Parabolic Singular Integrals in Probability Theory

Abstract

The final aim of this work is to prove the Central Limit Theorem described in the motivations given below. The key for that is a Resolvant estimate, of the type of Theorem 1.1 in [21], adapted for the Parabolic Green function G(X, Y) which is the heat diffusion kernel in some domain Ω in time-space: i.e. we must estimate \({\int_{\Omega}\nabla_{Y}G(X, Y)\nabla_{Y}^{2}G(Y,Z)\;dY}\). Exactly as the estimate in [21] is based on [10] our estimate here is based on the main Theorem of this paper. This main theorem refers to rough singular integrals on the Gaussian potential on ∂Ω.