Stochastic resonance in stochastic PDEs. (English) Zbl 1517.60072
Summary: 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 \(s<\frac{1}{2}\). The results generalise to an infinite-dimensional setting those obtained for 1-dimensional SDEs in [N. Berglund and B. Gentz, Ann. Appl. Probab. 12, No. 4, 1419–1470 (2002; Zbl 1023.60052)].
MSC:
| 60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
| 60G17 | Sample path properties |
| 34F15 | Resonance phenomena for ordinary differential equations involving randomness |
| 37H20 | Bifurcation theory for random and stochastic dynamical systems |
| 60H40 | White noise theory |
Keywords:
stochastic PDEs; stochastic resonance; sample-path estimates; slow-fast systems; transcritical bifurcationCitations:
Zbl 1023.60052References:
| [1] | Alkhayuon, H., Tyson, R.C., Wieczorek, S.: Phase-sensitive tipping: How cyclic ecosystems respond to contemporary climate. arXiv:2101.12107 |
| [2] | Benzi, R.; Sutera, A.; Vulpiani, A., The mechanism of stochastic resonance, J. Phys. A, 14, 11, L453-L457 (1981) · doi:10.1088/0305-4470/14/11/006 |
| [3] | Berglund, N.: An Eyring-Kramers law for slowly oscillating bistable diffusions. Probab. Math. Phys. (2020) Accepted · Zbl 1492.60159 |
| [4] | Berglund, N.; Gentz, B., Pathwise description of dynamic pitchfork bifurcations with additive noise, Probab. Theory Related Fields, 122, 3, 341-388 (2002) · Zbl 1008.37031 · doi:10.1007/s004400100174 |
| [5] | Berglund, N.; Gentz, B., A sample-paths approach to noise-induced synchronization: stochastic resonance in a double-well potential, Ann. Appl. Probab., 12, 4, 1419-1470 (2002) · Zbl 1023.60052 · doi:10.1214/aoap/1037125869 |
| [6] | Berglund, N.; Gentz, B., Geometric singular perturbation theory for stochastic differential equations, J. Differ. Equ., 191, 1-54 (2003) · Zbl 1053.34048 · doi:10.1016/S0022-0396(03)00020-2 |
| [7] | Berglund, N., Gentz, B.: Noise-induced phenomena in slow-fast dynamical systems. In: A sample-paths approach. Probability and its Applications (New York). Springer-Verlag London Ltd, London (2006) · Zbl 1098.37049 |
| [8] | Berglund, N.; Gentz, B., Sharp estimates for metastable lifetimes in parabolic SPDEs: Kramers’ law and beyond, Electron. J. Probab., 18, 24, 58 (2013) · Zbl 1285.60060 |
| [9] | Berglund, N.; Gentz, B., On the Noise-Induced Passage through an Unstable Periodic Orbit II: General Case, SIAM J. Math. Anal., 46, 1, 310-352 (2014) · Zbl 1291.60114 · doi:10.1137/120887965 |
| [10] | Bourdaud, G.: Le calcul symbolique dans certaines algèbres de type Sobolev. In Recent developments in fractals and related fields, Appl. Numer. Harmon. Anal., pages 131-144. Birkhäuser Boston, Boston, MA, (2010) · Zbl 1234.46033 |
| [11] | Bourdaud, G.; Meyer, Y., Fonctions qui opèrent sur les espaces de Sobolev, J. Funct. Anal., 97, 2, 351-360 (1991) · Zbl 0737.46011 · doi:10.1016/0022-1236(91)90006-Q |
| [12] | Eisenman, I.; Wettlaufer, JS, Nonlinear threshold behavior during the loss of Arctic sea ice, PNAS, 106, 1, 28-32 (2009) · Zbl 1355.86002 · doi:10.1073/pnas.0806887106 |
| [13] | Fenichel, N., Geometric singular perturbation theory for ordinary differential equations, J. Differ. Equ., 31, 1, 53-98 (1979) · Zbl 0476.34034 · doi:10.1016/0022-0396(79)90152-9 |
| [14] | Fox, RF, Stochastic resonance in a double well, Phys. Rev. A, 39, 4148-4153 (1989) · doi:10.1103/PhysRevA.39.4148 |
| [15] | Gammaitoni, L.; Menichella-Saetta, E.; Santucci, S.; Marchesoni, F.; Presilla, C., Periodically time-modulated bistable systems: Stochastic resonance, Phys. Rev. A, 40, 2114-2119 (1989) · Zbl 0872.70015 · doi:10.1103/PhysRevA.40.2114 |
| [16] | Gammaitoni, L.; Hänggi, P.; Jung, P.; Marchesoni, F., Stochastic resonance, Rev. Mod. Phys., 70, 223-287 (1998) · doi:10.1103/RevModPhys.70.223 |
| [17] | Gnann, MV; Kuehn, C.; Pein, A., Towards sample path estimates for fast-slow stochastic partial differential equations, Eur. J. Appl. Math., 30, 5, 1004-1024 (2019) · Zbl 1423.60099 · doi:10.1017/S095679251800061X |
| [18] | Hänggi, P., Stochastic resonance in biology: How noise can enhance detection of weak signals and help improve biological information processing, Chemphyschem, 3, 3, 285-290 (2002) · doi:10.1002/1439-7641(20020315)3:3<285::AID-CPHC285>3.0.CO;2-A |
| [19] | Herrmann, S.; Imkeller, P.; Pavlyukevich, I.; Peithmann, D., Stochastic resonance. A mathematical approach in the small noise limit (2014), Providence, RI: American Mathematical Society, Providence, RI · Zbl 1416.60008 |
| [20] | Jetschke, G., On the equivalence of different approaches to stochastic partial differential equations, Math. Nachr., 128, 315-329 (1986) · Zbl 0604.60056 · doi:10.1002/mana.19861280127 |
| [21] | Jung, P.; Hänggi, P., Amplification of small signals via stochastic resonance, Phys. Rev. A, 44, 8032-8042 (1991) · doi:10.1103/PhysRevA.44.8032 |
| [22] | Muratov, CB; Vanden-Eijnden, E.; E., W., Self-induced stochastic resonance in excitable systems, Physica D, 210, 227-240 (2005) · Zbl 1109.34040 · doi:10.1016/j.physd.2005.07.014 |
| [23] | Nicolis, C.; Nicolis, G., Stochastic aspects of climatic transitions-additive fluctuations, Tellus, 33, 3, 225-234 (1981) |
| [24] | Tihonov, AN, Systems of differential equations containing small parameters in the derivatives, Mat. Sbornik N. S., 31, 575-586 (1952) · Zbl 0048.07101 |
| [25] | Vélez-Belchí, P.; Alvarez, A.; Colet, P.; Tintoré, J.; Haney, RL, Stochastic resonance in the thermohaline circulation, Geophys. Res. Lett., 28, 10, 2053-2056 (2001) · doi:10.1029/2000GL012091 |
| [26] | Wiesenfeld, K.; Jaramillo, F., Minireview of stochastic resonance, Chaos, 8, 539-548 (1998) · doi:10.1063/1.166335 |
| [27] | Wiesenfeld, K.; Moss, F., Stochastic resonance and the benefits of noise: From ice ages to crayfish and SQUIDs, Nature, 373, 33-36 (1995) · doi:10.1038/373033a0 |
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