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We present probabilistic representations for some systems of elliptic equations. They are constructed as expectations of functionals of some specific Markov chains, in particular the walk on spheres processes. These representations rely on converse mean value theorems that we show for the equations under analysis, especially the Lame equation of elasticity theory. We deduce Monte Carlo procedures, whose convergence is proven and cost is analyzed.

Integral Formulation of the Boundary Value Problems and the Method of Random Walk on Spheres