A forward-backward SDE from the 2D nonlinear stochastic heat equation. (English) Zbl 1487.35463

Summary: We consider a nonlinear stochastic heat equation in spatial dimension \(d=2\), forced by a white-in-time multiplicative Gaussian noise with spatial correlation length \(\varepsilon > 0\) but divided by a factor of \(\sqrt{\log{\varepsilon^{-1}}} \). We impose a condition on the Lipschitz constant of the nonlinearity so that the problem is in the “weak noise” regime. We show that, as \(\varepsilon \downarrow 0\), the one-point distribution of the solution converges, with the limit characterized in terms of the solution to a forward-backward stochastic differential equation (FBSDE). We also characterize the limiting multipoint statistics of the solution, when the points are chosen on appropriate scales, in similar terms. Our approach is new even for the linear case, in which the FBSDE can be solved explicitly and we recover results of F. Caravenna et al. [Ann. Appl. Probab. 27, No. 5, 3050–3112 (2017; Zbl 1387.82032)].


35R60 PDEs with randomness, stochastic partial differential equations
35K15 Initial value problems for second-order parabolic equations
35K58 Semilinear parabolic equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)


Zbl 1387.82032
Full Text: DOI arXiv


[1] Alberts, T., Khanin, K. and Quastel, J. (2014). The intermediate disorder regime for directed polymers in dimension \[1+1\]. Ann. Probab. 42 1212-1256. · Zbl 1292.82014 · doi:10.1214/13-AOP858
[2] BERTINI, L. and CANCRINI, N. (1995). The stochastic heat equation: Feynman-Kac formula and intermittence. J. Stat. Phys. 78 1377-1401. · Zbl 1080.60508 · doi:10.1007/BF02180136
[3] BERTINI, L. and CANCRINI, N. (1998). The two-dimensional stochastic heat equation: Renormalizing a multiplicative noise. J. Phys. A 31 615-622. · Zbl 0976.82035 · doi:10.1088/0305-4470/31/2/019
[4] CANNIZZARO, G., ERHARD, D. and SCHÖNBAUER, P. (2021). 2D anisotropic KPZ at stationarity: Scaling, tightness and nontriviality. Ann. Probab. 49 122-156. · Zbl 1457.60112 · doi:10.1214/20-AOP1446
[5] CANNIZZARO, G., ERHARD, D. and TONINELLI, F. The stationary AKPZ equation: Logarithmic superdiffusivity. Preprint. Available at arXiv:2007.12203v3.
[6] CANNIZZARO, G., ERHARD, D. and TONINELLI, F. Weak coupling limit of the anisotropic KPZ equation. Preprint. Available at arXiv:2108.09046v1.
[7] CARAVENNA, F., SUN, R. and ZYGOURAS, N. The critical 2D stochastic heat flow. Preprint. Available at arXiv:2109.03766. · Zbl 1427.82063
[8] CARAVENNA, F., SUN, R. and ZYGOURAS, N. (2017). Universality in marginally relevant disordered systems. Ann. Appl. Probab. 27 3050-3112. · Zbl 1387.82032 · doi:10.1214/17-AAP1276
[9] CARAVENNA, F., SUN, R. and ZYGOURAS, N. (2019). On the moments of the \[(2+1)\]-dimensional directed polymer and stochastic heat equation in the critical window. Comm. Math. Phys. 372 385-440. · Zbl 1427.82063 · doi:10.1007/s00220-019-03527-z
[10] CARAVENNA, F., SUN, R. and ZYGOURAS, N. (2020). The two-dimensional KPZ equation in the entire subcritical regime. Ann. Probab. 48 1086-1127. · Zbl 1444.60061 · doi:10.1214/19-AOP1383
[11] CHATTERJEE, S. and DUNLAP, A. (2020). Constructing a solution of the \[(2+1)\]-dimensional KPZ equation. Ann. Probab. 48 1014-1055. · Zbl 1434.60148 · doi:10.1214/19-AOP1382
[12] CHEN, L. and HUANG, J. (2019). Comparison principle for stochastic heat equation on \[{\mathbb{R}^d} \]. Ann. Probab. 47 989-1035. · Zbl 1433.60049 · doi:10.1214/18-AOP1277
[13] CHEN, L. and KIM, K. (2019). Nonlinear stochastic heat equation driven by spatially colored noise: Moments and intermittency. Acta Math. Sci. Ser. B Engl. Ed. 39 645-668. · Zbl 1499.60215 · doi:10.1007/s10473-019-0303-6
[14] CHEN, L. and KIM, K. (2020). Stochastic comparisons for stochastic heat equation. Electron. J. Probab. 25 Paper No. 140, 38 pp. · Zbl 1468.60078 · doi:10.1214/20-ejp541
[15] COSCO, C., NAKAJIMA, S. and NAKASHIMA, M. Law of large numbers and fluctuations in the sub-critical and \[{L^2}\] regions for SHE and KPZ equation in dimension \[d\ge 3\]. Preprint. Available at arXiv:2005.12689v1.
[16] 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. · Zbl 0853.60080 · doi:10.1007/BF01191911
[17] 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. · Zbl 0922.60056 · doi:10.1214/EJP.v4-43
[18] DALANG, R. C. and QUER-SARDANYONS, L. (2011). Stochastic integrals for spde’s: A comparison. Expo. Math. 29 67-109. · Zbl 1234.60064 · doi:10.1016/j.exmath.2010.09.005
[19] Dawson, D. A. and Salehi, H. (1980). Spatially homogeneous random evolutions. J. Multivariate Anal. 10 141-180. · Zbl 0439.60051 · doi:10.1016/0047-259X(80)90012-3
[20] DING, J. and DUNLAP, A. (2020). Subsequential scaling limits for Liouville graph distance. Comm. Math. Phys. 376 1499-1572. · Zbl 1441.60076 · doi:10.1007/s00220-020-03684-6
[21] DUNLAP, A., GU, Y., RYZHIK, L. and ZEITOUNI, O. (2020). Fluctuations of the solutions to the KPZ equation in dimensions three and higher. Probab. Theory Related Fields 176 1217-1258. · Zbl 1445.35345 · doi:10.1007/s00440-019-00938-w
[22] DUNLAP, A., GU, Y., RYZHIK, L. and ZEITOUNI, O. (2021). The random heat equation in dimensions three and higher: The homogenization viewpoint. Arch. Ration. Mech. Anal. 242 827-873. · Zbl 1481.35031 · doi:10.1007/s00205-021-01694-9
[23] FROMM, A. (2014). Theory and applications of decoupling fields for forward-backward stochastic differential equations. Ph.D. thesis, Humboldt-Universität zu Berlin.
[24] GU, Y. (2020). Gaussian fluctuations from the 2D KPZ equation. Stoch. Partial Differ. Equ. Anal. Comput. 8 150-185. · Zbl 1431.35257 · doi:10.1007/s40072-019-00144-8
[25] GU, Y. and LI, J. (2020). Fluctuations of a nonlinear stochastic heat equation in dimensions three and higher. SIAM J. Math. Anal. 52 5422-5440. · Zbl 1456.60138 · doi:10.1137/19M1296380
[26] GU, Y., QUASTEL, J. and TSAI, L.-C. (2021). Moments of the 2D SHE at criticality. Probab. Math. Phys. 2 179-219. · Zbl 1483.60093
[27] GU, Y., RYZHIK, L. and ZEITOUNI, O. (2018). The Edwards-Wilkinson limit of the random heat equation in dimensions three and higher. Comm. Math. Phys. 363 351-388. · Zbl 1400.82131 · doi:10.1007/s00220-018-3202-0
[28] Hairer, M. (2014). A theory of regularity structures. Invent. Math. 198 269-504. · Zbl 1332.60093 · doi:10.1007/s00222-014-0505-4
[29] HAIRER, M. and PARDOUX, É. (2015). A Wong-Zakai theorem for stochastic PDEs. J. Math. Soc. Japan 67 1551-1604. · Zbl 1341.60062 · doi:10.2969/jmsj/06741551
[30] HAIRER, M. and QUASTEL, J. (2018). A class of growth models rescaling to KPZ. Forum Math. Pi 6 e3, 112 pp. · Zbl 1429.60057 · doi:10.1017/fmp.2018.2
[31] IHARA, S. (1993). Information Theory for Continuous Systems. World Scientific Co., Inc., River Edge, NJ. · Zbl 0798.94001 · doi:10.1142/9789814355827
[32] KHOSHNEVISAN, D. (2014). Analysis of Stochastic Partial Differential Equations. CBMS Regional Conference Series in Mathematics 119. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the Amer. Math. Soc., Providence, RI. · Zbl 1304.60005 · doi:10.1090/cbms/119
[33] LYGKONIS, D. and ZYGOURAS, N. Edwards-Wilkinson fluctuations for the directed polymer in the full \[{L^2}\]-regime for dimensions \[d\ge 3\]. Ann. Inst. Henri Poincaré Probab. Stat. To appear. Available at arXiv:2005.12706. · Zbl 1484.82076
[34] MA, J., PROTTER, P. and YONG, J. M. (1994). Solving forward-backward stochastic differential equations explicitly—A four step scheme. Probab. Theory Related Fields 98 339-359. · Zbl 0794.60056 · doi:10.1007/BF01192258
[35] MA, J., WU, Z., ZHANG, D. and ZHANG, J. (2015). On well-posedness of forward-backward SDEs—A unified approach. Ann. Appl. Probab. 25 2168-2214. · Zbl 1319.60132 · doi:10.1214/14-AAP1046
[36] MA, J. and YONG, J. (1999). Forward-Backward Stochastic Differential Equations and Their Applications. Lecture Notes in Math. 1702. Springer, Berlin. · Zbl 0927.60004
[37] MAGNEN, J. and UNTERBERGER, J. (2018). The scaling limit of the KPZ equation in space dimension 3 and higher. J. Stat. Phys. 171 543-598. · Zbl 1394.35508 · doi:10.1007/s10955-018-2014-0
[38] MAO, X. (1994). Stochastic stabilization and destabilization. Systems Control Lett. 23 279-290. · Zbl 0820.93071 · doi:10.1016/0167-6911(94)90050-7
[39] MUKHERJEE, C., SHAMOV, A. and ZEITOUNI, O. (2016). Weak and strong disorder for the stochastic heat equation and continuous directed polymers in \[d\ge 3\]. Electron. Commun. Probab. 21 Paper No. 61, 12 pp. · Zbl 1348.60094 · doi:10.1214/16-ECP18
[40] PESZAT, S. and ZABCZYK, J. (1997). Stochastic evolution equations with a spatially homogeneous Wiener process. Stochastic Process. Appl. 72 187-204. · Zbl 0943.60048 · doi:10.1016/S0304-4149(97)00089-6
[41] Stroock, D. W. and Varadhan, S. R. S. (2006). Multidimensional Diffusion Processes. Classics in Mathematics. Springer, Berlin. Reprint of the 1997 edition.
[42] TESSITORE, G. and ZABCZYK, J. (1998). Invariant measures for stochastic heat equations. Probab. Math. Statist. 18 271-287. · Zbl 0986.60057
[43] WATANABE, S. and YAMADA, T. (1971). On the uniqueness of solutions of stochastic differential equations. II. J. Math. Kyoto Univ. 11 553-563. · Zbl 0229.60039 · doi:10.1215/kjm/1250523620
[44] YAMADA, T. and WATANABE, S. (1971). On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. 11 155-167 · Zbl 0236.60037 · doi:10.1215/kjm/1250523691
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.