Osaka Journal of Mathematics

The Cauchy problem and the martingale problem for integro-differential operators with non-smooth kernels

Helmut Abels and Moritz Kassmann

Full-text: Open access


We consider the linear integro-differential operator $L$ defined by \begin{equation*} Lu(x) =\int_{\mathbb{R}^{n}}(u(x+y)-u(x) -\mathbbm{1}_{[1,2]}(\alpha)\mathbbm{1}_{\{|y|\leq 2\}}(y)y \cdot \nabla u(x))k(x,y)\, dy. \end{equation*} Here the kernel $k(x,y)$ behaves like $|y|^{-n-\alpha}$, $\alpha \in (0,2)$, for small $y$ and is Hölder-continuous in the first variable, precise definitions are given below. We study the unique solvability of the Cauchy problem corresponding to $L$. As an application we obtain well-posedness of the martingale problem for $L$. Our strategy follows the classical path of Stroock-Varadhan. The assumptions allow for cases that have not been dealt with so far.

Article information

Osaka J. Math., Volume 46, Number 3 (2009), 661-683.

First available in Project Euclid: 26 October 2009

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 47G20: Integro-differential operators [See also 34K30, 35R09, 35R10, 45Jxx, 45Kxx]
Secondary: 47G30: Pseudodifferential operators [See also 35Sxx, 58Jxx] 60J75: Jump processes 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60G07: General theory of processes 35K99: None of the above, but in this section 35B65: Smoothness and regularity of solutions 47A60: Functional calculus


Abels, Helmut; Kassmann, Moritz. The Cauchy problem and the martingale problem for integro-differential operators with non-smooth kernels. Osaka J. Math. 46 (2009), no. 3, 661--683.

Export citation


  • H. Abels: Bounded imaginary powers and $H_{\infty}$-calculus of the Stokes operator in unbounded domains; in Nonlinear Elliptic and Parabolic Problems, Progr. Nonlinear Differential Equations Appl. 64, Birkhäuser, Basel, 2005, 1--15.
  • H. Abels: Pseudodifferential boundary value problems with non-smooth coefficients, Comm. Partial Differential Equations 30 (2005), 1463--1503.
  • H. Abels: Reduced and generalized Stokes resolvent equations in asymptotically flat layers, part II: $H\sb \infty$-calculus, J. Math. Fluid Mech. 7 (2005), 223--260.
  • R.F. Bass: Uniqueness in law for pure jump Markov processes, Probab. Theory Related Fields 79 (1988), 271--287.
  • R.F. Bass: Stochastic differential equations with jumps, Probab. Surv. 1 (2004), 1--19 (electronic).
  • R.F. Bass and H. Tang: The martingale problem for a class of stable-like processes, Stochastic Process. Appl. 119 (2009), 1144--1167.
  • J. Bertoin: Lévy Processes, Cambridge Tracts in Mathematics 121, Cambridge Univ. Press, Cambridge, 1996
  • P. Billingsley: Convergence of Probability Measures, Wiley, New York, 1968.
  • P. Billingsley: Convergence of Probability Measures, Wiley Series in Probability and Statistics: Probability and Statistics, Second edition, Wiley, New York, 1999.
  • V. Bogachev, P. Lescot and M. Röckner: The martingale problem for pseudo-differential operators on infinite-dimensional spaces, Nagoya Math. J. 153 (1999), 101--118.
  • B. Böttcher: A parametrix construction for the fundamental solution of the evolution equation associated with a pseudo-differential operator generating a Markov process, Math. Nachr. 278 (2005), 1235--1241.
  • Ph. Courrège: Sur la forme intégro-differentielle des opérateurs de $c^{\infty}_{k}$ dans $c$ satisfaisant au princie du maximum, Séminaire Brelot-Choquet-Deny, Theorie du potentiel 10, 1--38, 1965--1966.
  • S.D. Eidelman, S.D. Ivasyshen and A.N. Kochubei: Analytic Methods in the Theory of Differential and Pseudo-Differential Equations of Parabolic Type, Operator Theory: Advances and Applications 152, Birkhäuser, Basel, 2004.
  • S.N. Ethier and T.G. Kurtz: Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, Markov Processes, Wiley, New York, 1986.
  • W. Hoh: The martingale problem for a class of pseudo-differential operators, Math. Ann. 300 (1994), 121--147.
  • W. Hoh: Pseudo differential operators with negative definite symbols of variable order, Rev. Mat. Iberoamericana 16 (2000), 219--241.
  • N. Jacob: Pseudo Differential Operators and Markov Processes, III, Imp. Coll. Press, London, 2005.
  • N. Jacob and H.-G. Leopold: Pseudo-differential operators with variable order of differentiation generating Feller semigroups, Integral Equations Operator Theory 17 (1993), 544--553.
  • N. Jacob and R.L. Schilling: Lévy-type processes and pseudodifferential operators; in Lévy Processes, Birkhäuser, Boston, Boston, MA, 2001, 139--168.
  • J. Jacod and A.N. Shiryaev: Limit Theorems for Stochastic Processes, second edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288, Springer, Berlin, 2003.
  • O. Kallenberg: Foundations of Modern Probability, Springer, New York, 1997.
  • M. Kassmann: $\mathcal{L}$-harmonische Funktionen und Sprungprozesse, Mitt. Dtsch. Math.-Ver. 14 (2006), 80--87.
  • H. Kumano-go: Pseudodifferential Operators, Translated from the Japanese by the author, Rémi Vaillancourt and Michihiro Nagase, MIT Press, Cambridge, Mass., 1981.
  • H. Kumano-go and M. Nagase: Pseudo-differential operators with non-regular symbols and applications, Funkcial. Ekvac. 21 (1978), 151--192.
  • K. Kikuchi and A. Negoro: On Markov process generated by pseudodifferential operator of variable order, Osaka J. Math. 34 (1997), 319--335.
  • V. Kolokoltsov: Symmetric stable laws and stable-like jump-diffusions, Proc. London Math. Soc. (3) 80 (2000), 725--768.
  • T. Komatsu: Markov processes associated with certain integro-differential operators, Osaka J. Math. 10 (1973), 271--303.
  • T. Komatsu: On the martingale problem for generators of stable processes with perturbations, Osaka J. Math. 21 (1984), 113--132.
  • T. Komatsu: Pseudodifferential operators and Markov processes, J. Math. Soc. Japan 36 (1984), 387--418.
  • J.-P. Lepeltier and B. Marchal: Problème des martingales et équations différentielles stochastiques associées à un opérateur intégro-différentiel, Ann. Inst. H. Poincaré Sect. B (N.S.) 12 (1976), 43--103.
  • J. Marschall: Pseudodifferential operators with nonregular symbols of the class $S^{m}_{\rho,\delta}$, Comm. Partial Differential Equations 12 (1987), 921--965.
  • R. Mikulevičius and H. Pragarauskas: On the Cauchy problem for certain integro-differential operators in Sobolev and Hölder spaces, Liet. Mat. Rink. 32 (1992), 299--331.
  • R. Mikulyavichyus and G. Pragarauskas: The martingale problem related to nondegenerate Lévy operators, Liet. Mat. Rink. 32 (1992), 377--396.
  • R. Mikulyavichyus and G. Pragarauskas: On the uniqueness of solutions of the martingale problem that is associated with degenerate Lévy operators, Liet. Mat. Rink. 33 (1993), 455--475.
  • A. Negoro and M. Tsuchiya: Stochastic processes and semigroups associated with degenerate Lévy generating operators, Stochastics Stochastics Rep. 26 (1989), 29--61.
  • O. Okitaloshima and J.A. van Casteren: On the uniqueness of the martingale problem, Internat. J. Math. 7 (1996), 775--810.
  • A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
  • K. Sato: Lévy Processes and Infinitely Divisible Distributions, Cambridge Studies in Advanced Mathematics 68, Translated from the 1990 Japanese original; Revised by the author, Cambridge Univ. Press, Cambridge, 1999.
  • E.M. Stein: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, with the assistance of T.S. Murphy; Monographs in Harmonic Analysis, III, Princeton Mathematical Series 43, Princeton Univ. Press, Princeton, NJ, 1993.
  • D.W. Stroock: Diffusion processes associated with Lévy generators, Z. Wahrsch. Verw. Gebiete 32 (1975), 209--244.
  • D.W. Stroock and S.R.S. Varadhan: Multidimensional Diffusion Processes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 233, Springer, Berlin, 1979.
  • M.E. Taylor: Pseudodifferential Operators and Nonlinear PDE, Birkhäuser Boston, Boston, MA, 1991.
  • M.E. Taylor: Tools for PDE, Mathematical Surveys and Monographs, Amer. Math. Soc., Providence, RI, 2000.
  • H. Triebel: Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amsterdam, 1978.
  • M. Tsuchiya: Lévy measure with generalized polar decomposition and the associated SDE with jumps, Stochastics Stochastics Rep. 38 (1992), 95--117.