Initial measures for the stochastic heat equation

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

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 1, pp. 136-153

We consider a family of nonlinear stochastic heat equations of the form ${\partial }_{t}u=ℒu+\sigma \left(u\right)\dot{W}$, where $\dot{W}$ denotes space-time white noise, $ℒ$ the generator of a symmetric Lévy process on $𝐑$, 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 $ℒf=c{f}^{\text{'}\text{'}}$ 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 ${\partial }_{t}u=ℒu+\sigma \left(u\right)\dot{W}$, où $\dot{W}$ est un bruit-blanc espace-temps, $ℒ$ est le générateur d’un processus de Lévy symétrique sur $𝐑$, 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 $ℒf=c{f}^{\text{'}\text{'}}$ 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 $t>0$.

Classification : 60H15, 35R60
Keywords: The stochastic heat equation, singular initial data

[1] L. Bertini and N. Cancrini. The stochastic heat equation: Feynman-Kac formula and intermittence. J. Stat. Phys. 78 (1994) 1377-1402. | Zbl | MR

[2] A. Borodin and I. Corwin. Macdonald processes. Preprint, 2012. Available at http://arxiv.org/abs/1111.4408. | MR

[3] D. L. Burkholder. Martingale transforms. Ann. Math. Statist. 37 (1966) 1494-1504. | Zbl | MR

[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. | Zbl | MR

[5] D. L. Burkholder and R. F. Gundy. Extrapolation and interpolation of quasi-linear operators on martingales. Acta Math. 124 (1970) 249-304. | Zbl | MR

[6] E. Carlen and P. Kree. $L^p$ estimates for multiple stochastic integrals. Ann. Probab. 19 (1991) 354-368. | Zbl | MR

[7] R. A. Carmona and S. A. Molchanov. Parabolic Anderson problem and intermittency. Mem. Amer. Math. Soc. 108 (1994) vii + 129. | Zbl | MR

[8] L. Chen and R. C. Dalang. Parabolic Anderson model driven by space-time white noise in ${𝐑}^{1+1}$ 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. | Zbl | MR

[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 http://arxiv.org/abs/1104.0189. | Zbl | MR

[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). | Zbl | MR

[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). | Zbl | MR

[13] B. Davis. On the $L^p$ norms of stochastic integrals and other martingales. Duke Math. J. 43 (1976) 697-704. | Zbl | MR

[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. | Zbl | MR | Numdam

[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). | Zbl | MR

[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. | Zbl | MR

[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. | Zbl | MR

[18] I. Gyöngy and D. Nualart. On the stochastic Burgers' equation in the real line. Ann. Probab. 27 (1999) 782-802. | Zbl | MR

[19] N. Jacob. Pseudo Differential Operators and Markov Processes, Vol. III. Imperial College Press, London, 2005. | Zbl | MR

[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. | Zbl

[22] C. Mueller. On the support of solutions to the heat equation with noise. Stochastics Stochastics Rep. 37 (1991) 225-245. | Zbl | MR

[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. | Zbl | MR