Seminário LisMath 

24/04/2015, 16:00 — 17:00 — Sala B3-01, Complexo Interdisciplinar da Universidade de Lisboa
Alexandra Symeonides, LisMath Programme, Universidade de Lisboa

Time reversible stochastic processes and the relevant Feynman-Kac formula

We will present a family of time reversible stochastic processes known (among other names) as Bernstein processes [1,2]. These processes are much closer, in their properties, to the solutions of deterministic (Lagrangian or Hamiltonian) dynamical equations, they are in fact absolutely continuous with respect to the Wiener process, i.e. they are processes with a drift term. The relevant Feynman-Kac formula for this class of processes will be proved. Some particular cases of the formula will be given as examples: the well-known Feynman-Kac formula and Doob's relation between Wiener and Ornstein-Uhlenbeck processes [3,4].

The consequences of this probabilistic perturbation theory and the underlying time reversible processes should go beyond stochastic analysis. We will give a hint of two applications, one motivated by mathematical quantum physics and the other by a stochastic version of geometric mechanics.

The first concerns a rigorous probabilistic interpretation of Feynman informal perturbation theory in his Path Integral approach [7], which needs the development of a rigorous integration by parts formula, also inspired by Feynman, done in terms of special reversible probability measures on path space [6]. Furthermore, to investigate the consequences of this probabilistic perturbation theory in the more general context of stochasttic analysis.

The second application concerns Geometric Mechanics, that in the recent years started to investigate perturbations under random noise, preserving as much as possible the geometric content. Roughly, what has been done by this community up to now is to add noise to the Hamiltonian equation for the momentum (i.e random force) [8], or add noise to the configuration (i.e the trajectories are random processes) [5]. This goal is close to the one of stochastic perturbation, whose foundations lie in the dynamics of the above mentioned Bernstein processes, e.g. [10]. In this context, a stochastic Euler-Poincaré reduction has been recently proved [7]: the corresponding equations of motion are dissipative perturbations of Hamiltonian systems, and should be relevant, in particular, in Hydrodynamics. So far the main application regards the Navier-Stokes equations (as perturbations of Euler equations) but many other systems can be considered and studied in this perspective. This study is still in its beginnings and many mathematical questions remain to be solved.

Bibliography

1. J. C. Zambrini, Variational processes and stochastic versions of mechanics, Journal of Math. Physics 27, p. 2307-2330 (1986).
2. Albeverio, Yasue and Zambrini, Euclidean quantum mechanics: analytical approach Annales de l'I.H.P., section A, tome 50, no3 (1989), p. 259-308.
3. P. Lescot, J. C. Zambrini. Probabilistic deformation of contact geometry, difusion processes and their quadratures. Progress in Probability 59, Birkhauser Verlag Basel (2007), 203-226.
4. A. B. Cruzeiro and J. C. Zambrini. Ornstein-Uhlenbeck processes as Bernstein difusions, Proceedings of the Conference on Stochastic Analysis (Barcelona), Birkhauser, Boston, Inc. (1993) (P.P. no 32).
5. J. C. Zambrini, The research program of Stochastic Deformation (with a view toward Geometric Mechanics).
6. A. B. Cruzeiro and J. C. Zambrini, Malliavin Calculus and Euclidean Quantum Mechanics, 1. Functional calculus, Journal of Functional Analysis,91,1,p.62 (1991).
7. Richard P. Feynman, Albert R. Hibbs. Quantum Mechanics and Path Integrals: Emended Edition. Dover Publications, Incorporated, 2012.
8. A. B. Cruzeiro, M. Arnaudon and X. Chen, Stochastic Euler-Poincaré reduction.
9. Joan-Andreu, Lazaro-Cami, Juan-Pablo Ortega. Stochastic Hamiltonian dynamical systems.
10. Fernanda Cipriano, A Stochastic Variational Principle for Burgers Equation and its Symmetries. Stochastic Analysis and Mathematical Physics II, 4th International ANESTOC Workshop in Santiago, Chile. Birkhauser, R. Rebolledo (2003), p.29..

The first concerns a rigorous probabilistic interpretation of Feynman informal perturbation theory in his Path Integral approach [7], which needs the development of a rigorous integration by parts formula, also inspired by Feynman, done in terms of special reversible probability measures on path space [6]. Furthermore, to investigate the consequences of this probabilistic perturbation theory in the more general context of stochastic analysis.

The second application concerns Geometric Mechanics, that in the recent years started to investigate perturbations under random noise, preserving as much as possible the geometric content. Roughly, what has been done by this community up to now is to add noise to the Hamiltonian equation for the momentum (i.e random force) [8], or add noise to the configuration (i.e the trajectories are random processes) [5]. This goal is close to the one of stochastic perturbation, whose foundations lie in the dynamics of the above mentioned Bernstein processes, e.g. [10]. In this context, a stochastic Euler-Poincaré reduction has been recently proved [7]: the corresponding equations of motion are dissipative perturbations of Hamiltonian systems, and should be relevant, in particular, in Hydrodynamics. So far the main application regards the Navier-Stokes equations (as perturbations of Euler equations) but many other systems can be considered and studied in this perspective. This study is still in its beginnings and many mathematical questions remain to be solved.

The consequences of this probabilistic perturbation theory and the underlying time reversible processes should go beyond stochastic analysis. We will give a hint of two applications, one motivated by mathematical quantum physics and the other by a stochastic version of geometric mechanics.

The first concerns a rigorous probabilistic interpretation of Feynman informal perturbation theory in his Path Integral approach [7], which needs the development of a rigorous integration by parts formula, also inspired by Feynman, done in terms of special reversible probability measures on path space [6]. Furthermore, to investigate the consequences of this probabilistic perturbation theory in the more general context of stochastic analysis.

The second application concerns Geometric Mechanics, that in the recent years started to investigate perturbations under random noise, preserving as much as possible the geometric content. Roughly, what has been done by this community up to now is to add noise to the Hamiltonian equation for the momentum (i.e random force) [8], or add noise to the configuration (i.e the trajectories are random processes) [5]. This goal is close to the one of stochastic perturbation, whose foundations lie in the dynamics of the above mentioned Bernstein processes, e.g. [10]. In this context, a stochastic Euler-Poincaré reduction has been recently proved [7]: the corresponding equations of motion are dissipative perturbations of Hamiltonian systems, and should be relevant, in particular, in Hydrodynamics. So far the main application regards the Navier-Stokes equations (as perturbations of Euler equations) but many other systems can be considered and studied in this perspective. This study is still in its beginnings and many mathematical questions remain to be so

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