Stochastic analysis and conservation laws

Stochastic analysis is applied to a Cauchy problem for a scalar conservation law u(t) + f(u)(x) = 0 by using " vanishing viscosity " method. It is proved that the stochastic characteristic of equation u(t) + f(u)(x) + mu u(xx) = 0 converges to the characteristic of equation u(t) + f(u)(x) = 0. In this way a well known result is recovered, that is, the solution of u(t) + f(u)(x) + mu u(xx) = 0 converges to the entropy solution of u(t) + f(u)(x) = 0 as mu --> 0 naturally. Entropy condition is crucial to the proofs. the result suggests that the new method, stochastic analysis, may be useful for solving difficult problems of quasi-linear partial differential equations.