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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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