Two‐sided Green function estimates for killed subordinate Brownian motions

By Panki Kim, Renming Song, Zoran Vondraček. 2010

60 citations recorded by the source corpus.

A subordinate Brownian motion is a Lévy process that can be obtained by replacing the time of the Brownian motion by an independent subordinator. The infinitesimal generator of a subordinate Brownian motion is−ϕ(−Δ), where ϕ is the Laplace exponent of the subordinator. In this paper, we consider a large class of subordinate Brownian motions without diffusion component and with ϕ comparable to a regularly varying function at infinity. This class of processes includes symmetric stable processes, relativistic stable processes, sums of independent symmetric stable processes, sums of independent relativistic stable processes, and much more. We give sharp two‐sided estimates on the Green functions of these subordinate Brownian motions in any bounded κ‐fat open set D. When D is a bounded C1, 1 open set, we establish an explicit form of the estimates in terms of the distance to the boundary. As a consequence of such sharp Green function estimates, we obtain a boundary Harnack principle in C1, 1 open sets with explicit rate of decay.

Subfields: Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/339ddc9e06e526101d388e8f9ce9f39e82f538a0

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Citations in the corpus

Boundary Harnack inequality for Markov processes with jumps (2012, unknown): “The estimates are now extended to subordinate Brownian motions, and the renewal function of the subordinator is used in the corresponding formulations ([66]).”
Dirichlet heat kernel estimates for rotationally symmetric Lévy processes (2013, unknown): “) When Ψ(|ξ|) = φ(|ξ|2) and φ is a complete Bernstein function satisfying condition (A), the following boundary Harnack principle on a C(1,1) open subset with explicit decay rate is established in [21] (see also [23]).”
Suprema of Lévy processes (2011, background): “By [30], Lemma 4, if ψ(ξ) is a CBF, then also ψ(ξ) is a CBF (this was independently proved in [24], Proposition 2.”
Gradient estimates of harmonic functions and transition densities for Levy processes (2013, background): “ δD(x) ∧ |x− y|∧1 , x∈ D, (3) where δD(x) = dist(x,∂D). The potential theory of L´evy processes, especially subordinate Brownian motions, has attracted a lot of attention during recent years see e.g. [6, 10, 9, 18, 19]. Our study is in the scope of this type of research. The assertion of Theorem 1.1”
A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations on Bounded Domains (2013, background): “In [33, 36] analogous estimates are derived for nontrivial perturbation of the above mentioned operators.”