Stochastic partial differential equations with reflection and applications

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Project objective

In 1992, D. Nualart and E. Pardoux introduced a Stochastic Partial Differential Equation with Reflection. In our PhD Thesis, we gave a connection between this equation and the Theory of Dirichlet Processes. This connection opens a number of problems. The Nualart-Pardoux (NP) equation admits an interpretation as an infinite-dimensional Skorokhod problem; the related geometrical objects, i.e. the state space, the boundary, the boundary measure and the normal vector field, are dictated by the unique invariant measure of the solution of the NP equation. The Local Time of the solution at the boundary, uniquely determined by the boundary measure, appears as a natural time scale for the solution. A deep understanding of the properties and the structure of this Local Time should allow in particular to obtain strong information about the geometrical structure of the contact set, i.e. the subset of space-time where the solution hits 0. The NP equation is an explicit example of a stochastic system with infinitely many degrees of freedom and with a reflecting boundary: a general theory of such systems is still in its infancy. Moreover, the interest in a fine analysis of the NP equation is motivated by the possibility of modeling several interesting phenomena from Physics, Biology and Mathematical Finance: e.g. a recent result of T. Funaki and S. Olla, connects the NP equation with the fluctuations around the hydro dynamical limit of an Interacting Particle System near a wall.

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Participants

• UNIVERSITY OF BIELEFELD · BIELEFELDCoordinatorGermany

Links

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