Regularization by noise and flows of solutions for a stochastic heat equation. (English) Zbl 1447.60091

The authors investigate the problem of uniqueness of the solution to the stochastic heat equations of the form \(\frac{\partial u}{\partial t}=\frac{1}{2}\frac{\partial ^2 u}{\partial z^2}+b(u(t,z))+\dot W(t,z)\), \(t\ge0, z\in \mathbb R\), \(u(0,z)=q(z)\), where \(\dot W\) is a Gaussian space-time white noise on \(\mathbb R_{+}\times \mathbb R\), \(b\) is a bounded Borel measurable function on \(\mathbb R\) and \(q\) is a Borel measurable function on \(\mathbb R\). Under some growth condition on \(q\), the path-by-path uniqueness and continuity with respect to initial conditions of the solution to the considered heat equation are proved. The authors develop a new method that extends A. M. Davie’s approach [Int. Math. Res. Not. 2007, No. 24, Article ID rnm124, 26 p. (2007; Zbl 1139.60028)] to the infinite-dimensional case and prove the existence and uniqueness of the flow of solutions to the considered equation.


60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
60H25 Random operators and equations (aspects of stochastic analysis)


Zbl 1139.60028
Full Text: DOI arXiv Euclid


[1] Bally, V., Gyöngy, I. and Pardoux, É. (1994). White noise driven parabolic SPDEs with measurable drift. J. Funct. Anal.120 484-510. · Zbl 0801.60049
[2] Butkovsky, O. and Mytnik, L. (2019). Supplement to “Regularization by noise and flows of solutions for a stochastic heat equation.” DOI:10.1214/18-AOP1259SUPP. · Zbl 1447.60091
[3] Catellier, R. and Gubinelli, M. (2016). Averaging along irregular curves and regularisation of ODEs. Stochastic Process. Appl.126 2323-2366. · Zbl 1348.60083 · doi:10.1016/j.spa.2016.02.002
[4] Cerrai, S. (2003). Stochastic reaction-diffusion systems with multiplicative noise and non-Lipschitz reaction term. Probab. Theory Related Fields125 271-304. · Zbl 1027.60064 · doi:10.1007/s00440-002-0230-6
[5] Davie, A. M. (2007). Uniqueness of solutions of stochastic differential equations. Int. Math. Res. Not. IMRN2007 Art. ID rnm124, 26. · Zbl 1139.60028
[6] Fedrizzi, E. and Flandoli, F. (2013). Hölder flow and differentiability for SDEs with nonregular drift. Stoch. Anal. Appl.31 708-736. · Zbl 1281.60055 · doi:10.1080/07362994.2012.628908
[7] Flandoli, F. (1995). Regularity Theory and Stochastic Flows for Parabolic SPDEs. Stochastics Monographs9. Gordon and Breach Science Publishers, Yverdon. · Zbl 0838.60054
[8] Flandoli, F. (2011). Random Perturbation of PDEs and Fluid Dynamic Models. Lecture Notes in Math.2015. Springer, Heidelberg. · Zbl 1221.35004
[9] Flandoli, F., Gubinelli, M. and Priola, E. (2010). Well-posedness of the transport equation by stochastic perturbation. Invent. Math.180 1-53. · Zbl 1200.35226 · doi:10.1007/s00222-009-0224-4
[10] Folland, G. B. (1999). Real Analysis: Modern Techniques and Their Applications, 2nd ed. Wiley, New York. · Zbl 0924.28001
[11] Goldys, B. and Zhang, X. (2011). Stochastic flows for nonlinear SPDEs driven by linear multiplicative space-time white noises. In Stochastic Analysis with Financial Applications. Progress in Probability65 83-97. Birkhäuser/Springer Basel AG, Basel. · Zbl 1255.60107
[12] Gyöngy, I. and Pardoux, É. (1993). On quasi-linear stochastic partial differential equations. Probab. Theory Related Fields94 413-425. · Zbl 0791.60047
[13] Gyöngy, I. and Pardoux, É. (1993). On the regularization effect of space-time white noise on quasi-linear parabolic partial differential equations. Probab. Theory Related Fields97 211-229. · Zbl 0793.60064
[14] Hairer, M. and Pardoux, É. (2015). A Wong-Zakai theorem for stochastic PDEs. J. Math. Soc. Japan67 1551-1604. · Zbl 1341.60062 · doi:10.2969/jmsj/06741551
[15] Hu, Y. and Le, K. (2013). A multiparameter Garsia-Rodemich-Rumsey inequality and some applications. Stochastic Process. Appl.123 3359-3377. · Zbl 1300.60051 · doi:10.1016/j.spa.2013.04.019
[16] Khoshnevisan, D. (2009). A primer on stochastic partial differential equations. In A Minicourse on Stochastic Partial Differential Equations. Lecture Notes in Math.1962 1-38. Springer, Berlin. · Zbl 1168.60027
[17] Krylov, N. V. and Röckner, M. (2005). Strong solutions of stochastic equations with singular time dependent drift. Probab. Theory Related Fields131 154-196. · Zbl 1072.60050
[18] Kunita, H. (1997). Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics24. Cambridge Univ. Press, Cambridge. · Zbl 0865.60043
[19] Mohammed, S.-E. A., Nilssen, T. K. and Proske, F. N. (2015). Sobolev differentiable stochastic flows for SDEs with singular coefficients: Applications to the transport equation. Ann. Probab.43 1535-1576. · Zbl 1333.60127 · doi:10.1214/14-AOP909
[20] Priola, E. (2018). Davie’s type uniqueness for a class of SDEs with jumps. Ann. Inst. H. Poincaré Probab. Statist.54 694-725. · Zbl 1391.60143 · doi:10.1214/16-AIHP818
[21] Rezakhanlou, F. (2014). Regular flows for diffusions with rough drifts. Preprint. Available at arXiv:1405.5856.
[22] Shaposhnikov, A. V. (2016). Some remarks on Davie’s uniqueness theorem. Proc. Edinb. Math. Soc. (2) 59 1019-1035. · Zbl 1367.60075 · doi:10.1017/S0013091515000589
[23] Veretennikov, A. J. (1980). Strong solutions and explicit formulas for solutions of stochastic integral equations. Mat. Sb. (N.S.) 111(153) 434-452, 480. · Zbl 0431.60061
[24] 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. · Zbl 0608.60060
[25] Wresch, L. (2017). Path-by-path uniqueness of infinite-dimensional stochastic differential equations. Preprint. Available at arXiv:1706.07720.
[26] Zhang, X. (2011). Stochastic homeomorphism flows of SDEs with singular drifts and Sobolev diffusion coefficients. Electron. J. Probab.16 1096-1116. · Zbl 1225.60099 · doi:10.1214/EJP.v16-887
[27] Zvonkin, A. · Zbl 0306.60049 · doi:10.1070/SM1974v022n01ABEH001689
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