Non-local dirichlet forms and symmetric jump processes

By Martin T. Barlow, Richard F. Bass, Zhen-Qing Chen, M. Kassmann. 2006

296 citations recorded by the source corpus.

We consider the non-local symmetric Dirichlet form (E,F) given by with F the closure with respect to E 1 of the set of C 1 functions on R d with compact support, where E 1 (f, f):= E(f, f) + f Rd f(x) 2 dx, and where the jump kernel J satisfies for 0 < α < β < 2, |x - y| < 1. This assumption allows the corresponding jump process to have jump intensities whose sizes depend on the position of the process and the direction of the jump. We prove upper and lower estimates on the heat kernel. We construct a strong Markov process corresponding to (E,F). We prove a parabolic Harnack inequality for non-negative functions that solve the heat equation with respect to E. Finally we construct an example where the corresponding harmonic functions need not be continuous.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ebdeea6c33fc45fd465ba879f1e4654f2d71570a

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Citations in the corpus

A priori estimates for integro-differential operators with measurable kernels (2009, background): “In [ BBCK ] u is known to be positive almost everywhere which simplifies the argument significantly.”
Corrigendum to “Global heat kernel estimates for symmetric jump processes” (2011, methodology): “8 of [1], we have that for every r ≥ t 0 ,”
Gradient flows of the entropy for jump processes (2012, background): “To motivate our interest in such a link between jump processes and optimal transport, let us highlight two observations.”
Heat Kernels on Metric Measure Spaces (2014, background): “1 Upper bounds for non-local Dirichlet forms The techniques for obtaining heat kernel bounds for non-local Dirichlet forms has been developed by a number of authors, see for example [7, 10, 14, 17, 18] and the references therein.”
On heat kernel estimates and parabolic Harnack inequality for jump processes on metric measure spaces (2009, unknown): “5 of [8]).”
Transition Density Estimates for a Class of Lévy and Lévy-Type Processes (2009, background): “Further results on two-sided heat kernel estimates and a version of the parabolic Harnack inequality for symmetric jump processes are contained in [2] and [3].”
The weak Harnack inequality for the Boltzmann equation without cut-off (2016, background): “THE WEAK HARNACK INEQUALITY FOR THE BOLTZMANN EQUATION WITHOUT CUT-OFF 9 is made in [44], [14], [41], [29] and [21].”
Estimates of heat kernels for non-local regular Dirichlet forms (2014, methodology): “The techniques for obtaining heat kernel bounds for non-local Dirichlet forms has been developed by a number of authors, see for example [4, 6, 7, 9, 10] and the references therein.”
Local Regularity for Parabolic Nonlocal Operators (2012, unknown): “18) we will choose two different sequences (rm) in the cases α ∈ [1, 2) and α ∈ (α0, 1): • α ∈ [1, 2) : Set rm = r + (R−r) 2m .”
Mixed local and nonlocal Sobolev inequalities with extremal and associated quasilinear singular elliptic problems (2021, methodology): “The question of regularity related to the problem (1.10) has been settled by Barlow-BassChen-Kassmann [9], Chen-Kim-Song-Vondraček [26, 27], Garain-Kinnunen [46, 47] and the references therein.”