David Weisbart

University of California, Riverside | UCR · Department of Mathematics

A p-adic Brownian motion is a continuous time stochastic process in a p-adic state space that has a Vladimirov operator as its infinitesimal generator. The current work shows that any such process is the scaling limit of a discrete time random walk on a discrete group. Earlier work required the exponent of the Vladimirov operator to be in (1,∞) , a...

A version of the classical Buffon problem in the plane naturally extends to the setting of any Riemannian surface with constant Gaussian curvature. The Buffon probability determines a Buffon deficit. The relationship between Gaussian curvature and the Buffon deficit is similar to the relationship that the Bertrand–Diguet–Puiseux theorem establishes...

The fundamental solution to a pseudo-differential equation for functions defined on the d-fold product of the p-adic numbers, Qp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{doc...

The fundamental solution of a pseudo-differential equation for functions defined on the $d$-fold product of the $p$-adic numbers, $\mathbb{Q}_p$, induces an analogue of the Wiener process in $\mathbb{Q}_p^d$. As in the real setting, the components are $1$-dimensional $p$-adic Brownian motions with the same diffusion constant and exponent as the ori...

For each prime p, a Vladimirov operator with a positive exponent specifies a p-adic diffusion equation and a measure on the Skorokhod space of p-adic paths. The product, P, of these measures with a fixed exponent is a probability measure on the product of the p-adic path spaces. The adelic paths have full measure if and only if the sum, σ, of the d...

For each prime p, a diffusion constant together with a positive exponent specify a Vladimirov operator and an associated p-adic diffusion equation. The fundamental solution of this pseudo-differential equation gives rise to a measure on the Skorokhod space of p-adic valued paths that is concentrated on the paths originating at the origin. We calcul...

Generalized span categories provide a framework for formalizing mathematical models of open systems in classical mechanics. We introduce categories LagSy and HamSy that, respectively, provide a categorical framework for the Lagrangian and Hamiltonian descriptions of open classical mechanical systems. The morphisms of LagSy and HamSy correspond to s...

In his famous work, “Measurement of a Circle,” Archimedes described a procedure for measuring both the circumference of a circle and the area it bounds. Implicit in his work is the idea that his procedure defines these quantities. Modern approaches for defining π eschew his method and instead use arguments that are easier to justify, but they invol...

We prove that $p$-adic Brownian motion is a scaling limit of an unscaled discrete time random walk. Our proof extends the classical result in the real setting to any Brownian motion that has a Vladimirov operator as its infinitesimal generator.

In his famous work, "Measurement of a Circle," Archimedes described a procedure for measuring both the circumference of a circle and the area it bounds. Implicit in his work is the idea that his procedure defines these quantities. Modern approaches for defining $\pi$ eschew his method and instead use arguments that are easier to justify, but they i...

The classical Buffon problem requires a precise presentation in order to be meaningful. We reinterpret the classical problem in the planar setting with a needle whose length is equal to the grating width and find analogs of this problem in the settings of the sphere and the Poincar\'e disk. We show that the probability that the needle intersects th...

Span categories provide an abstract framework for formalizing mathematical models of certain systems. The mathematical descriptions of some systems, such as classical mechanical systems, require categories that do not have pullbacks and this limits the utility of span categories as a formal framework. Given categories $\mathscr{C}$ and $\mathscr{C}...

A prime $p$, an exponent, and a diffusion constant together specify a $p$-adic diffusion equation and a measure on the Skorokhod space of $p$-adic valued paths. The product, $P$, taken over the prime numbers of these measures with a fixed exponent is a probability measure on the product of the $p$-adic path spaces. Bounds on the exit probabilities...

For each prime $p$, a diffusion constant together with a positive exponent specify a Vladimirov operator and an associated $p$-adic diffusion equation. The fundamental solution of this pseudo-differential equation gives rise to a measure on the Skorokhod space of $p$-adic valued paths that is concentrated on the paths originating at the origin. We...

The p-adic diffusion equation is a pseudo differential equation that is formally analogous to the real diffusion equation. The fundamental solutions to pseudo differential equations that generalize the p-adic diffusion equation give rise to p-adic Brownian motions. We show that these stochastic processes are similar to real Brownian motion in that...

The fundamental solutions to a large class of pseudo-differential equations that generalize the formal analogy of the diffusion equation in \(\mathbb {R}\) to the groups \(p^{-n}\mathbb {Z}_p/p^{n} \mathbb {Z}_p\) give rise to probability measures on the space of Skorokhod paths on these finite groups. These measures induce probability measures on...

We give a stochastic proof of the finite approximability of a class of Schr\"odinger operators over a local field, thereby completing a program of establishing in a non-Archimedean setting corresponding results and methods from the Archimedean (real) setting. A key ingredient of our proof is to show that Brownian motion over a local field can be o...

We investigate the asymptotic growth of the canonical measures on the fibers of morphisms between vector spaces over local fields of arbitrary characteristic. For non-archimedean local fields we use a version of the {\L}ojasiewicz inequality (\cite{lojasiewicz1959}, \cite{hormander1958division}) which follows from Greenberg \cite{greenberg1966ratio...

We extend the method of L. Schwartz [1] to classify elementary scalar particles in p-adic space time. Schwartz obtained the states of the elementary particles over real spacetime as tempered distributions on spacetime itself. We obtain the analogous description in p-adic spacetime. We introduce a natural notion of temperedness similar to one introd...

We show that the p-adic Schrödinger operator, as defined in [7], can be approximated in a very strong sense by finite Schröinger operators.

We consider quantum systems that have as their configuration spaces finite dimensional vector spaces over local fields. The quantum Hilbert space is taken to be a space with complex coefficients and we include in our model particles with internal symmetry. The Hamiltonian operator is a pseudo-differential operator that is initially only formally de...

Airy integrals are very classical but in recent years they have been generalized to higher dimensions and these generalizations have proved to be very useful in studying the topology of the moduli spaces of curves. We study a natural generalization of these integrals when the ground field is a non-archimedean local field such as the field of p-adic...

We consider quantum systems that have as their configuration spaces finite-dimensional vector spaces over local fields. The quantum Hilbert space is taken to be a space with complex coefficients and we include in our model particles with internal symmetry. The Hamiltonian operator is a pseudo-differential operator that is initially only formally de...

We discuss in this article some basic features of quantum systems whose configuration space is the ring of rational adeles. We also propose a probabilistic method for specifying the free hamiltonian operators in a quantum dynamical theory over such a configuration space and introduce a natural class of free adelic hamiltonians that generate a measu...

We discuss here the convergence of quantum systems on grids embedded in Rd and generalize the earlier results found for scalar-valued potentials to the case of matrix-valued potentials. We also discuss the essential self-adjointness of Schrödinger operators for a large class of matrix potentials and give a Feynman–Kac formula for their associated i...