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Stochastic resonance in stochastic PDEs

Nils Berglund, Rita Nader

Abstract

We consider stochastic partial differential equations (SPDEs) on the one-dimensional torus, driven by space-time white noise, and with a time-periodic drift term, which vanishes on two stable and one unstable equilibrium branches. Each of the stable branches approaches the unstable one once per period. We prove that there exists a critical noise intensity, depending on the forcing period and on the minimal distance between equilibrium branches, such that the probability that solutions of the SPDE make transitions between stable equilibria is exponentially small for subcritical noise intensity, while they happen with probability exponentially close to $1$ for supercritical noise intensity. Concentration estimates of solutions are given in the $H^s$ Sobolev norm for any $s<\frac12$. The results generalise to an infinite-dimensional setting those obtained for $1$-dimensional SDEs in [Nils Berglund and Barbara Gentz. A sample-paths approach to noise-induced synchronization: stochastic resonance in a double-well potential. Ann. Appl. Probab., 12(4):1419-1470, 2002].

Stochastic resonance in stochastic PDEs

Abstract

We consider stochastic partial differential equations (SPDEs) on the one-dimensional torus, driven by space-time white noise, and with a time-periodic drift term, which vanishes on two stable and one unstable equilibrium branches. Each of the stable branches approaches the unstable one once per period. We prove that there exists a critical noise intensity, depending on the forcing period and on the minimal distance between equilibrium branches, such that the probability that solutions of the SPDE make transitions between stable equilibria is exponentially small for subcritical noise intensity, while they happen with probability exponentially close to for supercritical noise intensity. Concentration estimates of solutions are given in the Sobolev norm for any . The results generalise to an infinite-dimensional setting those obtained for -dimensional SDEs in [Nils Berglund and Barbara Gentz. A sample-paths approach to noise-induced synchronization: stochastic resonance in a double-well potential. Ann. Appl. Probab., 12(4):1419-1470, 2002].

Paper Structure

This paper contains 15 sections, 16 theorems, 192 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. The set-up
  4. The stable case
  5. Bifurcations and avoided bifurcations
  6. Discussion
  7. Proofs: the stable case
  8. Deterministic case
  9. Stochastic case
  10. Linear case
  11. Nonlinear case
  12. Proofs: bifurcations
  13. Deterministic case
  14. Stochastic case
  15. Some useful inequalities in Sobolev spaces

Key Result

Proposition 2.3

There exist constants $C, \varepsilon_0>0$ such that for $0 < \varepsilon < \varepsilon_0$, the equation eq:SPDE_det admits a particular solution $\bar{\phi}(t,x)$ satisfying

Figures (4)

  • Figure 1: Equilibrium branches and associated adiabatic solutions near the avoided bifurcation point $(0,0)$.
  • Figure 2: Weak noise regime $\sigma\ll(\delta\vee\varepsilon)^{3/4}$. The equilibrium branches $\phi^*_\pm(t)$, as well as the deterministic solution $\bar{\phi}_0(t)$, belong to the hyperplane $\{\phi_\perp = 0\}$, while $\phi_0(t)$ denotes the projection of the solution $\phi(t,x)$ on this hyperplane.
  • Figure 3: Strong noise regime $\sigma\gg(\delta\vee\varepsilon)^{3/4}$. Solutions are likely to cross the unstable equilibrium branch $\phi^*_-(t)$.
  • Figure 4: Strong noise regime, behaviour after reaching level $-d$. If the drift term is negative, bounded away from zero, in an interval $[-d_0,-d]$, solutions are likely to reach $-d_0$ after another time of order $\varepsilon$.

Theorems & Definitions (35)

  • Proposition 2.3: Deterministic dynamics in the stable case
  • Theorem 2.4: Stochastic dynamics in the stable case
  • Remark 1
  • Remark 2
  • Proposition 2.6: Deterministic dynamics near the origin
  • Theorem 2.7: Transverse stochastic dynamics for $\phi_\perp$
  • Theorem 2.8: Stochastic dynamics near $\bar{\phi}_0(t)$
  • Theorem 2.9: Strong-noise regime
  • Remark 3
  • Proposition 2.10: Reaching level $-d_0 < -d$
  • ...and 25 more