Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Initial measures for the stochastic heat equation

Daniel Conus, Mathew Joseph, Davar Khoshnevisan, and Shang-Yuan Shiu

Full-text: Open access


We consider a family of nonlinear stochastic heat equations of the form $\partial_{t}u=\mathcal{L}u+\sigma(u)\dot{W}$, where $\dot{W}$ denotes space–time white noise, $\mathcal{L}$ the generator of a symmetric Lévy process on $\mathbf{R} $, and $\sigma$ is Lipschitz continuous and zero at 0. We show that this stochastic PDE has a random-field solution for every finite initial measure $u_{0}$. Tight a priori bounds on the moments of the solution are also obtained.

In the particular case that $\mathcal{L}f=cf''$ for some $c>0$, we prove that if $u_{0}$ is a finite measure of compact support, then the solution is with probability one a bounded function for all times $t>0$.


Nous considérons une famille d’équations de la chaleur stochastique de la forme $\partial_{t}u=\mathcal{L}u+\sigma(u)\dot{W}$, où $\dot{W}$ est un bruit-blanc espace–temps, $\mathcal{L}$ est le générateur d’un processus de Lévy symétrique sur $\mathbf{R} $, et $\sigma$ est une fonction lipschizienne s’annulant en $0$. Nous montrons que cette équation aux dérivées partielles stochastique a une solution de type champ aléatoire pour toute mesure initiale finie $u_{0}$. Nous obtenons également des bornes a priori sur les moments de la solution.

Dans le cas particulier où $\mathcal{L}f=cf''$ pour un $c>0$, nous montrons que si $u_{0}$ est une mesure finie à support compact, la solution est presque sûrement une fonction bornée pour tout $t>0$.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 50, Number 1 (2014), 136-153.

First available in Project Euclid: 1 January 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H15: Stochastic partial differential equations [See also 35R60]
Secondary: 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15]

The stochastic heat equation Singular initial data


Conus, Daniel; Joseph, Mathew; Khoshnevisan, Davar; Shiu, Shang-Yuan. Initial measures for the stochastic heat equation. Ann. Inst. H. Poincaré Probab. Statist. 50 (2014), no. 1, 136--153. doi:10.1214/12-AIHP505.

Export citation


  • [1] L. Bertini and N. Cancrini. The stochastic heat equation: Feynman–Kac formula and intermittence. J. Stat. Phys. 78 (1994) 1377–1402.
  • [2] A. Borodin and I. Corwin. Macdonald processes. Preprint, 2012. Available at
  • [3] D. L. Burkholder. Martingale transforms. Ann. Math. Statist. 37 (1966) 1494–1504.
  • [4] D. L. Burkholder, B. J. Davis and R. F. Gundy. Integral inequalities for convex functions of operators on martingales. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability II 223–240. Univ. California Press, Berkeley, CA, 1972.
  • [5] D. L. Burkholder and R. F. Gundy. Extrapolation and interpolation of quasi-linear operators on martingales. Acta Math. 124 (1970) 249–304.
  • [6] E. Carlen and P. Kree. $L^{p}$ estimates for multiple stochastic integrals. Ann. Probab. 19 (1991) 354–368.
  • [7] R. A. Carmona and S. A. Molchanov. Parabolic Anderson problem and intermittency. Mem. Amer. Math. Soc. 108 (1994) vii + 129.
  • [8] L. Chen and R. C. Dalang. Parabolic Anderson model driven by space–time white noise in $\mathbf{R}^{1+1}$ with Schwartz distribution-valued initial data: Solutions and explicit formula for second moments. Preprint, 2011.
  • [9] D. Conus and D. Khoshnevisan. Weak nonmild solutions to some SPDEs. Illinois J. Math. 54(4) (2010) 1329–1341.
  • [10] D. Conus, M. Joseph and D. Khoshnevisan. On the chaotic character of the stochastic heat equation, before the onset of intermittency. Ann. Probab. To appear. Available at
  • [11] R. C. Dalang. Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.’s. Electron. J. Probab. 4 (1999) Paper no. 6, 29 (electronic).
  • [12] R. C. Dalang and C. Mueller. Some non-linear S.P.D.E.’s that are second order in time. Electron. J. Probab. 8 (2003) Paper no. 1, 21 (electronic).
  • [13] B. Davis. On the $L^{p}$ norms of stochastic integrals and other martingales. Duke Math. J. 43 (1976) 697–704.
  • [14] M. Foondun and D. Khoshnevisan. On the global maximum of the solution to a stochastic heat equation with compact-support initial data, Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010) 895–907.
  • [15] M. Foondun and D. Khoshnevisan. Intermittence and nonlinear parabolic stochastic partial differential equations. Electron J. Probab. 14 (2009) Paper no. 12, 548–568 (electronic).
  • [16] M. Foondun, D. Khoshnevisan and E. Nualart. A local time correspondence for stochastic partial differential equations. Trans. Amer. Math. Soc. 363 (2011) 2481–2515.
  • [17] I. M. Gel’fand and N. Y. Vilenkin. Generalized Functions, Vol. 4: Applications of harmonic analysis. Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1964 [1977]. Translated from the Russian by Amiel Feinstein.
  • [18] I. Gyöngy and D. Nualart. On the stochastic Burgers’ equation in the real line. Ann. Probab. 27 (1999) 782–802.
  • [19] N. Jacob. Pseudo Differential Operators and Markov Processes, Vol. III. Imperial College Press, London, 2005.
  • [20] M. Kardar. Roughening by impurities at finite temperatures. Phys. Rev. Lett. 55 (1985) 2923.
  • [21] M. Kardar, G. Parisi and Y.-C. Zhang. Dynamic scaling of growing interfaces. Phys. Rev. Lett. 56 (1986) 889–892.
  • [22] C. Mueller. On the support of solutions to the heat equation with noise. Stochastics Stochastics Rep. 37 (1991) 225–245.
  • [23] J. B. Walsh. An introduction to stochastic partial differential equations. In École d’été de probabilités de Saint-Flour, XIV – 1984 265–439. Lecture Notes in Math. 1180. Springer, Berlin, 1986.