A general class of invariant diffusion processes in one dimension

TL;DR

This paper provides a method to determine the existence of symmetries in one-dimensional diffusion processes and offers canonical forms and generators, simplifying analysis and aiding in understanding physically relevant solutions.

Contribution

It introduces a new test for symmetry existence in diffusion processes and classifies their canonical forms, streamlining the analysis of such models.

Findings

A new criterion for symmetry detection in diffusion processes.

Classification of canonical forms with symmetry groups.

Analysis of physically relevant solutions in six models.

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.