Quadratic and rate-independent limits for a large-deviations functional

TL;DR

This paper develops a stochastic model linking noise, gradient flows, and rate-independent systems, demonstrating how different limits produce a spectrum of generalized gradient flows from quadratic to rate-independent, using large deviations and Mosco-convergence.

Contribution

It introduces a stochastic birth-death process model that captures the transition from quadratic to rate-independent gradient flows through large deviations analysis.

Findings

Derivation of a family of generalized gradient flows from stochastic models.

Connection between noise levels and types of gradient flows.

Application of Mosco-convergence to large deviations rate functionals.

Abstract

We construct a stochastic model showing the relationship between noise, gradient flows and rate-independent systems. The model consists of a one-dimensional birth-death process on a lattice, with rates derived from Kramers' law as an approximation of a Brownian motion on a wiggly energy landscape. Taking various limits we show how to obtain a whole family of generalized gradient flows, ranging from quadratic to rate-independent ones, connected via '$L \log L$' gradient flows. This is achieved via Mosco-convergence of the renormalized large-deviations rate functional of the stochastic process.