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Hölder Regularity for Generalized Master Equations with Rough Kernels

Hölder Regularity for Generalized Master Equations with Rough Kernels

Chapter:
(p.63) Chapter Four Hölder Regularity for Generalized Master Equations with Rough Kernels
Source:
Advances in Analysis
Author(s):
Luis CaffarelliLuis Silvestre, Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, Stephen Wainger, Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, Stephen Wainger
Publisher:
Princeton University Press
DOI:10.23943/princeton/9780691159416.003.0004

This chapter studies evolution problems that are related to continuous time random walks (CTRW), having a discontinuous path for which both the jumps and the time elapsed in between them are random. These processes are governed by a generalized master equation which is nonlocal both in space and time. To illustrate, the chapter considers kernels K(t, x, s, y) in a particular function. Here, studying correlated kernels provides a more flexible framework where more interesting physical phenomena can be observed, and more subtle mathematical questions appear. The regularity estimates are in fact more interesting (harder mathematically) when the jumps in space and the waiting times are strongly correlated.

Keywords:   evolution problems, continuous time random walks, CTRW, rough kernels, correlated kernels, holder regularity

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