A probabilistic interpretation to the symmetries of a discrete heat equation

Abstract

A probabilistic interpretation is constructed for the symmetry group G of the finite difference-differential equation ∂t η(x, t) = η(x, t) - η(x + 1, t) using the Doob transform for Markov (jump) processes. While the first three generators of G correspond to the identity and to space and time shifts, we show that in this interpretation the fourth generator of G is associated to time dilations and is linked to a creation operator on the Poisson space. © 2008 Springer-Verlag Berlin Heidelberg.