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A class of infinite dimensional stochastic processes with unbounded diffusion
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, The Institute of Technology.
2015 (English)In: Stochastics: An International Journal of Probablitiy and Stochastic Processes, ISSN 1744-2508, E-ISSN 1744-2516, Vol. 87, no 3, 424-457 p.Article in journal (Refereed) Published
Abstract [en]

The paper studies Dirichlet forms on the classical Wiener space and the Wiener space over non-compact complete Riemannian manifolds. The diffusion operator is almost everywhere an unbounded operator on the Cameron-Martin space. In particular, it is shown that under a class of changes of the reference measure, quasi-regularity of the form is preserved. We also show that under these changes of the reference measure, derivative and divergence are closable with certain closable inverses. We first treat the case of the classical Wiener space and then we transfer the results to the Wiener space over a Riemannian manifold.

Place, publisher, year, edition, pages
Taylor and Francis: STM, Behavioural Science and Public Health Titles , 2015. Vol. 87, no 3, 424-457 p.
Keyword [en]
Dirichlet form on Wiener space; Dirichlet form on Wiener space over non-compact manifold; closability; weighted Wiener measure; quasi-regularity
National Category
Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-118070DOI: 10.1080/17442508.2014.959952ISI: 000353580300004OAI: oai:DiVA.org:liu-118070DiVA: diva2:812854
Available from: 2015-05-20 Created: 2015-05-20 Last updated: 2015-10-26
In thesis
1. A class of infinite dimensional stochastic processes with unbounded diffusion and its associated Dirichlet forms
Open this publication in new window or tab >>A class of infinite dimensional stochastic processes with unbounded diffusion and its associated Dirichlet forms
2015 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of two papers which focuses on a particular diffusion type Dirichlet form

 

where  Here  is the basis in the Cameron-Martin space, H, consisting of the Schauder functions, and ν denotes the Wiener measure.

In Paper I, we let  vary over the space of wiener trajectories in a way that the diffusion operator A is almost everywhere an unbounded operator on the Cameron–Martin space. In addition we put a weight function  on theWiener measure  and show that under these changes of the reference measure, the Malliavin derivative and divergence are closable operators with certain closable inverses. It is then shown that under certain conditions on , and these changes of reference measure, the Dirichlet form is quasi-regular. This is done first in the classical Wiener space and then the results are transferred to the Wiener space over a Riemannian manifold.

Paper II focuses on the case when  is a sequence of non-decreasing real numbers. The process X associated to  is then an infinite dimensional Ornstein-Uhlenbeck process. In this case we show that the distributions of a sequence of certain finite dimensional Ornstein-Uhlenbeck processes converge weakly to the distribution of the infinite dimensional Ornstein-Uhlenbeck process. We also investigate the quadratic variation for this process, both in the classical sense and in the recent framework of stochastic calculus via regularization. Since the process is Banach space valued, the tensor quadratic variation is an appropriate tool to establish the Itô formula for the infinite dimensional Ornstein-Uhlenbeck process X. Sufficient conditions are presented for the scalar as well as the tensor quadratic variation to exist.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2015. 34 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1699
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:liu:diva-121636 (URN)10.3384/diss.diva-121636 (DOI)978-91-7685-966-7 (print) (ISBN)
Public defence
2015-12-10, C3, C-huset, Campus Valla, Linköping, 13:30 (English)
Opponent
Supervisors
Available from: 2015-10-26 Created: 2015-09-29 Last updated: 2015-11-30Bibliographically approved

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Karlsson, JohnLöbus, Jörg-Uwe
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Mathematical Statistics The Institute of Technology
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