The Stratonovich heat equation: a continuity result and weak approximations

Aurélien Deya (Institut Élie Cartan, Nancy)
Maria Jolis (Universitat Autònoma de Barcelona)
Lluís Quer-Sardanyons (Universitat Autònoma de Barcelona)


We consider a Stratonovich heat equation in $(0,1)$ with a nonlinear multiplicative noise driven by a trace-class Wiener process. First, the equation is shown to have a unique mild solution. Secondly, convolutional rough paths techniques are used to provide an almost sure continuity result for the solution with respect to the solution of the 'smooth' equation obtained by replacing the noise with an absolutely continuous process. This continuity result is then exploited to prove weak convergence results based on Donsker and Kac-Stroock type approximations of the noise.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1-34

Publication Date: January 6, 2013

DOI: 10.1214/EJP.v18-2004


  • Bal, Guillaume. Convergence to SPDEs in Stratonovich form. Comm. Math. Phys. 292 (2009), no. 2, 457--477. MR2544739
  • Bally, Vlad; Millet, Annie; Sanz-Solé, Marta. Approximation and support theorem in Hölder norm for parabolic stochastic partial differential equations. Ann. Probab. 23 (1995), no. 1, 178--222. MR1330767
  • Bardina, Xavier; Jolis, Maria; Quer-Sardanyons, Lluís. Weak convergence for the stochastic heat equation driven by Gaussian white noise. Electron. J. Probab. 15 (2010), no. 39, 1267--1295. MR2678391
  • Brézis, H. Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert. (French) North-Holland Mathematics Studies, No. 5. Notas de Matemática (50). North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973. vi+183 pp. MR0348562
  • Brzeźniak, Zdzisław; Flandoli, Franco. Almost sure approximation of Wong-Zakai type for stochastic partial differential equations. Stochastic Process. Appl. 55 (1995), no. 2, 329--358. MR1313027
  • Buckdahn, Rainer; Ma, Jin. Stochastic viscosity solutions for nonlinear stochastic partial differential equations. I. Stochastic Process. Appl. 93 (2001), no. 2, 181--204. MR1828772
  • Buckdahn, Rainer; Ma, Jin. Stochastic viscosity solutions for nonlinear stochastic partial differential equations. II. Stochastic Process. Appl. 93 (2001), no. 2, 205--228. MR1831830
  • Carmona, René A.; Fouque, Jean-Pierre. A diffusion approximation result for two parameter processes. Probab. Theory Related Fields 98 (1994), no. 3, 277--298. MR1262967
  • Da Prato, Giuseppe; Zabczyk, Jerzy. Stochastic equations in infinite dimensions. Encyclopedia of Mathematics and its Applications, 44. Cambridge University Press, Cambridge, 1992. xviii+454 pp. ISBN: 0-521-38529-6 MR1207136
  • Deya, Aurélien. A discrete approach to rough parabolic equations. Electron. J. Probab. 16 (2011), no. 54, 1489--1518. MR2827468
  • Deya, A.; Gubinelli, M.; Tindel, S. Non-linear rough heat equations. Probab. Theory Related Fields 153 (2012), no. 1-2, 97--147. MR2925571
  • Florit, Carme; Nualart, David. Diffusion approximation for hyperbolic stochastic differential equations. Stochastic Process. Appl. 65 (1996), no. 1, 1--15. MR1422876
  • Friz, Peter K.; Victoir, Nicolas B. Multidimensional stochastic processes as rough paths. Theory and applications. Cambridge Studies in Advanced Mathematics, 120. Cambridge University Press, Cambridge, 2010. xiv+656 pp. ISBN: 978-0-521-87607-0 MR2604669
  • Griego, Richard J.; Heath, David; Ruiz-Moncayo, Alberto. Almost sure convergence of uniform transport processes to Brownian motion. Ann. Math. Statist. 42 1971 1129--1131. MR0278389
  • Gubinelli, M. Controlling rough paths. J. Funct. Anal. 216 (2004), no. 1, 86--140. MR2091358
  • Gubinelli, Massimiliano; Tindel, Samy. Rough evolution equations. Ann. Probab. 38 (2010), no. 1, 1--75. MR2599193
  • Hu, Yaozhong; Nualart, David. Stochastic heat equation driven by fractional noise and local time. Probab. Theory Related Fields 143 (2009), no. 1-2, 285--328. MR2449130
  • Kac, Mark. A stochastic model related to the telegrapher's equation. Reprinting of an article published in 1956. Papers arising from a Conference on Stochastic Differential Equations (Univ. Alberta, Edmonton, Alta., 1972). Rocky Mountain J. Math. 4 (1974), 497--509. MR0510166
  • 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 MR1121940
  • Manthey, Ralf. Weak convergence of solutions of the heat equation with Gaussian noise. Math. Nachr. 123 (1985), 157--168. MR0809342
  • Manthey, Ralf. Weak approximation of a nonlinear stochastic partial differential equation. Random partial differential equations (Oberwolfach, 1989), 139--148, Internat. Ser. Numer. Math., 102, Birkhäuser, Basel, 1991. MR1185745
  • Mörters, Peter; Peres, Yuval. Brownian motion. With an appendix by Oded Schramm and Wendelin Werner. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2010. xii+403 pp. ISBN: 978-0-521-76018-8 MR2604525
  • Pazy, A. Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. viii+279 pp. ISBN: 0-387-90845-5 MR0710486
  • Pinsky, Mark. Differential equations with a small parameter and the central limit theorem for functions defined on a finite Markov chain. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 9 1968 101--111. MR0228067
  • Rosenthal, Haskell P. On the subspaces of $L^{p}$ $(p>2)$ spanned by sequences of independent random variables. Israel J. Math. 8 1970 273--303. MR0271721
  • Runst, Thomas; Sickel, Winfried. Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations. de Gruyter Series in Nonlinear Analysis and Applications, 3. Walter de Gruyter & Co., Berlin, 1996. x+547 pp. ISBN: 3-11-015113-8 MR1419319
  • Skorokhod, A. V. Studies in the theory of random processes. Translated from the Russian by Scripta Technica, Inc. Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965 viii+199 pp. MR0185620
  • Sussmann, Héctor J. On the gap between deterministic and stochastic ordinary differential equations. Ann. Probability 6 (1978), no. 1, 19--41. MR0461664
  • Tessitore, Gianmario; Zabczyk, Jerzy. Wong-Zakai approximations of stochastic evolution equations. J. Evol. Equ. 6 (2006), no. 4, 621--655. MR2267702
  • Tindel, Samy. Diffusion approximation for elliptic stochastic differential equations. Stochastic analysis and related topics, V (Silivri, 1994), 255--268, Progr. Probab., 38, Birkhäuser Boston, Boston, MA, 1996. MR1396335
  • Tindel, Samy. Stochastic parabolic equations with anticipative initial condition. Stochastics Stochastics Rep. 62 (1997), no. 1-2, 1--20. MR1489179
  • Walsh, John B. A stochastic model of neural response. Adv. in Appl. Probab. 13 (1981), no. 2, 231--281. MR0612203

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.