Non-Archimedean Pseudodifferential Operators and Feller Semigroups

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

doi:10.1134/s2070046618010041

Cited by 22 publications

(13 citation statements)

References 24 publications

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“…where [37], [40,Chapter 2] and the references therein. We pick the Taibleson operator due to the fact that the corresponding fundamental solutions are well-known.…”

Section: The Taibleson Operatormentioning

confidence: 99%

Non-Archimedean Coulomb Gases

Torba¹,

Zúñiga–Galindo²

2019

Preprint

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This article aims to study the Coulomb gas model over the d-dimensional p-adic space. We establish the existence of equilibria measures and the Γ-limit for the Coulomb energy functional when the number of configurations tends to infinity. For a cloud of charged particles confined into the unit ball, we compute the equilibrium measure and the minimum of its Coulomb energy functional. In the p-adic setting the Coulomb energy is the continuum limit of the minus a hierarchical Hamiltonian attached to a spin glass model with a p-adic coupling.

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“…where [37], [40,Chapter 2] and the references therein. We pick the Taibleson operator due to the fact that the corresponding fundamental solutions are well-known.…”

Section: The Taibleson Operatormentioning

confidence: 99%

Non-Archimedean Coulomb Gases

Torba¹,

Zúñiga–Galindo²

2019

Preprint

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“…The study of p-adic heat equations and the attached Markov processes is a relevant mathematical matter [5], [9], [11], [25], [36], [37], [39], [42], [43], [44], and the references therein. The study of p-adic heat equations on p-adic manifolds is an open research area.…”

Section: Introductionmentioning

confidence: 99%

Ultrametric Diffusion, Rugged Energy Landscapes and Transition Networks

Zúñiga-Galindo

2022

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In this article we introduce the ultrametric networks which are p-adic continuous analogues of the standard Markov state models constructed using master equations. A p-adic transition network (or an ultrametric network) is a model of a complex system consisting of a hierarchical energy landscape, a Markov process on the energy landscape, and a master equation. The energy landscape consists of a finite number of basins. Each basin is formed by infinitely many network configurations organized hierarchically in a infinite regular tree. The transitions between the basins are determined by a transition density matrix, whose entries are functions defined on the energy landscape. The Markov process in the energy landscape encodes the temporal evolution of the network as random transitions between configurations from the energy landscape. The master equation describe the time evolution of the density of the configurations. We focus on networks where the transition rates between two different basins are constant functions, and the jumping process inside of each basin is controlled by a p-adic radial function. We solve explicitly the Cauchy problem for the master equation attached to this type of networks. The solution of this problem is the network response to a given initial concentration. If the Markov process attached to the network is conservative, the long term response of the network is controlled by Markov chain, If the process is not conservative the network has absorbing states. We define an absorbing time, which depends on the initial concentration, if this time is finite the network reaches an absorbing state in a finite time. We identify in the response, the terms that produced this fast response, we call them, the fast transition modes. The existence of the fast transition modes is a consequence of assumption that the energy landscape is ultrametric (hierarchical), and to the best our understanding this result cannot be obtained by using standard of Markov state models. Nowadays, it is widely accepted that protein native states are kinetic hubs that can be reached quickly from any other state. The existence of fast transition modes implies that certain states on an ultrametric network work as kinetic hubs.

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“…[1], [2], [3], [6], [7], [10], [13], [14], [15], [19], [21], and the references therein. Particularly, pseudo-differential operators whose symbols are associated with negative definite functions on the padic numbers, see [8], [9], [17], [18]. In this article, we introduce a new class of non-archimedean pseudo-differential operators (called the generalized Bessel potentials) associated with negative definite functions in the p-adic context and in arbitrary dimension.…”

Section: Introductionmentioning

confidence: 99%

“…The functions ψ 1 , ψ 2 : Q n p → C are continuous, negative definite and radials. Taking, ψ 1 = 1 and ψ 2 = || • || p , then by [4,] and [17,Example 3.4] we have that A α correspond to the Bessel potential studied at [9] and [16]. We are interested in the pseudo-differential operators A α due to our interest for contemporary physical theories, in particular, study new Cauchy problems (or padic heat equations), which governs the temperature distribution in an object over time.…”

Section: Introductionmentioning

confidence: 99%

Non-archimedean generalized Bessel potentials and their applications

Torresblanca-Badillo¹

2020

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This article describes a class of pseudo-differential operatorsis the symbol of the operator A α . These operators can be seen as a generalization of the Bessel potentials in the p-adic context. We show that the family (Kα) α>0 of convolution kernels attached to generalized Bessel potentials A α , α > 0, determine a convolution semigroup on Q n p . Imposing certain conditions we have that Kα, α > 0, is a probability measure on Q n p . Moreover, we will study certain properties corresponding to the Green function of the operator A α and we show that heat equations, naturally associated to these operators, describes the cooling (or loss of heat) in a given region over time.

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Non-Archimedean Pseudodifferential Operators and Feller Semigroups

Cited by 22 publications

References 24 publications

Non-Archimedean Coulomb Gases

Non-Archimedean Coulomb Gases

Ultrametric Diffusion, Rugged Energy Landscapes and Transition Networks

Non-archimedean generalized Bessel potentials and their applications

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