A general class of invariant diffusion processes in one dimension

Abstract

This paper improves a previously established test involving only coefficients to decide a priori whether or not non-trivial symmetries of a large class of space-time dependent diffusion processes on the real line exist. When the existence of these symmetries are ensured, the transformation to canonical forms admitting either four- or six-dimensional symmetry groups and the full list of their infinitesimal generators are then immediately at our disposal without any cumbersome calculations that happen when at least one of the coefficients is arbitrarily chosen. We study in depth symmetry and reducibility properties and physically important solutions of six models arising in applications.