Local Stability of Dynamical Processes in Random Media

TL;DR

This paper develops a method of local multipliers to analyze the local stability of infinite-dimensional random dynamical systems with fluctuating parameters, illustrated through several key examples.

Contribution

Introduces a novel method of local multipliers for assessing local stability in infinite-dimensional random dynamical systems, with applications to various physical equations.

Findings

Stationary solutions are unstable under small random perturbations.

The method is demonstrated on random diffusion, wave, and Schrödinger equations.

The concept of random structural stability is introduced.

Abstract

A particular type of random dynamical processes is considered, in which the stochasticity is introduced through randomly fluctuating parameters. A method of local multipliers is developed for treating the local stability of such dynamical processes corresponding to infinite--dimensional dynamical systems. The method is illustrated by several examples, by the random diffusion equation, random wave equation, and random Schrodinger equation. The evolution equation for the density matrix of a quasiopen statistical system subjest to the action of random surrounding is considered. The stationary solutions to this equation are found to be unstable against arbitrary small finite random perturbations. The notion of random structural stability is introduced.