Gradinaru, Mihai; Haugomat, Tristan Locally Feller processes and martingale local problems. (English) Zbl 1466.60148 Stochastic Processes Appl. 133, 129-165 (2021). This paper is devoted to the study of a certain type of martingale problems associated to general operators corresponding to processes, which have finite lifetime. Heuristically, a Lévy-type process \(X\) with symbol \(q:\mathbb R^d \times \mathbb R^d \to C\) is a Markov process, which behaves locally like a Lévy process with characteristic exponent \(q(a, \cdot )\), in a neighbourhood of each point \(a \in \mathbb R^d\) . The pseudo-differential operator \(L\) associated to a Lévy-type process is given by, for \(f \in C_c^\infty (\mathbb R^d)\),\(Lf(a): = - \int_{\mathbb R^d} {{e^{ia \cdot \alpha }}q(a,\alpha )\hat f(\alpha )d\alpha } \), where\(\hat f(\alpha ): = {(2\pi )^{ - d}}\int_{\mathbb R^d} {{e^{ - \alpha \cdot a}}f(a)da} \). The authors investigate the following questions, which naturally appear in studying the Lévy-type processes. 1. Does a sequence \({X^{(n)}}\) of Lévy-type processes, having symbols \({q_n}\), converges towards some process, when \({q_n}\) converges to a symbol \(q\)? 2. What can we say about the sequence \({X^{(n)}}\) when the corresponding sequence of pseudo-differential operators \({L_n}\) converges to an operator \(L\)? 3. What could be the appropriate setting when one wants to approximate a Lévy-type process by a family of discrete Markov chains? In the present paper, the authors describe a general method, which serves as the main tool to tackle the difficulties, which appear in applying earlier obtained results. Moreover, it relaxes some of technical restrictions. For this purpose, they analyze sequences of martingale problems associated to large class of operators acting on continuous functions. Reviewer: Oleg K. Zakusilo (Kyïv) MSC: 60J25 Continuous-time Markov processes on general state spaces 60G44 Martingales with continuous parameter 60J35 Transition functions, generators and resolvents 60B10 Convergence of probability measures 47D07 Markov semigroups and applications to diffusion processes Keywords:martingale problem; Feller processes; weak convergence of probability measures; Skorokhod topology; generators; localisation PDFBibTeX XMLCite \textit{M. Gradinaru} and \textit{T. Haugomat}, Stochastic Processes Appl. 133, 129--165 (2021; Zbl 1466.60148) Full Text: DOI arXiv References: [1] Billingsley, P., (Convergence of Probability Measures. Convergence of Probability Measures, Wiley Series in Probability and Statistics (1999), John Wiley & Sons, Inc.: John Wiley & Sons, Inc. New York) · Zbl 0944.60003 [2] Böttcher, B.; Schilling, R. L.; Wang, J., (Lévy Matters. III. Lévy Matters. 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