Limit theorems for time-dependent averages of nonlinear stochastic heat equations. (English) Zbl 1482.60086

Summary: We study limit theorems for time-dependent averages of the form \(X_t:=\frac{1}{2L(t)} \int_{-L(t)}^{L(t)} u(t,x)dx\), as \(t\to \infty \), where \(L(t)=\exp (\lambda t)\) and \(u(t,x)\) is the solution to a stochastic heat equation on \(\mathbb{R}_+\times \mathbb{R}\) driven by space-time white noise with \(u_0(x)=1\) for all \(x\in \mathbb{R} \). We show that for \(X_t \)
the weak law of large numbers holds when \(\lambda >\lambda_1 \),
the strong law of large numbers holds when \(\lambda >\lambda_2 \),
the central limit theorem holds when \(\lambda >\lambda_3 \), but fails when \(\lambda<\lambda_4\le\lambda_3 \),
the quantitative central limit theorem holds when \(\lambda >\lambda_5 \),
where \(\lambda_i \)’s are positive constants depending on the moment Lyapunov exponents of \(u(t,x)\).


60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60F05 Central limit and other weak theorems
60F15 Strong limit theorems
Full Text: DOI arXiv Link


[1] Ben Arous, G., Molchanov, S. and Ramírez, A.F. (2005). Transition from the annealed to the quenched asymptotics for a random walk on random obstacles. Ann. Probab. 33 2149-2187. · Zbl 1099.82003 · doi:10.1214/009117905000000404
[2] Carmona, R.A. and Molchanov, S.A. (1994). Parabolic Anderson problem and intermittency. Mem. Amer. Math. Soc. 108 viii+125. · Zbl 0925.35074 · doi:10.1090/memo/0518
[3] Chen, L., Khoshnevisan, D., Nualart, D. and Pu, F. (2019). Poincaré inequality, and central limit theorems for parabolic stochastic partial differential equations. Preprint. Available at arXiv:1912.01482.
[4] Chen, L., Khoshnevisan, D., Nualart, D. and Pu, F. (2019). Spatial ergodicity for SPDEs via Poincaré-type inequalities. Preprint. Available at arXiv:1912.01482.
[5] Chen, L., Khoshnevisan, D., Nualart, D. and Pu, F. (2020). Central limit theorems for spatial averges of the stochastic heat equation via Malliavin-Stein’s method. Preprint. Available at arXiv:2008.02408.
[6] 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. · Zbl 1361.60049 · doi:10.1214/15-AIHP719
[7] Chen, L. and Kim, K. (2020). Stochastic comparisons for stochastic heat equation. Electron. J. Probab. 25 Paper No. 140, 1-38. · Zbl 1468.60078 · doi:10.1214/20-ejp541
[8] 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. · Zbl 1333.60136 · doi:10.1214/15-AIHP673
[9] 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. · Zbl 1286.60060 · doi:10.1214/11-AOP717
[10] Cranston, M. and Molchanov, S. (2007). Quenched to annealed transition in the parabolic Anderson problem. Probab. Theory Related Fields 138 177-193. · Zbl 1136.60016 · doi:10.1007/s00440-006-0020-7
[11] Cranston, M., Mountford, T.S. and Shiga, T. (2002). Lyapunov exponents for the parabolic Anderson model. Acta Math. Univ. Comenian. (N.S.) 71 163-188. · Zbl 1046.60057
[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, 1-29. · Zbl 0922.60056 · doi:10.1214/EJP.v4-43
[13] Das, S. and Tsai, L.C. (2019). Fractional moments of the stochastic heat equation. Preprint. Available at arXiv:1910.09271. · Zbl 1472.60051
[14] Dharmadhikari, S.W. and Jogdeo, K. (1969). Bounds on moments of certain random variables. Ann. Math. Stat. 40 1506-1509. · Zbl 0208.44903 · doi:10.1214/aoms/1177697526
[15] Foondun, M. and Khoshnevisan, D. (2009). Intermittence and nonlinear parabolic stochastic partial differential equations. Electron. J. Probab. 14 548-568. · Zbl 1190.60051 · doi:10.1214/EJP.v14-614
[16] Gärtner, J. and Schnitzler, A. (2015). Stable limit laws for the parabolic Anderson model between quenched and annealed behaviour. Ann. Inst. Henri Poincaré Probab. Stat. 51 194-206. · Zbl 1321.60039 · doi:10.1214/13-AIHP574
[17] Ghosal, P. and Lin, Y. (2020). Lyapunov exponents of the SHE for general initial data. Preprint. Available at arXiv:2007.06505.
[18] Huang, J., Nualart, D. and Viitasaari, L. (2020). A central limit theorem for the stochastic heat equation. Stochastic Process. Appl. 130 7170-7184. · Zbl 1458.60072 · doi:10.1016/j.spa.2020.07.010
[19] Joseph, M., Khoshnevisan, D. and Mueller, C. (2017). Strong invariance and noise-comparison principles for some parabolic stochastic PDEs. Ann. Probab. 45 377-403. · Zbl 1367.60082 · doi:10.1214/15-AOP1009
[20] Khoshnevisan, D. (2014). Analysis of Stochastic Partial Differential Equations. CBMS Regional Conference Series in Mathematics 119. Providence, RI: Amer. Math. Soc.. · Zbl 1304.60005 · doi:10.1090/cbms/119
[21] Khoshnevisan, D., Kim, K. and Xiao, Y. (2017). Intermittency and multifractality: A case study via parabolic stochastic PDEs. Ann. Probab. 45 3697-3751. · Zbl 1418.60081 · doi:10.1214/16-AOP1147
[22] Khoshnevisan, D., Kim, K. and Xiao, Y. (2018). A macroscopic multifractal analysis of parabolic stochastic PDEs. Comm. Math. Phys. 360 307-346. · Zbl 1454.60095 · doi:10.1007/s00220-018-3136-6
[23] Mueller, C. (1991). On the support of solutions to the heat equation with noise. Stoch. Stoch. Rep. 37 225-245. · Zbl 0749.60057 · doi:10.1080/17442509108833738
[24] Nourdin, I. and Peccati, G. (2012). Normal Approximations with Malliavin Calculus: From Stein’s Method to Universality. Cambridge Tracts in Mathematics 192. Cambridge: Cambridge Univ. Press. · Zbl 1266.60001 · doi:10.1017/CBO9781139084659
[25] von Bahr, B. and Esseen, C.-G. (1965). Inequalities for the \(r\) th absolute moment of a sum of random variables, \[1\le r\le 2\]. Ann. Math. Stat. 36 299-303. · Zbl 0134.36902 · doi:10.1214/aoms/1177700291
[26] 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. Berlin: Springer · Zbl 0608.60060 · doi:10.1007/BFb0074920
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.