An operatorial approach to stochastic partial differential equations driven by linear multiplicative noise
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Publication:496465
DOI10.4171/JEMS/545zbMATH Open1327.60122arXiv1402.4940OpenAlexW1807107269MaRDI QIDQ496465
Author name not available (Why is that?)
Publication date: 21 September 2015
Published in: (Search for Journal in Brave)
Abstract: 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)xH, 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 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 the one 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, as e.g. stochastic transport equations.
Full work available at URL: https://arxiv.org/abs/1402.4940
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