Some Estimates on the Transition Probability Functions

Abstract

In the derivation of our basic uniqueness theorem, Theorem 7.2.1, we made essential use of an analytic representation for the transition probability function associated with diffusion coefficients which are nearly independent of the spatial variables. We have already seen that this representation not only allows us to prove uniqueness, but also leads to regularity properties of the transition probability function as a function of the “backwards” variables. The purpose of the present chapter is to exploit this same representation to conclude properties about the transition probability function as a function of its “forward ” variables. More specifically, we will show that the transition probability function has a density with respect to Lebesgue measure and that this density satisfies certain L p -estimates.