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

Johanna Dettweiler; J.M.A.M. van Neerven

Czechoslovak Mathematical Journal (2006)

  • Volume: 56, Issue: 2, page 579-586
  • ISSN: 0011-4642

Abstract

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Let A = d / d θ denote the generator of the rotation group in the space C ( Γ ) , where Γ denotes the unit circle. We show that the stochastic Cauchy problem d U ( t ) = A U ( t ) + f d b t , U ( 0 ) = 0 , ( 1 ) where b is a standard Brownian motion and f ∈ C ( Γ ) is fixed, has a weak solution if and only if the stochastic convolution process t ↦ ( 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 ∈ C ( Γ ) outside a set of the first category.

How to cite

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Dettweiler, Johanna, and Neerven, J.M.A.M. van. "Continuity versus nonexistence for a class of linear stochastic Cauchy problems driven by a Brownian motion." Czechoslovak Mathematical Journal 56.2 (2006): 579-586. <http://eudml.org/doc/31049>.

@article{Dettweiler2006,
abstract = {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.},
author = {Dettweiler, Johanna, Neerven, J.M.A.M. van},
journal = {Czechoslovak Mathematical Journal},
keywords = {stochastic linear Cauchy problems; nonexistence of weak solutions; continuous modifications; $C_0$-groups of linear operators; nonexistence of weak solutions; continuous modifications; -groups of linear operators},
language = {eng},
number = {2},
pages = {579-586},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Continuity versus nonexistence for a class of linear stochastic Cauchy problems driven by a Brownian motion},
url = {http://eudml.org/doc/31049},
volume = {56},
year = {2006},
}

TY - JOUR
AU - Dettweiler, Johanna
AU - Neerven, J.M.A.M. van
TI - Continuity versus nonexistence for a class of linear stochastic Cauchy problems driven by a Brownian motion
JO - Czechoslovak Mathematical Journal
PY - 2006
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 56
IS - 2
SP - 579
EP - 586
AB - 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.
LA - eng
KW - stochastic linear Cauchy problems; nonexistence of weak solutions; continuous modifications; $C_0$-groups of linear operators; nonexistence of weak solutions; continuous modifications; -groups of linear operators
UR - http://eudml.org/doc/31049
ER -

References

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