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A UNIFIED EXISTENCE AND UNIQUENESS THEOREM FOR STOCHASTIC EVOLUTION EQUATIONS

Published online by Cambridge University Press:  05 October 2009

A. JENTZEN
Affiliation:
Institut für Mathematik, Johann Wolfgang Goethe-Universität, D-60054 Frankfurt am Main, Germany (email: jentzen@math.uni-frankfurt.de)
P. E. KLOEDEN*
Affiliation:
Institut für Mathematik, Johann Wolfgang Goethe-Universität, D-60054 Frankfurt am Main, Germany (email: kloeden@math.uni-frankfurt.de)
*
For correspondence; e-mail: kloeden@math.uni-frankfurt.de
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Abstract

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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.

Type
Research Article
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

Footnotes

This work has been supported by the DFG project ‘Pathwise numerics and dynamics of stochastic evolution equations’.

References

[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.CrossRefGoogle Scholar
[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.Google Scholar
[3]Da Prato, G. and Zabczyk, J., Stochastic Equations in Infinite Dimensions (Cambridge University Press, Cambridge, 1992).CrossRefGoogle Scholar
[4]Da Prato, G. and Zabczyk, J., Ergodicity for Infinite Dimensional Systems, London Mathematical Society Lecture Notes Series, 229 (Cambridge University Press, Cambridge, 1996).CrossRefGoogle Scholar
[5]Jentzen, A., ‘Taylor expansions of solutions of stochastic partial differential equations’, Preprint, 2009.Google Scholar
[6]Jentzen, A. and Kloeden, P. E., ‘The numerical approximation of stochastic partial differential equations’, Milan J. Math., (2009) to appear.CrossRefGoogle Scholar
[7]Manthey, R. and Zausinger, T., ‘Stochastic evolution equations in L2νρ’, Stoch. Stoch. Rep. 66 (1999), 3785.CrossRefGoogle Scholar
[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.CrossRefGoogle Scholar
[9]Prévot, C. and Röckner, M., A Concise Course on Stochastic Partial Differential Equations (Springer, Berlin, 2007).Google Scholar
[10]Sell, G. R. and You, Y., Dynamics of Evolutionary Equations (Springer, New York, 2002).CrossRefGoogle Scholar