## Electronic Journal of Probability

### New results on pathwise uniqueness for the heat equation with colored noise

#### Abstract

We consider strong uniqueness and thus also existence of strong solutions for the stochastic heat equation with  a multiplicative colored noise term. Here, the noise is white in time and colored in $q$ dimensional space ($q \geq 1$) with a singular correlation kernel. The noise coefficient is Hölder continuous in the solution. We discuss improvements of the sufficient conditions obtained in Mytnik, Perkins and Sturm (2006) that relate the Hölder coefficient with the singularity of the correlation kernel of the noise. For this we use new ideas of Mytnik and Perkins (2011) who treat the case of strong uniqueness for the stochastic heat equation with multiplicative white noise in one dimension. Our main result on pathwise uniqueness confirms a conjecture that was put forward in their paper.

#### Article information

Source
Electron. J. Probab., Volume 18 (2013), paper no. 77, 46 pp.

Dates
Accepted: 24 August 2013
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465064302

Digital Object Identifier
doi:10.1214/EJP.v18-2506

Mathematical Reviews number (MathSciNet)
MR3101643

Zentralblatt MATH identifier
06247246

Rights

#### Citation

Rippl, Thomas; Sturm, Anja. New results on pathwise uniqueness for the heat equation with colored noise. Electron. J. Probab. 18 (2013), paper no. 77, 46 pp. doi:10.1214/EJP.v18-2506. https://projecteuclid.org/euclid.ejp/1465064302

#### References

• Adams, Robert A.; Fournier, John J. F. Sobolev spaces. Second edition. Pure and Applied Mathematics (Amsterdam), 140. Elsevier/Academic Press, Amsterdam, 2003. xiv+305 pp. ISBN: 0-12-044143-8
• Dalang, Robert C. Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.'s. Electron. J. Probab. 4 (1999), no. 6, 29 pp. (electronic).
• Jacod, Jean. Weak and strong solutions of stochastic differential equations. Stochastics 3, no. 3, 171–191. (1980),
• Karatzas, Ioannis; Shreve, Steven E. Brownian motion and stochastic calculus. Second edition. Graduate Texts in Mathematics, 113. Springer-Verlag, New York, 1991. xxiv+470 pp. ISBN: 0-387-97655-8
• Konno, N.; Shiga, T. Stochastic partial differential equations for some measure-valued diffusions. Probab. Theory Related Fields 79 (1988), no. 2, 201–225.
• Kurtz, Thomas G. The Yamada-Watanabe-Engelbert theorem for general stochastic equations and inequalities. Electron. J. Probab. 12 (2007), 951–965.
• Meyer, Paul-A. Probability and potentials. Blaisdell Publishing Co. Ginn and Co., Waltham, Mass.-Toronto, Ont.-London 1966 xiii+266 pp.
• Carl Mueller, Leonid Mytnik, and Edwin Perkins, Nonuniqueness for a parabolic SPDE with 3/4 - Îµ-Hölder diffusion coefficients, preprint (2012), arXiv:1201.2767v1 [math.PR].
• Mytnik, Leonid. Superprocesses in random environments. Ann. Probab. 24 (1996), no. 4, 1953–1978.
• Mytnik, Leonid; Perkins, Edwin. Pathwise uniqueness for stochastic heat equations with Hölder continuous coefficients: the white noise case. Probab. Theory Related Fields 149 (2011), no. 1-2, 1–96.
• Mytnik, Leonid; Perkins, Edwin; Sturm, Anja. On pathwise uniqueness for stochastic heat equations with non-Lipschitz coefficients. Ann. Probab. 34 (2006), no. 5, 1910–1959.
• Perkins, Edwin. Dawson-Watanabe superprocesses and measure-valued diffusions. Lectures on probability theory and statistics (Saint-Flour, 1999), 125–324, Lecture Notes in Math., 1781, Springer, Berlin, 2002.
• Peszat, Szymon; Zabczyk, Jerzy. Nonlinear stochastic wave and heat equations. Probab. Theory Related Fields 116 (2000), no. 3, 421–443.
• Reimers, Mark. One-dimensional stochastic partial differential equations and the branching measure diffusion. Probab. Theory Related Fields 81 (1989), no. 3, 319–340.
• Rippl, Thomas. Pathwise uniqueness of the stochastic heat equation with Hölder continuous diffusion coefficient and colored noise, rlhttp://hdl.handle.net/11858/00-1735-0000-000D-F0ED-A, 2012.
• Sanz-Solé, M.; Sarrà, M. Hölder continuity for the stochastic heat equation with spatially correlated noise. Seminar on Stochastic Analysis, Random Fields and Applications, III (Ascona, 1999), 259–268, Progr. Probab., 52, BirkhÃ¤user, Basel, 2002.
• Sturm, Anja. On convergence of population processes in random environments to the stochastic heat equation with colored noise. Electron. J. Probab. 8 (2003), no. 6, 39 pp. (electronic).
• Walsh, John B. An introduction to stochastic partial differential equations. École d'été de probabilités de Saint-Flour, XIV–1984, 265–439, Lecture Notes in Math., 1180, Springer, Berlin, 1986.
• Xiong, Jie. Super-Brownian motion as the unique strong solution to an SPDE, Ann. Probab. 41 (2013), no. 2, 1030–1054.
• Yamada, Toshio; Watanabe, Shinzo. On the uniqueness of solutions of stochastic differential equations. J. Math. Kyoto Univ. 11 1971 155–167.
• Zähle, Henryk. Approximation of SDEs by population-size-dependent Galton-Watson processes. Stoch. Anal. Appl. 28 (2010), no. 2, 377–388.