Abstract
Over the last couple of years there has been a growing interest in stochastic partial differential equations (SPDEs). Various methods have been used to study SPDEs. Here we apply white noise analysis to obtain abstract existence and uniqueness theorems. More specifically we combine the ideas of Kondratiev spaces with the variational formulation for elliptic partial differential equations to study elliptic SPDEs. Hyperbolic and parabolic SPDEs can be treated similarly (see [10]).
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References
Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions, Dover Publications, New York, 1972.
Takeyuki Hida, Hui-Hsiung Kuo, Jürgen Potthoff, and Ludwig Streit, White Noise, Kluwer Academic Publishing, Dordrecht, 1993.
Helge Holden, Tom Lindstrøm, Bernt Øksendal, Jan Ubøe, and Tu- Sheng Zhang, The Pressure Equation for Fluid Flow in a Stochastic Medium, Potential Analysis, in print.
Stochastic Boundary Value Problems. A White Noise Functional Approach, Probab. Theory Relat. Fields 95 (1993), 391–419.
The Burgers Equation with a Noisy Force and the Stochastic Heat Equation, Comm. PDE 19 (1& 2) (1994), 119–141.
Yuri G. Kondratiev, Peter Leukert, and Ludwig Streit, Wick Calculus in Gaussian Analysis, Manuscript (1994).
Tom Lindstrøm, Bernt Øksendal, and Jan Ubøe, Stochastic Differential Equations Involving Positive Noise, in M. Barlow and N. Bingham (eds): Stochastic Analysis, Cambridge University Press (1991), 261–303.
, Stochastic Modelling of Fluid Flow in Porous Media, in S. Chen & J. Young (eds): Control Theory, Stochastic Analysis and Applications, World Scientific (1991), 156–172.
Bernt Øksendal and Tu-Sheng Zhang, The Stochastic Volterra Equation, D. Nualart and Sang Solé: Barcelona Seminar on Stochastic Analysis (Birkhäuser 1993 ), 168–202.
Gjermund Våge, Hilbert Space Methods for Stochastic Partial Differential Equations, Manuscript (1995).
Joseph Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987.
Tu-Sheng Zhang, Characterizations of White Noise Test Functions and Hida Distributions, Stochastics 41 (1992), 71–87.
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© 1996 Birkhäuser Boston
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Våge, G. (1996). Hilbert Space Methods Applied to Elliptic Stochastic Partial Differential Equations. In: Körezlioğlu, H., Øksendal, B., Üstünel, A.S. (eds) Stochastic Analysis and Related Topics V. Progress in Probability, vol 38. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-2450-1_15
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DOI: https://doi.org/10.1007/978-1-4612-2450-1_15
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