On stochastic differential equations with random delay

TL;DR

This paper studies stochastic differential equations with random delays, revealing their equivalence to higher-order deterministic equations and exploring unusual behaviors like energy growth and effects of discretization.

Contribution

It establishes the equivalence between stochastic delay equations and deterministic higher-order equations, and analyzes their behaviors and discretization effects.

Findings

Energy of harmonic oscillator grows as exp((3/2) t^{2/3})

Discrete time steps introduce intrinsic fluctuations

Crossover from stochastic to deterministic behavior as epsilon approaches zero

Abstract

We consider stochastic dynamical systems defined by differential equations with a uniform random time delay. The latter equations are shown to be equivalent to deterministic higher-order differential equations: for an $n$-th order equation with random delay, the corresponding deterministic equation has order $n+1$. We analyze various examples of dynamical systems of this kind, and find a number of unusual behaviors. For instance, for the harmonic oscillator with random delay, the energy grows as $\exp((3/2)\,t^{2/3})$ in reduced units. We then investigate the effect of introducing a discrete time step $\epsilon$. At variance with the continuous situation, the discrete random recursion relations thus obtained have intrinsic fluctuations. The crossover between the fluctuating discrete problem and the deterministic continuous one as $\epsilon$ goes to zero is studied in detail on the example of a first-order linear differential equation.