Hörmander’s Hypoelliptic Theorem for Nonlocal Operators

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 $$\partial _t-{{\mathscr {K}}}$$ ∂ t - K , where $${{\mathscr {K}}}$$ K is the following nonlocal kinetic operator: $$\begin{aligned} {{\mathscr {K}}}f(x,\mathrm{v})= & {} \mathrm{p.v.}\int _{{{\mathbb {R}}}^d}(f(x,\mathrm{v}+w)-f(x,\mathrm{v}))\frac{\kappa (x,\mathrm{v},w)}{|w|^{d+\alpha }}\, {\mathord {\mathrm{d}}}w \\&+\mathrm{v}\cdot \nabla _x f(x,\mathrm{v})+b(x,\mathrm{v})\cdot \nabla _\mathrm{v} f(x,\mathrm{v}). \end{aligned}$$ K f ( x , v ) = p . v . ∫ R d ( f ( x , v + w ) - f ( x , v ) ) κ ( x , v , w ) | w | d + α d w + v · ∇ x f ( x , v ) + b ( x , v ) · ∇ v f ( x , v ) . Here $$\kappa _0^{-1}\leqslant \kappa (x,\mathrm{v},w)\leqslant \kappa _0$$ κ 0 - 1 ⩽ κ ( x , v , w ) ⩽ κ 0 belongs to $$C^\infty _b({{\mathbb {R}}}^{3d})$$ C b ∞ ( R 3 d ) and is symmetric in w, p.v. stands for the Cauchy principal value, and $$b\in C^\infty _b({{\mathbb {R}}}^{2d};{{\mathbb {R}}}^d)$$ b ∈ C b ∞ ( R 2 d ; R d ) .