Intrinsic scaling properties for nonlocal operators

By M. Kassmann, Ante Mimica. 2013

84 citations recorded by the source corpus.

We study growth lemmas and questions of regularity for generators of Markov processes. The generators are allowed to have an arbitrary order of differentiability less than 2. In general, this order is represented by a function and not by a number. The approach enables a careful study of regularity issues up to the phase boundary between integro-differential (positive order of differentiability) and integral operators (nonnegative order of differentiability). The proof is based on intrinsic scaling properties of the underlying operators and stochastic processes.

Subfields: Computational Theory and Mathematics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8b5086299956d28dd803d36191b786ebe4124183

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Theory and Mathematics

1. Russell W. Schwab
2. Sven Jarohs
3. Luis Silvestre
4. Tobias Weth
5. Alberto Saldaña
6. Xavier Ros-Oton
7. Enrico Valdinoci
8. J. Serra

In Finance

1. Russell W. Schwab
2. Luis Silvestre
3. Xavier Ros-Oton
4. Luis Caffarelli
5. Enrico Valdinoci
6. D. Kriventsov

In Mathematical Physics

1. Russell W. Schwab
2. Luis Silvestre
3. Tobias Weth
4. Sven Jarohs
5. Alberto Saldaña
6. Pablo Raúl Stinga
7. Enrico Valdinoci
8. C. Imbert

Citations in the corpus

• A new look at the fractional Poisson problem via the logarithmic Laplacian (2019, background): “(ii) The interior regularity property has been proven in [23, 24, 31, 32].”
• Regularity for parabolic integro-differential equations with very irregular kernels (2014, methodology): “The question of what happens as α → 0 is interesting for the nonlocal theory, and some results are obtained in [23] (note, there they do not use the typical normalization constant as in potential theory, where Cd,α ≈ α as α→ 0, so the limit operator is not a multiple of the identity).”
• 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].”
• Boundary regularity for fully nonlinear integro-differential equations (2014, unknown): “Then, v satisfies { M+v ≥ 0 and M−v ≤ 0 in {xn > 0}, v = 0 in {xn 0}.”