Noise-induced stabilization of planar flows II

Abstract

We continue the work started in Part I [6], showing how the addition of noise can stabilize an otherwise unstable system. The analysis makes use of nearly optimal Lyapunov functions. In this continuation, we remove the main limiting assumption of Part I by an inductive procedure as well as establish a lower bound which shows that our construction is radially sharp. We also prove a version of Peskir's [7] generalized Tanaka formula adapted to patching together Lyapunov functions. This greatly simplifies the analysis used in previous works.

Geometric ergodicity of two-dimensional Hamiltonian systems with a Lennard–Jones-like repulsive potential

“…That is, the resulting solution Ψ will only be globally continuous and not globally C 2 . It will however be piecewise C 2 and the ideas from [HM15a,HM15b] will be exploited to nonetheless prove H + Ψ is a Lyapunov function for the time t dynamics. 4.3.2.…”

“…In particular, we will see that extracting the asymptotic behavior is more difficult than [HM09] as our potentials do not strictly scale homogeneously. To overcome this we will use the idea of approximating the dynamics near the point at infinity from [AKM12, HM15a,HM15b] as well as techniques for joining together peicewise-defined Lyapunov functions in an analytically simple way from [HM15a,HM15b].…”

Seemingly stable chemical kinetics can be stable, marginally stable, or unstable

“…In a recent series of papers [13,14], the problem of assembling local, homogeneously scaling Lyapunov functions was studied to prove stability of a certain family of diffusion processes. In that paper the authors show, under geometric assumptions related to the convexity of the (continuously assembled) Lyapunov function across the boundary separating two contiguous regions of phase space, that the assembled Lyapunov function automatically satisfies the desired Foster-Lyapunov condition on the union of those regions, and in particular on their common boundary.…”

Noise-induced strong stabilization

“…It is known that the addition of noise can stabilize an explosive ordinary differential equation (ODE) such that it becomes a non-explosive stochastic differential equation (SDE). For examples, see [Sch93], [BD12], [AKM12], [BHW12], [HM15a], [HM15b] and [KCSW19]. This phenomenon is often called noiseinduced stability or noise-induced stabilization, if, in addition, the corresponding Markov process admits an invariant probability measure.…”