Infinite dimensional weak Dirichlet processes and convolution type processes.

january, 2017
Publication type:
Paper in peer-reviewed journals
Journal:
Stochastic Processes and its Applications, vol. 127 (1), pp. 325-357
arXiv:
images/icons/icon_arxiv.png 1606.03828
Keywords :
Covariation and Quadratic variation; Calculus via regularization; Infinite dimensional analysis; Tensor analysis; Dirichlet processes; Generalized Fukushima decomposition; Convolution type processes; Stochastic partial differential equations.
Abstract:
The present paper continues the study of infinite dimensional calculus via regularization, started by C. Di Girolami and the second named author, introducing the notion of \emph{weak Dirichlet process} in this context. Such a process $\X$, taking values in a Banach space $H$, is the sum of a local martingale and a suitable {\it orthogonal} process. The concept of weak Dirichlet process fits the notion of \emph{convolution type processes}, a class including mild solutions for stochastic evolution equations on infinite dimensional Hilbert spaces and in particular of several classes of stochastic partial differential equations (SPDEs). In particular the mentioned decomposition appears to be a substitute of an It\^o's type formula applied to $f(t, \X(t))$ where $f:[0,T] \times H \rightarrow \R$ is a $C^{0,1}$ function and $\X$ a convolution type processes.
BibTeX:
@article{Fab-Rus-2017,
    author={Giorgio Fabbri and Francesco Russo },
    title={Infinite dimensional weak Dirichlet processes and convolution 
           type processes. },
    doi={10.1016/j.spa.2016.06.010 },
    journal={Stochastic Processes and its Applications },
    year={2017 },
    month={1},
    volume={127 (1) },
    pages={325--357},
}