Large deviations for Markov processes with switching and homogenisation via Hamilton-Jacobi-Bellman equations

TL;DR

This paper establishes a large deviation principle for a molecular motor model with internal switching, using Hamilton-Jacobi-Bellman equations that depend on position and momentum, advancing understanding of stochastic homogenization.

Contribution

It introduces a novel approach to large deviations for Markov processes with internal states by analyzing a Hamilton-Jacobi-Bellman equation with position- and momentum-dependent Hamiltonian.

Findings

Proves large deviation principles for molecular motor models.

Develops a Hamilton-Jacobi-Bellman framework with position and momentum dependence.

Provides a new method for analyzing stochastic processes with internal degrees of freedom.

Abstract

We consider the context of molecular motors modelled by a diffusion process driven by the gradient of a weakly periodic potential that depends on an internal degree of freedom. The switch of the internal state, that can freely be interpreted as a molecular switch, is modelled as a Markov jump process that depends on the location of the motor. Rescaling space and time, the limit of the trajectory of the diffusion process homogenizes over the periodic potential as well as over the internal degree of freedom. Around the homogenized limit, we prove the large deviation principle of trajectories with a method developed by Feng and Kurtz based on the analysis of an associated Hamilton--Jacobi--Bellman equation with an Hamiltonian that here, as an innovative fact, depends on both position and momenta.