  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.
