Skip to main content Accessibility help
×
Home

ORNSTEIN–UHLENBECK PROCESSES IN BANACH SPACES AND THEIR SPECTRAL REPRESENTATIONS

Published online by Cambridge University Press:  17 June 2002


James S. Groves
Affiliation:
Department of Mathematics and Statistics, University of Lancaster, Lancaster LA1 4YF, UK (j.groves@lancaster.ac.uk)
Corresponding
E-mail address:

Abstract

For Q the variance of some centred Gaussian random vector in a separable Banach space it is shown that, necessarily, Q factors through $\ell^2$ as a product of 2-summing operators. This factorization condition is sufficient when the Banach space is of Gaussian type 2. The stochastic integral of a deterministic family of operators with respect to a Q-Wiener process is shown to exist under a continuity condition involving the 2-summing norm. A Langevin equation

$$ \rd\bm{Z}_t+\sLa\bm{Z}_t\,\rd t=\rd\bm{B}_t, $$

with values in a separable Banach space, is studied. The operator $\sLa$ is closed and densely defined. A weak solution $(\bm{Z}_t,\bm{B}_t)$, where $\bm{Z}_t$ is centred, Gaussian and stationary, while $\bm{B}_t$ is a Q-Wiener process, is given when $\ri\sLa$ and $\ri\sLa^*$ generate $C_0$ groups and the resolvent of $\sLa$ is uniformly bounded on the imaginary axis. Both $\bm{Z}_t$ and $\bm{B}_t$ are stochastic integrals with respect to a spectral Q-Wiener process.

AMS 2000 Mathematics subject classification: Primary 60G15. Secondary 46E40; 47B10; 47D03; 60H10


Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 2002

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 73 *
View data table for this chart

* Views captured on Cambridge Core between September 2016 - 1st December 2020. This data will be updated every 24 hours.

Access
Hostname: page-component-6d4bddd689-w4gz7 Total loading time: 0.344 Render date: 2020-12-01T16:10:04.914Z Query parameters: { "hasAccess": "1", "openAccess": "0", "isLogged": "0", "lang": "en" } Feature Flags last update: Tue Dec 01 2020 15:57:01 GMT+0000 (Coordinated Universal Time) Feature Flags: { "metrics": true, "metricsAbstractViews": false, "peerReview": true, "crossMark": true, "comments": true, "relatedCommentaries": true, "subject": true, "clr": false, "languageSwitch": true }

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

ORNSTEIN–UHLENBECK PROCESSES IN BANACH SPACES AND THEIR SPECTRAL REPRESENTATIONS
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

ORNSTEIN–UHLENBECK PROCESSES IN BANACH SPACES AND THEIR SPECTRAL REPRESENTATIONS
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

ORNSTEIN–UHLENBECK PROCESSES IN BANACH SPACES AND THEIR SPECTRAL REPRESENTATIONS
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *