Electronic Journal of Probability

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

Franziska Kühn

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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

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

Received: 4 March 2019
Accepted: 26 January 2020
First available in Project Euclid: 5 February 2020

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Digital Object Identifier

Primary: 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]
Secondary: 60J25: Continuous-time Markov processes on general state spaces 60H10: Stochastic ordinary differential equations [See also 34F05] 60J75: Jump processes 45K05: Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20] 35S05: Pseudodifferential operators 60G51: Processes with independent increments; Lévy processes

martingale problem pseudo-differential operator Markovian selection existence result discontinuous coefficents Krylov estimate jump process Lévy-driven stochastic differential equation Harnack inequality viscosity solution

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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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