Is the stochastic parabolicity condition dependent on $p$ and $q$?

Zdzislaw Brzezniak (University of York)
Mark Veraar (Delft University of Technology)


In this paper we study well-posedness of a second order SPDE with multiplicative noise on the torus $\mathbb{T} = [0,2\pi]$. The equation is considered in $L^p((0,T)\times\Omega;L^q(\mathbb{T}))$ for $p,q\in (1, \infty)$. It is well-known that if the noise is of gradient type, one needs a stochastic parabolicity condition on the coefficients for well-posedness with $p=q=2$. In this paper we investigate whether the well-posedness depends on $p$ and $q$. It turns out that this condition does depend on $p$, but not on $q$. Moreover, we show that if $1<p<2$ the classical stochastic parabolicity condition can be weakened.

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Pages: 1-24

Publication Date: July 22, 2012

DOI: 10.1214/EJP.v17-2186


  • Brzeźniak, Zdzisław. Stochastic partial differential equations in M-type $2$ Banach spaces. Potential Anal. 4 (1995), no. 1, 1--45. MR1313905
  • Brzeźniak, Z.; Capiński, M.; Flandoli, F. A convergence result for stochastic partial differential equations. Stochastics 24 (1988), no. 4, 423--445. MR0972973
  • Brzeźniak, Z.; van Neerven, J. M. A. M.; Veraar, M. C.; Weis, L. Itô's formula in UMD Banach spaces and regularity of solutions of the Zakai equation. J. Differential Equations 245 (2008), no. 1, 30--58. MR2422709
  • 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.; Iannelli, M.; Tubaro, L. Some results on linear stochastic differential equations in Hilbert spaces. Stochastics 6 (1981/82), no. 2, 105--116. MR0665246
  • 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
  • E. Dettweiler, Stochastic integration of Banach space valued functions., Stochastic space-time models and limit theorems, Math. Appl., D. Reidel Publ. Co. 19, 53-59 (1985)., 1985.
  • Edwards, R. E.; Gaudry, G. I. Littlewood-Paley and multiplier theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 90. Springer-Verlag, Berlin-New York, 1977. ix+212 pp. ISBN: 3-540-07726-X MR0618663
  • Engel, Klaus-Jochen; Nagel, Rainer. One-parameter semigroups for linear evolution equations. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt. Graduate Texts in Mathematics, 194. Springer-Verlag, New York, 2000. xxii+586 pp. ISBN: 0-387-98463-1 MR1721989
  • Grafakos, Loukas. Classical Fourier analysis. Second edition. Graduate Texts in Mathematics, 249. Springer, New York, 2008. xvi+489 pp. ISBN: 978-0-387-09431-1 MR2445437
  • Kallenberg, Olav. Foundations of modern probability. Second edition. Probability and its Applications (New York). Springer-Verlag, New York, 2002. xx+638 pp. ISBN: 0-387-95313-2 MR1876169
  • Krylov, N. V. An analytic approach to SPDEs. Stochastic partial differential equations: six perspectives, 185--242, Math. Surveys Monogr., 64, Amer. Math. Soc., Providence, RI, 1999. MR1661766
  • Krylov, N. V. SPDEs in $L_ Q((0,\tau]\!],L_ P)$ spaces. Electron. J. Probab. 5 (2000), Paper no. 13, 29 pp. (electronic). MR1781025
  • 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
  • Kunstmann, Peer C.; Weis, Lutz. Maximal $L_ p$-regularity for parabolic equations, Fourier multiplier theorems and $H^ \infty$-functional calculus. Functional analytic methods for evolution equations, 65--311, Lecture Notes in Math., 1855, Springer, Berlin, 2004. MR2108959
  • Kwapień, Stanisław; Woyczyński, Wojbor A. Random series and stochastic integrals: single and multiple. Probability and its Applications. Birkhäuser Boston, Inc., Boston, MA, 1992. xvi+360 pp. ISBN: 0-8176-3572-6 MR1167198
  • Lunardi, Alessandra. Analytic semigroups and optimal regularity in parabolic problems. Progress in Nonlinear Differential Equations and their Applications, 16. Birkhäuser Verlag, Basel, 1995. xviii+424 pp. ISBN: 3-7643-5172-1 MR1329547
  • bysame, Interpolation Theory, Appunti, Scuola Normale Superiore Pisa, 1999.
  • J. Marcinkiewicz, Sur les multiplicateurs des séries de Fourier., Stud. Math. 8 (1939), 78--91.
  • van Neerven, J. M. A. M.; Veraar, M. C.; Weis, L. Stochastic integration in UMD Banach spaces. Ann. Probab. 35 (2007), no. 4, 1438--1478. MR2330977
  • bysame, Maximal L^p-regularity for stochastic evolution equations, SIAM J. Math. Anal. 44 (2012), no. 3, 1372--1414. MR?
  • bysame, Stochastic maximal L^p-regularity, Ann. Probab. 40 (2012), no. 2, 788--812. MR?
  • A.L. Neidhardt, Stochastic Integrals in 2-Uniformly Smooth Banach Spaces, Ph.D. thesis, University of Wisconsin, 1978.
  • Ondreját, Martin. Uniqueness for stochastic evolution equations in Banach spaces. Dissertationes Math. (Rozprawy Mat.) 426 (2004), 63 pp. MR2067962
  • Pardoux, E. Stochastic partial differential equations and filtering of diffusion processes. Stochastics 3, no. 2, 127--167. (1979), MR0553909
  • Prévôt, Claudia; Röckner, Michael. 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
  • Reed, Michael; Simon, Barry. Methods of modern mathematical physics. I. Functional analysis. Second edition. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. xv+400 pp. ISBN: 0-12-585050-6 MR0751959
  • 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
  • Schmeisser, Hans-Jürgen; Triebel, Hans. Topics in Fourier analysis and function spaces. A Wiley-Interscience Publication. John Wiley & Sons, Ltd., Chichester, 1987. 300 pp. ISBN: 0-471-90895-9 MR0891189
  • Seidler, Jan. Exponential estimates for stochastic convolutions in 2-smooth Banach spaces. Electron. J. Probab. 15 (2010), no. 50, 1556--1573. MR2735374
  • Stein, Elias M. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. With the assistance of Timothy S. Murphy. Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993. xiv+695 pp. ISBN: 0-691-03216-5 MR1232192
  • Triebel, Hans. Interpolation theory, function spaces, differential operators. Second edition. Johann Ambrosius Barth, Heidelberg, 1995. 532 pp. ISBN: 3-335-00420-5 MR1328645

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