Optimal regularity for semilinear stochastic partial differential equations with multiplicative noise

Raphael Kruse (Bielefeld University)
Stig Larsson (Chalmers University of Technology and University of Gothenburg)


This paper deals with the spatial and temporal regularity of the unique Hilbert space valued mild solution to a semilinear stochastic parabolic partial differential equation with nonlinear terms that satisfy global Lipschitz conditions and certain linear growth bounds. It is shown that the mild solution has the same optimal regularity properties as the stochastic convolution. The proof is elementary and makes use of existing results on the regularity of the solution, in particular, the Hölder continuity with a non-optimal exponent.

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Pages: 1-19

Publication Date: August 18, 2012

DOI: 10.1214/EJP.v17-2240


  • Alt, H. W. Lineare Funktionalanalysis, 5., revised ed., Springer-Verlag, Berlin, 2006.
  • Brzeźniak, Z.; Hausenblas, E. Maximal regularity for stochastic convolutions driven by Lévy processes. Probab. Theory Related Fields 145 (2009), no. 3-4, 615--637. MR2529441
  • Da Prato, G. Regularity results of a convolution stochastic integral and applications to parabolic stochastic equations in a Hilbert space. Confer. Sem. Mat. Univ. Bari No. 182 (1982), 17 pp. MR0679566
  • Da Prato, G.; Kwapień, S.; Zabczyk, J. Regularity of solutions of linear stochastic equations in Hilbert spaces. Stochastics 23 (1987), no. 1, 1--23. MR0920798
  • Da Prato, G.; Lunardi, A.. Maximal regularity for stochastic convolutions in $L^ p$ spaces. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 9 (1998), no. 1, 25--29. MR1669252
  • Da Prato, G.; Zabczyk, J. 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
  • Jentzen, A.; Röckner, M. Regularity analysis for stochastic partial differential equations with nonlinear multiplicative trace class noise. J. Differential Equations 252 (2012), no. 1, 114--136. MR2852200
  • Kruse, R. Strong and Weak Galerkin Approximation of Stochastic Evolution Equations, PhD thesis, Bielefeld University, 2012.
  • Krylov, N. V.; Rozovskiĭ, B. L. Stochastic evolution equations. (Russian) Current problems in mathematics, Vol. 14 (Russian), pp. 71--147, 256, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Informatsii, Moscow, 1979. MR0570795
  • Larsson, S.; Thomée, V. Partial differential equations with numerical methods. Texts in Applied Mathematics, 45. Springer-Verlag, Berlin, 2003. x+259 pp. ISBN: 3-540-01772-0 MR1995838
  • 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
  • Prévôt, C.; Röckner, M. A concise course on stochastic partial differential equations. Lecture Notes in Mathematics, 1905. Springer, Berlin, 2007. vi+144 pp. ISBN: 978-3-540-70780-6; 3-540-70780-8 MR2329435
  • Printems, J. On the discretization in time of parabolic stochastic partial differential equations. M2AN Math. Model. Numer. Anal. 35 (2001), no. 6, 1055--1078. MR1873517
  • Rozovskiĭ, B. L. Stochastic evolution systems. Linear theory and applications to nonlinear filtering. Translated from the Russian by A. Yarkho. Mathematics and its Applications (Soviet Series), 35. Kluwer Academic Publishers Group, Dordrecht, 1990. xviii+315 pp. ISBN: 0-7923-0037-8 MR1135324
  • Sell, G. R.; You, Y. Dynamics of evolutionary equations. Applied Mathematical Sciences, 143. Springer-Verlag, New York, 2002. xiv+670 pp. ISBN: 0-387-98347-3 MR1873467
  • Thomée, V. Galerkin finite element methods for parabolic problems. Second edition. Springer Series in Computational Mathematics, 25. Springer-Verlag, Berlin, 2006. xii+370 pp. ISBN: 978-3-540-33121-6; 3-540-33121-2 MR2249024
  • van Neerven, J.; Veraar, M.; Weis, L. Maximal $L^p$-regularity for stochastic evolution equations, SIAM J. Math. Anal. 44 (2012), no. 3, 1372--1414.
  • van Neerven, J.; Veraar, M.; Weis, L. Stochastic maximal $L^p$-regularity, Annals Probab. 40 (2012), no. 2, 788--812.
  • Walsh, J. 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. MR0876085
  • Zhang, X. Regularities for semilinear stochastic partial differential equations. J. Funct. Anal. 249 (2007), no. 2, 454--476. MR2345340

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