An analytic approach to the ergodic theory of stochastic variational inequalities

TL;DR

This paper introduces a new way to characterize the invariant measure of a stochastic variational inequality related to elasto-plastic oscillators, linking nonlocal and local PDEs through Markov process cycles.

Contribution

It provides a novel characterization of the invariant measure by connecting nonlocal PDEs with local PDEs in the context of stochastic variational inequalities.

Findings

Established a new link between nonlocal and local PDEs.

Provided a characterization of the invariant measure.

Demonstrated the connection through short cycles of the Markov process.

Abstract

In an earlier work made by the first author with J. Turi (Degenerate Dirichlet Problems Related to the Invariant Measure of Elasto-Plastic Oscillators, AMO, 2008), the solution of a stochastic variational inequality modeling an elasto-perfectly-plastic oscillator has been studied. The existence and uniqueness of an invariant measure have been proven. Nonlocal problems have been introduced in this context. In this work, we present a new characterization of the invariant measure. The key finding is the connection between nonlocal PDEs and local PDEs which can be interpreted with short cycles of the Markov process solution of the stochastic variational inequality.