The probabilistic machinery (Central Limit Theorem, Feynman-Kac formula and Girsanov Theorem) is used to study the homogenization property for PDE with second-order partial differential operator in divergence-form whose coefficients are stationary, ergodic random fields. Furthermore, we use the theory of Dirichlet forms, so that the only conditions required on the coefficients are non degenerac...

Motivated by application to quantum physics, anticommuting analogues of Wiener measure and Brownian motion are constructed. The corresponding Itô integrals are defined and the existence and uniqueness of solutions to a class of stochastic differential equations is established. This machinery is used to provide a Feynman-Kac formula for a class of Hamiltonians. Several specific examples are cons...

Three topics featuring bilinear integration are described: the noncommutative Feynman-Kac formula, the connection between stationary state and time-dependent scattering theory and the stochastic integration of vector valued processes. Mathematics Subject Classification (2000). Primary 28B05; Secondary 46B28 46G10 46N50 60H05.

We present an algorithm for the numerical solution of nonlinear parabolic partial differential equations. This extends classical Feynman–Kac formula to fully equations, by using random trees that carry information on nonlinearities their branches. It applies functional, non-polynomial are not treated standard branching arguments, and deals with derivative terms arbitrary orders. A Monte Carlo i...

The classical Feynman-Kac formula states the connection between linear parabolic partial differential equations (PDEs), like the heat equation, and expectation of stochastic processes driven by Brownian motion. It gives then a method for solving linear PDEs by Monte Carlo simulations of random processes. The extension to (fully)nonlinear PDEs led in the recent years to important developments in...

In a canonical way, one can think of B as “two-parameter Brownian motion”. In this article, we address the following question: “Given a measurable function υ : R → R+, what can be said about the distribution of ∫ [0,1]2 υ(Bs) ds?” The one-parameter variant of this question is both easy-to-state and well understood. Indeed, if b designates standard Brownian motion, the Laplace transform of ∫ 1 0...