The Energy-Dissipation Principle for stochastic parabolic equations

TL;DR

This paper extends the Energy-Dissipation Principle, a variational approach for parabolic equations, to stochastic cases, enabling new analysis tools for stability and optimal control in stochastic parabolic evolution problems.

Contribution

It introduces a stochastic extension of the Energy-Dissipation Principle, broadening its applicability to stochastic parabolic equations and related control and stability analyses.

Findings

Established a variational characterization for stochastic parabolic solutions.

Applied the principle to stability analysis of stochastic systems.

Demonstrated the utility in optimal control scenarios.

Abstract

The Energy-Dissipation Principle provides a variational tool for the analysis of parabolic evolution problems: solutions are characterized as so-called null-minimizers of a global functional on entire trajectories. This variational technique allows for applying the general results of the calculus of variations to the underlying differential problem and has been successfully applied in a variety of deterministic cases, ranging from doubly nonlinear flows to curves of maximal slope in metric spaces. The aim of this note is to extend the Energy-Dissipation Principle to stochastic parabolic evolution equations. Applications to stability and optimal control are also presented.