| Title:
|
Continuity versus nonexistence for a class of linear stochastic Cauchy problems driven by a Brownian motion (English) |
| Author:
|
Dettweiler, Johanna |
| Author:
|
Neerven, Jan van |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
56 |
| Issue:
|
2 |
| Year:
|
2006 |
| Pages:
|
579-586 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
stochastic linear Cauchy problems |
| Keyword:
|
nonexistence of weak solutions |
| Keyword:
|
continuous modifications |
| Keyword:
|
$C_0$-groups of linear operators |
| MSC:
|
34F05 |
| MSC:
|
34G10 |
| MSC:
|
35R15 |
| MSC:
|
47D05 |
| MSC:
|
47D06 |
| MSC:
|
47N20 |
| MSC:
|
60H15 |
| idZBL:
|
Zbl 1164.35520 |
| idMR:
|
MR2291757 |
| . |
| Date available:
|
2009-09-24T11:35:38Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128087 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
[3] Z. Brzeźniak, Sz. Peszat, and J. Zabczyk: Continuity of stochastic convolutions.Czechoslovak Math. J. 51 (2001), 679–684. MR 1864035, 10.1023/A:1013752526625 |
| Reference:
|
[4] G. Da Prato, J. Zabczyk: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications.Cambridge University Press, Cambridge, 1992. MR 1207136 |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
|
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| . |
