Abstract
Let α(x) be a measurable function taking values in [α1,α2] for 0<α1⩽α2<2, and κ(x,z) be a positive measurable function that is symmetric in z and bounded between two positive constants. Under uniform Hölder continuous assumptions on α(x) and x↦κ(x,z), we obtain existence, upper and lower bounds, and regularity properties of the heat kernel associated with the following non-local operator of variable order [Formula presented]. In particular, we show that the operator L generates a conservative Feller process on Rd having strong Feller property, which is usually assumed a priori in the literature to study analytic properties of L via probabilistic approaches. Our near-diagonal estimates and lower bound estimates of the heat kernel depend on the local behavior of index function α(x). When α(x)≡α∈(0,2), our results recover some results by Chen and Kumagai (2003) and Chen and Zhang (2016).
Original language | English |
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Pages (from-to) | 3574-3647 |
Number of pages | 74 |
Journal | Stochastic Processes and their Applications |
Volume | 130 |
Issue number | 6 |
DOIs | |
Publication status | Published - Jun 2020 |
Externally published | Yes |
Keywords
- Heat kernel
- Levi's method
- Non-local operator with variable order
- Stable-like process