# Applied Math Colloquium

### Computational methods for the study of stochastic dynamics with small noise

### Abstract

Nongradient Stochastic Differential Equations with small noise often arise in modeling biological and ecological systems.

An effective description of the dynamics of such systems can be given by comparing the stability of different attractors, finding a low-dimensional manifold to which the dynamics is virtually restricted (if any), finding transition rates between different attractors and the maximum likelihood transition paths. Addressing these questions by means of direct simulations may be difficult or impossible due to long waiting times. Alternatively, one can use asymptotic analysis tools for the vanishing noise limit offered by the Large Deviation Theory (Freidlin and Wentzell, 1970s). The key function of the Large Deviation Theory is the quasi-potential that is somewhat analogous to the potential for gradient systems. It gives estimates for transition rates, transition paths, and the invariant probability measure.

In this talk, I will introduce a family of Dijkstra-like Ordered Line Integral Methods (OLIMs) for computing the quasi-potential on 2D and 3D meshes. A number of technical innovations allowed us to make them accurate and fast. I will demonstrate what one can find out about stochastic systems once the quasi-potential is computed. Applications to the Lorenzâ€™63 model perturbed by small white noise and to two genetic switch models will be presented.