Electronic Journal of Probability

Stochastic evolution equations with Wick-polynomial nonlinearities

Tijana Levajković, Stevan Pilipović, Dora Seleši, and Milica Žigić

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Abstract

We study nonlinear parabolic stochastic partial differential equations with Wick-power and Wick-polynomial type nonlinearities set in the framework of white noise analysis. These equations include the stochastic Fujita equation, the stochastic Fisher-KPP equation and the stochastic FitzHugh-Nagumo equation among many others. By implementing the theory of $C_0-$semigroups and evolution systems into the chaos expansion theory in infinite dimensional spaces, we prove existence and uniqueness of solutions for this class of SPDEs. In particular, we also treat the linear nonautonomous case and provide several applications featured as stochastic reaction-diffusion equations that arise in biology, medicine and physics.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 116, 25 pp.

Dates
Received: 8 June 2018
Accepted: 30 October 2018
First available in Project Euclid: 24 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1543028704

Digital Object Identifier
doi:10.1214/18-EJP241

Subjects
Primary: 60H15: Stochastic partial differential equations [See also 35R60] 60H40: White noise theory 60G20: Generalized stochastic processes 37L55: Infinite-dimensional random dynamical systems; stochastic equations [See also 35R60, 60H10, 60H15] 47J35: Nonlinear evolution equations [See also 34G20, 35K90, 35L90, 35Qxx, 35R20, 37Kxx, 37Lxx, 47H20, 58D25] 11B83: Special sequences and polynomials

Keywords
Hida–Kondratiev spaces stochastic nonlinear evolution equations Wick product $C_0-$semigroup infinitesimal generator Catalan numbers

Rights
Creative Commons Attribution 4.0 International License.

Citation

Levajković, Tijana; Pilipović, Stevan; Seleši, Dora; Žigić, Milica. Stochastic evolution equations with Wick-polynomial nonlinearities. Electron. J. Probab. 23 (2018), paper no. 116, 25 pp. doi:10.1214/18-EJP241. https://projecteuclid.org/euclid.ejp/1543028704


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References

  • [1] Albeverio, S., Di Persio, L. S.: Some stochastic dynamical models in neurobiology: Recent developments. Europena Communications in Mathematical and Theoretical Biology 14, (2011), 44–53.
  • [2] Aronson, D., Weinberger, H.: Nonlinear diffusion in population genetics, combustion and nerve propagation. In J. A. Goldstein, editor, Partial Differential Equations and Related Topics, number 466 in Lecture Notes in Mathematics. Springer–Verlag, New York, 1975.
  • [3] Barbu, V., Cordoni, F., Di Persio, L.S.: Optimal control of stochastic FitzHugh–Nagumo equation. International Journal of Control 89(4), (2016), 746–756.
  • [4] Fisher, R.A.: The wave of advance of advantageous genes. Ann. Eug. 7, (1937), 353–369.
  • [5] FitzHugh, R.: Impulses and physiological states in theoretical models of nerve membrane. Biophysical Journal 1, (1961), 445–466.
  • [6] Fujita, H., Watanabe, S.: On the uniqueness and non-uniqueness of solutions of initial value problems for some quasi-linear parabolic equations. Comm. Pure. Appl. Math 21, (1968), 631–652.
  • [7] Fujita, H., Chen, Y. G.: On the set of blow-up points and asymptotic behaviours of blow-up solutions to a semilinear parabolic equation. Analyse mathématique et applications, 181–201, Gauthier–Villars, Montrouge, 1988.
  • [8] Hida, T., Kuo, H.-H., Pothoff, J., Streit, L.: White Noise. An Infinite-dimensional Calculus. Kluwer Academic Publishers Group, Dordrecht, 1993.
  • [9] Holden, H., Øksendal, B., Ubøe, J., Zhang, T.: Stochastic Partial Differential Equations. A Modeling, White Noise Functional Approach. Second Edition. Springer, New York, 2010.
  • [10] Huang, Z., Liu, Z.: Stochastic travelling wave solution to stochastic generalized KPP equation. Nonlinear Differ. Equ. Appl. 22, (2015), 143–173.
  • [11] Kaligotla, S.; Lototsky, S. V.: Wick product in the stochastic Burgers equation: a curse or a cure? Asymptot. Anal. 75(3-4), (2011), 145–168.
  • [12] Kato, T.: Linear evolution equations of “hyperbolic” type. II. J. Math. Soc. Japan 25, (1973), 648–666.
  • [13] Kolmogorov, A., Petrovskii, I., Piskunov, N.: Study of the diffusion equation with increase in the amount of substance and its application to a biological problem. Bull. State Univ. Mos. 1, (1937), 1–25.
  • [14] Levajković, T., Pilipović, S., Seleši, D., Žigić, M.: Stochastic evolution equations with multiplicative noise. Electron. J. Probab. 20(19), (2015), 1–23.
  • [15] Meneses, R., Quaas, A.: Fujita type exponent for fully nonlinear parabolic equations and existence results, Journal of Mathematical Analysis and Applications 376(2), (2011), 514–527.
  • [16] Mikulevicius, R., Rozovskii, B.: On unbiased stochastic Navier-Stokes equations. Probab. Theory Related Fields 154, (2012), 787–834.
  • [17] Nagumo, J., Arimoto, S., Yoshizawa, S.: An active pulse transmission line simulating nerve axon. Proceedings of the Institute of Radio Engineers 50(10), (1962), 2061–2070.
  • [18] Neidhardt, H., Zagrebnov, V. A.: Linear non-autonomous Cauchy problems and evaluation semigroups. Advan. Diff. Equat. 14, (2009), 289–340.
  • [19] Øksendal, B., Våge, G., Zhao, H. Z.: Asymptotic properties of the solutions to stochastic KPP equations. Proc. Roy. Soc. Edinburgh Sect. A 130(6), (2000), 1363–1381.
  • [20] Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences 44, Springer–Verlag, New York, 1983.
  • [21] Pilipović, S., Seleši, D.: Expansion theorems for generalized random processes, Wick products and applications to stochastic differential equations. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10(1), (2007), 79–110.
  • [22] Pilipović, S., Seleši, D.: On the generalized stochastic Dirichlet problem - Part I: The stochastic weak maximum principle. Potential Anal. 32, (2010), 363–387.
  • [23] Stanley, R.P.: Catalan Numbers. Cambridge University Press, New York, 2015.
  • [24] Yosida, K.: Time dependent evolution equations in a locally convex space. Math. Ann. 162, (1965/1966), 83–86.
  • [25] Zeidler, E.: Nonlinear functional analysis and its applications. I. Fixed-point theorems. Springer-Verlag, New York, 1986.