SPDEs with pseudodifferential generators: the existence of a density

Samy Tindel

Applicationes Mathematicae (2000)

  • Volume: 27, Issue: 3, page 287-308
  • ISSN: 1233-7234

Abstract

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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 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 ) ∈ [ 0 , T ] × 𝒪 is absolutely continuous with respect to the Lebesgue measure.

How to cite

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Tindel, Samy. "SPDEs with pseudodifferential generators: the existence of a density." Applicationes Mathematicae 27.3 (2000): 287-308. <http://eudml.org/doc/219274>.

@article{Tindel2000,
abstract = {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.},
author = {Tindel, Samy},
journal = {Applicationes Mathematicae},
keywords = {pseudodifferential operators; stochastic partial differential equations; Malliavin's calculus; Malliavin calculus},
language = {eng},
number = {3},
pages = {287-308},
title = {SPDEs with pseudodifferential generators: the existence of a density},
url = {http://eudml.org/doc/219274},
volume = {27},
year = {2000},
}

TY - JOUR
AU - Tindel, Samy
TI - SPDEs with pseudodifferential generators: the existence of a density
JO - Applicationes Mathematicae
PY - 2000
VL - 27
IS - 3
SP - 287
EP - 308
AB - 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.
LA - eng
KW - pseudodifferential operators; stochastic partial differential equations; Malliavin's calculus; Malliavin calculus
UR - http://eudml.org/doc/219274
ER -

References

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  1. [1] V. Bally and E. Pardoux, Malliavin calculus for white noise driven parabolic SPDEs, Potential Anal. 9 (1998), 27-64. Zbl0928.60040
  2. [2] Z. Brzeźniak, On stochastic convolution in Banach spaces and applications, Stochastics Stochastics Rep. 61 (1997), 245-295. Zbl0891.60056
  3. [3] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, 1992. Zbl0761.60052
  4. [4] C. Donati et E. Pardoux, EDPS réfléchies et calcul de Malliavin, Bull. Sci. Math. 121 (1997), 405-422. 
  5. [5] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer, Berlin, 1981. Zbl0456.35001
  6. [6] N. Jacob, Feller semigroups, Dirichlet forms, and pseudo-differential operators, Forum Math. 4 (1992), 433-446. Zbl0759.60078
  7. [7] G. Kallianpur and J. Xiong, Large deviation for a class of stochastic differential equations, Ann. Probab. 24 (1996), 320-345. Zbl0854.60026
  8. [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. [9] P. Kotelenez, A class of function and density valued stochastic partial differential equations driven by space-time white noise, to appear. 
  10. [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. Zbl0936.60055
  11. [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. Zbl0939.60067
  12. [12] A. Millet and M. Sanz, A stochastic wave equation in two space dimension: smoothness of the law, Ann. Probab. 27 (1999), 803-844. Zbl0944.60067
  13. [13] D. Nualart, The Malliavin Calculus and Related Topics, Springer, Berlin, 1995. Zbl0837.60050
  14. [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. Zbl0777.60046
  15. [15] S. Peszat and J. Zabczyk, Stochastic evolution equations with a spatially homogeneous Wiener process, Stochastic Process. Appl. 72 (1997), 187-204. Zbl0943.60048
  16. [16] J. Seidler, Da Prato-Zabczyk's maximal inequality revisited I, Math. Bohem. 118 (1993), 67-106. Zbl0785.35115
  17. [17] F. Treves, Introduction to Pseudodifferential and Fourier Integral Operators, Plenum, New York, 1982. Zbl0453.47027
  18. [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. 

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