Abstract. In this article we introduce a new type of nonlocal operators and study the Cauchy problem for certain parabolic-type pseudodifferential equations naturally associated to these operators. Some of these equations are the p-adic master equations of certain models of complex systems introduced by Avetisov et al. The fundamental solutions of these parabolic-type equations are transition functions of random walks on the n-dimensional vector space over the field of p-adic numbers. We study some properties of these random walks, including the first passage time.
IntroductionDuring the last twenty-five years there have been a strong interest on random walks on ultrametric spaces mainly due its connections with models of complex systems, such as glasses and proteins. Random walks on ultrametric spaces are very convenient for describing phenomena whose space of states display a hierarchical structure, see e.g. . These models can be applied, among other things, to the study the relaxation of biological complex systems [5]. From a mathematical point view, in these models the time-evolution of a complex system is described by a p-adic master equation (a parabolic-type pseudodifferential equation) which controls the time-evolution of a transition function of a random walk on an ultrametric space, and the random walk describes the dynamics of the system in the space of configurational states which is approximated by an ultrametric space (Q p ).This article continues and extends some of the mathematical results given in [5], [3]. We introduced a new class of nonlocal operators which includes the Vladimirov operator. These operators are determined by a radial function which determines the structure of the energy landscape being used. In [5] several solvable models for degenerated landscapes of types linear, logarithmic and exponential were studied. In this article we study a large class of solvable models, we have called them polynomial and exponential landscapes, which includes the linear and exponential landscapes considered in [5], see Section 3.2. We attach to each of these operators 2000 Mathematics Subject Classification. Primary 82B41, 82C44; Secondary 26E30.
In this article, we introduce a new class of parabolic-type pseudo differential equations with variable coefficients over the p-adics. We establish the existence and uniqueness of solutions for the Cauchy problem associated with these equations. The fundamental solutions of these equations are connected with Markov processes. Some of these equations are related to new models of complex systems.2000 Mathematics Subject Classification. Primary 35K90, 60J25; Secondary 26E30.
In this article we initiate the study of the heat traces and spectral zeta functions for certain p-adic Laplacians. We show that the heat traces are given by p-adic integrals of Laplace type, and that the spectral zeta functions are p-adic integrals of Igusa-type. We find good estimates for the behaviour of the heat traces when the time tends to infinity, and for the asymptotics of the function counting the eigenvalues less than or equal to a given quantity.
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