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September 2009 The Cauchy problem and the martingale problem for integro-differential operators with non-smooth kernels
Helmut Abels, Moritz Kassmann
Osaka J. Math. 46(3): 661-683 (September 2009).

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.

Citation

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Helmut Abels. Moritz Kassmann. "The Cauchy problem and the martingale problem for integro-differential operators with non-smooth kernels." Osaka J. Math. 46 (3) 661 - 683, September 2009.

Information

Published: September 2009
First available in Project Euclid: 26 October 2009

zbMATH: 1196.47037
MathSciNet: MR2583323

Subjects:
Primary: 47G20
Secondary: 35B65 , 35K99 , 47A60 , 47G30 , 60G07 , 60J35 , 60J75

Rights: Copyright © 2009 Osaka University and Osaka City University, Departments of Mathematics

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Vol.46 • No. 3 • September 2009
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