A Non-Archimedean Wave Equation

Let K be a non-Archimedean local field with the normalized absolute value $|\cdot |$. It is shown that a ``plane wave'' $f(t+\omega_1 x_1+... +\omega_nx_n)$, where f is a Bruhat-Schwartz complex-valued test function on K, $(t,x_1,..., x_n)\in K^{n+1}$, $\max\limits_{1\le j\le n}|\omega_j|=1$, satisfies, for any f, a certain homogeneous pseudo-differential equation, an analog of the classical wave equation. A theory of the Cauchy problem for this equation is developed.Comment: 17 pages; the final version, to appear in Pacif. J. Mat

Non-Archimedean Group Algebras with Baer Reductions

Within the concept of a non-Archimedean operator algebra with the Baer reduction (A. N. Kochubei, On some classes of non-Archimedean operator algebras, Contemporary Math. 596 (2013), 133--148), we consider algebras of operators on Banach spaces over non-Archimedean fields generated by regular representations of discrete groups.Comment: Final version, to appear in Algebras and Representation Theor

Evolution Equations and Functions of Hypergeometric Type over Fields of Positive Characteristic

We consider a class of partial differential equations with Carlitz derivatives over a local field of positive characteristic, for which an analog of the Cauchy problem is well-posed. Equations of such type correspond to quasi-holonomic modules over the ring of differential operators with Carlitz derivatives. The above class of equations includes some equations of hypergeometric type. Building on the work of Thakur, we develop his notion of the hypergeometric function of the first kind (whose parameters belonged initially to $\mathbb Z$) in such a way that it becomes fully an object of the function field arithmetic, with the variable, parameters and values from the field of positive characteristic

Fractional-Hyperbolic Systems

We describe a class of evolution systems of linear partial differential equations with the Caputo-Dzhrbashyan fractional derivative of order $\alpha \in (0,1)$ in the time variable $t$ and the first order derivatives in spatial variables $x=(x_1,...,x_n)$, which can be considered as a fractional analogue of the class of hyperbolic systems. For such systems, we construct a fundamental solution of the Cauchy problem having exponential decay outside the fractional light cone $\{(t,x):\ |t^{-\alpha}x|\le 1\}$.Comment: Final version. available at http://link.springer.com/journal/1354

Dwork-Carlitz Exponential and Overconvergence for Additive Functions in Positive Characteristic

We study overconvergence phenomena for $\mathbb F_q$-linear functions on a function field over a finite field $\mathbb F_q$. In particular, an analog of the Dwork exponential is introduced

Hausdorff Measure for a Stable-Like Process over an Infinite Extension of a Local Field

We consider an infinite extension $K$ of a local field of zero characteristic which is a union of an increasing sequence of finite extensions. $K$ is equipped with an inductive limit topology; its conjugate $\bar{K}$ is a completion of $K$ with respect to a topology given by certain explicitly written seminorms. The semigroup of measures, which defines a stable-like process $X(t)$ on $\bar{K}$, is concentrated on a compact subgroup $S\subset \bar{K}$. We study properties of the process $X_S(t)$, a part of $X(t)$ in $S$. It is shown that the Hausdorff and packing dimensions of the image of an interval equal 0 almost surely. In the case of tamely ramified extensions a correct Hausdorff measure for this set is found.Comment: The final version, to appear in Journal of Theoretical Probabilit

Polylogarithms and a Zeta Function for Finite Places of a Function Field

We introduce and study new versions of polylogarithms and a zeta function on a completion of $\mathbb F_q (x)$ at a finite place. The construction is based on the use of the Carlitz differential equations for $\mathbb F_q$-linear functions.Comment: 15 pages, LaTeX-2

General Fractional Calculus, Evolution Equations, and Renewal Processes

We develop a kind of fractional calculus and theory of relaxation and diffusion equations associated with operators in the time variable, of the form $(Du)(t)=\frac{d}{dt}\int\limits_0^tk(t-\tau)u(\tau)\,d\tau -k(t)u(0)$ where $k$ is a nonnegative locally integrable function. Our results are based on the theory of complete Bernstein functions. The solution of the Cauchy problem for the relaxation equation $Du=-\lambda u$, $\lambda >0$, proved to be (under some conditions upon $k$) continuous on $[(0,\infty)$ and completely monotone, appears in the description by Meerschaert, Nane, and Vellaisamy of the process $N(E(t))$ as a renewal process. Here $N(t)$ is the Poisson process of intensity $\lambda$, $E(t)$ is an inverse subordinator.Comment: To appear in Integral Equations and Operator Theor

Analysis and Probability over Infinite Extensions of a Local Field

We consider an infinite extension $K$ of a local field of zero characteristic which is a union of an increasing sequence of finite extensions. $K$ is equipped with an inductive limit topology; its conjugate $\bar{K}$ is a completion of $K$ with respect to a topology given by certain explicitly written seminorms. We construct and study a Gaussian measure, a Fourier transform, a fractional differentiation operator and a cadlag Markov process on $\bar{K}$. If we deal with Galois extensions then all these objects are Galois-invariant.Comment: 24 pages, LaTex; to appear in Potential Analysi

Analysis and Probability over Infinite Extensions of a Local Field, II: A Multiplicative Theory

Let $V$ be a projective limit, with respect to the renormalized norm mappings, of the groups of principal units corresponding to a strictly increasing sequence of finite separable totally and tamely ramified Galois extensions of a local field. We study the structure of the dual group $V'$, introduce and investigate a fractional differentiation operator on $V$, and the corresponding L\'evy process. Part I: Potential Anal., 10 (1999), 305-325.Comment: 11 pages, LaTe