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A class of infinite dimensional stochastic processes with unbounded diffusion and its associated Dirichlet forms
Linköping University, Department of Mathematics, Mathematical Statistics . Linköping University, Faculty of Science & Engineering.
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

$\varepsilon(F,G) = \int \langle ADF,DG \rangle \mathbb_{H} d \nu,$

where $A = \small\sum\nolimits_{i=1}^\infty \ \lambda_i \langle S_i, \cdot \rangle \mathbb{H} S_i.$ Here $\small {S_i}, \ i \ \varepsilon \ \mathbb{N},$ 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 $\small\lambda_i \, i \, \varepsilon \, \mathbb{N}$ 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 $\small\varphi$ on theWiener measure $\small\nu$ 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 $\small\lambda_i \ i \ \varepsilon \ N$, 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 $\small\lambda_i \ i \ \varepsilon \ \mathbb{N}$ is a sequence of non-decreasing real numbers. The process X associated to $\small(\varepsilon, \ D(\varepsilon))$ 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.

##### Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1699
##### National Category
Probability Theory and Statistics
##### Identifiers
ISBN: 978-91-7685-966-7 (print)OAI: oai:DiVA.org:liu-121636DiVA: diva2:857512
##### Public defence
2015-12-10, C3, C-huset, Campus Valla, Linköping, 13:30 (English)
##### Supervisors
Available from: 2015-10-26 Created: 2015-09-29 Last updated: 2015-11-30Bibliographically approved
##### List of papers
1. A class of infinite dimensional stochastic processes with unbounded diffusion
Open this publication in new window or tab >>A class of infinite dimensional stochastic processes with unbounded diffusion
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
##### Keyword
Dirichlet form on Wiener space; Dirichlet form on Wiener space over non-compact manifold; closability; weighted Wiener measure; quasi-regularity
Mathematics
##### Identifiers
urn:nbn:se:liu:diva-118070 (URN)10.1080/17442508.2014.959952 (DOI)000353580300004 ()
Available from: 2015-05-20 Created: 2015-05-20 Last updated: 2017-12-04
2. Infinite dimensional Ornstein-Uhlenbeck processes with unbounded diffusion: Approximation, quadratic variation, and Itô formula
Open this publication in new window or tab >>Infinite dimensional Ornstein-Uhlenbeck processes with unbounded diffusion: Approximation, quadratic variation, and Itô formula
2016 (English)In: Mathematische Nachrichten, ISSN 0025-584X, E-ISSN 1522-2616, Vol. 289, no 17-18, 2192-2222 p.Article in journal (Refereed) Published
##### Abstract [en]

The paper studies a class of Ornstein-Uhlenbeck processes on the classical Wiener space. These processes are associated with a diffusion type Dirichlet form whose corresponding diffusion operator is unbounded in the Cameron- Martin space. It is shown that the distributions of certain finite dimensional Ornstein-Uhlenbeck processes converge weakly to the distribution of such an infinite dimensional Ornstein-Uhlenbeck process. For the infinite dimensional processes, the ordinary scalar quadratic variation is calculated. Moreover, relative to the stochastic calculus via regularization, the scalar as well as the tensor quadratic variation are derived. A related Itô formula is presented.

##### Place, publisher, year, edition, pages
Wiley-VCH Verlagsgesellschaft, 2016
##### Keyword
Infinite dimensional Ornstein-Uhlenbeck process, quadratic variation, Itô formula, weak approximation
Mathematics
##### Identifiers
urn:nbn:se:liu:diva-122181 (URN)10.1002/mana.201500146 (DOI)000389128100008 ()
##### Note

At the time for thesis presentation publication was in status: Manuscript.

Available from: 2015-10-23 Created: 2015-10-23 Last updated: 2017-12-01Bibliographically approved

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