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Finite-Time Blowup and Existence of Global Positive Solutions of a Semi-linear Stochastic Partial Differential Equation with Fractional Noise

In: Modern Stochastics and Applications

Author

Listed:
  • M. Dozzi

    (IECN, Université de Lorraine)

  • E. T. Kolkovska

    (Centro de Investigación en Matemáticas)

  • J. A. López-Mimbela

    (Centro de Investigación en Matemáticas)

Abstract

We consider stochastic equations of the prototype $$\displaystyle{\mathrm{d}u(t,x) = \left (\Delta u(t,x) +\gamma u(t,x) + u{(t,x)}^{1+\beta }\right )\mathrm{d}t +\kappa u(t,x)\,\mathrm{d}B_{ t}^{H}}$$ on a smooth domain $$D \subset {\mathbb{R}}^{d}$$ , with Dirichlet boundary condition, where β > 0, γ and κ are constants and $$\{B_{t}^{H}$$ , t ≥ 0} is a real-valued fractional Brownian motion with Hurst index H > 1∕2. By means of the associated random partial differential equation, obtained by the transformation $$v(t,x) = u(t,x)\exp \{\kappa B_{t}^{H}\}$$ , lower and upper bounds for the blowup time of u are given. Sufficient conditions for blowup in finite time and for the existence of a global solution are deduced in terms of the parameters of the equation. For the case H = 1∕2 (i.e. for Brownian motion), estimates for the probability of blowup in finite time are given in terms of the laws of exponential functionals of Brownian motion.

Suggested Citation

  • M. Dozzi & E. T. Kolkovska & J. A. López-Mimbela, 2014. "Finite-Time Blowup and Existence of Global Positive Solutions of a Semi-linear Stochastic Partial Differential Equation with Fractional Noise," Springer Optimization and Its Applications, in: Volodymyr Korolyuk & Nikolaos Limnios & Yuliya Mishura & Lyudmyla Sakhno & Georgiy Shevchenko (ed.), Modern Stochastics and Applications, edition 127, pages 95-108, Springer.
    • Handle: RePEc:spr:spochp:978-3-319-03512-3_6
    DOI: 10.1007/978-3-319-03512-3_6
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    Cited by:

    1. Dung, Nguyen Tien, 2019. "The probability of finite-time blowup of a semi-linear SPDE with fractional noise," Statistics & Probability Letters, Elsevier, vol. 149(C), pages 86-92.
    2. Nguyen, Tien Dung, 2018. "Tail estimates for exponential functionals and applications to SDEs," Stochastic Processes and their Applications, Elsevier, vol. 128(12), pages 4154-4170.

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