Ergodicity for SPDEs driven by divergence-free transport noise

TL;DR

The paper investigates ergodicity for a McKean–Vlasov SPDE on the torus driven by divergence-free transport noise. By constructing a flow transformation and developing a random dynamical system framework, it proves that for d≥2 and sufficiently mixing, strong noise, the SPDE admits a unique invariant measure, namely the uniform state δ1, even when the deterministic system has multiple steady states. The proof hinges on showing a negative top Lyapunov exponent at the uniform state, establishing a stable manifold, and combining reachability and regularisation to rule out nontrivial invariant measures. Key tools include a support theorem for the SPDE, Wong–Zakai approximations for stochastic characteristics and transport, and explicit Lyapunov- and energy-based arguments. The results illuminate how mixing transport noise can induce ergodicity and enhanced dissipation in nonlinear, nonlocal SPDEs, with explicit phase-transition–type examples confirming the sharpness of the approach.

Abstract

We study the ergodic behaviour of the McKean-Vlasov equations driven by common, divergence-free transport noise. In particular, we show that in dimension $d\geq 2$, if the noise is mixing and sufficiently strong it can enforce the uniqueness of invariant probability measures, even if the deterministic part of equation has multiple steady states.

Figures (4)

• Figure 1: An illustration of the stabilisation effect induced by the skew-symmetric noise for the SDE \ref{['eq:finite_dimensional_example']}. The red outward-facing arrows represent the weaker, unstable directions while the blue inward-facing arrows represent the stronger, stable directions. The black circular arrow indicates the rapid rotation induced by the noise.
• Figure 2: The typical spectrum of the operator $L$ in the unstable regime $\kappa\ll1$. The red crosses represent unstable modes while the blue dots represent stable modes. The red vertical line at $|k|=R$ denotes the threshold after which all modes are stable. The black arrow indicates the effect of the mixing noise $\xi$ which moves the solution to higher frequencies as time progresses.
• Figure 3: Schematic depiction of the free-energy landscape for the single-mode interaction potential in Section \ref{['subsec:single_mode']} as it undergoes a second-order phase transition with critical parameter $\kappa_{\textnormal{c}}$. The blue dots indicate the stable steady states, while the red cross indicates an unstable one. The central point in each panel corresponds to the uniform state $\boldsymbol{1}$.
• Figure 4: Schematic depiction of the free-energy landscape for the two-mode interaction potential in Section \ref{['subsec:two_mode']} as it undergoes a first-order phase transition with critical parameter $\kappa_{\textnormal{c}}$. The blue dots indicate stable steady states, while the red crosses indicate unstable ones. The central point in each panel corresponds to the uniform state $\boldsymbol{1}$.

Theorems & Definitions (69)

• Proposition 1.1: chayes_panferov_10
• Theorem 1.2
• Theorem 2.1
• Remark 2.2
• Definition 4.1: Weak Solution to \ref{['eq:path_by_path_PDE']}
• Lemma 4.2
• proof
• Lemma 4.3
• proof
• Definition 5.1: Weak Solution to \ref{['eq:SPDE']}