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A UNIFIED EXISTENCE AND UNIQUENESS THEOREM FOR STOCHASTIC EVOLUTION EQUATIONS

  • A. JENTZEN (a1) and P. E. KLOEDEN (a2)
Abstract
Abstract

An existence and uniqueness theorem for mild solutions of stochastic evolution equations is presented and proved. The diffusion coefficient is handled in a unified way which allows a unified theorem to be formulated for different cases, in particular, of multiplicative space–time white noise and trace-class noise.

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Copyright
Corresponding author
For correspondence; e-mail: kloeden@math.uni-frankfurt.de
Footnotes
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This work has been supported by the DFG project ‘Pathwise numerics and dynamics of stochastic evolution equations’.

Footnotes
References
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[1]Chojnowska-Michalik A. and Goldys B., ‘Existence, uniqueness and invariant measures for stochastic semilinear equations on Hilbert spaces’, Probab. Theory Related Fields 102 (1995), 331356.
[2]Da Prato G., Debussche A. and Goldys B., ‘Some properties of invariant measures of non symmetric dissipative stochastic systems’, Probab. Theory Related Fields 123 (2002), 355380.
[3]Da Prato G. and Zabczyk J., Stochastic Equations in Infinite Dimensions (Cambridge University Press, Cambridge, 1992).
[4]Da Prato G. and Zabczyk J., Ergodicity for Infinite Dimensional Systems, London Mathematical Society Lecture Notes Series, 229 (Cambridge University Press, Cambridge, 1996).
[5]Jentzen A., ‘Taylor expansions of solutions of stochastic partial differential equations’, Preprint, 2009.
[6]Jentzen A. and Kloeden P. E., ‘The numerical approximation of stochastic partial differential equations’, Milan J. Math., (2009) to appear.
[7]Manthey R. and Zausinger T., ‘Stochastic evolution equations in L2νρ’, Stoch. Stoch. Rep. 66 (1999), 3785.
[8]Müller-Gronbach T. and Ritter K., ‘Lower bounds and nonuniform time discretization for approximation of stochastic heat equations’, Found. Comput. Math. 7 (2007), 135181.
[9]Prévot C. and Röckner M., A Concise Course on Stochastic Partial Differential Equations (Springer, Berlin, 2007).
[10]Sell G. R. and You Y., Dynamics of Evolutionary Equations (Springer, New York, 2002).
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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