Osaka Journal of Mathematics

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

Helmut Abels and Moritz Kassmann

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Abstract

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

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

Dates
First available in Project Euclid: 26 October 2009

Permanent link to this document
https://projecteuclid.org/euclid.ojm/1256564200

Mathematical Reviews number (MathSciNet)
MR2583323

Zentralblatt MATH identifier
1196.47037

Subjects
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

Citation

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. https://projecteuclid.org/euclid.ojm/1256564200


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