Large deviations for Gaussian diffusions with delay

TL;DR

This paper develops an explicit large deviations framework for Gaussian delay SDEs, enabling efficient computation of rare event probabilities and most likely transition paths, with applications to genetic regulatory circuits.

Contribution

It introduces a fully explicit large deviations method for linear delay SDEs, facilitating fast numerical analysis of rare events and metastable transitions in delayed stochastic systems.

Findings

Explicit action functional for Gaussian delay SDEs derived

Numerical methods for most likely paths developed

Application to genetic toggle switch demonstrates practical utility

Abstract

Dynamical systems driven by nonlinear delay SDEs with small noise can exhibit important rare events on long timescales. When there is no delay, classical large deviations theory quantifies rare events such as escapes from metastable fixed points. Near such fixed points, one can approximate nonlinear delay SDEs by linear delay SDEs. Here, we develop a fully explicit large deviations framework for (necessarily Gaussian) processes $X_t$ driven by linear delay SDEs with small diffusion coefficients. Our approach enables fast numerical computation of the action functional controlling rare events for $X_t$ and of the most likely paths transiting from $X_0 = p$ to $X_T=q$. Via linear noise local approximations, we can then compute most likely routes of escape from metastable states for nonlinear delay SDEs. We apply our methodology to the detailed dynamics of a genetic regulatory circuit, namely the co-repressive toggle switch, which may be described by a nonlinear chemical Langevin SDE with delay.