Large Deviations for Dynamical Systems with Small Noise

Dynamical systems with small noise can exhibit important rare events on long timescales. For systems driven by stochastic differential equations (SDEs) with small noise and no delay, classical large deviations theory quantifies rare events such as escapes from nominally stable fixed points, which are difficult to evaluate through direct simulation. Near such fixed points, one can approximate nonlinear SDEs by linear SDEs. When delay is introduced, the situation is quite similar where nonlinear \emph{delay} SDEs can be approximated by linear \emph{delay} SDEs near metastable states. For genetic evolution of bacterial populations of \emph{Escherichia coli}, commonly called \emph{E. coli}, modeled by discrete Markov chains with small mutation rates and random dilution, radical shifts in the genetic composition of large cell populations are \emph{rare events} with quite low probabilities. Direct simulations generally fail to evaluate these events accurately. Large deviations theory then becomes a natural approach in order to quantify transition pathways linking a fixed initial population state to a desired target state. In this dissertation, we first develop a fully explicit large deviations framework for (necessarily Gaussian) processes $Xt$ driven by linear delay SDEs with small diffusion coefficients. Our approach enables fast numerical computation of the action functional controlling rare events for $Xt$ and the most likely paths transitioning from $X0=p$ to $XT=q$. 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. Second, we develop an applicable large deviations framework for a class of Markov chains used to model genetic evolution of \emph{E. coli} bacteria. Finally, we apply this framework using realistic parameter sets in order to solve several difficult numerical and mathematical questions of high biological interest such as computing the most likely evolutionary path linking two given population states in the fitness landscape and evaluating transition probabilities between successive genotype fixations.