Cauchy-Dirichlet problems for a class of hypoelliptic equation in $\R^d$: q new probabilistic representation formula for the gradient of the solutions

We are concerned with an Ornstein–Uhlenbeck process X(t, x) = etAx + ∫ t 0 e (t−s)A √ C dW (s) in Rd, d ≥ 1, where A and C are d × d matrices, C being semidefinite positive. Our basic assumption is that the matrix Qt = ∫ t 0 e sACesA ∗ ds is non singular for all t > 0; this implies that the corresponding Kolmogorov operator is hypoellyptic. Then we consider the stopped semigroup Rr T φ(x) = E [ φ(X(T, x))1lT≤τr x ] , T ≥ 0 where Or = {g < r} is bounded, g is convex, and τ r x = inf{t > 0 : X(t, x) ∈ Or }. We prove the existence and a new representation formula for the gradient of Rr T φ, where T > 0 and φ is bounded and Borel. 2000 Mathematics Subject Classification AMS: 35J15, 60G53, 60H99, 60J65.