Path Diffusion, Part I

TL;DR

This paper explores the distribution of position in a binomial process with a Markovian velocity process, transforming it into hyperbolic PDEs like the Telegraph and Klein-Gordon equations, with applications to Newtonian mechanics.

Contribution

It introduces a novel approach modeling velocity as a Markov process, linking discrete stochastic models to hyperbolic PDEs and Newtonian dynamics.

Findings

Probability converges to hyperbolic functions for small grids.

The process can be transformed into Telegraph and Klein-Gordon equations.

Numerical examples illustrate the connection to Newton's laws.

Abstract

This paper investigates the position (state) distribution of the single step binomial (multi-nomial) process on a discrete state / time grid under the assumption that the velocity process rather than the state process is Markovian. In this model the particle follows a simple multi-step process in velocity space which also preserves the proper state equation of motion. Many numerical numerical examples of this process are provided. For a smaller grid the probability construction converges into a correlated set of probabilities of hyperbolic functions for each velocity at each state point. It is shown that the two dimensional process can be transformed into a Telegraph equation and via transformation into a Klein-Gordon equation if the transition rates are constant. In the last Section there is an example of multi-dimensional hyperbolic partial differential equation whose numerical average satisfies Newton's equation. There is also a momentum measure provided both for the two-dimensional case as for the multi-dimensional rate matrix.