Monotonicity Property of Non-Archimedean Heat Equation
We prove the monotone dependence of the heat kernel corresponding to the Vladimirov-Taibleson non-Archimedean fractional differentiation operator $D^α$ on the order $α$.
math.AP↗
arXiv subjects
Anatoly N. Kochubei
Publications and source records attributed to Anatoly N. Kochubei.
A Pseudo-Differential Operator of the $p$-Adic Multiplicative Calculus
We introduce and study a pseudo-differential operator $W^α$, defined via the $p$-adic Mellin transformation, acting on real- or complex-valued functions on the multiplicative group of the field of $p$-adic numbers. The operator $W^α$ is maximally dissipative. An estimate of its heat kernel is given.
math.AP↗
Non-Archimedean Kelvin Transformation
We introduce and study an analog of the Kelvin transformation connected with the Vladimirov-Taibleson operator acting on real- or complex-valued functions on a space $K^n$ over a non-Archimedean local field $K$.
math.NT↗
Non-Archimedean Neumann problem: weak and strong solutions
We consider the Neumann problem for the equation with the Vladimirov-Taibleson fractional differentiation operator over a non-Archimedean local field. We study weak solutions following the method by Dipierro, Ros-Oton and Valdinoci (2017). Our investigation of strong solutions is based on the ultrametric identities for the operator under consideration.
math.AP↗
Fundamental Solution for a New Class of Non-Archimedean Pseudo-Differential Equations
In this paper we study a new class of pseudo-differential equations on functions of two $p$-adic variables. It is proved that the correspondent Cauchy problem has a unique solution. Some properties of this solution are studied, in particular, the finite dependence property and an $L^1$-estimate.
math.AP↗
The Vladimirov-Taibleson Operator: Inequalities, Dirichlet Problem, Boundary Hölder Regularity
We study the Vladimirov-Taibleson operator, a model example of a pseudo-differential operator acting on real- or complex-valued functions defined on a non-Archimedean local field. We prove analogs of classical inequalities for fractional Laplacian, study the counterpart of the Dirichlet problem including the property of boundary Hölder regularity of solutions.
math.AP↗
Pseudo-Differential Equations with Weak Degeneration for Radial Functions of $p$-adic Argument
In earlier papers (A. N. Kochubei, Pacif. J. Math., 269 (2014), 355-369; J. Math. Anal. Appl.483 (2020), Article 123609), one of the authors developed a theory of pseudo-differential equations for radial real-valued functions on a non-Archimedean local field, with some features resembling those of classical ordinary differential equations. Here we consider equations of this kind, but with a weak degeneration. Under various assumptions, we prove the local existence and uniqueness of mild solutions, existence of their global extensions and a regularity property.
math.CA↗
$L^p$ properties of non-Archimedean fractional differentiation operators
Let $D^α, α>0$, be the Vladimirov-Taibleson fractional differentiation operator acting on complex-valued functions on a non-Archimedean local field. The identity $D^αD^{-α}f=f$ was known only for the case where $f$ has a compact support. Following a result by Samko about the fractional Laplacian of real analysis, we extend the above identity in terms of $L^p$-convergence of truncated integrals. Differences between real and non-Archimedean cases are discussed.
math.FA↗
Non-Archimedean Radial Calculus: Volterra Operator and Laplace Transform
In an earlier paper (A. N. Kochubei, {\it Pacif. J. Math.} 269 (2014), 355--369), the author considered a restriction of Vladimirov's fractional differentiation operator $D^α$, $α>0$, to radial functions on a non-Archimedean field. In particular, it was found to possess such a right inverse $I^α$ that the appropriate change of variables reduces equations with $D^α$ (for radial functions) to integral equations whose properties resemble those of classical Volterra equations. In other words, we found, in the framework of non-Archimedean pseudo-differential operators, a counterpart of ordinary differential equations. In the present paper, we begin an operator-theoretic investigation of the operator $I^α$, and study a related analog of the Laplace transform.
math.FA↗
Representations of the Infinite-Dimensional $p$-Adic Affine Group
We introduce an infinite-dimensional $p$-adic affine group and construct its irreducible unitary representation. Our approach follows the one used by Vershik, Gelfand and Graev for the diffeomorphism group, but with modifications made necessary by the fact that the group does not act on the phase space. However it is possible to define its action on some classes of functions.
math.RT↗
Ground States for Nonlocal Schrödinger Type Operators on Locally Compact Abelian Groups
We find classes of nonlocal operators of Schrödinger type on a locally compact noncompact Abelian group $G$, for which there exists a ground state. In particular, such a result is obtained for the case where the principal part of our operator generates a recurrent random walk. Explicit conditions for the existence of a ground state are obtained for the case $G =\mathbb Q_p^n$ where $\mathbb Q_p$ is the field of $p$-adic numbers.
math.FA↗
Multidimensional nonlinear pseudo-differential evolution equation with p-adic spatial variables
We study the Cauchy problem for $p$-adic nonlinear evolutionary pseudo-differential equations for complex-valued functions of a real positive time variable and p-adic spatial variables. Among the equations under consideration there is the p-adic analog of the porous medium equation (or more generally, the nonlinear filtration equation) which arise in numerous application in mathematical physics and mathematical biology. Our approach is based on the construction of a linear Markov semigroup on a p-adic ball and the proof of m-accretivity of the appropriate nonlinear operator. The latter result is equivalent to the existence and uniqueness of a mild solution of the Cauchy problem of a nonlinear equation of the porous medium type.
math.AP↗
Random Time Change and Related Evolution Equations: Time Asymptotic Behavior
In this paper we investigate the long time behavior of solutions to fractional in time evolution equations which appear as results of random time changes in Markov processes. We consider inverse subordinators as random times and use the subordination principle for the solutions to forward Kolmogorov equations. The class of subordinators for which asymptotic analysis may be realized is described.
math.PR↗
Nonlinear Pseudo-Differential Equations for Radial Real Functions on a Non-Archimedean Field
In an earlier paper (A. N. Kochubei, {\it Pacif. J. Math.} 269 (2014), 355--369), the author considered a restriction of Vladimirov's fractional differentiation operator $D^α$, $α>0$, to radial functions on a non-Archimedean field. In particular, it was found to possess such a right inverse $I^α$ that the change of an unknown function $u=I^αv$ reduces the Cauchy problem for a linear equation with $D^α$ (for radial functions) to an integral equation whose properties resemble those of classical Volterra equations. In other words, we found, in the framework of non-Archimedean pseudo-differential operators, a counterpart of ordinary differential equations. In the present paper, we study nonlinear equations of this kind, find conditions of their local and global solvability.
math.CA↗
Growth Equation of the General Fractional Calculus
We consider the Cauchy problem $(\mathbb D_{(k)} u)(t)=λu(t)$, $u(0)=1$, where $\mathbb D_{(k)}$ is the general convolutional derivative introduced in the paper (A. N. Kochubei, Integral Equations Oper. Theory {\bf 71} (2011), 583--600), $λ>0$. The solution is a generalization of the function $t\mapsto E_α(λt^α)$ where $0<α<1$, $E_α$ is the Mittag-Leffler function. The asymptotics of this solution, as $t\to \infty$, is studied.
math.CA↗
Heat Kernel Estimates for Fractional Heat Equation
We study the long-time behavior of the Cesaro means of fundamental solutions for fractional evolution equations corresponding to random time changes in the Brownian motion and other Markov processes. We consider both stable subordinators leading to equations with the Caputo-Djrbashian fractional derivative and more general cases corresponding to differential-convolution operators, in particular, distributed order derivatives.
math-ph↗
From Random Times to Fractional Kinetics
In this paper we study the effect of the subordination by a general random time-change to the solution of a model on spatial ecology in terms of its evolution density. In particular on traveling waves for a non-local spatial logistic equation. We study the Cesaro limit of the subordinated dynamics in a number of particular cases related to the considered fractional derivative making use of the Karamata-Tauberian theorem.
math.PR↗
On the p-Adic Navier-Stokes Equation
We prove the local solvability of the p-adic analog of the Navier-Stokes equation. This equation describes, within the p-adic model of porous medium, the flow of a fluid in capillaries.
math-ph↗