# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Anulova, S., Pragarauskas, H.: Weak Markov solutions of stochastic equations. Lit. Math. J. 17 (1978), 141-155. · Zbl 0404.60065 [2] Barles, G., Chasseigne, E., Imbert, C.: Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations. J. Eur. Math. Soc. 13 (2011), 1-26. · Zbl 1207.35277 [3] Bass, R. F., Kaßmann, M.: Harnack inequalities for non-local operators of variable order. Trans. Amer. Math. Soc. 357 (2005), 837-850. · Zbl 1052.60060 [4] Bass, R. F., Levin, D. A.: Harnack inequalities for jump processes. Potential Anal. 17 (2002), 375-388. · Zbl 0997.60089 [5] Böttcher, B., Schilling, R. L., Wang, J.: Lévy-Type Processes: Construction, Approximation and Sample Path Properties. Springer Lecture Notes in Mathematics vol. 2099, (vol. III of the “Lévy Matters” subseries). Springer, 2014. · Zbl 1384.60004 [6] Costantini, C., Kurtz, T. G.: Viscosity methods giving uniqueness for martingale problems. Electron. J. Probab. 20 (2015), 1-27. · Zbl 1341.60030 [7] Denk, R., Kupper, M., Nendel, M.: A semigroup approach to nonlinear Lévy processes. To appear: Stoch. Proc. Appl. Preprint arXiv:1710.08130. · Zbl 07173248 [8] Ethier, S. N., Kurtz, T. G.: Markov Processes: Characterization and Convergence. Wiley, New York, 1986. · Zbl 0592.60049 [9] Hoh, W.: The martingale problem for a class of pseudo differential operators. Math. Ann. 300 (1994), 121-148. · Zbl 0805.47045 [10] Hoh, W.: Pseudo differential operators with negative definite symbols and the martingale problem. Stoch. Stoch. Rep. 55 (1995), 225-252. · Zbl 0880.47029 [11] Hoh, W.: Pseudo-Differential Operators Generating Markov Processes. Habilitationsschrift. Universität Bielefeld, Bielefeld 1998. · Zbl 0922.47045 [12] Hollender, J.: Lévy-Type Processes under Uncertainty and Related Nonlocal Equations. CreateSpace Independent Publishing Platform, 2016. [13] Imkeller, P., Willrich, N.: Solutions of martingale problems for Lévy-type operators with discontinuous coefficients and related SDEs. Stoch. Proc. Appl. 126 (2016), 703-734. · Zbl 1334.60068 [14] Jacob, N.: Pseudo Differential Operators and Markov Processes II. Imperial College Press/World Scientific, London 2002. · Zbl 1005.60004 [15] Jacob, N.: Pseudo Differential Operators and Markov Processes III. Imperial College Press/World Scientific, London 2005. · Zbl 1076.60003 [16] Kaßmann, M., Mimica, A.: Analysis of jump processes with nondegenerate jumping kernels. Stoch. Proc. Appl. 123 (2013), 629-650. · Zbl 1259.60100 [17] Kolokoltsov, V.: On Markov processes with decomposable pseudo-differential generators. Stoch. Stoch. Rep. 76 (2004), 1045-1129. · Zbl 1091.60014 [18] Krylov, N. V.: On the selection of a Markov process from a system of processes and the construction of quasi-diffusion processes. Math. USSR Izv. 7 (1973), 691-709. · Zbl 0295.60057 [19] Kühn, F.: Probability and Heat Kernel Estimates for Lévy(-Type) Processes. PhD Thesis, Technische Universität Dresden 2016. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-214839 [20] Kühn, F.: Lévy-Type Processes: Moments, Construction and Heat Kernel Estimates. Springer Lecture Notes in Mathematics vol. 2187 (vol. VI of the “Lévy Matters” subseries). Springer, 2017. · Zbl 1442.60002 [21] Kühn, F.: On Martingale Problems and Feller Processes. Eletron. J. Probab. 23 (2018), 1-18. · Zbl 1390.60278 [22] Kühn, F.: Solutions of Lévy-driven SDEs with unbounded coefficients as Feller processes. Proc. Amer. Math. Soc. 146 (2018), 3591-3604. · Zbl 1391.60192 [23] Kühn, F.: Perpetual integrals via random time changes. Bernoulli 25 (2019), 1755-1769. · Zbl 07066238 [24] Kühn, F.: Random time changes of Feller processes. Preprint arXiv:1705.02830. [25] Kühn, F., Schilling, R. L.: On the domain of fractional Laplacians and related generators of Feller processes. J. Funct. Anal. 276 (2019), 2397-2439. · Zbl 07030902 [26] Kurenok, V.: Stochastic equations with multidimensional drift driven by Lévy processes. Random Oper. Stoch. Equ. 14 (2006), 311-324. · Zbl 1119.60048 [27] Kurenok, V.: A note on $$L^2$$ estimates of stable integrals with drift. Trans. Amer. Math. Soc. 360 (2008), 925-938. · Zbl 1137.60029 [28] Kurenok, V.: On Stochastic Equations with Measurable Coefficients Driven by Symmetric Stable Processes. Int. J. Stoch. Anal. vol. 2012, Article ID 258415. · Zbl 1241.60027 [29] Kurtz, T. G.: Equivalence of stochastic equations and martingale problems. In: Stochastic Analysis 2010. Springer, 2011, pp. 113-130. · Zbl 0931.41017 [30] Schilling, R. L.: Growth and Hölder condtions for the sample paths of Feller processes. Probab. Theory Relat. Fields 112 (1998), 565-611. · Zbl 0930.60013 [31] Schilling, R. L., Schnurr, A.: The Symbol Associated with the Solution of a Stochastic Differential Equation. Electron. J. Probab. 15 (2010), 1369-1393. · Zbl 1226.60116 [32] Schilling, R. L., Wang, J.: Some theorems on Feller processes: Transience, local times and ultracontractivity. Trans. Amer. Math. Soc. 365 (2013), 3255-3286. · Zbl 1282.60070 [33] Wada, M.: Continuity of Harmonic Functions for Non-local Markov Generators. Potential Anal. 39 (2013), 1-11. · Zbl 1271.45003 [34] Zanzotto, P. A.: On stochastic differential equations driven by a Cauchy process and other stable Lévy motions. Ann. Probab. 30 (2002), 802-825. · Zbl 1017.60058 [35] Zhao, G.: Weak uniqueness for SDEs driven by supercritical stable processes with Hölder drifts. Proc. Amer. Math. Soc. 147 (2019), 849-860. · Zbl 1451.60049
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.