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2000 | 27 | 3 | 287-308
Tytuł artykułu

SPDEs with pseudodifferential generators: the existence of a density

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We consider the equation du(t,x)=Lu(t,x)+b(u(t,x))dtdx+σ(u(t,x))dW(t,x) where t belongs to a real interval [0,T], x belongs to an open (not necessarily bounded) domain $\mathcal O$, and L is a pseudodifferential operator. We show that under sufficient smoothness and nondegeneracy conditions on L, the law of the solution u(t,x) at a fixed point $(t,x)\in [0,T] \times \mathcal O$ is absolutely continuous with respect to the Lebesgue measure.
Rocznik
Tom
27
Numer
3
Strony
287-308
Opis fizyczny
Daty
wydano
2000
otrzymano
1999-03-09
poprawiono
1999-11-10
Twórcy
autor
  • Département de Mathématiques-Institut Galilée, Université Paris 13, Avenue J. B. Clément, 93430 Villetaneuse, France
Bibliografia
  • [1] V. Bally and E. Pardoux, Malliavin calculus for white noise driven parabolic SPDEs, Potential Anal. 9 (1998), 27-64.
  • [2] Z. Brzeźniak, On stochastic convolution in Banach spaces and applications, Stochastics Stochastics Rep. 61 (1997), 245-295.
  • [3] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, 1992.
  • [4] C. Donati et E. Pardoux, EDPS réfléchies et calcul de Malliavin, Bull. Sci. Math. 121 (1997), 405-422.
  • [5] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer, Berlin, 1981.
  • [6] N. Jacob, Feller semigroups, Dirichlet forms, and pseudo-differential operators, Forum Math. 4 (1992), 433-446.
  • [7] G. Kallianpur and J. Xiong, Large deviation for a class of stochastic differential equations, Ann. Probab. 24 (1996), 320-345.
  • [8] P. Kotelenez, Existence, uniqueness and smoothness for a class of function valued stochastic partial differential equations driven by space-time white noise, Stochastics Stochastics Rep. 41 (1992), 177-199.
  • [9] P. Kotelenez, A class of function and density valued stochastic partial differential equations driven by space-time white noise, to appear.
  • [10] N. Lanjri and D. Nualart, Burgers equation driven by space-time white noise: absolute continuity of the solution, Stochastics Stochastics Rep. 66 (1999), 273-292.
  • [11] D. Márquez and M. Sanz, Taylor expansion of the density of the law in a stochastic heat equation, Collect. Math. 49 (1998), 399-415.
  • [12] A. Millet and M. Sanz, A stochastic wave equation in two space dimension: smoothness of the law, Ann. Probab. 27 (1999), 803-844.
  • [13] D. Nualart, The Malliavin Calculus and Related Topics, Springer, Berlin, 1995.
  • [14] E. Pardoux and T. Zhang, Absolute continuity of the law of the solution of a parabolic SPDE, J. Funct. Anal. 112 (1993), 447-458.
  • [15] S. Peszat and J. Zabczyk, Stochastic evolution equations with a spatially homogeneous Wiener process, Stochastic Process. Appl. 72 (1997), 187-204.
  • [16] J. Seidler, Da Prato-Zabczyk's maximal inequality revisited I, Math. Bohem. 118 (1993), 67-106.
  • [17] F. Treves, Introduction to Pseudodifferential and Fourier Integral Operators, Plenum, New York, 1982.
  • [18] J. Walsh, An introduction to stochastic partial differential equations, in: Ecole d'été de Probabilité de Saint-Flour XIV, Lecture Notes in Math. 1180, Springer, Berlin, 1986, 265-439.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-zmv27i3p287bwm
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