Construction of the quasi-potential for linear SDEs using false quasi-potentials and a geometric recursion

Abstract

The quasi-potential is a key concept of the Large Deviation Theory for Stochastic Differential Equations (SDEs). Once the quasi-potential with respect to an attractor of the corresponding deterministic system is found, one can readily obtain maximum likelihood exit paths and estimate the exit rate from the basin of attraction and the invariant probability density near the attractor. The quasi-potential for a linear SDE with asymptotically stable equilibrium at the origin is a quadratic form whose matrix was found by Z. Chen and M. Freidlin (2005). Their formula involves an integral of certain matrix exponentials and is inconvenient for numerical evaluation. In this work, I propose a different approach for constructing the quasi-potential matrix for linear SDEs based on a certain easy-to-obtain hierarchy of orthogonal decompositions inspired by those used by M. Tao (2018). A number of properties of these decompositions, named false quasi-potential decompositions, is established. A geometric recursive algorithm implementing this approach is presented. It involves only numerically stable sub-algorithms such as the computation of Schur decompositions and solving systems of linear equations or quadratic least squares problems. This work is motivated by the need to initialize the quasi-potential solvers near asymptotically stable equilibria. An accurate way to do it is to linearize the SDE near the equilibrium and set the values of the quasi-potential at the neighboring mesh points to the exact quasi-potential of the linearized system.