# PDE and probability in mathematical physics

PTDC/MAT/120354/2010

## Researchers

## Description

The idea is to join together specialists of PDE with mathematical physicists using Stochastic Analysis as a tool for the study of specific Dynamical Systems.

On one hand, we shall study parabolic PDE with time dependent coefficients whose associated two parameters propagators appear in various probabilistic contexts. A class of time reversible stochastic processes introduced by us in the mid eighties ("Bernstein" or "Reciprocal") provide such a natural context.

On the other hand, we shall use such processes for a new approach to some aspects of Aubry-Mather theory, where they are appropriate for the stochastic deformation (parabolic perturbation of the Hamilton-Jacobi (HJ) equation) of the classical dynamical system under investigation. The viscosity solutions of HJ appear, then, naturally associated with various problems of stochastic control theory, a probabilistic extension of the calculus of variation. Along the way we shall study some PDE whose solutions are expressed in terms of conditional expectations of functionals of those stochastic processes, regarded as stochastic deformations of properties of the underlying deterministic dynamical systems.

Time span: 01/01/2012-31/12/2014

Funding institution: FCT

Budget: € 77600.00

## Some relevant publications

(for other publications by the researchers involved, see the respective homepages)

- P.A. Vuillermot, W.F. Wreszinski, V.A. Zagrebnov
- A general Trotter-Kato formula for a class of evolution
operators

*Journal of Functional Analysis***257**(2009), 2246-2290 - P.A. Vuillermot
- A generalization of Chernoff's product formula for time-dependent
operators

*Journal of Functional Analysis***259**(2010), 2923-2938 - K.L. Chung, J.C. Zambrini
- Introduction to random time and quantum randomness

World Scientific (2003) - N. Privault, J.C. Zambrini
- Stochastic deformation of integrable dynamical systems and random
time symmetry

*J. Math. Physics***51**(2010) - A.B. Cruzeiro; C.J.S. Alves
- Monte-Carlo simulation of stochastic differential systems - A
geometrical approach

*Stochastic Processes and Applications***118**(2008), 346-367