Mark Kac introduced a method for calculating the distribution of the integral Av= ∫ T 0 v(Xt) dt for a function v of a Markov process (Xt; t¿0) and a suitable random time T , which yields the Feynman–Kac formula for the moment-generating function of Av. We review Kac’s method, with emphasis on an aspect often overlooked. This is Kac’s formula for moments of Av, which may be stated as follows. F...

Let V be a multiplication operator, whose negative part, V-(V-< 0) obeys-A + (1 + c)V->-c for some c, c > 0. Let W = Vx where x is the characteristic function of the exterior of a ball. Our main result asserts that the scattering for-A + V is complete if and only if that for-A + W is complete. Our technical estimates exploit Wiener integrals and the Feynman-Kac formula. We also make an applicat...

Abstract We consider the Parabolic Anderson Model ∂tu = Lu + uẆ , where L is the generator of a Lévy Process and Ẇ is a white noise in time, possibly correlated in space. We present an alternate proof and an extension to a result by Hu and Nualart ([14]) giving explicit expressions for moments of the solution. We do not consider a Feynman-Kac representation, but rather make a recursive use of I...

A class of functions is speciied which give rise to semibounded quadratic forms on weighted Bergman spaces and thus can be interpreted as symbols of self-adjoint Berezin-Toeplitz operators. A similar class admits a probabilistic expression of the sesqui-analytic integral kernel for the associated semigroups. Both results are the consequence of a relation of Berezin-Toeplitz operators to Schrr o...

Applying the well-known Feynman-Kac formula of inhomogeneous case, an interesting and rigorous mathematical proof of generalized Jarzynski’s equality of inhomogeneous multidimensional diffusion processes is presented, followed by an extension of the second law of thermodynamics. Then, we explain its physical meaning and applications, extending Hummer and Szabo’s work (Proc. Natl. Acad. Sci. USA...

A scalar quantum field model defined on a pseudo Riemann manifold is considered. The model is unitarily transformed to the one with a variable mass. By means of a Feynman-Kac-type formula, it is shown that when the variable mass is short range, the Hamiltonian has no ground state. Moreover the infrared divergence of the expectation values of the number of bosons in the ground state is discussed.