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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.

MSC:

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)

Citations:

Zbl 1139.60028
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References:

[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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