Solvability of dichotomous flows, dichotomous diffusion, and generalizations

Abstract

We first consider the one-dimensional stochastic flow dx/dt = f(x) + g(x) xi(t), where xi(t) is a dichotomous Markov noise. A procedure involving the algebra of the relevant differential operators is used to identify the conditions under which the integro-differential equation satisfied by the total probability density P(x,t) of the driven variable can be reduced to a differential equation of finite order. This systematizes the enumeration of the "solvable" cases, of which the case of linear drift and additive noise is a notable one.We then revisit the known formula for the stationary density thatexists under suitable conditions in dichotomous flow, and indicate howthis expression may be derived and interpreted on direct physicalgrounds. Finally, we consider a diffusion process driven by an N-level extension of dichotomous noise, and explicitly derive the higher-order partial differential equation satisfied by P(x,t) in this case. This multi-level noise driven diffusion is a process that interpolates between the usual extremes of dichotomous diffusion and Brownian motion. We comment on the possible use of certain algebraic techniques to solve the master equation for this generalized diffusion.