On the causality of real-valued semigroups and diffusion

TL;DR

This paper demonstrates that classical diffusion models cannot be causal due to their semigroup properties, and introduces a discrete-time, non-smooth causal diffusion model that satisfies a wave equation with a time-dependent coefficient.

Contribution

It presents a causal diffusion model with discrete time points, contrasting classical models, and analyzes its properties and relation to wave equations.

Findings

Classical diffusion models cannot be causal due to semigroup properties.

A new causal diffusion model with discrete time points is proposed.

Diffusion with constant speed satisfies an inhomogeneous wave equation.

Abstract

In this paper we show that a process modeled by a strongly continuous real-valued semigroup (that has a space convolution operator as infinitesimal generator) cannot satisfy causality. We present and analyze a causal model of diffusion that satisfies the semigroup property at a discrete set of time points $M:=\{\tau_m\,|\,m\in\N_0\}$ and that is in contrast to the classical diffusion model not smooth. More precisely, if $v$ denotes the concentration of a substance diffusing with constant speed, then $v$ is continuous but its time derivative is discontinuous at the discrete set $M$ of time points. It is this property of diffusion that forbids the classical limit procedure that leads to the noncausal diffusion model in Stochastics. Furthermore, we show that diffusion with constant speed satisfies an inhomogeneous wave equation with a time dependent coefficient.