Minimising the expected commute time.

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.