Abstract
We investigate an example of noise-induced stabilization in the plane that was also considered in (Gawedzki, Herzog, Wehr 2010) and (Birrell, Herzog, Wehr 2011). We show that despite the deterministic system not being globally stable, the addition of additive noise in the vertical direction leads to a unique invariant probability measure to which the system converges at a uniform, exponential rate. These facts are established primarily through the construction of a Lyapunov function which we generate as the solution to a sequence of Poisson equations. Unlike a number of other works, however, our Lyapunov function is constructed in a systematic way, and we present a meta-algorithm we hope will be applicable to other problems. We conclude by proving positivity properties of the transition density by using Malliavin calculus via some unusually explicit calculations.
Highlights
Stabilization by noise is a mathematically intriguing phenomenon
We describe a general methodology for building a Lyapunov function in a setting where the global stability of the systems requires flux of probability into regions which are clearly dissipative from the rest of phase space
We are most interested in problems, like the example considered here, where the noise plays an essential role in creating this transport is some regions
Summary
Stabilization by noise is a mathematically intriguing phenomenon. For instance, in the classic example of the inverted pendulum, the addition of noise opens up a small neighborhood of local stability around a deterministically unstable fixed point [2, 17]; in the striking examples of [28], the addition of noise leads to global stabilization. Our general approach is to build local Lyapunov functions as solutions to associated partial differential equations (PDEs), where the PDEs are defined in regions delineated by different asymptotic behaviors of the flow. The boundary data g for the Poisson equation is given by the dominant behavior/scaling of the priming Lyapunov function on the boundary between the priming and adjacent region.
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