arXiv subjects

David Weisbart

Publications and source records attributed to David Weisbart.

14 recordsLinked to original sources

arXiv2026

A compositional framework for open classical kinematic systems

Our aim is to introduce a framework sufficiently general to describe the kinematics of a wide variety of open systems in classical mechanics while uniquely characterizing systems with specified simplest components. The data describing a physical system are local, so the construction of a global configuration space requires compatibility among local interactions. We model open systems as morphisms in a category $\mathsf{Kin}(\mathcal{F})$, where composition encodes how subsystems attach to one another and embed into larger systems. The framework supports a precise treatment of geometric constraints and clarifies when locally specified subsystems are compatible. It also yields a structural approach to the study of linkages: we prove the nonconstructibility of sliding hinges and universal joints from two rigid bodies attached by a surface constraint compatible with rigid-motion symmetries.

math-ph↗

arXiv2025

Geometry-induced criticality in $p$-adic scaling limits of random walks

An anisotropy parameter $h$ in $(0,1]$ induces on $\mathbb{Q}_p^2$ a duality-compatible, two-scale filtration that collapses to one scale at the right endpoint. This filtration defines shell-uniform transition laws for hierarchical random walks on a discrete group whose scaling limits are L\'{e}vy processes on $\mathbb{Q}_p^2$. The diffusion constants of the coordinate processes jump at the right endpoint, even though the radial jump law depends continuously on $h$. This instance of geometry-induced criticality isolates a structural mechanism that should extend to locally compact abelian groups and suggests a route to studying critical behavior in ultrametric models.

math.PR↗

arXiv2024

Brownian Motion in the $p$-Adic Integers is a Limit of Discrete Time Random Walks

Vladimirov defined an operator on balls in $\mathbb Q_p$, the $p$-adic numbers, that is analogous to the Laplace operator in the real setting. Kochubei later provided a probabilistic interpretation of the operator. This Vladimirov-Kochubei operator generates a real-time diffusion process in the ring of $p$-adic integers, a Brownian motion in $\mathbb Z_p$. The current work shows that this process is a limit of discrete time random walks. It motivates the construction of the Vladimirov-Kochubei operator, provides further intuition about the properties of ultrametric diffusion, and gives an example of the weak convergence of stochastic processes in a profinite group.

math.PR↗

arXiv2024

Brownian Motion in a Vector Space over a Local Field is a Scaling Limit

For any natural number $d$, the Vladimirov-Taibleson operator is a natural analogue of the Laplace operator for complex-valued functions on a $d$-dimensional vector space $V$ over a local field $K$. Just as the Laplace operator on $L^2(\mathbb R^d)$ is the infinitesimal generator of Brownian motion with state space $\mathbb R^d$, the Vladimirov-Taibleson operator on $L^2(V)$ is the infinitesimal generator of real-time Brownian motion with state space $V$. This study deepens the formal analogy between the two types of diffusion processes by demonstrating that both are scaling limits of discrete-time random walks on a discrete group. It generalizes the earlier works, which restricted $V$ to be the $p$-adic numbers.

math.PR↗

arXiv2022

Components and Exit Times of Brownian Motion in two or more $p$-Adic Dimensions

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 original process. Asymptotic analysis of the conditional probabilities shows that the vector components are dependent for all time. Exit time probabilities for the higher dimensional processes reveal a concrete effect of the component dependency.

math.PR↗

arXiv2020

$p$-Adic Brownian Motion is a Scaling Limit

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 $(1, \infty)$, and the convergence was the weak convergence of probability measures on the Skorohod space of paths on a compact time interval. The current approach simplifies the earlier approach, allows for any positive exponent, eliminates the restriction to compact time intervals, and establishes some moment estimates for the discrete time processes that are of independent interest.

math.PR↗

arXiv2020

Modernizing Archimedes' Construction of $\pi$

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 involve ideas that are not elementary. This paper makes Archimedes' measurement procedure rigorous from a modern perspective. In so doing, it brings a rigorous and geometric treatment of the differential properties of the trigonometric functions into the purview of an introductory calculus course.

math.HO↗

arXiv2020

Buffon's Problem determines Gaussian Curvature in three Geometries

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 between Gaussian curvature and both circumference and area deficits.

math.PR↗

arXiv2020

On Infinitesimal Generators and Feynman-Kac Integrals of Adelic Diffusion

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 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, $\sigma$, of the diffusion constants is finite. Finiteness of $\sigma$ implies that there is an adelic Vladimirov operator, $\Delta_{\mathbb A}$, and an associated diffusion equation whose fundamental solution gives rise to the measure induced by $P$ on an adelic Skorokhod space. For a wide class of potentials, the dynamical semigroups associated to adelic Schr\"{o}dinger operators with free part $\Delta_{\mathbb A}$ have path integral representations.

math.PR↗

arXiv2020

Constructing Span Categories From Categories Without Pullbacks

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 $\mathscr{C}^\prime$ and a functor $\mathcal F$ from $\mathscr{C}$ to $\mathscr{C}^\prime$, we introduce the notion of an $\mathcal F$ pullback of a cospan in $\mathscr{C}$, as well as the notion of span tightness of $\mathcal F$. If $\mathcal F$ is span tight, then we can form a generalized span category ${\rm Span}(\mathscr{C},\mathcal F)$ and circumvent the technical difficulty of $\mathscr{C}$ failing to have pullbacks. Composition in ${\rm Span}(\mathscr{C},\mathcal F)$ uses $\mathcal F$-pullbacks rather than pullbacks and in this way differs from the category ${\rm Span}(\mathscr{C})$, but reduces to it when both $\mathscr{C}$ has pullbacks and $\mathcal F$ is the identity functor.

math.CT↗

arXiv2020

Estimates of Certain Exit Probabilities for $p$-Adic Brownian Bridges

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 calculate the first exit probabilities of paths from balls and estimate these probabilities for the brownian bridges.

math.PR↗

arXiv2017

Open Systems in Classical Mechanics

Generalized span categories provide a framework for formalizing mathematical models of open systems in classical mechanics. We introduce categories $\mathsf{LagSy}$ and $\mathsf{HamSy}$ that respectively provide a categorical framework for the Lagrangian and Hamiltonian descriptions of open classical mechanical systems. The morphisms of $\mathsf{LagSy}$ and $\mathsf{HamSy}$ correspond to such open systems, and composition of morphisms models the construction of systems from subsystems. The Legendre transformation gives rise to a functor from $\mathsf{LagSy}$ to $\mathsf{HamSy}$ that translates from the Lagrangian to the Hamiltonian perspective.

math-ph↗

arXiv2016

Brownian Motion and Finite Approximations of Quantum Systems over Local Fields

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 obtained as a limit of random walks over finite grids. Also, we prove a Feynman-Kac formula for the finite systems, and show that the propagator at the finite level converges to the propagator at the infinite level.

math-ph↗

arXiv2009

Airy functions over local fields

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 numbers. We prove that the p-adic Airy integrals are locally constant functions of moderate growth and present evidence that the Airy integrals associated to compact p-adic Lie groups also have these properties.

math-ph↗