Abstract
Suppose that \(d\ge 1\) and \(0<\beta <2\). We establish the existence and uniqueness of the fundamental solution \(q^b(t, x, y)\) to the operator \({\mathcal {L}}^b=\Delta +{\mathcal {S}}^b\), where
and \(b(x, z)\) is a bounded measurable function on \({\mathbb {R}}^d\times {\mathbb {R}}^d\) with \(b(x, z)=b(x, -z)\) for \(x, z\in {\mathbb {R}}^d\). We show that if for each \(x\in {\mathbb {R}}^d, b(x, z) \ge 0\) for a.e. \(z\in {\mathbb {R}}^d\), then \(q^b(t, x, y)\) is a strictly positive continuous function and it uniquely determines a conservative Feller process \(X^b\), which has strong Feller property. Furthermore, sharp two-sided estimates on \(q^b(t, x, y)\) are derived.
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Acknowledgments
The author is grateful to Professor Z.-Q. Chen for his valuable comments. The author thanks the referee for careful reading and useful suggestions for improving the quality of this paper.
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Research partially supported by NNSFC Grant 11371054 and NNSFC Grant 11171024.
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Wang, JM. Laplacian perturbed by non-local operators. Math. Z. 279, 521–556 (2015). https://doi.org/10.1007/s00209-014-1380-9
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DOI: https://doi.org/10.1007/s00209-014-1380-9
Keywords
- Brownian motion
- Laplacian
- Perturbation
- Non-local operator
- Integral kernel
- Positivity
- Lévy system
- Feller semigroup