Probabilistic models of parabolic conservation and balance laws and systems with switching regimes

We construct a probabilistic representation of the Cauchy problem classical solution for systems of semilinear parabolic equations such that their second order terms enter in a diagonal way with different coefficients while lower order term enter in nondiagonal way. Systems of this kind arise as mathematical models of parabolic conservation and balance laws and as mathematical models of dynamical systems with switching regimes.