A probabilistic representation for systems of elliptic equations

New probabilistic representations for systems of elliptic equations are constructed in the form of expectations over the Markov chains. It is shown that this approach gives the effective Monte Carlo algorithms even in the cases, where the classical probabilistic representation based on the Wiener and diffusion processes fails. As an important example, we consider the system of Lame equations. Construction of the method is based on the direct and inverse mean value theorems which we prove, in particular, for the Lame equation. We derive accurate estimations for the exponential moments for the case of ε-spherical process (the "walk on spheres" process).