Exponentially Stable Stationary Solutions for Stochastic Evolution Equations and Their Perturbation

Abstract

Abstract. We consider the exponential stability of stochastic evolution equations with Lipschitz continuous non-linearities when zero is not a solution for these equations. We prove the existence of a non-trivial stationary solution which is exponentially stable, where the stationary solution is generated by the composition of a random variable and the Wiener shift. We also construct stationary solutions with the stronger property of attracting bounded sets uniformly. The existence of these stationary solutions follows from the theory of random dynamical systems and their attractors. In addition, we prove some perturbation results and formulate conditions for the existence of stationary solutions for semi-linear stochastic partial differential equations with Lipschitz continuous non-linearities.

Attractors for stochastic lattice dynamical systems with a multiplicative noise

“…This is not true in general for infinite dimensional equations. However, for particular kinds of noise, as will be our case, we can apply the following simple lemma to obtain a random dynamical system (see Caraballo et al [9]). …”

“…This equation has a random fixed point in the sense of random dynamical systems generating a stationary solution known as the stationary Ornstein-Uhlenbeck process (see Caraballo et al [9] for more details). In fact, we have…”

A Random Model for Immune Response to Virus in Fluctuating Environments

“…In the paper we will study the properties of solutions to (6) - (8). In particular, we are interested in the long term behavior of solutions to (6) - (8), characterized by a global random attractor.…”

“…In particular, we are interested in the long term behavior of solutions to (6) - (8), characterized by a global random attractor. The rest of the paper is organized as follows.…”

“…In Section 3 we prove the existence and uniqueness of a positive bounded solution to (6) - (8), and show that the solution generates a random dynamical system. In section 4 we prove the existence and uniqueness of a global random attractor to the random dynamical system generated by the solution to (6) - (8), and also investigate the conditions under which the global random attractor consists of a singleton axial solution (endemic), or nontrivial component sets (pandemic). Numerical simulations are provided in Section 6, to illustrate the conditions for the endemic and pandemic of system (6) -(8).…”