Regularity and representation of viscosity solutions of partial differential equations via backward stochastic differential equations

We study the regularity of the viscosity solution of a quasilinear parabolic partial differential equation with Lipschitz coefficients by using its connection with a forward backward stochastic differential equation (in short FBSDE) and we give a probabilistic representation of the generalized gradient (derivative in the distribution sense) of the viscosity solution. This representation is a kind of nonlinear Feynman–Kac formula. The main idea is to show that the FBSDE admits a unique linearized version interpreted as its distributional derivative with respect to the initial condition. If the diffusion coefficient of the forward equation is uniformly elliptic, we approximate the FBSDE by smooth ones and use Krylov’s estimate to prove the convergence of the derivatives. In the degenerate case, we use techniques of Bouleau–Hirsch on absolute continuity of probability measures.