Solvability of dichotomous ows, dichotomous diusion and generalizations

We rst consider the one-dimensional stochastic ow _x = f(x) + g(x) (t), where (t) is a dichotomous Markov noise. A procedure involving the algebra of the relevant dierential operators is used to identify the conditions under which the integro-dierential equation satised by the total probability density P (x; t) of the driven vari-able can be reduced to a dierential equation of nite 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 that exists under suitable conditions in dichotomous ow, and indicate how this expression may be derived and interpreted on direct physical grounds. Finally, we consider a diusion process driven by an N-level extension of dichotomous noise, and explicitly derive the higher-order partial dierential equation satis ed by P (x; t) in this case. This multi-level noise driven diusion is a process that interpolates between the usual extremes of dichotomous diusion and Brownian motion. We comment on the possible use of certain algebraic techniques to solve the master equation for this generalized diusion