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Original Articles

Stochastic Heat and Wave Equations on a Lie Group

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Pages 662-695 | Received 27 Jul 2009, Accepted 31 Mar 2010, Published online: 01 Jun 2010

References

  • Bojdecki , T. , and J. Jakubowski , J. 1989 . Ito stochastic integral in the dual of a nuclear space . Journal of Multivariate Analysis 31 : 40 – 58 .
  • Brzeźniak , Z. 1995 . Stochastic partial differential equations in M-type 2 Banach spaces . Potential Analysis 4 : 1 – 45 .
  • Brzeźniak , Z. 1997 . On stochastic convolutions in Banach spaces and applications . Stochastics Stochastics Reports 61 : 245 – 295 .
  • Brzeźniak , Z. , and Peszat , S. 1999 . Space-time continuous solutions to SPDEs driven by a homogeneous Wiener process . Studia Mathematica 137 : 261 – 299 .
  • Corwin , L. , and Greenleaf , F. 1990 . Representation of Nilpotent Lie Groups and Their Applications , Cambridge University Press , Cambridge , UK .
  • Curtain , R.F. , and Pritchard , A.J. 1978 . Infinite Dimensional Linear Systems Theory , Lecture Notes in Control and Information Science 8 , Springer-Verlag , Berlin .
  • Coulhon , T , Saloff-Coste , L. , and Varopoulos , N. 1992 . Analysis and Geometry on Groups , Cambridge University Press , Cambridge , UK .
  • Dalang , R.C. 1999 . Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.'s . Electronic Journal of Probability 4 : 1 – 29 .
  • Dalang , R. , and Frangos , N. 1998 . The stochastic wave equation in two spatial dimensions . Annals of Probability 26 : 187 – 212 .
  • Dettweiler , E. 1991 . Stochastic integration relative to Brownian motion on a general Banach space . Dogˇa Matematika 15 : 6 – 44 .
  • Da Prato , G. , and Zabczyk , J. 1992 . Stochastic Equations in Infinite Dimensions , Cambridge University Press , Cambridge , UK .
  • Folland , G. 1995 . A Course in Abstract Harmonic Analysis , Studies in Advanced Mathematics, CRC Press , Boca Raton , FL .
  • Funaki , T. 1991 . Regularity properties for stochastic partial differential equations of parabolic type . Osaka Journal of Mathematics 28 : 495 – 516 .
  • Gel'fand , I.M. , and Vilenkin , N. Ya. 1964 . Generalized Functions IV Applications of Harmonic Analysis , Academic Press , New York .
  • Itô , K. 1984 . Foundations of Stochastic Differential Equations in Infinite Dimensional Spaces , SIAM , Philadelphia .
  • Kallianpur , G. , and Xiong , J. 1995 . Stochastic Differential Equations in Infinite Dimensional Spaces , Lecture Notes—Monograph Series 26, Institute of Mathematical Statistics .
  • Karczewska , A. , and Zabczyk , J. 2001 . A note on stochastic wave equations . In Evolution Equations and their Applications in Physical and Life Sciences. Bad Herrenhalb 1998 . Marcel Dekker , New York .
  • Karczewska , A. , and Zabczyk , J. 2000 . Stochastic PDEs with function-valued solutions . In Infinite Dimensional Stochastic Analysis Amsterdam 1999 . Verh. Afd. Natuurkd. 1. Reeks. K. Ned. Acad. Wet., 52. R. Neth. Acad. Arts Sci.
  • Malliavin , M.P. , and Malliavin , P. 1974. Factorisations et lois limites de la diffusion horizontale au-dessus d'un espace riemannien symetrique. In Théorie du potentiel et analyse harmonique . Lecture Notes in Mathematics 404. Springer , Berlin.
  • Millet , A. , and Morien , P.L. 2000 . On a stochastic wave equation in two space dimensions: regularity of the solution and its density . Stochastic Processes and Applications 86 : 141 – 162 .
  • Millet , A. , and Sanz-Solé , M. 1999 . A stochastic wave equation in two space dimension: Smootheness of the law . Annals of Probability 27 : 803 – 844 .
  • Neidhardt , A.L. 1978 . Stochastic Integrals in 2-Uniformly Smooth Banach Spaces, Ph.D. Thesis, University of Wisconsin.
  • Peszat , S. 1995 . Existence and uniqueness of the solution for stochastic equations on Banach spaces . Stochastics Stochastics Reports 55 : 167 – 193 .
  • Peszat , S. 2002 . The Cauchy problem for a nonlinear stochastic wave equation in any dimension . Journal of Evolutionary Equations 2 : 383 – 394 .
  • Peszat , S. 2001 . SPDEs driven by a spatially homogeneous Wiener process . In SPDE and Applications , Marcel Dekker , New York .
  • Peszat , S. , and Zabczyk. J. 1997 . Stochastic evolution equations with a spatially homogeneous Wiener process . Stochastic Processes and Applications 72 : 187 – 204 .
  • Peszat , S. , and Zabczyk. J. 2000 . Nonlinear stochastic wave and heat equations . Probability Theory and Related Fields 116 : 421 – 443 .
  • Peszat , S. , and Zabczyk , J. 2007 . Stochastic Partial Differential Equations with Lévy Noise (an Evolution Equation Approach) , in Encyclopedia of Mathematics and its Applications . Cambridge University Press , Cambridge , UK .
  • Sanz-Solé , M. , and Sarrà , M. 2000 . Path properties of a class of martingale measures with applications to spde's . In Stochastic Processes, Physics and Geometry: New Interplays . CMS Conference Proceeding, 28 . American Mathematical Society Providence , RI .
  • Thangavelu , S. 1998 . Harmonic Analysis on the Heisenberg Group , Progress in Mathematics 159, Birkhäuser .
  • Tindel , S. , and Viens , F. 1999 . On space-time regularity for the stochastic heat equations on a Lie groups . Journal of Functional Analysis 169 : 559 – 603 .
  • Walsh , J.B. 1986 . An introduction to stochastic partial differential equations . In École d'été de probabilités de Saint-Flour, XIV–1984 , Lecture Notes in Mathematics 1180 . Springer , Berlin .

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