Potential theory of truncated stable processes

By Panki Kim, Renming Song. 2006

48 citations recorded by the source corpus.

For any $$\alpha \in (0, 2)$$, a truncated symmetric α-stable process is a symmetric Lévy process in $$\mathbb{R}^{d}$$ with a Lévy density given by $$c|x|^{-d-\alpha} 1_{\{|x| < 1\}}$$ for some constant c. In this paper we study the potential theory of truncated symmetric stable processes in detail. We prove a Harnack inequality for nonnegative harmonic functions of these processes. We also establish a boundary Harnack principle for nonnegative functions which are harmonic with respect to these processes in bounded convex domains. We give an example of a non-convex domain for which the boundary Harnack principle fails.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/17d6a3c817c144cffb67ed998dde1a0780945893

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Weighted Poincaré inequality and heat kernel estimates for finite range jump processes (2008, background): “1 extends the result on Harnack principle in [24] to the case that c(x, y) is not necessarily constant.”
Strong rate of convergence for the Euler-Maruyama approximation of SDEs with H\"older continuous drift coefficient (2015, unknown): “For more information on Lévy processes, we refer the reader to [1, 2, 10, 20] and references therein.”
Boundary Harnack principle for subordinate Brownian motions (2007, background): “This comparison of Green functions played a crucial role in the arguments of [12,16,17].”
Boundary Harnack principle for Δ+Δ^{/2} (2012, background): “In [29, 30], the potential theory of truncated symmetric stable processes including BHP was studied.”
Boundary Harnack inequality for Markov processes with jumps (2012, unknown): “A different version of boundary Harnack inequality was proved for Xt under some regularity assumptions on the domain of harmonicity in [61, 62].”