Travelling Waves in Monostable and Bistable Stochastic Partial Differential Equations

  • Christian KuehnEmail author
Survey Article


In this review, we provide a concise summary of several important mathematical results for stochastic travelling waves generated by monostable and bistable reaction-diffusion stochastic partial differential equations (SPDEs). In particular, this survey is intended for readers new to the topic but who have some knowledge in any sub-field of differential equations. The aim is to bridge different backgrounds and to identify the most important common principles and techniques currently applied to the analysis of stochastic travelling wave problems. Monostable and bistable reaction terms are found in prototypical dissipative travelling wave problems, which have already guided the deterministic theory. Hence, we expect that these terms are also crucial in the stochastic setting to understand effects and to develop techniques. The survey also provides an outlook, suggests some open problems, and points out connections to results in physics as well as to other active research directions in SPDEs.


Travelling wave Reaction-diffusion equation Stochastic partial differential equation Monostable nonlinearity Bistable nonlinearity Stability Wave speed 



I would like to thank the VolkswagenStiftung for support via a Lichtenberg Professorship. I also acknowledge the very helpful comments of two anonymous referees, of the editor and of Christian Hamster, who all helped to improve the presentation of the manuscript.


  1. 1.
    Achleitner, F., Kuehn, C.: Analysis and numerics of travelling waves for asymmetric fractional reaction-diffusion equations. Commun. Appl. Ind. Math. 6(2), 1–25 (2015) Google Scholar
  2. 2.
    Achleitner, F., Kuehn, C.: Traveling waves for a bistable equation with nonlocal-diffusion. Adv. Differ. Equ. 20(9), 887–936 (2015) MathSciNetzbMATHGoogle Scholar
  3. 3.
    Allen, S.M., Cahn, J.W.: A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall. 27(6), 1085–1905 (1979) CrossRefGoogle Scholar
  4. 4.
    Antonopoulou, D.C., Blömker, D., Karali, G.D.: Front motion in the one-dimensional stochastic Cahn-Hilliard equation. SIAM J. Math. Anal. 44(5), 3242–3280 (2012) MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Antonopoulou, D.C., Bates, P.W., Blömker, D., Karali, G.D.: Motion of a droplet for the stochastic mass-conserving Allen-Cahn equation. SIAM J. Math. Anal. 48(1), 670–708 (2016) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Armero, J., Sancho, J.M., Casademunt, J., Lacasta, A.M., Ramirez-Piscina, L., Sagués, F.: External fluctuations in front propagation. Phys. Rev. Lett. 76(17), 3045–3048 (1996) CrossRefGoogle Scholar
  7. 7.
    Armero, J., Casademunt, J., Ramirez-Piscina, L., Sancho, J.M.: Ballistic and diffusive corrections to front propagation in the presence of multiplicative noise. Phys. Rev. E 58(5), 5494 (1998) CrossRefGoogle Scholar
  8. 8.
    Arnold, L.: Stochastic Differential Equations: Theory and Applications. Wiley, New York (1974) zbMATHGoogle Scholar
  9. 9.
    Aronson, D.G., Weinberger, H.F.: Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In: Partial Differential Equations and Related Topics. Lecture Notes in Mathematics, vol. 446, pp. 5–49. Springer, Berlin (1974) CrossRefGoogle Scholar
  10. 10.
    Assing, S.: Comparison of systems of stochastic partial differential equations. Stoch. Process. Appl. 82(2), 259–282 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Barkley, D.: A model for fast computer simulation of waves in excitable media. Physica D 49, 61–70 (1991) CrossRefGoogle Scholar
  12. 12.
    Benguria, R.D., Depassier, M.C.: Speed of fronts of the reaction-diffusion equation. Phys. Rev. Lett. 77(6), 1171–1173 (1996) zbMATHCrossRefGoogle Scholar
  13. 13.
    Benguria, R.D., Depassier, M.C., Haikala, V.: Effect of a cutoff on pushed and bistable fronts of the reaction-diffusion equation. Phys. Rev. E 76(5), 051101 (2007) CrossRefGoogle Scholar
  14. 14.
    Bérard, J., Gouéré, J.: Brunet-Derrida behavior of branching-selection particle systems on the line. Commun. Math. Phys. 298(2), 323–342 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Berglund, N., Kuehn, C.: Regularity structures and renormalisation of FitzHugh-Nagumo SPDEs in three space dimensions. Electron. J. Probab. 21(18), 1–48 (2016) MathSciNetzbMATHGoogle Scholar
  16. 16.
    Birzu, G., Hallatschek, O., Korolev, K.S.: Fluctuations uncover a distinct class of traveling waves. Proc. Natl. Acad. Sci. USA 115(6), E3645–E3654 (2018) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Bonaccorsi, S., Mastrogiacomo, E.: Analysis of the stochastic Fitzhugh-Nagumo system. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 11(3), 427–446 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Brassesco, S., De Masi, A., Presutti, E.: Brownian fluctuations of the interface in the \(D=1\) Ginzburg-Landau equation with noise. Ann. Inst. Henri Poincaré Probab. Stat. 31(1), 81–118 (1995) MathSciNetzbMATHGoogle Scholar
  19. 19.
    Bressloff, P.C.: Stochastic neural field theory and the system-size expansion. SIAM J. Appl. Math. 70(5), 1488–1521 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Bressloff, P.C.: Spatiotemporal dynamics of continuum neural fields. J. Phys. A, Math. Theor. 45, 033001 (2012) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Bressloff, P.C., Webber, M.A.: Front propagation in stochastic neural fields. SIAM J. Appl. Dyn. Syst. 11(2), 708–740 (2012) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Bressloff, P.C., Wilkerson, J.: Traveling pulses in a stochastic neural field model of direction selectivity. Front. Comput. Neurosci. 6(90), 1–14 (2012) Google Scholar
  23. 23.
    Breuer, H.P., Huber, W., Petruccione, F.: Fluctuation effects on wave propagation in a reaction-diffusion process. Physica D 73(3), 259–273 (1994) zbMATHCrossRefGoogle Scholar
  24. 24.
    Brockmann, D., Hufnagel, L.: Front propagation in reaction-superdiffusion dynamics: taming Lévy flights with fluctuations. Phys. Rev. Lett. 98(17), 178301 (2007) CrossRefGoogle Scholar
  25. 25.
    Brunet, E., Derrida, B.: Shift in the velocity front due to a cutoff. Phys. Rev. E 56(3), 2597–2604 (1997) MathSciNetCrossRefGoogle Scholar
  26. 26.
    Brunet, E., Derrida, B.: Effect of microscopic noise on front propagation. J. Stat. Phys. 103(1), 269–282 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Brunet, E., Derrida, B.: Exactly soluble noisy traveling-wave equation appearing in the problem of directed polymers in a random medium. Phys. Rev. E 70(1), 016106 (2004) MathSciNetCrossRefGoogle Scholar
  28. 28.
    Brunet, E., Derrida, B., Mueller, A.H., Munier, S.: Noisy traveling waves: effect of selection on genealogies. Europhys. Lett. 76(1), 1 (2006) MathSciNetCrossRefGoogle Scholar
  29. 29.
    Brunet, E., Derrida, B., Mueller, A.H., Munier, S.: Phenemenological theory giving full statistics of the position of fluctuating fronts. Phys. Rev. E 73, 056126 (2006) CrossRefGoogle Scholar
  30. 30.
    Brunet, E., Derrida, B., Mueller, A.H., Munier, S.: Effect of selection on ancestry: an exactly soluble case and its phenomenological generalization. Phys. Rev. E 76(4), 0411 (2007) MathSciNetCrossRefGoogle Scholar
  31. 31.
    Brzeźniak, Z., Gatarek, D.: Martingale solutions and invariant measures for stochastic evolution equations in Banach spaces. Stoch. Process. Appl. 84(2), 187–225 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Cabré, X., Roquejoffre, J.M.: The influence of fractional diffusion in Fisher-KPP equations. Commun. Math. Phys. 320(3), 679–722 (2013) MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Carr, J., Pego, R.L.: Metastable patterns in solutions of \(u_{t}= \epsilon ^{2} u_{xx}-f(u)\). Commun. Pure Appl. Math. 42(5), 523–576 (1989) zbMATHCrossRefGoogle Scholar
  34. 34.
    Cartwright, M.C., Gottwald, G.A.: A collective coordinate framework to study the dynamics of travelling waves in stochastic partial differential equations, pp. 1–20 (2018). arXiv:1806.07194
  35. 35.
    Chen, X.: Existence, uniqueness, and asymptotic stability of travelling waves in nonlocal evolution equations. Adv. Differ. Equ. 2, 125–160 (1997) zbMATHGoogle Scholar
  36. 36.
    Chmaj, A.: Existence of traveling waves in the fractional bistable equation. Arch. Math. 100(5), 473–480 (2013) MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Chow, P.-L.: Stochastic Partial Differential Equations. Chapman & Hall/CRC, Boca Raton (2007) zbMATHGoogle Scholar
  38. 38.
    Cohen, E., Kessler, D.A., Levine, H.: Recombination dramatically speeds up evolution of finite populations. Phys. Rev. Lett. 94(9), 098102 (2005) CrossRefGoogle Scholar
  39. 39.
    Conlon, J.G., Doering, C.R.: On travelling waves for the stochastic Fisher-Kolmogorov-Petrovsky-Piscunov equation. J. Stat. Phys. 120(3), 421–477 (2005) MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Crauel, H., Flandoli, F.: Attractors for random dynamical systems. Probab. Theory Relat. Fields 100(3), 365–393 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992) zbMATHCrossRefGoogle Scholar
  42. 42.
    Da Prato, G., Jentzen, A., Röckner, M.: A mild Itô formula for SPDEs, pp. 1–39 (2012). arXiv:1009.3526v4
  43. 43.
    Davies, I.M., Truman, A., Zhao, H.Z.: Stochastic generalised KPP equations. Proc. R. Soc. Edinb. A 126(5), 957–983 (1996) MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    De Pasquale, F., Gorecki, J., Popielawski, J.: On the stochastic correlations in a randomly perturbed chemical front. J. Phys. A 25(2), 433 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Debussche, A., Hofmanová, M., Vovelle, J.: Degenerate parabolic stochastic partial differential equations: quasilinear case. Ann. Probab. 44(3), 1916–1955 (2016) MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    del Castillo-Negrete, D., Carreras, B.A., Lynch, V.E.: Front dynamics in reaction-diffusion systems with Levy flights: a fractional diffusion approach. Phys. Rev. Lett. 91(1), 018302 (2003) CrossRefGoogle Scholar
  47. 47.
    Dierckx, H., Panfilov, A.V., Verschelde, H., Biktashev, V.N., Biktasheva, I.V.: A response function framework for the dynamics of meandering or large-core spiral waves and modulated traveling waves, pp. 1–23 (2019). arXiv:1901.05530
  48. 48.
    Doering, C.R., Mueller, C., Smereka, P.: Interacting particles,the stochastic Fisher–Kolmogorov–Petrovsky–Piscounov equation, and duality. Physica A 325, 243–259 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Donati-Martin, C., Pardoux, E.: White noise driven SPDEs with reflection. Probab. Theory Relat. Fields 95(1), 1–24 (1993) MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Duan, J., Wang, W.: Effective Dynamics of Stochastic Partial Differential Equations. Elsevier, New York (2014) zbMATHGoogle Scholar
  51. 51.
    Dumortier, F., Popovic, N., Kaper, T.J.: The critical wave speed for the Fisher-Kolmogorov-Petrowskii-Piscounov equation with cut-off. Nonlinearity 20(4), 855–877 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Dumortier, F., Popovic, N., Kaper, T.J.: A geometric approach to bistable front propagation in scalar reaction-diffusion equations with cut-off. Physica D 239(20), 1984–1999 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Elworthy, K.D., Zhao, H.Z.: Approximate travelling waves for generalized and stochastic KPP equations. In: Probability Theory and Mathematical Statistics, St. Petersburg, 1993, pp. 141–154 (1996) Google Scholar
  54. 54.
    Elworthy, K.D., Zhao, H.Z., Gaines, J.G.: The propagation of travelling waves for stochastic generalized KPP equations. Math. Comput. Model. 20(4), 131–166 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Engler, H.: On the speed of spread for fractional reaction-diffusion equations. Int. J. Differ. Equ. 2010, 315421 (2010) MathSciNetzbMATHGoogle Scholar
  56. 56.
    Evans, L.C.: Partial Differential Equations. AMS, Providence (2002) Google Scholar
  57. 57.
    Faugeras, O., Inglis, J.: Stochastic neural field equations: a rigorous footing. J. Math. Biol. 71(2), 259–300 (2015) MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Fife, P., McLeod, J.B.: The approach of solutions nonlinear diffusion equations to travelling front solutions. Arch. Ration. Mech. Anal. 65, 335–361 (1977) MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Fisher, R.A.: The wave of advance of advantageous genes. Annu. Eugen. 7, 353–369 (1937) zbMATHGoogle Scholar
  60. 60.
    FitzHugh, R.: Mathematical models of threshold phenomena in the nerve membrane. Bull. Math. Biophys. 17, 257–269 (1955) CrossRefGoogle Scholar
  61. 61.
    Funaki, T.: The scaling limit for a stochastic PDE and the separation of phases. Probab. Theory Relat. Fields 102(2), 221–288 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    Funaki, T.: Singular limit for stochastic reaction-diffusion equation and generation of random interfaces. Acta Math. Sin. 15(3), 407–438 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Furutsu, K.: On the statistical theory of electromagnetic waves in a fluctuating medium (I). Journal of Research of the National Bureau of Standards-D. Radio Propagation 67, 303–323 (1963) zbMATHCrossRefGoogle Scholar
  64. 64.
    Gaines, J.G.: Numerical experiments with S(P)DEs. In: Etheridge, A. (ed.) Stochastic Partial Differential Equations. LMS Lecture Note Series, vol. 216, pp. 55–71. Cambridge University Press, Cambridge (1995) CrossRefGoogle Scholar
  65. 65.
    Garcia-Ojalvo, J., Sancho, J.: Noise in Spatially Extended Systems. Springer, Berlin (1999) zbMATHCrossRefGoogle Scholar
  66. 66.
    Ginzburg, V.L., Landau, L.D.: On the theory of superconductivity. In: On Superconductivity and Superfluidity, pp. 113–137. Springer, Berlin (2009) CrossRefGoogle Scholar
  67. 67.
    Glimm, J.: Nonlinear and stochastic phenomena: the grand challenge for partial differential equations. SIAM Rev. 33(4), 626–643 (1991) MathSciNetzbMATHCrossRefGoogle Scholar
  68. 68.
    Gowda, K., Kuehn, C.: Warning signs for pattern-formation in SPDEs. Commun. Nonlinear Sci. Numer. Simul. 22(1), 55–69 (2015) MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    Gubinelli, M., Perkowski, N.: KPZ reloaded. Commun. Math. Phys. 349(1), 165–269 (2017) MathSciNetzbMATHCrossRefGoogle Scholar
  70. 70.
    Gubinelli, M., Imkeller, P., Perkowski, N.: Paracontrolled distributions and singular PDEs. Forum Math. Pi 3, e6 (2015) MathSciNetzbMATHCrossRefGoogle Scholar
  71. 71.
    Gui, C., Zhao, M.: Traveling wave solutions of Allen-Cahn equation with a fractional Laplacian. Ann. Inst. Henri Poincaré C 32(4), 785–812 (2015) MathSciNetzbMATHCrossRefGoogle Scholar
  72. 72.
    Gyöngy, I.: Existence and uniqueness results for semilinear stochastic partial differential equations. Stoch. Process. Appl. 73(2), 271–299 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  73. 73.
    Hairer, M.: Solving the KPZ equation. Ann. Math. 178(2), 559–664 (2013) MathSciNetzbMATHCrossRefGoogle Scholar
  74. 74.
    Hairer, M.: A theory of regularity structures. Invent. Math. 198(2), 269–504 (2014) MathSciNetzbMATHCrossRefGoogle Scholar
  75. 75.
    Hairer, M., Maas, J.: A spatial version of the Itô-Stratonovich correction. Ann. Probab. 40(4), 1675–1714 (2012) MathSciNetzbMATHCrossRefGoogle Scholar
  76. 76.
    Hallatschek, O.: The noisy edge of traveling waves. Proc. Natl. Acad. Sci. USA 108(5), 1783–1787 (2011) CrossRefGoogle Scholar
  77. 77.
    Hallatschek, O., Korolev, K.S.: Fisher waves in the strong noise limit. Phys. Rev. Lett. 103, 108103 (2009) CrossRefGoogle Scholar
  78. 78.
    Hamster, C.H.S., Hupkes, H.J.: Stability of travelling waves for reaction-diffusion equations with multiplicative noise, p. 1 (2017). arXiv:1712.00266
  79. 79.
    Hamster, C.H.S., Hupkes, H.J.: Stability of travelling waves for systems of reaction-diffusion equations with multiplicative noise, p. 1 (2018). arXiv:1808.04283
  80. 80.
    Hamster, C.H.S., Hupkes, H.J.: Travelling waves for reaction-diffusion equations forced by translation invariant noise, p. 1 (2019). arXiv:1906.01844
  81. 81.
    Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Springer, Berlin (1981) zbMATHCrossRefGoogle Scholar
  82. 82.
    Hohenberg, P.C., Halperin, B.I.: Theory of dynamic critical phenomena. Rev. Mod. Phys. 49(3), 435 (1977) CrossRefGoogle Scholar
  83. 83.
    Horridge, P., Tribe, R.: On stationary distributions for the KPP equation with branching noise. Ann. Inst. Henri Poincaré Probab. Stat. 40(6), 759–770 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  84. 84.
    Huang, Z., Liu, Z.: Random traveling wave and bifurcations of asymptotic behaviors in the stochastic KPP equation driven by dual noises. J. Differ. Equ. 261(2), 1317–1356 (2016) MathSciNetzbMATHCrossRefGoogle Scholar
  85. 85.
    Huang, Z., Liu, Z., Wang, Z.: Stochastic traveling wave solution to a stochastic KPP equation. J. Dyn. Differ. Equ. 28(2), 389–417 (2016) MathSciNetzbMATHCrossRefGoogle Scholar
  86. 86.
    Inglis, J., MacLaurin, J.: A general framework for stochastic traveling waves and patterns, with application to neural field equations. SIAM J. Appl. Dyn. Syst. 15(1), 195–234 (2016) MathSciNetzbMATHCrossRefGoogle Scholar
  87. 87.
    Kallenberg, O.: Foundations of Modern Probability, 2nd edn. Springer, New York (2002) zbMATHCrossRefGoogle Scholar
  88. 88.
    Kapitula, T., Promislow, K.: Spectral and Dynamical Stability of Nonlinear Waves. Springer, Berlin (2013) zbMATHCrossRefGoogle Scholar
  89. 89.
    Karazi, M.A., Lemarchand, A., Mareschal, M.: Fluctuation effects on chemical wave fronts. Phys. Rev. E 54(5), 4888 (1996) CrossRefGoogle Scholar
  90. 90.
    Kardar, M., Parisi, G., Zhang, Y.C.: Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56(9), 889–892 (1986) zbMATHCrossRefGoogle Scholar
  91. 91.
    Kessler, D.A., Ner, Z., Sander, L.M.: Front propagation: precursors, cutoffs, and structural stability. Phys. Rev. E 58(1), 107 (1998) CrossRefGoogle Scholar
  92. 92.
    Khain, E., Meerson, B.: Velocity fluctuations of noisy reaction fronts propagating into a metastable state. J. Phys. A 46(12), 125002 (2013) zbMATHCrossRefGoogle Scholar
  93. 93.
    Khain, E., Lin, Y.T., Sander, L.M.: Fluctuations and stability in front propagation. Europhys. Lett. 93(2), 28001 (2011) CrossRefGoogle Scholar
  94. 94.
    Kilpatrick, Z.P., Ermentrout, B.: Wandering bumps in stochastic neural fields. SIAM J. Appl. Dyn. Syst. 12(1), 61–94 (2013) MathSciNetzbMATHCrossRefGoogle Scholar
  95. 95.
    Kliem, S.: Convergence of rescaled competing species processes to a class of SPDEs. Electron. J. Probab. 16, 618–657 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  96. 96.
    Kliem, S.: Travelling wave solutions to the KPP equation with branching noise arising from initial conditions with compact support. Stoch. Process. Appl. 127(2), 385–418 (2017) MathSciNetzbMATHCrossRefGoogle Scholar
  97. 97.
    Kliem, S.: Right marker speeds of solutions to the KPP equation with noise, p. 1 (2018). arXiv:1806.05915
  98. 98.
    Kolmogorov, A., Petrovskii, I., Piscounov, N.: A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem. In: Tikhomirov, V.M. (ed.) Selected Works of A.N. Kolmogorov I, pp. 248–270. Kluwer, Dordrecht (1991). Translated by V.M. Volosov from Bull. Moscow Univ., Math. Mech. 1, 1–25 (1937) Google Scholar
  99. 99.
    Konotop, V.K., Vazquez, L.: Nonlinear Random Waves. World Scientific, Singapore (1994) zbMATHCrossRefGoogle Scholar
  100. 100.
    Kotelenez, P.: Comparison methods for a class of function valued stochastic partial differential equations. Probab. Theory Relat. Fields 93(1), 1–19 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  101. 101.
    Kotelenez, P.: Stochastic Ordinary and Stochastic Partial Differential Equations: Transition from Microscopic to Macroscopic Equations. Springer, Berlin (2007) zbMATHGoogle Scholar
  102. 102.
    Krüger, J., Stannat, W.: Front propagation in stochastic neural fields: a rigorous mathematical framework. SIAM J. Appl. Dyn. Syst. 13(3), 1293–1310 (2014) MathSciNetzbMATHCrossRefGoogle Scholar
  103. 103.
    Krüger, J., Stannat, W.: A multiscale-analysis of stochastic bistable reaction–diffusion equations. Nonlinear Anal. 162, 197–223 (2017) MathSciNetzbMATHCrossRefGoogle Scholar
  104. 104.
    Kuehn, C.: Warning signs for wave speed transitions of noisy Fisher-KPP invasion fronts. Theor. Ecol. 6(3), 295–308 (2013) MathSciNetCrossRefGoogle Scholar
  105. 105.
    Kuehn, C.: The curse of instability. Complexity 20(6), 9–14 (2015) MathSciNetCrossRefGoogle Scholar
  106. 106.
    Kuehn, C.: Multiple Time Scale Dynamics. Springer, Berlin (2015) zbMATHCrossRefGoogle Scholar
  107. 107.
    Kuehn, C.: Moment closure—a brief review. In: Schöll, E., Klapp, S., Hövel, P. (eds.) Control of Self-Organizing Nonlinear Systems, pp. 253–271. Springer, Berlin (2016) CrossRefGoogle Scholar
  108. 108.
    Kuehn, C.: PDE Dynamics: An Introduction. SIAM, Philadelphia (2019) zbMATHGoogle Scholar
  109. 109.
    Kuehn, C., Neamtu, A.: Pathwise mild solutions for quasilinear stochastic partial differential equations, pp. 1–41 (2018). arXiv:1802.10016
  110. 110.
    Kuehn, C., Riedler, M.G.: Large deviations for nonlocal stochastic neural fields. J. Math. Neurosci. 4(1), 1 (2014) MathSciNetzbMATHCrossRefGoogle Scholar
  111. 111.
    Kuehn, C., Tölle, J.M.: A gradient flow formulation for the stochastic Amari neural field model, p. 1 (2018). arXiv:1807.02575
  112. 112.
    Kuzovkov, V.N., Mai, J., Sokolov, I.M., Blumen, A.: Front propagation in the one-dimensional autocatalytic \(A+ B\rightarrow 2 A\) reaction with decay. Phys. Rev. E 59(3), 2561 (1999) CrossRefGoogle Scholar
  113. 113.
    Laing, C., Lord, G. (eds.): Stochastic Methods in Neuroscience. Oxford University Press, Oxford (2009) zbMATHGoogle Scholar
  114. 114.
    Laing, C.R., Troy, W.C.: PDE methods for nonlocal models. SIAM J. Appl. Dyn. Syst. 2(3), 487–516 (2003) MathSciNetzbMATHCrossRefGoogle Scholar
  115. 115.
    Lang, E.: A multiscale analysis of traveling waves in stochastic neural fields. SIAM J. Appl. Dyn. Syst. 15(3), 1581–1614 (2016) MathSciNetzbMATHCrossRefGoogle Scholar
  116. 116.
    Lee, K.: Generation and motion of interfaces in one-dimensional stochastic Allen-Cahn equation. J. Theor. Probab. 31(1), 268–293 (2018) MathSciNetzbMATHCrossRefGoogle Scholar
  117. 117.
    Lemarchand, A., Lesne, A., Mareschal, M.: Langevin approach to a chemical wave front: selection of the propagation velocity in the presence of internal noise. Phys. Rev. E 51(5), 4457–4465 (1995) CrossRefGoogle Scholar
  118. 118.
    Lin, J., Andreasen, V., Casagrandi, R., Levin, S.A.: Traveling waves in a model of influenza A drift. J. Theor. Biol. 222(4), 437–445 (2003) MathSciNetCrossRefGoogle Scholar
  119. 119.
    Lord, G.J., Thümmler, V.: Computing stochastic traveling waves. SIAM J. Sci. Comput. 34(1), B24–B43 (2012) MathSciNetzbMATHCrossRefGoogle Scholar
  120. 120.
    Lord, G.J., Powell, C.E., Shardlow, T.: An Introduction to Computational Stochastic PDEs. Cambridge University Press, Cambridge (2014) zbMATHCrossRefGoogle Scholar
  121. 121.
    Mai, J., Sokolov, I.M., Blumen, A.: Discreteness effects on the front propagation in the \(A+ B\rightarrow 2A\) reaction in 3 dimensions. Europhys. Lett. 44(1), 7 (1998) CrossRefGoogle Scholar
  122. 122.
    Manthey, R., Zausinger, T.: Stochastic evolution equations in \(L^{2\nu }_{p}\). Stochastics 66(1), 37–85 (1999) MathSciNetzbMATHGoogle Scholar
  123. 123.
    Meerson, B., Sasorov, P.V.: Negative velocity fluctuations of pulled reaction fronts. Phys. Rev. E 84(3), 030101 (2011) CrossRefGoogle Scholar
  124. 124.
    Meerson, B., Sasorov, P.V., Kaplan, Y.: Velocity fluctuations of population fronts propagating into metastable states. Phys. Rev. E 84(1), 011147 (2011) CrossRefGoogle Scholar
  125. 125.
    Méndez, V., Campos, D., Zemskov, E.P.: Variational principles and the shift in the front speed due to a cutoff. Phys. Rev. E 72(5), 056113 (2005) MathSciNetCrossRefGoogle Scholar
  126. 126.
    Mikhailov, A.S., Schimansky-Geier, L., Ebeling, W.: Stochastic motion of the propagating front in bistable media. Phys. Lett. A 96(9), 453–456 (1983) MathSciNetCrossRefGoogle Scholar
  127. 127.
    Moro, E.: Internal fluctuations effects on Fisher waves. Phys. Rev. Lett. 87(23), 238303 (2001) CrossRefGoogle Scholar
  128. 128.
    Mörters, P., Peres, Y.: Brownian Motion. Cambridge University Press, Cambridge (2010) zbMATHCrossRefGoogle Scholar
  129. 129.
    Mueller, C.: On the support of solutions to the heat equation with noise. Stochastics 37(4), 225–245 (1991) MathSciNetzbMATHGoogle Scholar
  130. 130.
    Mueller, C., Perkins, E.A.: The compact support property for solutions to the heat equation with noise. Probab. Theory Relat. Fields 93(3), 325–358 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  131. 131.
    Mueller, C., Sowers, R.B.: Random travelling waves for the KPP equation with noise. J. Funct. Anal. 128(2), 439–498 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  132. 132.
    Mueller, C., Tribe, R.: A phase transition for a stochastic PDE related to the contact process. Probab. Theory Relat. Fields 100, 131–156 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  133. 133.
    Mueller, C., Tribe, R.: Stochastic PDEs arising from the long range contact and long range voter processes. Probab. Theory Relat. Fields 102, 519–545 (1995) zbMATHCrossRefGoogle Scholar
  134. 134.
    Mueller, C., Tribe, R.: A phase diagram for a stochastic reaction diffusion system. Probab. Theory Relat. Fields 149, 561–637 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  135. 135.
    Mueller, C., Mytnik, L., Quastel, J.: Effect of noise on front propagation in reaction-diffusion equations of KPP type. Invent. Math. 184(2), 405–453 (2011) MathSciNetzbMATHCrossRefGoogle Scholar
  136. 136.
    Mueller, C., Mytnik, L., Ryzhik, L.: The speed of a random front for stochastic reaction-diffusion equations with strong noise, p. 1 (2019). arXiv:1903.03645
  137. 137.
    Nagumo, J., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proc. IRE 50, 2061–2070 (1962) CrossRefGoogle Scholar
  138. 138.
    Novikov, E.A.: Functionals and the random-force method in turbulence theory. Sov. Phys. JETP 20(5), 1290–1294 (1965) MathSciNetGoogle Scholar
  139. 139.
    Novikov, A.A.: On an identity for stochastic integrals. Theory Probab. Appl. 217(4), 717–720 (1973) zbMATHCrossRefGoogle Scholar
  140. 140.
    Øksendal, B.: Stochastic Differential Equations, 5th edn. Springer, Berlin (2003) zbMATHCrossRefGoogle Scholar
  141. 141.
    Øksendal, B., Våge, H., Zhao, H.Z.: Asymptotic properties of the solutions to stochastic KPP equations. Proc. R. Soc. Edinb. A 130(6), 1363–1381 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  142. 142.
    Øksendal, B., Våge, H., Zhao, H.Z.: Two properties of stochastic KPP equations: ergodicity and pathwise property. Nonlinearity 14(3), 639–662 (2001) MathSciNetzbMATHCrossRefGoogle Scholar
  143. 143.
    Panja, D.: Asymptotic scaling of the diffusion coefficient of fluctuating pulled fronts. Phys. Rev. E 68(6), 065202 (2003) MathSciNetCrossRefGoogle Scholar
  144. 144.
    Panja, D.: Effects of fluctuations on propagating fronts. Phys. Rep. 393(2), 87–174 (2004) CrossRefGoogle Scholar
  145. 145.
    Panja, D., van Saarloos, W.: Fronts with a growth cutoff but with speed higher than the linear spreading speed. Phys. Rev. E 66(1), 015206 (2002) CrossRefGoogle Scholar
  146. 146.
    Panja, D., van Saarloos, W.: Weakly pushed nature of “pulled” fronts with a cutoff. Phys. Rev. E 65(5), 057202 (2002) CrossRefGoogle Scholar
  147. 147.
    Pechenik, L., Levine, H.: Interfacial velocity corrections due to multiplicative noise. Phys. Rev. E 59(4), 3893–3900 (1999) CrossRefGoogle Scholar
  148. 148.
    Péseli, H.L.: Fluctuations in Physical Systems. Cambridge University Press, Cambridge (2000) Google Scholar
  149. 149.
    Prévot, C., Röckner, M.: A Concise Course on Stochastic Partial Differential Equations. Lecture Notes in Mathematics, vol. 1905. Springer, Berlin (2008) zbMATHGoogle Scholar
  150. 150.
    Protter, P.: Stochastic Integration and Differential Equations—Version 2.1. Springer, Berlin (2005) CrossRefGoogle Scholar
  151. 151.
    Riordan, J., Doering, C.R., Ben-Avraham, D.: Fluctuations and stability of Fisher waves. Phys. Rev. Lett. 75(3), 565 (1995) CrossRefGoogle Scholar
  152. 152.
    Robinson, J.C.: Infinite-Dimensional Dynamical Systems. Cambridge University Press, Cambridge (2001) CrossRefGoogle Scholar
  153. 153.
    Rocco, A., Ebert, U., van Saarloos, W.: Subdiffusive fluctuations of “pulled” fronts with multiplicative noise. Phys. Rev. E 62(1), R13 (2000) CrossRefGoogle Scholar
  154. 154.
    Rocco, A., Casademunt, J., Ebert, U., van Saarloos, W.: Diffusion coefficient of propagating fronts with multiplicative noise. Phys. Rev. E 65, 012102 (2001) CrossRefGoogle Scholar
  155. 155.
    Rocco, A., Ramirez-Piscina, L., Casademunt, J.: Kinematic reduction of reaction-diffusion fronts with multiplicative noise: derivation of stochastic sharp-interface equations. Phys. Rev. E 65, 056116 (2002) CrossRefGoogle Scholar
  156. 156.
    Romano, F., Kuehn, C.: Analysis and predictability for tipping points with leading-order nonlinear terms. Int. J. Bifurc. Chaos 28(8), 1850103 (2018) MathSciNetzbMATHCrossRefGoogle Scholar
  157. 157.
    Sandstede, B.: Stability of travelling waves. In: Fiedler, B. (ed.) Handbook of Dynamical Systems, vol. 2, pp. 983–1055. Elsevier, Amsterdam (2001) zbMATHGoogle Scholar
  158. 158.
    Santos, M.A., Sancho, J.M.: Noise-induced fronts. Phys. Rev. E 59(1), 98 (1999) CrossRefGoogle Scholar
  159. 159.
    Sattinger, D.H.: On the stability of waves of nonlinear parabolic systems. Adv. Math. 22(3), 312–355 (1976) MathSciNetzbMATHCrossRefGoogle Scholar
  160. 160.
    Sauer, M., Stannat, W.: Analysis and approximation of stochastic nerve axon equations. Math. Comput. 85(301), 2457–2481 (2016) MathSciNetzbMATHCrossRefGoogle Scholar
  161. 161.
    Sauer, M., Stannat, W.: Reliability of signal transmission in stochastic nerve axon equations. J. Comput. Neurosci. 40(1), 103–111 (2016) MathSciNetzbMATHCrossRefGoogle Scholar
  162. 162.
    Schimansky-Geier, L., Zülicke, C.: Kink propagation induced by multiplicative noise. Z. Phys. B 82(1), 157–162 (1991) CrossRefGoogle Scholar
  163. 163.
    Schimansky-Geier, L., Mikhailov, A.S., Ebeling, W.: Effect of fluctuation on plane front propagation in bistable nonequilibrium systems. Ann. Phys. 495(4), 277–286 (1983) CrossRefGoogle Scholar
  164. 164.
    Schlögl, F.: Chemical reaction models for non-equilibrium phase transitions. Z. Phys. 253(2), 147–161 (1972) MathSciNetCrossRefGoogle Scholar
  165. 165.
    Schlögl, F., Berry, R.S.: Small roughness fluctuations in the layer between two phases. Phys. Rev. A 21(6), 2078 (1980) MathSciNetCrossRefGoogle Scholar
  166. 166.
    Schneider, G., Uecker, H.: Nonlinear PDEs: A Dynamical Systems Approach. AMS, Providence (2017) zbMATHCrossRefGoogle Scholar
  167. 167.
    Sendina-Nadal, I., Alonso, S., Perez-Munuzuri, V., Gomez-Gesteira, M., Perez-Miller, V., Ramirez-Piscina, L., Casademunt, J., Sancho, J.M., Sagues, F.: Brownian motion of spiral waves driven by spatiotemporal structured noise. Phys. Rev. Lett. 84(12), 2734–2737 (2000) CrossRefGoogle Scholar
  168. 168.
    Shiga, T.: Two contrasting properties of solutions for one-dimensional stochastic partial differential equations. Can. J. Math. 46, 415–437 (1994) MathSciNetzbMATHCrossRefGoogle Scholar
  169. 169.
    Sieber, M., Malchow, H., Petrovskii, S.V.: Noise-induced suppression of periodic travelling waves in oscillatory reaction-diffusion systems. Proc. R. Soc. A 466(2119), 1903–1917 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  170. 170.
    Snyder, R.E.: How demographic stochasticity can slow biological invasions. Ecology 84(5), 1333–1339 (2003) CrossRefGoogle Scholar
  171. 171.
    Stannat, W.: Stability of travelling waves in stochastic Nagumo equations, pp. 1–22 (2013). arXiv:1301.6378
  172. 172.
    Stannat, W.: Stability of travelling waves in stochastic bistable reaction diffusion equations, pp. 1–28 (2014). arXiv:1404.3853
  173. 173.
    Temam, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics. Springer, Berlin (1997) zbMATHCrossRefGoogle Scholar
  174. 174.
    Tessitore, G., Zabczyk, J.: Strict positivity for stochastic heat equations. Stoch. Process. Appl. 77(1), 83–98 (1998) MathSciNetzbMATHCrossRefGoogle Scholar
  175. 175.
    Tribe, R.: Large time behavior of interface solutions to the heat equation with Fisher-Wright white noise. Probab. Theory Relat. Fields 102(3), 289–311 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  176. 176.
    Tribe, R.: A travelling wave solution to the Kolmogorov equation with noise. Stochastics 56(3), 317–340 (1996) MathSciNetzbMATHGoogle Scholar
  177. 177.
    Tribe, R., Woodward, N.: Stochastic order methods applied to stochastic travelling waves. Electron. J. Probab. 16(16), 436–469 (2013) MathSciNetzbMATHGoogle Scholar
  178. 178.
    Tripathy, G., van Saarloos, W.: Fluctuation and relaxation properties of pulled fronts: a scenario for nonstandard Kardar-Parisi-Zhang scaling. Phys. Rev. Lett. 85(17), 3556 (2000) CrossRefGoogle Scholar
  179. 179.
    Tripathy, G., Rocco, A., Casademunt, J., van Saarloos, W.: Universality class of fluctuating pulled fronts. Phys. Rev. Lett. 86(23), 5215 (2001) CrossRefGoogle Scholar
  180. 180.
    Tuckwell, H.C.: Analytical and simulation results for the stochastic spatial Fitzhugh-Nagumo model neuron. Neural Comput. 20(12), 3003–3033 (2008) MathSciNetzbMATHCrossRefGoogle Scholar
  181. 181.
    Tuckwell, H.C.: Stochastic partial differential equations in neurobiology: linear and nonlinear models for spiking neurons. In: Stochastic Biomathematical Models, pp. 149–173. Springer, Berlin (2013) CrossRefzbMATHGoogle Scholar
  182. 182.
    van Saarloos, W.: Front propagation into unstable states. Phys. Rep. 386, 29–222 (2003) zbMATHCrossRefGoogle Scholar
  183. 183.
    Volpert, A.I., Volpert, V., Volpert, V.A.: Traveling Wave Solutions of Parabolic Systems. Amer. Math. Soc., Providence (1994) zbMATHCrossRefGoogle Scholar
  184. 184.
    Volpert, V.A., Nec, Y., Nepomnyashchy, A.A.: Exact solutions in front propagation problems with superdiffusion. Physica D 239(3), 134–144 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  185. 185.
    Warren, C.P., Mikus, G., Somfai, E., Sander, L.M.: Fluctuation effects in an epidemic model. Phys. Rev. E 63(5), 056103 (2001) CrossRefGoogle Scholar
  186. 186.
    Xin, J.: An Introduction to Fronts in Random Media. Springer, Berlin (2009) zbMATHCrossRefGoogle Scholar
  187. 187.
    Zanette, D.H.: Wave fronts in bistable reactions with anomalous Lévy-flight diffusion. Phys. Rev. E 55, 1181–1184 (1997) CrossRefGoogle Scholar

Copyright information

© Deutsche Mathematiker-Vereinigung and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of MathematicsTechnical University of MunichGarching b. MünchenGermany

Personalised recommendations