## The Annals of Probability

### Strong uniqueness for SDEs in Hilbert spaces with nonregular drift

#### Abstract

We prove pathwise uniqueness for a class of stochastic differential equations (SDE) on a Hilbert space with cylindrical Wiener noise, whose nonlinear drift parts are sums of the sub-differential of a convex function and a bounded part. This generalizes a classical result by one of the authors to infinite dimensions. Our results also generalize and improve recent results by N. Champagnat and P. E. Jabin, proved in finite dimensions, in the case where their diffusion matrix is constant and nondegenerate and their weakly differentiable drift is the (weak) gradient of a convex function. We also prove weak existence, hence obtain unique strong solutions by the Yamada–Watanabe theorem. The proofs are based in part on a recent maximal regularity result in infinite dimensions, the theory of quasi-regular Dirichlet forms and an infinite dimensional version of a Zvonkin-type transformation. As a main application, we show pathwise uniqueness for stochastic reaction diffusion equations perturbed by a Borel measurable bounded drift. Hence, such SDE have a unique strong solution.

#### Article information

Source
Ann. Probab., Volume 44, Number 3 (2016), 1985-2023.

Dates
Revised: January 2015
First available in Project Euclid: 16 May 2016

https://projecteuclid.org/euclid.aop/1463410037

Digital Object Identifier
doi:10.1214/15-AOP1016

Mathematical Reviews number (MathSciNet)
MR3502599

Zentralblatt MATH identifier
1347.60077

#### Citation

Da Prato, G.; Flandoli, F.; Röckner, M.; Veretennikov, A. Yu. Strong uniqueness for SDEs in Hilbert spaces with nonregular drift. Ann. Probab. 44 (2016), no. 3, 1985--2023. doi:10.1214/15-AOP1016. https://projecteuclid.org/euclid.aop/1463410037

#### References

• [1] Albeverio, S. and Röckner, M. (1989). Classical Dirichlet forms on topological vector spaces—The construction of the associated diffusion process. Probab. Theory Related Fields 83 405–434.
• [2] Albeverio, S. and Röckner, M. (1990). Classical Dirichlet forms on topological vector spaces—Closability and a Cameron–Martin formula. J. Funct. Anal. 88 395–436.
• [3] Albeverio, S. and Röckner, M. (1991). Stochastic differential equations in infinite dimensions: Solutions via Dirichlet forms. Probab. Theory Related Fields 89 347–386.
• [4] Aronszajn, N. (1976). Differentiability of Lipschitzian mappings between Banach spaces. Studia Math. 57 147–190.
• [5] Barbu, V. (2010). Nonlinear Differential Equations of Monotone Types in Banach Spaces. Springer, New York.
• [6] Bogachev, V., Da Prato, G. and Röckner, M. (2010). Existence and uniqueness of solutions for Fokker–Planck equations on Hilbert spaces. J. Evol. Equ. 10 487–509.
• [7] Bogachev, V. I. (2010). Differentiable Measures and the Malliavin Calculus. Mathematical Surveys and Monographs 164. Amer. Math. Soc., Providence, RI.
• [8] Cerrai, S. and Da Prato, G. (2013). Pathwise uniqueness for stochastic reaction-diffusion equations in Banach spaces with an Hölder drift component. Stoch. PDE: Anal. Comp. 1 507–551.
• [9] Champagnat, N. and Jabin, P. E. (2013). Strong solutions to stochastic differential equations with rough coefficients. Available at arXiv:1303:2611v1.
• [10] Da Prato, G. (2004). Kolmogorov Equations for Stochastic PDEs. Birkhäuser, Basel.
• [11] Da Prato, G., Flandoli, F., Priola, E. and Röckner, M. (2013). Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift. Ann. Probab. 41 3306–3344.
• [12] Da Prato, G., Flandoli, F., Priola, E. and Röckner, M. (2016). Strong uniqueness for stochastic evolution equations with unbounded measurable drift term. J. Theoret. Probab. To appear. DOI:10.1007/s10959-014-0545-0.
• [13] Da Prato, G. and Lunardi, A. (2014). Sobolev regularity for a class of second order elliptic PDE’s in infinite dimension. Ann. Probab. 42 2113–2160.
• [14] Da Prato, G. and Röckner, M. (2002). Singular dissipative stochastic equations in Hilbert spaces. Probab. Theory Related Fields 124 261–303.
• [15] Da Prato, G., Röckner, M. and Wang, F.-Y. (2009). Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups. J. Funct. Anal. 257 992–1017.
• [16] Da Prato, G. and Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and Its Applications 44. Cambridge Univ. Press, Cambridge.
• [17] Ma, Z. M. and Röckner, M. (1992). Introduction to the Theory of (nonsymmetric) Dirichlet Forms. Springer, Berlin.
• [18] Marinelli, C. and Röckner, M. (2010). On uniqueness of mild solutions for dissipative stochastic evolution equations. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13 363–376.
• [19] Phelps, R. R. (1978). Gaussian null sets and differentiability of Lipschitz map on Banach spaces. Pacific J. Math. 77 523–531.
• [20] Phelps, R. R. (1993). Convex Functions, Monotone Operators and Differentiability, 2nd ed. Lecture Notes in Math. 1364. Springer, Berlin.
• [21] Röckner, M. and Schmuland, B. (1992). Tightness of general $C_{1,p}$ capacities on Banach space. J. Funct. Anal. 108 1–12.
• [22] Röckner, M., Schmuland, B. and Zhang, X. (2008). Yamada–Watanabe theorem for stochastic evolution equations in infinite dimensions. Cond. Matt. Phys 11 247–259.
• [23] Veretennikov, A. Y. (1980). Strong solutions and explicit formulas for solutions of stochastic integral equations. Mat. Sb. 111 434–452, 480.