Gaussian and non-Gaussian Behaviour of Diffusion Processes

Abstract

We survey some recent results on Gaussian and non-Gaussian behaviour for the solutions of second-order diffusion equations on R^d. Our emphasis is on non-Gaussian aspects of the diffusion corresponding to degenerate operators. In particular we describe

• the equivalence of strong ellipticity and Gaussian upper and lower bounds,

• the deduction of non-ergodic behaviour from integrated Gaussian upper bounds, and

• the relationships between volume doubling, the Poincaré inequality and Gaussian estimates.

To place these results into context we also summarize some well-established structural properties of diffusion phenomena.

Mathematics Subject Classification (2010). 47B25, 47D07, 35J70.