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Existence of (Markovian) solutions to martingale problems associated with Lévy-type operators. (English) Zbl 1448.60162
A Lévy-type operator is defined on the smooth functions with compact support $$C_c^{\infty}(\mathbb{R}^d)$$ and has a representation of the form \begin{multline*} Af(x)=b(x)\cdot \nabla f(x)+\frac{1}{2}\operatorname{tr}(Q(x)\cdot \nabla^2f(x))\\ +\int_{\mathbb{R}^d\backslash \{0\}}\left(f(x+y)-f(x)-\nabla f(x)\cdot y\ \mathbf{1}_{(0,1)}(|y|)\right)\nu(x,dy), \end{multline*} where $$(b(x),Q(x),\nu(x,dy))$$ is for each fixed $$x\in \mathbb{R}^d$$ a Lévy triplet. Equivalently, $$A$$ can be written as a pseudo-differential operator $Af(x)=-\int_{\mathbb{R}^d}e^{ix\cdot \xi}q(x,\xi)\hat{f}(\xi)d\xi$ with symbol $$q$$, $q(x,\xi):=-ib(x)\cdot \xi+\frac{1}{2}\xi\cdot Q(x)\xi+\int_{\mathbb{R}^d\backslash\{0\}}\left(1-e^{iy\cdot \xi}+iy\cdot \xi\ \mathbf{1}_{(0,1)}(|y|)\right)\nu(x,dy).$
The paper under review studies the existence of (Markovian) solutions to the $$(A,C_c^{\infty}(\mathbb{R}^d))$$-martingales problem associated with the Lévy-type operator $$A$$ with symbol $$q(x,\xi)$$. The contributions contain two parts. The first part is the existence result which allows for discontinuity in $$x\mapsto q(x,\xi)$$, with applications to the existence of weak solutions to a class of Lévy-driven SDEs with Borel measurable coefficients and on the existence of stable-like processes with discontinuous coefficients. The second part is a Markovian selection theorem which shows that – under mild assumptions – the $$(A,C_c^{\infty}(\mathbb{R}^d))$$-martingales problem gives rise to a strong Markov process. Some applications were given, one of which is to build a Harnack inequality for non-local operators of variable order.

##### MSC:
 60J35 Transition functions, generators and resolvents 60J25 Continuous-time Markov processes on general state spaces 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60J76 Jump processes on general state spaces 45K05 Integro-partial differential equations 35S05 Pseudodifferential operators as generalizations of partial differential operators 60G51 Processes with independent increments; Lévy processes
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