An operatorial approach to stochastic partial differential equations driven by linear multiplicative noise

Barbu V, Röckner M (2015)
Journal of the European Mathematical Society 17(7): 1789-1815.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Autor*in
Barbu, Viorel; Röckner, MichaelUniBi
Abstract / Bemerkung
In this paper, we develop a new general approach to the existence and uniqueness theory of infinite-dimensional stochastic equations of the form dX + A(t)Xdt = XdW in (0, T) x H, where A(t) is a nonlinear monotone and demicontinuous operator from V to V', coercive and with polynomial growth. Here, V is a reflexive Banach space continuously and densely embedded in a Hilbert space H of (generalized) functions on a domain Omicron subset of R-d, and V' is the dual of V in the duality induced by H as pivot space. Furthermore, W is a Wiener process in H. The new approach is based on an operatorial reformulation of the stochastic equation which is quite robust under perturbation of A(t). This leads to new existence and uniqueness results of a larger class of equations with linear multiplicative noise than those treatable by the known approaches. In addition, we obtain regularity results for the solutions with respect to both the time and spatial variable which are sharper than the classical ones. New applications include stochastic partial differential equations, e.g. stochastic transport equations.
Stichworte
operatorial equations; Maximal monotone operator; stochastic integral
Erscheinungsjahr
2015
Zeitschriftentitel
Journal of the European Mathematical Society
Band
17
Ausgabe
7
Seite(n)
1789-1815
ISSN
1435-9855
Page URI
https://pub.uni-bielefeld.de/record/2780571

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Barbu V, Röckner M. An operatorial approach to stochastic partial differential equations driven by linear multiplicative noise. Journal of the European Mathematical Society. 2015;17(7):1789-1815.
Barbu, V., & Röckner, M. (2015). An operatorial approach to stochastic partial differential equations driven by linear multiplicative noise. Journal of the European Mathematical Society, 17(7), 1789-1815. doi:10.4171/JEMS/545
Barbu, V., and Röckner, M. (2015). An operatorial approach to stochastic partial differential equations driven by linear multiplicative noise. Journal of the European Mathematical Society 17, 1789-1815.
Barbu, V., & Röckner, M., 2015. An operatorial approach to stochastic partial differential equations driven by linear multiplicative noise. Journal of the European Mathematical Society, 17(7), p 1789-1815.
V. Barbu and M. Röckner, “An operatorial approach to stochastic partial differential equations driven by linear multiplicative noise”, Journal of the European Mathematical Society, vol. 17, 2015, pp. 1789-1815.
Barbu, V., Röckner, M.: An operatorial approach to stochastic partial differential equations driven by linear multiplicative noise. Journal of the European Mathematical Society. 17, 1789-1815 (2015).
Barbu, Viorel, and Röckner, Michael. “An operatorial approach to stochastic partial differential equations driven by linear multiplicative noise”. Journal of the European Mathematical Society 17.7 (2015): 1789-1815.