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Large Deviations for Gaussian Diffusions with Delay

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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.

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Acknowledgements

Funding was provided by Directorate for Mathematical and Physical Sciences (Grant Nos. 1413437 and 1412927.

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  1. University of Houston, Houston, TX, USA

    Robert Azencott, Brett Geiger & William Ott

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  1. Robert Azencott
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  2. Brett Geiger
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  3. William Ott
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Correspondence to William Ott.

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Azencott, R., Geiger, B. & Ott, W. Large Deviations for Gaussian Diffusions with Delay. J Stat Phys 170, 254–285 (2018). https://doi.org/10.1007/s10955-017-1909-5

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