Taibleson operators,p-adic parabolic equations and ultrametric diffusion

Parabolic Equations and Markov Processes Over p−adic Fields

In this paper we construct and study a fundamental solution of Cauchy’s problem for p−adic parabolic equations of the type ∂u (x, t) ∂t + (f (D, β)u) (x, t) = 0, x ∈ Qnp , n ≥ 1, t ∈ (0, T ] , where f (D, β), β > 0, is an elliptic pseudo-differential operator. We also show that the fundamental solution is the transition density of a Markov process with state space Qp .

, interbasin kinetics and ultrametric diffusion

We discuss the interbasin kinetics approximation for random walk on a complex landscape. We show that for a generic landscape the corresponding model of interbasin kinetics is equivalent to an ultrametric diffusion, generated by an ultrametric pseudodifferential operator on the ultrametric space related to the tree of basins. The simplest example of ultrametric diffusion of this kind is describ...

Non Degenerate Ultrametric Diffusion

General non-degenerate p-adic operators of ultrametric diffusion are introduced. Bases of eigenvectors for the introduced operators are constructed and the corresponding eigenvalues are computed. Properties of the corresponding dynamics (i.e. of the ultrametric diffusion) are investigated.

The balance between diffusion and absorption in semilinear parabolic equations

Let h : [0,∞) 7→ [0,∞) be continuous and nondecreasing, h(t) > 0 if t > 0, and m,q be positive real numbers. We investigate the behavior when k → ∞ of the fundamental solutions u = uk of ∂tu − ∆u m + h(t)u = 0 in Ω × (0, T ) satisfying uk(x, 0) = kδ0. The main question is wether the limit is still a solution of the above equation with an isolated singularity at (0, 0), or a solution of the asso...

Nonlinear Elliptic and Parabolic Equations with Fractional Diffusion