Ultrametric Diffusion, Exponential Landscapes, and the First Passage Time Problem

Torresblanca-Badillo, Anselmo; Zúñiga–Galindo, W. A.

doi:10.48550/arxiv.1511.08757

Cited by 2 publications

(2 citation statements)

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“…they are p-adic counterparts of the classical heat equations. For instance, J( x p ) = x γ p e − x p , with γ > −n, corresponds to an exponential landscape in the sense of [6], in this case, the fundamental solution of (1.5) is the transition density of a bounded right-continuous Markov process without second kind discontinuities, see [34], [14] and the references therein. As a consequence of the work of many people, among them, Vlamimirov, Volovich, Zelenov, Avetisov, Kozyrev, Kochubei, Khrennikov, Albeverio, and Zúñiga-Galindo, we have now a good theory of p-adic 'linear' reaction-diffusion equations which has emerged motivated by connections between p-adic analysis and models of complex systems.…”

Section: Introductionmentioning

confidence: 99%

“…As a consequence of the work of many people, among them, Vlamimirov, Volovich, Zelenov, Avetisov, Kozyrev, Kochubei, Khrennikov, Albeverio, and Zúñiga-Galindo, we have now a good theory of p-adic 'linear' reaction-diffusion equations which has emerged motivated by connections between p-adic analysis and models of complex systems. For further details the reader may consult [38], [23], see also [34], as well as the classics, [36], [22].…”

Section: Introductionmentioning

confidence: 99%

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Non-Archimedean reaction-ultradiffusion equations and complex hierarchic systems

Zúñiga–Galindo

2018

Nonlinearity

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We initiate the study of non-Archimedean reaction-ultradiffusion equations and their connections with models of complex hierarchic systems. From a mathematical perspective, the equations studied here are the p-adic counterpart of the integro-differential models for phase separation introduced by Bates and Chmaj. Our equations are also generalizations of the ultradiffusion equations on trees studied in the 80s by Ogielski, Stein, Bachas, Huberman, among others, and also generalizations of the master equations of the Avetisov et al. models, which describe certain complex hierarchic systems. From a physical perspective, our equations are gradient flows of non-Archimedean free energy functionals and their solutions describe the macroscopic density profile of a bistable material whose space of states has an ultrametric structure. Some of our results are p-adic analogs of some well-known results in the Archimedean setting, however, the mechanism of diffusion is completely different due to the fact that it occurs in an ultrametric space.2000 Mathematics Subject Classification. Primary 80A22, 45K05; Secondary 46S10.

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Section: Introductionmentioning

confidence: 99%