Continuity versus nonexistence for a class of linear stochastic Cauchy problems driven by a Brownian motion

Title:

Continuity versus nonexistence for a class of linear stochastic Cauchy problems driven by a Brownian motion

Creator:

Dettweiler, Johanna and van Neerven, Jan

Identifier:

https://cdk.lib.cas.cz/client/handle/uuid:c67bc401-e466-47e4-a005-5e68f0d7989f
uuid:c67bc401-e466-47e4-a005-5e68f0d7989f

Subject:

stochastic linear Cauchy problems, nonexistence of weak solutions, continuous modifications, and $C_0$-groups of linear operators

Type:

model:article and TEXT

Format:

bez média and svazek

Description:

Let $A={\mathrm d}/{\mathrm d}\theta $ denote the generator of the rotation group in the space $C(\Gamma )$, where $\Gamma $ denotes the unit circle. We show that the stochastic Cauchy problem \[ {\mathrm d}U(t) = AU(t)+ f\mathrm{d}b_t, \quad U(0)=0, \qquad \mathrm{(1)}\] where $b$ is a standard Brownian motion and $f\in C(\Gamma )$ is fixed, has a weak solution if and only if the stochastic convolution process $t\mapsto (f * b)_t$ has a continuous modification, and that in this situation the weak solution has a continuous modification. In combination with a recent result of Brzeźniak, Peszat and Zabczyk it follows that (1) fails to have a weak solution for all $f\in C(\Gamma )$ outside a set of the first category.

Language:

English

Rights:

http://creativecommons.org/publicdomain/mark/1.0/
policy:public

Coverage:

579-586

Source:

Czechoslovak Mathematical Journal | 2006 Volume:56 | Number:2

Harvested from:

CDK

Metadata only:

false