Stochastic evolution equations driven by Liouville fractional Brownian motion

Zdzisław Brzeźniak; Jan van Neerven; Donna Salopek

Czechoslovak Mathematical Journal (2012)

  • Volume: 62, Issue: 1, page 1-27
  • ISSN: 0011-4642

Abstract

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Let H be a Hilbert space and E a Banach space. We set up a theory of stochastic integration of ℒ ( H , E ) -valued functions with respect to H -cylindrical Liouville fractional Brownian motion with arbitrary Hurst parameter 0 < β < 1 . For 0 < β < 1 2 we show that a function Φ : ( 0 , T ) → ℒ ( H , E ) is stochastically integrable with respect to an H -cylindrical Liouville fractional Brownian motion if and only if it is stochastically integrable with respect to an H -cylindrical fractional Brownian motion. We apply our results to stochastic evolution equations d U ( t ) = A U ( t ) d t + B d W H β ( t ) driven by an H -cylindrical Liouville fractional Brownian motion, and prove existence, uniqueness and space-time regularity of mild solutions under various assumptions on the Banach space E , the operators A : 𝒟 ( A ) → E and B : H → E , and the Hurst parameter β . As an application it is shown that second-order parabolic SPDEs on bounded domains in ℝ d , driven by space-time noise which is white in space and Liouville fractional in time, admit a mild solution if 1 4 d < β < 1 .

How to cite

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Brzeźniak, Zdzisław, van Neerven, Jan, and Salopek, Donna. "Stochastic evolution equations driven by Liouville fractional Brownian motion." Czechoslovak Mathematical Journal 62.1 (2012): 1-27. <http://eudml.org/doc/246136>.

@article{Brzeźniak2012,
abstract = {Let $H$ be a Hilbert space and $E$ a Banach space. We set up a theory of stochastic integration of $\{\mathcal \{L\}\}(H,E)$-valued functions with respect to $H$-cylindrical Liouville fractional Brownian motion with arbitrary Hurst parameter $0<\beta <1$. For $0<\beta <\frac\{1\}\{2\}$ we show that a function $\Phi \colon (0,T)\rightarrow \{\mathcal \{L\}\}(H,E)$ is stochastically integrable with respect to an $H$-cylindrical Liouville fractional Brownian motion if and only if it is stochastically integrable with respect to an $H$-cylindrical fractional Brownian motion. We apply our results to stochastic evolution equations \[ \{\rm d\}U(t) = AU(t) \{\rm d\}t + B \{\rm d\}W\_H^\beta (t) \] driven by an $H$-cylindrical Liouville fractional Brownian motion, and prove existence, uniqueness and space-time regularity of mild solutions under various assumptions on the Banach space $E$, the operators $A\colon \mathcal \{D\}(A)\rightarrow E$ and $B\colon H\rightarrow E$, and the Hurst parameter $\beta $. As an application it is shown that second-order parabolic SPDEs on bounded domains in $\mathbb \{R\}^d$, driven by space-time noise which is white in space and Liouville fractional in time, admit a mild solution if $\frac\{1\}\{4\}d<\beta <1$.},
author = {Brzeźniak, Zdzisław, van Neerven, Jan, Salopek, Donna},
journal = {Czechoslovak Mathematical Journal},
keywords = {(Liouville) fractional Brownian motion; fractional integration; stochastic evolution equations; Liouville fractional Brownian motion; stochastic evolution equation; fractional Ornstein-Uhlenbeck process},
language = {eng},
number = {1},
pages = {1-27},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Stochastic evolution equations driven by Liouville fractional Brownian motion},
url = {http://eudml.org/doc/246136},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Brzeźniak, Zdzisław
AU - van Neerven, Jan
AU - Salopek, Donna
TI - Stochastic evolution equations driven by Liouville fractional Brownian motion
JO - Czechoslovak Mathematical Journal
PY - 2012
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 62
IS - 1
SP - 1
EP - 27
AB - Let $H$ be a Hilbert space and $E$ a Banach space. We set up a theory of stochastic integration of ${\mathcal {L}}(H,E)$-valued functions with respect to $H$-cylindrical Liouville fractional Brownian motion with arbitrary Hurst parameter $0<\beta <1$. For $0<\beta <\frac{1}{2}$ we show that a function $\Phi \colon (0,T)\rightarrow {\mathcal {L}}(H,E)$ is stochastically integrable with respect to an $H$-cylindrical Liouville fractional Brownian motion if and only if it is stochastically integrable with respect to an $H$-cylindrical fractional Brownian motion. We apply our results to stochastic evolution equations \[ {\rm d}U(t) = AU(t) {\rm d}t + B {\rm d}W_H^\beta (t) \] driven by an $H$-cylindrical Liouville fractional Brownian motion, and prove existence, uniqueness and space-time regularity of mild solutions under various assumptions on the Banach space $E$, the operators $A\colon \mathcal {D}(A)\rightarrow E$ and $B\colon H\rightarrow E$, and the Hurst parameter $\beta $. As an application it is shown that second-order parabolic SPDEs on bounded domains in $\mathbb {R}^d$, driven by space-time noise which is white in space and Liouville fractional in time, admit a mild solution if $\frac{1}{4}d<\beta <1$.
LA - eng
KW - (Liouville) fractional Brownian motion; fractional integration; stochastic evolution equations; Liouville fractional Brownian motion; stochastic evolution equation; fractional Ornstein-Uhlenbeck process
UR - http://eudml.org/doc/246136
ER -

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