## The Annals of Probability

### Regularization by noise and flows of solutions for a stochastic heat equation

#### Abstract

Motivated by the regularization by noise phenomenon for SDEs, we prove existence and uniqueness of the flow of solutions for the non-Lipschitz stochastic heat equation

$\frac{\partial u}{\partial t}=\frac{1}{2}\frac{\partial^{2}u}{\partial z^{2}}+b\bigl(u(t,z)\bigr)+\dot{W}(t,z),$ where $\dot{W}$ is a space-time white noise on $\mathbb{R}_{+}\times\mathbb{R}$ and $b$ is a bounded measurable function on $\mathbb{R}$. As a byproduct of our proof, we also establish the so-called path-by-path uniqueness for any initial condition in a certain class on the same set of probability one. To obtain these results, we develop a new approach that extends Davie’s method (2007) to the context of stochastic partial differential equations.

#### Article information

Source
Ann. Probab., Volume 47, Number 1 (2019), 165-212.

Dates
Revised: February 2018
First available in Project Euclid: 13 December 2018

https://projecteuclid.org/euclid.aop/1544691620

Digital Object Identifier
doi:10.1214/18-AOP1259

Mathematical Reviews number (MathSciNet)
MR3909968

Zentralblatt MATH identifier
07036336

#### Citation

Butkovsky, Oleg; Mytnik, Leonid. Regularization by noise and flows of solutions for a stochastic heat equation. Ann. Probab. 47 (2019), no. 1, 165--212. doi:10.1214/18-AOP1259. https://projecteuclid.org/euclid.aop/1544691620

#### References

• [1] Bally, V., Gyöngy, I. and Pardoux, É. (1994). White noise driven parabolic SPDEs with measurable drift. J. Funct. Anal. 120 484–510.
• [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.
• [3] Catellier, R. and Gubinelli, M. (2016). Averaging along irregular curves and regularisation of ODEs. Stochastic Process. Appl. 126 2323–2366.
• [4] Cerrai, S. (2003). Stochastic reaction–diffusion systems with multiplicative noise and non-Lipschitz reaction term. Probab. Theory Related Fields 125 271–304.
• [5] Davie, A. M. (2007). Uniqueness of solutions of stochastic differential equations. Int. Math. Res. Not. IMRN 2007 Art. ID rnm124, 26.
• [6] Fedrizzi, E. and Flandoli, F. (2013). Hölder flow and differentiability for SDEs with nonregular drift. Stoch. Anal. Appl. 31 708–736.
• [7] Flandoli, F. (1995). Regularity Theory and Stochastic Flows for Parabolic SPDEs. Stochastics Monographs 9. Gordon and Breach Science Publishers, Yverdon.
• [8] Flandoli, F. (2011). Random Perturbation of PDEs and Fluid Dynamic Models. Lecture Notes in Math. 2015. Springer, Heidelberg.
• [9] Flandoli, F., Gubinelli, M. and Priola, E. (2010). Well-posedness of the transport equation by stochastic perturbation. Invent. Math. 180 1–53.
• [10] Folland, G. B. (1999). Real Analysis: Modern Techniques and Their Applications, 2nd ed. Wiley, New York.
• [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 Probability 65 83–97. Birkhäuser/Springer Basel AG, Basel.
• [12] Gyöngy, I. and Pardoux, É. (1993). On quasi-linear stochastic partial differential equations. Probab. Theory Related Fields 94 413–425.
• [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 Fields 97 211–229.
• [14] Hairer, M. and Pardoux, É. (2015). A Wong–Zakai theorem for stochastic PDEs. J. Math. Soc. Japan 67 1551–1604.
• [15] Hu, Y. and Le, K. (2013). A multiparameter Garsia–Rodemich–Rumsey inequality and some applications. Stochastic Process. Appl. 123 3359–3377.
• [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.
• [17] Krylov, N. V. and Röckner, M. (2005). Strong solutions of stochastic equations with singular time dependent drift. Probab. Theory Related Fields 131 154–196.
• [18] Kunita, H. (1997). Stochastic Flows and Stochastic Differential Equations. Cambridge Studies in Advanced Mathematics 24. Cambridge Univ. Press, Cambridge.
• [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.
• [20] Priola, E. (2018). Davie’s type uniqueness for a class of SDEs with jumps. Ann. Inst. H. Poincaré Probab. Statist. 54 694–725.
• [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.
• [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.
• [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.
• [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.
• [27] Zvonkin, A. K. (1974). A transformation of the phase space of a diffusion process that will remove the drift. Mat. Sb. (N.S.) 93(135) 129–149, 152.

#### Supplemental materials

• Supplement to “Regularization by noise and flows of solutions for a stochastic heat equation”. The supplementary material provides proofs of auxiliary results related to the properties of the heat kernel.