Harnack inequality for some classes of Markov processes

By Renming Song, Zoran Vondraček. 2004

112 citations recorded by the source corpus.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f27fb6981320bb9d2f04a1da8ebd1341390fc3d0

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Harnack inequalities for non-local operators of variable order (2004, result): “The current paper is a major generalization of the results obtained in [BL02a] and [SV04] in that we remove the requirement α = β.”
Holder estimates for solutions of integra-differential equations like the fractional laplace (2006, background): “In [16] and [6] a Harnack inequality is obtained, but no Hölder estimate follows.”
Green Function Estimates and Harnack Inequality for Subordinate Brownian Motions (2006, methodology): “In their paper, Song and Vondraček [24] extracted the essential ingredients of the BassYLevin method by isolating three conditions that suffice to prove the Harnack inequality and showed that various classes of Markov processes, not necessarily having any scaling properties, satisfy the Harnack ineq”
Regularity for parabolic integro-differential equations with very irregular kernels (2014, result): “Other results using probabilistic techniques were [5] and [39], where different assumptions on the kernels are considered.”
Potential Theory of Special Subordinators and Subordinate Killed Stable Processes (2006, unknown): “1 of [24]) that there exists c1 > 0 such that Px(τB2 ≤ t) ≤ c1t, x ∈ B1.”
Potential Theory of Geometric Stable Processes (2006, methodology): “We are going to use the notation and the approach from [ 20 ], combined with some results and ideas from [17] and [22].”
Regularity theory for fully nonlinear integro‐differential equations (2007, result): “Even though there are some known results about Harnack inequalities and Hölder estimates for integro-differential equations with either analytical proofs [10] or probabilistic proofs [4], [3], [5], [12], the estimates in all these previous results blow up as the order of the equation approaches 2.”
Potential theory of truncated stable processes (2006, methodology): “The method of [1] was extended in [ 28 ]t o cover a large class of Markov processes.”
Estimates of the potential kernel and Harnack's inequality for the anisotropic fractional Laplacian (2005, background): “The class of considered operators gradually extended, see [47], [45], [4], [18], [5], and the references given there.”
Non-local dirichlet forms and symmetric jump processes (2006, background): “Harnack inequalities for non-local operators have been considered in [BL02a] and [SV04] for fixed order, non-symmetric operators, [BL02b] and [CK03] for fixed order, symmetric operators.”
Heat kernel estimates for jump processes of mixed types on metric measure spaces (2007, background): “See also [3, 4, 5, 7, 8, 21, 22] for related Harnack inequalities, Hölder continuity of harmonic functions and the Feller property for non-local operators.”