Abstract
We consider stochastic equations of the prototype
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.
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References
Arcones, M.A.: On the law of the iterated logarithm for Gaussian processes. J. Theor. Probab. 8, 877–903 (1995)
Bandle, C., Dozzi, M., Schott, R.: Blow up behaviour of a stochastic partial differential equation of reaction-diffusion type. In: Stochastic Analysis: Random Fields and Measure-Valued Processes. Israel Mathematical Conference Proceedings, vol. 10, pp. 27–33. Bar-Ilan University, Ramat Gan (1996)
Borodin, A.N., Salminen, P.: Handbook of Brownian Motion—Facts and Formulae, 2nd edn. Birkhäuser, Basel (2002)
Dozzi, M., López-Mimbela, J.A.: Finite-time blowup and existence of global positive solutions of a semi-linear SPDE. Stoch. Process. Appl. 120(6), 767–776 (2010)
Dufresne, D.: The distribution of a perpetuity, with applications to risk theory and pension funding. Scand. Actuarial J. 9, 39–79 (1990)
Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice-Hall Inc., Englewood Cliffs (1964)
Fujita, H.: On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations. In: Nonlinear Functional Analysis. American Mathematical Society, Providence, RI (1970). Proceedings of Symposia in Pure Mathematics, vol. 18, Part 1, pp. 105–113, Chicago (1968)
Maslowski, B., Nualart, D.: Evolution equations driven by a fractional Brownian motion. J. Funct. Anal. 202, 277–305 (2003)
Matsumoto, H., Yor, M.: Exponential functionals of Brownian motion. I. Probability laws at fixed time. Probab. Surv. 2, 312–347 (2005)
Mishura, Y.: Stochastic calculus for fractional Brownian motion and related processes. Springer Lecture Notes in Mathematics, vol. 1929. Springer, Berlin (2008)
Nualart, D., Vuillermot, P.: Variational solutions for partial differential equations driven by a fractional noise. J. Funct. Anal. 232, 390–454 (2006)
Ohashi, A.M.F.: Stochastic evolution equations driven by a fractional white noise. Stoch. Anal. Appl. 24, 555–578 (2006)
Ouhabaz, E.M., Wang, F.-Y.: Sharp estimates for intrinsic ultracontractivity on \({C}^{1,\alpha }\)-domains. Manuscripta Math. 122(2), 229–244 (2007)
Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences, vol. 44. Springer, New York (1983)
Pintoux, C., Privault, N.: A direct solution to the Fokker-Planck equation for exponential Brownian functionals. Anal. Appl. (Singapore) 8, 287–304 (2010)
Salminen, P., Yor, M.: Properties of perpetual integral functionals of Brownian motion with drift. Ann. Inst. H. Poincaré Probab. Stat. 41(3), 335–347 (2005)
Sanz-Solé, M., Vuillermot, P.: Mild solutions for a class of fractional SPDE’s and their sample paths. J. Evol. Equat. 9, 235–265 (2009)
Yor, M.: On some exponential functionals of Brownian motion. Adv. Appl. Probab. 24, 509–531 (1992)
Zähle, M.: Integration with respect to fractal functions and stochastic calculus I. Probab. Theory Relat. Fields 111, 333–374 (2001)
Zähle, M.: Integration with respect to fractal functions and stochastic calculus II. Math. Nachr. 225, 145–183 (2001)
Acknowledgements
This research was partially supported by the CNRS-CONACyT research grant “Blow-up of parabolic stochastic partial differential equations”. M. Dozzi acknowledges the European Commission for partial support by the grant PIRSES 230804 “Multifractionality”; he also acknowledges the hospitality of CIMAT, Guanajuato, where part of this work was done. E.T. Kolkovska and J.A. López-Mimbela acknowledge the hospitality of Institut Elie Cartan, Université de Lorraine.
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Dozzi, M., Kolkovska, E.T., López-Mimbela, J.A. (2014). Finite-Time Blowup and Existence of Global Positive Solutions of a Semi-linear Stochastic Partial Differential Equation with Fractional Noise. In: Korolyuk, V., Limnios, N., Mishura, Y., Sakhno, L., Shevchenko, G. (eds) Modern Stochastics and Applications. Springer Optimization and Its Applications, vol 90. Springer, Cham. https://doi.org/10.1007/978-3-319-03512-3_6
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