On generalized Malliavin calculus
type: Item
instance of: scholarly article
title: On generalized Malliavin calculus
zbMATH Open document ID: 1262.60049
author: Dora Seleši
author: S. V. Lototsky
author: B. Rozovskii
published in: Stochastic Processes and their Applications
publication date: 2012-03-22T00:00:00Z
review text: The existence of simple SPDEs for which any solution must have an infinite energy motivates the authors to consider a generalization of the Malliavin calculus to Sobolev weighted chaos spaces. Namely, given independent standard Gaussian variables \((\xi_j)\) and a sequence of (multi-indexed) positive weights \((r_\alpha)\), they consider the random elements \(u= \sum_\alpha u_\alpha\xi_\alpha\) with \(\sum_\alpha r^2_\alpha|u_\alpha|^2<\infty\) (a typical example \(r_\alpha= (\alpha!)^\rho q^\alpha\)). Then, the basic operators of Malliavin calculus read \[ \begin{aligned} D_u(v) &:= \sum_{\alpha,\beta} \sqrt{{\alpha+\beta\choose\beta}} (v_{\alpha+\beta}\otimes u_\beta)\xi_\alpha,\\ \delta_u(v) &:= \sum_{\beta\leq\alpha} \sqrt{{\alpha\choose \beta}}\langle v_\beta, u_{\alpha-\beta}\rangle \xi_\alpha,\\ L_u(v) &:= \sum_{\alpha,\beta,\gamma} \sqrt{{\alpha\choose \beta}{\beta+\gamma\choose \beta}}\langle u_\gamma, u_{\alpha-\beta}\rangle v_{\beta+ \gamma}\xi_\alpha,\end{aligned} \] as long as the series converge. The authors specify convergence conditions, and inequalities yielding continuity of the above operators (in suitable chaos spaces), mainly in the example quoted above. Finally, they address associated differential equations, and yield sufficient conditions for existence and uniqueness, in appropriate chaos spaces, and, further, discuss some examples.
reviewed by: Jacques Franchi
Mathematics Subject Classification ID: 60H07
Mathematics Subject Classification ID: 60H25
Mathematics Subject Classification ID: 60H15
zbMATH DE Number: 6017605
zbMATH Keywords: Malliavin operators
zbMATH Keywords: stochastc PDEs
zbMATH Keywords: generalized stochastic processes
zbMATH Keywords: Sobolev chaos spaces
MaRDI profile type: Publication
full work available at URL: j.spa.2011.11.003
OpenAlex ID: W2151059046
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DOI: 10.1016/J.SPA.2011.11.003
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description: scientific article; zbMATH DE number 6017605