Wiener measure for Heisenberg group

In this paper, we build Wiener measure for the path space on the Heisenberg group by using of the heat kernel corresponding to the sub-Laplacian and give the definition of the Wiener integral. Then we give the Feynman-Kac formula.Comment: 14 page

Solution of the Fokker-Planck equation with boundary conditions by Feynman-Kac integration.

In this paper, we apply the results about d and d-function perturbations in order to formulate within the Feynman-Kac integration the solution of the forward Fokker-Planck equation subject to Dirichlet or Neumann boundary conditions. We introduce the concept of convex order to derive upper and lower bounds for path integrals with d and d- functions in the integrand. We suggest the use of bounds as an approximation for the solution.Feynman-Kac integration; Functions; Integration; Path integral; Perturbations theory; SDE;

Functional Integral Representation of the Pauli-Fierz Model with Spin 1/2

A Feynman-Kac-type formula for a L\'evy and an infinite dimensional Gaussian random process associated with a quantized radiation field is derived. In particular, a functional integral representation of e^{-t\PF} generated by the Pauli-Fierz Hamiltonian with spin \han in non-relativistic quantum electrodynamics is constructed. When no external potential is applied \PF turns translation invariant and it is decomposed as a direct integral \PF = \int_\BR^\oplus \PF(P) dP. The functional integral representation of e^{-t\PF(P)} is also given. Although all these Hamiltonians include spin, nevertheless the kernels obtained for the path measures are scalar rather than matrix expressions. As an application of the functional integral representations energy comparison inequalities are derived.Comment: This is a revised version. This paper will be published from J. Funct. Ana