Lie symmetry methods for multi-dimensional parabolic PDEs and diffusions

In this paper we introduce new methods based upon integrating Lie symmetries for the construction of explicit fundamental solutions of multi-dimensional second order parabolic PDEs. We present applications to the problem of finding transition probability densities for multi-dimensional diffusions and to representation theory. Many explicit examples are given to illustrate the techniques. (C) 2011 Elsevier Inc. All rights reserved.