Minimising the expected commute time

TL;DR

This paper investigates how to optimally control a diffusion process to minimize the expected commute time between two points, introducing a novel stochastic control problem with static and dynamic drift strategies.

Contribution

It formulates and analyzes a new stochastic control problem for minimizing commute times in diffusions, including both static and dynamic control mechanisms.

Findings

Derived optimal control strategies for static and dynamic cases.

Identified the dynamic control problem as a novel stochastic control challenge.

Provided theoretical insights into minimizing expected commute times in diffusions.

Abstract

Motivated in part by a problem in simulated tempering (a form of Markov chain Monte Carlo) we seek to minimise, in a suitable sense, the time it takes a (regular) diffusion with instantaneous reflection at 0 and 1 to travel from the origin to $1$ and then return (the so-called commute time from 0 to 1). We consider the static and dynamic versions of this problem where the control mechanism is related to the diffusion\rq{}s drift via the corresponding scale function. In the static version the diffusion's drift can be chosen at each point in [0,1], whereas in the dynamic version, we are only able to choose the drift at each point at the time of first visiting that point. The dynamic version leads to a novel type of stochastic control problem.