Multiscale Expansion of Invariant Measures for SPDEs

Abstract

We derive the first two terms in an ε-expansion for the invariant measure of a class of semilinear parabolic SPDEs near a change of stability, when the noise strength and the linear instability are of comparable order ε 2 . This result gives insight into the stochastic bifurcation and allows to rigorously approximate correlation functions. The error between the approximate and the true invariant measure is bounded in both the Wasserstein and the total variation distance.

Modulation Equation for Stochastic Swift--Hohenberg Equation

“…So, this result is only the starting point for modulation equations on unbounded domains. The stochastic Swift-Hohenberg model was first studied in the context of amplitude equations with non-degenerate noise in [5] and later in [3].…”

Numerical study of amplitude equations for SPDEs with degenerate forcing

“…For space-time white noise Blömker & Hairer [2] provided an approximation result including higher order corrections. This is in this case just a fast OrnsteinUhlenbeck-process on S. Moreover, this approximation carries over to an invariant measure of (13), see [2].…”

“…This is in this case just a fast OrnsteinUhlenbeck-process on S. Moreover, this approximation carries over to an invariant measure of (13), see [2]. But here we focus on the transient approximation result, only.…”

Degenerate Nucleation in the Cahn--Hilliard--Cook Model

“…This is necessary since due to the degeneracy of the driving noise we cannot directly employ the theory of large deviations. This approach has previously been used for example by Debussche & Da Prato [9], Blömker & Hairer [3], or Da Prato & Zabczyck [10]. We believe that by extending the method of Wanner [35] or Desi, Sander & Wanner [12], both of which are based on results due to Kahane [21], it might be possible to improve our estimate by deriving exponential moments of the stochastic convolution.…”

“…While we delay a full comparison to our previous notation, for now recall that in Section 2 we studied the two-dimensional case d = 2 with domain Ω = (0, 1) 2 , and the eigenvalues η k are given by (3), where µ in the nucleation region implies that η k < 0. In fact, in the twodimensional case the κ k are the real numbers π 2 (j 2 + 2 ) for j, ∈ N 0 , and the corresponding eigenfunctions ψ k are precisely the forcing modes ϕ j, defined in (6).…”