On generalized Malliavin calculus
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.
scientific article; zbMATH DE number 6017605
scientific article