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The Cauchy problem and the martingale problem for integro-differential operators with non-smooth kernels. (English) Zbl 1196.47037

Let
\[ Lu(x)=\int_{\mathbb R^n} [u(x+y)-u(x)-1_{[1,2]}(\alpha)1_{|y|\leq 2} y \cdot \nabla u(x)] k(x,y)\,dy, \]
where \(k(x,y)\asymp |y|^{-n-\alpha}\), \(\alpha\in (0,2)\), for small \(y\), and \(k\) is Hölder continuous in \(x\). Denote by \({\mathcal C}^s(\mathbb R^n)\) the Hölder-Zygmund space, \({\mathcal C}_0^s(\mathbb R^n):=\overline{C_0^\infty(\mathbb R^n)}^{\|\cdot\|_{{\mathcal C}^s}}\), and denote by \(C^s(\mathbb R^n)\) the space of uniformly bounded Hölder continuous functions of order \(s\), with uniformly bounded constant. The main result of the paper under review states that, under some suitable conditions on \(k\), the Cauchy problem
\[ \begin{cases} \partial_t u-Lu =f &\text{in \((0,T)\times\mathbb R^n\)}, \\ u(0,\cdot)=0&\text{in \(\mathbb R^n\)}, \end{cases} \]
is uniquely solvable for any \(f\in C^\theta ([0,T]; {\mathcal C}_0^s (\mathbb R^n))\), \(f(0)=0\), \(0<\theta<1\), and
\[ u\in C^{1,\theta} ([0,T]; {\mathcal C}_0^s (\mathbb R^n))\cap C^\theta ([0,T]; {\mathcal C}_0^{s+\alpha} (\mathbb R^n)). \]
As corollary of the main result, the well-posedness of the martingale problem for \((L,C_0^\infty(\mathbb R^n))\) is proved.

MSC:

47G20 Integro-differential operators
47G30 Pseudodifferential operators
60J75 Jump processes (MSC2010)
60J35 Transition functions, generators and resolvents
60G07 General theory of stochastic processes
35K99 Parabolic equations and parabolic systems
35B65 Smoothness and regularity of solutions to PDEs
47A60 Functional calculus for linear operators
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References:

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