Harnack Inequalities for Jump Processes

By Richard F. Bass, David A. Levin. 2002

265 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/9049f70acf6f6c498368ad9b21612fb04e14bb52

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In Applied Mathematics

In Finance

Citations in the corpus

Harnack inequalities for subordinate Brownian motions (2012, methodology): “A new technique on Harnack inequalities for stable like jump processes was developed in [3] and generalized in [19].”
Schauder estimates for a class of non-local elliptic equations (2011, methodology): “ity, H¨older estimates, and non-local versions of the Aleksandrov–Bakelman–Pucci estimate, were studied ﬁrst by using probabilistic methods, and more recently, by analytic methods; see, for instance, [5, 3, 4] and [9, 14, 24, 7, 16]. In [11] by using a purely analytic method, we established Lp-estim”
Regularity results for nonlocal parabolic equations (2013, methodology): “A Harnack inequality and Hölder regularity results are obtained for linear equations by Bass and Levin in [7] not assuming regularity of the kernel with respect to the state variable.”
Fractional powers of second order partial differential operators: extension problem and regularity theory (2010, methodology): “1) la teor a se desarroll o en los ultimos a~ nos usando un enfoque probabil stico [6, 7, 10, 11, 12, 14, 44, 71], v ease tambi en el libro de K.”
Regularity theory for general stable operators (2014, background): “1), or related ones, has been widely investigated; see the works by Bass, Kassmann, Schwab, Silvestre, Sztonyk, and Bogdan, among others [1, 23, 22, 3, 39, 21, 2, 37, 5, 6, 7, 9, 24].”
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.”
Nonlocal elliptic equations in bounded domains: a survey (2015, unknown): “• Interior regularity of solutions: Bass–Levin [6], Kassmann [33], Caffarelli–Silvestre [13, 14, 15], Barles–Chasseigne–Imbert [4], Kassmann–Mimica [34, 35], Schwab–Silvestre [53], and Serra [54].”
Differentiability of spectral functions for symmetric $\alpha$-stable processes (2007, methodology): “To prove the continuity of h, we use a theorem of Bass and Levin [4].”
A priori estimates for integro-differential operators with measurable kernels (2009, background): “Note that these results based on a probabilistic approach are related to our result but do not contain it. On one hand, they use a different notion of harmonic functions and on the other hand, they are restricted to isotropic kernels, see the comment (1) in Section 5. Non-local operators that do not”
Regularity of Harmonic functions for a class of singular stable-like processes (2009, methodology): “By the Lévy system formula (see [3] or [5]),”
Boundary Harnack inequality for Markov processes with jumps (2012, unknown): “Hölder continuity of harmonic functions is crucial in the theory of partial differential equations [6, 16], and the existence of limits of ratios of nonnegative harmonic functions leads to the construction of the Martin kernel and to representation of nonnegative harmonic functions ([5, 26]).”
Transition Probabilities for Symmetric Jump Processes (2002, background): “It is fairly straightforward at this point to follow the argument of [ BL ] and obtain a Harnack inequality of Moser type for functions that are harmonic with respect to Xn. In this paper, however, we are primarily interested in transition probability estimates.”
Nonlinear Diffusion with Fractional Laplacian Operators (2012, unknown): “See also [11, 12].”
Potential Theory of Subordinate Brownian Motion (2009, methodology): “Similar estimates involving Lebesgue measure were obtained in [56] based on the work of Bass and Levin [4].”
Holder estimates for solutions of integra-differential equations like the fractional laplace (2006, background): “3 Kernels comparable to those of (−4)s In [8] they consider operators like (1.”