


[Contents]  [Index] 
Cooperation with: P. Dai Pra (Università degli Studi di Padova, Italy), D. Dereudre (École Polytechnique, Palaiseau, France), M. Thieullen (Université Paris VI, France), H. Zessin (Universität Bielefeld)
Description: The concept of reciprocal processes can be traced back to E. Schrödinger. They form a class of stochastic processes that generalize the notion of Markov processes and are defined via a temporal Markovian field property. They play an important role in the context of Quantum Diffusions, Nelson's stochastic mechanics, and several variational problems such as entropy maximization on path space.
In a collaboration with M. Thieullen ([1]) we characterize realvalued reciprocal processes associated to a Brownian diffusion as a solution of an integrationbyparts formula. In this characterization appears a function, called reciprocal characteristics, that turns out to play the same role as the Hamiltonian in statistical mechanics. This approach is systematically applied in the context of periodic OrnsteinUhlenbeck processes. The same approach was also carried through in the multidimensional setting. The next major goal of this project is an extension to infinitedimensional situations, where the reciprocal process is the solution of a stochastic partial differential equation.
In collaboration with P. Dai Pra and H. Zessin, we have developed a characterization of the stationary law of interacting diffusion processes indexed by the lattice as Gibbsian measures. Let be the weak solution of the Stochastic Differential Equation
where is a measurable bounded local functional on the path space, a priori non Markovian. Then the law Q of X can be interpreted as Gibbs distribution on , with a priori measure P the product of Wiener measures drifted by and a certain interaction potential . The main result in [2] shows the following equivalenceIn [3], the above results are completed by an existence result for the solution of (*). The authors use a spacetime cluster expansions method, well adapted when the coupling parameter is sufficiently small.
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