Integration and the Feynman-Kac formula

The Feynman-Kac formula

where ∆ is the Laplace operator. Here σ > 0 is a constant (the diffusion constant). It has dimensions of distance squared over time, so H0 has dimensions of inverse time. The operator exp(−tH0) for t > 0 is an self-adjoint integral operator, which gives the solution of the heat or diffusion equation. Here t is the time parameter. It is easy to solve for this operator by Fourier transforms. Sinc...

First Order Feynman-Kac Formula

We study the parabolic integral kernel associated with the weighted Laplacian with a potential. For manifold with a pole we deduce formulas and estimates for the derivatives of the Feynman-Kac kernels and their logarithms, these are in terms of a ‘Gaussian’ term and the semi-classical bridge. Assumptions are on the Riemannian data. AMS Mathematics Subject Classification : 60Gxx, 60Hxx, 58J65, 5...

A Feynman-Kac Formula for Unbounded Semigroups

We prove a Feynman-Kac formula for Schrödinger operators with potentials V (x) that obey (for all ε > 0) V (x) ≥ −ε|x| − Cε. Even though e is an unbounded operator, any φ, ψ ∈ L with compact support lie in D(e) and 〈φ, eψ〉 is given by a Feynman-Kac formula.

Feynman-Kac formula for stochastic hybrid systems.

We derive a Feynman-Kac formula for functionals of a stochastic hybrid system evolving according to a piecewise deterministic Markov process. We first derive a stochastic Liouville equation for the moment generator of the stochastic functional, given a particular realization of the underlying discrete Markov process; the latter generates transitions between different dynamical equations for the...

The Feynman-kac Formula in the Operator Setting