Hörmander's hypoelliptic theorem for nonlocal operators

Abstract

In this paper we show the Hörmander hypoelliptic theorem for nonlocal operators by a purely probabilistic method: the Malliavin calculus. Roughly speaking, under general Hörmander's Lie bracket conditions, we show the regularization effect of discontinuous Lévy 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 ${\partial }_t-\mathcal{K}$, where $\mathcal{K}$ is the nonlocal kinetic operator: $$\mathcal{K}f(x,\mathrm{v}):=p.v{\int }_{{\mathbb{R}}^d}(f(x,\mathrm{v}+w)-f(x,\mathrm{v}))\frac{\kappa (x,\mathrm{v},w)}{|w{|}^{d+\alpha }}\mathrm{d}w+\mathrm{v}\cdot {\nabla }_{x}f(x,\mathrm{v})+b(x,\mathrm{v})\cdot {\nabla }_{\mathrm{v}}f(x,\mathrm{v}).$$ Here ${\kappa }_0^{-1}\le \kappa (x,\mathrm{v},w)\le {\kappa }_0$ belongs to $C_b^{\infty }({\mathbb{R}}^{3d})$ and is symmetric in $w$, p.v. stands for the Cauchy principal value, and $b\in C_b^{\infty }({\mathbb{R}}^{2d};{\mathbb{R}}^d)$.

36pages

References in corpus (2)

• On the absolute continuity of Lévy processes with drift
• Smooth densities for stochastic differential equations with jumps

Cited by in corpus (1)

• Schauder's estimate for nonlocal kinetic equations and its applications