Annals of Applied Probability

An approximation result for a class of stochastic heat equations with colored noise

Mohammud Foondun, Mathew Joseph, and Shiu-Tang Li

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We show that a large class of stochastic heat equations can be approximated by systems of interacting stochastic differential equations. As a consequence, we prove various comparison principles extending earlier works of [Stoch. Stoch. Rep. 37 (1991) 225–245] and [Ann. Probab. 45 (2017) 377–403] among others. Among other things, our results enable us to obtain sharp estimates on the moments of the solution. A main technical ingredient of our method is a local limit theorem which is of independent interest.

Article information

Ann. Appl. Probab., Volume 28, Number 5 (2018), 2855-2895.

Received: December 2016
Revised: October 2017
First available in Project Euclid: 28 August 2018

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: 35K57: Reaction-diffusion equations

Stochastic PDEs comparison theorems colored noise


Foondun, Mohammud; Joseph, Mathew; Li, Shiu-Tang. An approximation result for a class of stochastic heat equations with colored noise. Ann. Appl. Probab. 28 (2018), no. 5, 2855--2895. doi:10.1214/17-AAP1376.

Export citation


  • [1] Bertini, L. and Cancrini, N. (1995). The stochastic heat equation: Feynman-Kac formula and intermittence. J. Stat. Phys. 78 1377–1401.
  • [2] Boulanba, L., Eddahbi, M. and Mellouk, M. (2010). Fractional SPDEs driven by spatially correlated noise: Existence of the solution and smoothness of its density. Osaka J. Math. 47 41–65.
  • [3] Carmona, R. A. and Molchanov, S. A. (1994). Parabolic Anderson problem and intermittency. Mem. Amer. Math. Soc. 108 viii$+$125.
  • [4] Chen, L. and Dalang, R. C. (2015). Moments and growth indices for the nonlinear stochastic heat equation with rough initial conditions. Ann. Probab. 43 3006–3051.
  • [5] Chen, L. and Kim, K. (2017). On comparison principle and strict positivity of solutions to the nonlinear stochastic fractional heat equations. Ann. Inst. Henri Poincaré Probab. Stat. 53 358–388.
  • [6] Chen, X. (2015). Precise intermittency for the parabolic Anderson equation with an $(1+1)$-dimensional time-space white noise. Ann. Inst. Henri Poincaré Probab. Stat. 51 1486–1499.
  • [7] Chen, X. (2016). Spatial asymptotics for the parabolic Anderson models with generalized time-space Gaussian noise. Ann. Probab. 44 1535–1598.
  • [8] Conus, D., Joseph, M. and Khoshnevisan, D. (2013). On the chaotic character of the stochastic heat equation, before the onset of intermitttency. Ann. Probab. 41 2225–2260.
  • [9] Conus, D., Joseph, M., Khoshnevisan, D. and Shiu, S.-Y. (2013). On the chaotic character of the stochastic heat equation, II. Probab. Theory Related Fields 156 483–533.
  • [10] Cox, J. T., Fleischmann, K. and Greven, A. (1996). Comparison of interacting diffusions and an application to their ergodic theory. Probab. Theory Related Fields 105 513–528.
  • [11] 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.
  • [12] Dalang, R. C. (1999). Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.’s. Electron. J. Probab. 4 no. 6, 29 pp.
  • [13] Davar, K. (2011). Topics in Probability: Lévy Processes. Springer, New York.
  • [14] Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd ed. Wiley, New York.
  • [15] Foondun, M. and Khoshnevisan, D. (2009). Intermittence and nonlinear parabolic stochastic partial differential equations. Electron. J. Probab. 14 548–568.
  • [16] Foondun, M. and Khoshnevisan, D. (2013). On the stochastic heat equation with spatially-colored random forcing. Trans. Amer. Math. Soc. 365 409–458.
  • [17] Funaki, T. (1983). Random motion of strings and related stochastic evolution equations. Nagoya Math. J. 89 129–193.
  • [18] Geiß, C. and Manthey, R. (1994). Comparison theorems for stochastic differential equations in finite and infinite dimensions. Stochastic Process. Appl. 53 23–35.
  • [19] Georgiou, N., Joseph, M., Khoshnevisan, D. and Shiu, S.-Y. (2015). Semi-discrete semi-linear parabolic SPDEs. Ann. Appl. Probab. 25 2959–3006.
  • [20] Grafakos, L. (2008). Classical Fourier Analysis, 2nd ed. Graduate Texts in Mathematics 249. Springer, New York.
  • [21] Hairer, M. (2013). Solving the KPZ equation. Ann. of Math. (2) 178 559–664.
  • [22] Huang, J., Lê, K. and Nualart, D. (2017). Large time asymptotics for the parabolic Anderson model driven by spatially correlated noise. Ann. Inst. Henri Poincaré Probab. Stat. 53 1305–1340.
  • [23] Joseph, M., Khoshnevisan, D. and Mueller, C. (2017). Strong invariance and noise-comparison principles for some parabolic stochastic PDEs. Ann. Probab. 45 377–403.
  • [24] Kim, K. On the large-scale structure of the tall peaks for stochastic heat equations with fractional Laplacian. Preprint.
  • [25] Kolokoltsov, V. (2000). Symmetric stable laws and stable-like jump-diffusions. Proc. Lond. Math. Soc. (3) 80 725–768.
  • [26] Li, S.-T. (2016). Comparison principles of solutions to stochastic heat equations. Ph.D. thesis, Univ. Utah, Salt Lake City, UT.
  • [27] Manthey, R. and Zausinger, T. (1999). Stochastic evolution equations in $L^{2\nu}_{\rho}$. Stoch. Stoch. Rep. 66 37–85.
  • [28] Mueller, C. (1991). On the support of solutions to the heat equation with noise. Stoch. Stoch. Rep. 37 225–245.
  • [29] Samorodnitsky, G. and Taqqu, M. S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Stochastic Modeling. Chapman & Hall, New York.
  • [30] Shiga, T. and Shimizu, A. (1980). Infinite-dimensional stochastic differential equations and their applications. J. Math. Kyoto Univ. 20 395–416.
  • [31] Stein, E. M. (1993). Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series 43. Princeton Univ. Press, Princeton, NJ.
  • [32] 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.