. We state our strategy as below.First, by using an abstract result given by Hairer and Mattingly, we prove that a type of Wasserstein distance is contracting for the dual operator of Markov semigroup associated with the limit equation which forced by Brownian motions. Then, through server priori uniform moment estimates and a convergence result on stochastic convolutions of cylindrical subordinated Brownian motions, we establish a strong convergence result on the solution of stochastic real Ginzburg-Landau equation driven by cylindrical subordinated Brownian motions conditioned on the initial data is distributed as an invariant measure, in the limit α→2. Finally, by the Monge-Kantorovich duality, we prove that the invariant probability measure of stochastic real Ginzburg-Landau equation driven by cylindrical α-stable Lévy processes converges to the invariant measure of stochastic real Ginzburg-Landau equation forced by Brownian motions under the Wasserstein distance.