Andrzej Roszkosz (Nic. Copern. Torun): On stochastic representation of solutions of parabolic perturbations to conservation laws

The theory of backward stochastic differential equations provides probabilistic formulas for solutions of systems of semilinear parabolic or elliptic partial differential equations which may be viewed as the nonlinear generalizations of the celebrated Feynman-Kac formula. However, at present satisfactory theory exists only in the case, where the right-hand side of the equation is Lipschitz continuous with respect to the solution and its gradient. In our talk we will consider Cauchy problem for systems of parabolic perturbations of conservation laws. Such systems are important theoretically and in applications, and on the other hand may serve as an example of systems which are not covered by the existing theory of backward stochastic differential equations. We will show that under standard assumptions on the flux function and initial condition a weak solution of the perturbed system of conservation laws admits three different but related stochastic representations: by means of solutions to systems of usual backward stochastic differential equations, by means of solutions to some stochastic backward systems and solutions to some forward-backward stochastic differential equations.