Diffusions with a nonlinear irregular drift coefficient and probabilistic interpretation of generalized Burgers’ equations

Abstract We prove existence and uniqueness for two classes of mar tingale problems involving a nonlinear but bounded drift coecient In the rst class this coecient depends on the time t the position x and the marginal of the solution at time t In the second it depends on t x and pt x the density of the time marginal wrt Lebesgue measure As far as the dependence on t and x is concerned no continuity assump tion is made The results rst proved for the identity di usion matrix are extended to bounded uniformly elliptic and Lipschitz continuous matrices As an application we show that within each class a partic ular choice of the coecients leads to a probabilistic interpretation of generalizations of Burgers equatio