Extension of explicit formulas in Poissonian white noise analysis using harmonic analysis on configuration spaces

Harmonic analysis on configuration spaces is used in order to extend explicit expressions for the images of creation, annihilation, and second quantization operators in L2-spaces with respect to Poisson point processes to a set of functions larger than the space obtained by directly using chaos expansion. This permits, in particular, to derive an explicit expression for the generator of the second quantization of a sub-Markovian contraction semigroup on a set of functions which forms a core of the generator.

The Brownian traveller on manifolds

We study the inuence of the intrinsic curvature on the large time behaviour of the heat equation in a tubular neighbourhood of an unbounded geodesic in a two-dimensional Riemannian manifold. Since we consider killing boundary conditions, there is always an exponential-type decay for the heat semigroup. We show that this exponential-type decay is slower for positively curved manifolds comparing to the at case. As the main result, we establish a sharp extra polynomial-type decay for the heat semigroup on negatively curved manifolds comparing to the at case. The proof employs the existence of Hardy-type inequalities for the Dirichlet Laplacian in the tubular neighbourhoods on negatively curved manifolds and the method of self-similar variables and weighted Sobolev spaces for the heat equation.