INTEGRAL AND PROBABILISTIC REPRESENTATIONS FOR SYSTEMS OF ELLIPTIC-EQUATIONS

We present probabilistic representations for some systems of elliptic equations constructed as expectations of functionals of some specific Markov chains, in particular, the walk on spheres processes; These rep resentations are deduced from converse mean value theorems that we pro ve for the equations under analysis, especially the Lame equation of e lasticity theory. We construct Monte Carlo algorithms, and estimate th e variances and the cost.