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 ) .

中文翻译：

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非局部算子的 Hörmander 亚椭圆定理

在本文中，我们通过纯概率方法展示了非局部算子的 Hörmander 亚椭圆定理：Malliavin 演算。粗略地说，在一般 Hörmander's Lie 括号条件下，我们展示了不连续 Lévy 噪声对可能具有跳跃的退化随机微分方程的正则化效果。为了处理大跳跃，我们将扰动参数与插值技术和半群的一些短时间渐近估计一起使用。作为一个应用，我们展示了算子 $$\partial _t-{{\mathscr {K}}}$$ ∂ t - K 的基本解的存在性，其中 $${{\mathscr {K}}}$$ K是以下非局部动力学算子： $$\begin{aligned} {{\mathscr {K}}}f(x,\mathrm{v})= & {} \mathrm{pv}\int _{{{\mathbb {R}}}^d}(f(x,\mathrm{v}+w)-f(x,\mathrm{v}))\frac{\kappa (x,\mathrm{v}, w)}{|w|^{d+\alpha }}\, {\mathord {\mathrm{d}}}w \\&+\mathrm{v}\cdot \nabla _x f(x,\mathrm{v })+b(x,\mathrm{v})\cdot \nabla _\mathrm{v} f(x,\mathrm{v})。\end{aligned}$$ K f ( x , v ) = p 。诉。∫ R d ( f ( x , v + w ) - f ( x , v ) ) κ ( x , v , w ) | | | d + α dw + v · ∇ xf ( x , v ) + b ( x , v ) · ∇ vf ( x , v ) 。这里 $$\kappa _0^{-1}\leqslant \kappa (x,\mathrm{v},w)\leqslant \kappa _0$$ κ 0 - 1 ⩽ κ ( x , v , w ) ⩽ κ 0 属于到 $$C^\infty _b({{\mathbb {R}}}^{3d})$$ C b ∞ ( R 3 d ) 并且在 w 中对称，pv 代表柯西主值，$$ b\in C^\infty _b({{\mathbb {R}}}^{2d};{{\mathbb {R}}}^d)$$ b ∈ C b ∞ ( R 2 d ; R d ) . v + w ) - f ( x , v ) ) κ ( x , v , w ) | | | d + α dw + v · ∇ xf ( x , v ) + b ( x , v ) · ∇ vf ( x , v ) 。这里 $$\kappa _0^{-1}\leqslant \kappa (x,\mathrm{v},w)\leqslant \kappa _0$$ κ 0 - 1 ⩽ κ ( x , v , w ) ⩽ κ 0 属于到 $$C^\infty _b({{\mathbb {R}}}^{3d})$$ C b ∞ ( R 3 d ) 并且在 w 中对称，pv 代表柯西主值，$$ b\in C^\infty _b({{\mathbb {R}}}^{2d};{{\mathbb {R}}}^d)$$ b ∈ C b ∞ ( R 2 d ; R d ) . v + w ) - f ( x , v ) ) κ ( x , v , w ) | | | d + α dw + v · ∇ xf ( x , v ) + b ( x , v ) · ∇ vf ( x , v ) 。这里 $$\kappa _0^{-1}\leqslant \kappa (x,\mathrm{v},w)\leqslant \kappa _0$$ κ 0 - 1 ⩽ κ ( x , v , w ) ⩽ κ 0 属于到 $$C^\infty _b({{\mathbb {R}}}^{3d})$$ C b ∞ ( R 3 d ) 并且在 w 中对称，pv 代表柯西主值，$$ b\in C^\infty _b({{\mathbb {R}}}^{2d};{{\mathbb {R}}}^d)$$ b ∈ C b ∞ ( R 2 d ; R d ) .