Sharp derivative bounds for solutions of degenerate semi-linear partial differential equations

The paper is a continuation of the Kusuoka–Stroock programme of establishing smoothness properties of solutions of (possibly) degenerate partial differential equations by using probabilistic methods. We analyze here a class of semi-linear parabolic partial differential equations for which the linear part is a second-order differential operator of the form , where V0,…,VN are first-order differential operators that satisfy the so-called UFG condition (see Kusuoka and Stroock, 1987, [16]), which is weaker than the Hörmander one. Specifically, we prove that the bounds of the higher-order derivatives of the solution along the vector fields coincide with those obtained in the linear case when the boundary condition is Lipschitz continuous, but that the asymptotic behavior of the derivatives may change because of the simultaneity of the nonlinearity and of the degeneracy when the boundary condition is of polynomial growth and measurable only.