Uniqueness of stable-like processes

Abstract

In this work we consider the following $\alpha$-stable-like operator (a class of pseudo-differential operator) $$\mathcal{L}f(x):={\int }_{{\mathbb{R}}^d}[f(x+{\sigma }_{x}y)-f(x)-1_{\alpha \in [1,2)}{1}_{|y|\le 1}{\sigma }_{x}y\cdot \nabla f(x)]{\nu }_x(dy),$$ where the Lévy measure ${\nu }_x(dy)$ is comparable with a non-degenerate $\alpha$-stable-type Lévy measure (possibly singular), and ${\sigma }_x$ is a bounded and nondegenerate matrix-valued function. Under Hölder assumption on $x\mapsto {\nu }_x(dy)$ and uniformly continuity assumption on $x\mapsto {\sigma }_x$, we show the well-posedness of martingale problem associated with the operator $\mathcal{L}$. Moreover, we also obtain the existence-uniqueness of strong solutions for the associated SDE when $\sigma$ belongs to the first order Sobolev space ${\mathbb{W}}^{1,p}({\mathbb{R}}^d)$ provided $p>d(1+\alpha \vee 1)$ and ${\nu }_x=\nu$ is a non-degenerate $\alpha$-stable-type Lévy measure.

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• Heat kernels for time-dependent non-symmetric stable-like operators
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• Heat kernel of supercritical SDEs with unbounded drifts
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• Quenched invariance principle for long range random walks in balanced random environments