The Annals of Probability

Global solutions to stochastic reaction–diffusion equations with super-linear drift and multiplicative noise

Robert C. Dalang, Davar Khoshnevisan, and Tusheng Zhang

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Let $\xi (t,x)$ denote space–time white noise and consider a reaction–diffusion equation of the form \[\dot{u}(t,x)=\frac{1}{2}u"(t,x)+b\big(u(t,x)\big)+\sigma \big(u(t,x)\big)\xi (t,x),\] on $\mathbb{R}_{+}\times [0,1]$, with homogeneous Dirichlet boundary conditions and suitable initial data, in the case that there exists $\varepsilon >0$ such that $\vert b(z)\vert \ge \vert z\vert (\log \vert z\vert )^{1+\varepsilon }$ for all sufficiently-large values of $\vert z\vert $. When $\sigma \equiv 0$, it is well known that such PDEs frequently have nontrivial stationary solutions. By contrast, Bonder and Groisman [Phys. D 238 (2009) 209–215] have recently shown that there is finite-time blowup when $\sigma $ is a nonzero constant. In this paper, we prove that the Bonder–Groisman condition is unimprovable by showing that the reaction–diffusion equation with noise is “typically” well posed when $\vert b(z)\vert =O(\vert z\vert \log_{+}\vert z\vert )$ as $\vert z\vert \to \infty $. We interpret the word “typically” in two essentially-different ways without altering the conclusions of our assertions.

Article information

Ann. Probab., Volume 47, Number 1 (2019), 519-559.

Received: June 2017
Revised: March 2018
First available in Project Euclid: 13 December 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H15: Stochastic partial differential equations [See also 35R60] 35K57: Reaction-diffusion equations
Secondary: 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 35B45: A priori estimates 35B33: Critical exponents

Stochastic partial differential equations reaction–diffusion equations blow-up logarithmic Sobolev inequality


Dalang, Robert C.; Khoshnevisan, Davar; Zhang, Tusheng. Global solutions to stochastic reaction–diffusion equations with super-linear drift and multiplicative noise. Ann. Probab. 47 (2019), no. 1, 519--559. doi:10.1214/18-AOP1270.

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  • [1] Bally, V., Millet, A. and Sanz-Solé, M. (1995). Approximation and support theorem in Hölder norm for parabolic stochastic partial differential equations. Ann. Probab. 23 178–222.
  • [2] Bass, R. F. (1998). Diffusions and Elliptic Operators. Probability and Its Applications (New York). Springer, New York.
  • [3] Bonder, J. F. and Groisman, P. (2009). Time-space white noise eliminates global solutions in reaction–diffusion equations. Phys. D 238 209–215.
  • [4] Cerrai, S. (2003). Stochastic reaction–diffusion systems with multiplicative noise and non-Lipschitz reaction term. Probab. Theory Related Fields 125 271–304.
  • [5] Cerrai, S. (2011). Averaging principle for systems of reaction–diffusion equations with polynomial nonlinearities perturbed by multiplicative noise. SIAM J. Math. Anal. 43 2482–2518.
  • [6] Chen, L. and Dalang, R. C. (2014). Hölder-continuity for the nonlinear stochastic heat equation with rough initial conditions. Stoch. Partial Differ. Equ. Anal. Comput. 2 316–352.
  • [7] Dalang, R., Khoshnevisan, D., Mueller, C., Nualart, D. and Xiao, Y. (2009). A Minicourse on Stochastic Partial Differential Equations. Lecture Notes in Math. 1962. Springer, Berlin.
  • [8] Dalang, R. C. (1999). Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.’s. Electron. J. Probab. 4 Article ID 6.
  • [9] Dalang, R. C., Khoshnevisan, D. and Nualart, E. (2007). Hitting probabilities for systems of non-linear stochastic heat equations with additive noise. ALEA Lat. Am. J. Probab. Math. Stat. 3 231–271.
  • [10] Dalang, R. C., Khoshnevisan, D. and Nualart, E. (2009). Hitting probabilities for systems for non-linear stochastic heat equations with multiplicative noise. Probab. Theory Related Fields 144 371–427.
  • [11] Donati-Martin, C. and Pardoux, É. (1993). White noise driven SPDEs with reflection. Probab. Theory Related Fields 95 1–24.
  • [12] Fang, S. and Zhang, T. (2005). A study of a class of stochastic differential equations with non-Lipschitzian coefficients. Probab. Theory Related Fields 132 356–390.
  • [13] Foondun, M. and Khoshnevisan, D. (2009). Intermittence and nonlinear parabolic stochastic partial differential equations. Electron. J. Probab. 14 548–568.
  • [14] Galaktionov, V. A. and Vázquez, J. L. (2002). The problem of blow-up in nonlinear parabolic equations. Discrete Contin. Dyn. Syst. 8 399–433. Current developments in partial differential equations (Temuco, 1999).
  • [15] Garsia, A. M. (1972). Continuity properties of Gaussian processes with multidimensional time parameter. In Proc. Sixth Berkeley Symp. on Math. Statist. and Probab. Vol. II: Probability Theory 369–374. Univ. California Press, Berkeley, CA.
  • [16] Garsia, A. M. and Rodemich, E. (1974). Monotonicity of certain functionals under rearrangement. Ann. Inst. Fourier (Grenoble) 24 67–116. Colloque International sur les Processus Gaussiens et les Distributions Aléatoires (Colloque Internat. du CNRS, No. 222, Strasbourg, 1973).
  • [17] Gross, L. (1993). Logarithmic Sobolev inequalities and contractivity properties of semigroups. In Dirichlet Forms (Varenna, 1992). Lecture Notes in Math. 1563 54–88. Springer, Berlin.
  • [18] Gyöngy, I. and Rovira, C. (2000). On $L^{p}$-solutions of semilinear stochastic partial differential equations. Stochastic Process. Appl. 90 83–108.
  • [19] Khoshnevisan, D. (2014). Analysis of Stochastic Partial Differential Equations. CBMS Regional Conference Series in Mathematics 119. Amer. Math. Soc., Providence, RI.
  • [20] Kunita, H. (1990). Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics 24. Cambridge Univ. Press, Cambridge.
  • [21] Liu, W. and Röckner, M. (2015). Stochastic Partial Differential Equations: An Introduction. Universitext. Springer, Cham.
  • [22] Mueller, C. (1991). On the support of solutions to the heat equation with noise. Stoch. Stoch. Rep. 37 225–245.
  • [23] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Probability and Its Applications (New York). Springer, Berlin.
  • [24] Nualart, D. and Pardoux, E. (1994). Markov field properties of solutions of white noise driven quasi-linear parabolic PDEs. Stoch. Stoch. Rep. 48 17–44.
  • [25] Pardoux, E. (1979). Stochastic partial differential equations and filtering of diffusion processes. Stochastics 3 127–167.
  • [26] Samarskii, A. A., Galaktionov, V. A., Kurdyumov, S. P. and Mikhailov, A. P. (1995). Blow-up in Quasilinear Parabolic Equations. De Gruyter Expositions in Mathematics 19. de Gruyter, Berlin. Translated from the 1987 Russian original by Michael Grinfeld and revised by the authors.
  • [27] Sharpe, M. (1988). General Theory of Markov Processes. Pure and Applied Mathematics 133. Academic Press, Boston, MA.
  • [28] Walsh, J. B. (1986). An introduction to stochastic partial differential equations. In École D’été de Probabilités de Saint-Flour XIV—1984. Lecture Notes in Math. 1180 265–439. Springer, Berlin.