Skip to main content Accessibility help
×
Hostname: page-component-7c8c6479df-hgkh8 Total loading time: 0 Render date: 2024-03-28T11:05:48.015Z Has data issue: false hasContentIssue false

Introduction: Motivating examples

Published online by Cambridge University Press:  21 March 2010

Jerzy Zabczyk
Affiliation:
Polish Academy of Sciences
Get access

Summary

As we have said in the Preface, stochastic evolution equations in infinite dimensions are natural generalizations of stochastic ordinary differential equations and their theory has motivations coming both from mathematics and the natural sciences: physics, chemistry and biology.

We present here several examples of stochastic equations of the form (0.1), together with some comments concerning their derivations. Examples 0.1 - 0.3 have purely mathematical motivations, Examples 0.4 - 0.6 come from physics, Example 0.7 from chemistry and 0.8 - 0.9 from biology.

Lifts of diffusion processes

Consider an ordinary stochastic differential equation on Rd of the form

where f and b1, …,bN are continuous mappings from Rd into Rd. Let us fix a closed subset KRd and let E be a Hilbert space of mappings from K into Rd contained in the space C(K;Rd) of continuous mappings from K into Rd. The following equation on E:

in which

is called the lift of (0.2) to E.

Note that if the identity mapping Id(ξ) : Id(ξ) = ξ ∈ K belongs to E and there exists a solution to (0.3) with x = Id then the formula:

defines a version of the solution to (0.2) continuously depending on the initial condition ξ. Such versions are called stochastic flows. If in addition the space E consists of diffeomorphisms then the stochastic flow (0.4) is the flow of diffeomorphisms. This way one can obtain basic results about stochastic flows from elementary facts on stochastic equations with values in infinite dimensional spaces and known results about Sobolev spaces. See [37] for a detailed exposition of the subject.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@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 saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved 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.

Available formats
×

Save book to Dropbox

To save content items to your account, please 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 account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please 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 account. Find out more about saving content to Google Drive.

Available formats
×