Blow-up of a Stable Stochastic Differential Equation

Abstract

We examine a 2-dimensional ODE which exhibits explosion in finite time. Considered as an SDE with additive white noise, it is known to be complete -in the sense that for each initial condition there is almost surely no explosion. Furthermore, the associated Markov process even admits an invariant probability measure. On the other hand, as we will show, the corresponding local stochastic flow will almost surely not be strongly complete, i.e. there exist (random) initial conditions for which the solutions explode in finite time.

Non-explosion by Stratonovich noise for ODEs

“…We expect the answer to be negative. Indeed, in [LS11], the authors construct a (driftless) SDE by applying a similar transformation to bring ∞ into 0 and vice versa and show the lack of a stochastic flow solution of that SDE; see also [LS17] for a similar phenomenon for the example in [HM15a].…”

Noise-induced strong stabilization

“…As already mentioned, it was shown in [LS15] that the planar systems analyzed in [BHW12, AKM12, HM15a, HM15b] do not possess a random attractor despite having stable one-point motions. The difference, between our current system and those, lies in the properties of the explosive trajectory.…”

“…In [LS15] the authors show that a certain family of SDEs, which exhibits noise-induced stabilization as shown in [HM15a] and [HM15b], is not strongly complete, i.e. there exist (random) initial conditions for which the solutions explode in finite time.…”

Ergodicity of a nonlinear stochastic reaction–diffusion equation with memory

“…There are many deterministic systems whose solutions only exist up to a finite time window. Interestingly, by adding a suitable stochastic forcing, it can be shown that these dynamics become non-explosive and even further stabilize as time tends to infinity [1,4,5,10,11,13,17,18,19,23]. This phenomenon is typically known as either noise-induced stability or noise-induced stabilization [1,18].…”