## Electronic Journal of Probability

### Existence of (Markovian) solutions to martingale problems associated with Lévy-type operators

Franziska Kühn

#### Abstract

We study the existence of (Markovian) solutions to the $(A,C_{c}^{\infty }(\mathbb{R} ^{d}))$-martingale problem associated with the Lévy-type operator $A$ with symbol $q(x,\xi )$. Firstly, we establish conditions which ensure the existence of a solution. The main contribution is that our existence result allows for discontinuity in $x \mapsto q(x,\xi )$. Applying the result, we obtain new insights on the existence of weak solutions to a class of Lévy-driven SDEs with Borel measurable coefficients and on the the existence of stable-like processes with discontinuous coefficients. Secondly, we prove a Markovian selection theorem which shows that – under mild assumptions – the $(A,C_{c}^{\infty }(\mathbb{R} ^{d}))$-martingale problem gives rise to a strong Markov process. The result applies, in particular, to Lévy-driven SDEs. We illustrate the Markovian selection theorem with applications in the theory of non-local operators and equations; in particular, we establish under weak regularity assumptions a Harnack inequality for non-local operators of variable order.

#### Article information

Source
Electron. J. Probab., Volume 25 (2020), paper no. 16, 26 pp.

Dates
Accepted: 26 January 2020
First available in Project Euclid: 5 February 2020

https://projecteuclid.org/euclid.ejp/1580871682

Digital Object Identifier
doi:10.1214/20-EJP424

#### Citation

Kühn, Franziska. Existence of (Markovian) solutions to martingale problems associated with Lévy-type operators. Electron. J. Probab. 25 (2020), paper no. 16, 26 pp. doi:10.1214/20-EJP424. https://projecteuclid.org/euclid.ejp/1580871682

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