Hormander's Hypoelliptic Theorem for Nonlocal Operators

In this paper we show the Hormander hypoelliptic theorem for nonlocal operators by a purely probabilistic method: the Malliavin calculus. Roughly speaking, under general Hormander's Lie bracket conditions, we show the regularization effect of discontinuous Levy noises for possibly degenerate stochastic differential equations with jumps. To treat the large jumps, we use the perturbation argument together with interpolation techniques and some short time asymptotic estimates of the semigroup. As an application, we show the existence of fundamental solutions for operator partial derivative(t)-K, where K is the following nonlocal kinetic operator: K f (x, v) = p.v integral(Rd) (f (x, v + w) - f (x, v)) kappa (x, v, w)/vertical bar w vertical bar(d+alpha) dw + v . del(x) f (x, v) + b (x, v) . del(v) f (x, v). Here kappa(-1)(0) <= kappa(x, v, w) <= kappa(0) belongs to C-b(infinity) (R-3d) and is symmetric in w, p.v. stands for the Cauchy principal value, and b is an element of C-b(infinity) (R-2d; R-d).