Non-autonomous stochastic Cauchy problems in Banach spaces

Mark Veraar, and Jan Zimmerschied. "Non-autonomous stochastic Cauchy problems in Banach spaces." Studia Mathematica 185.1 (2008): 1-34. <http://eudml.org/doc/285107>.

@article{MarkVeraar2008,
abstract = {We study the non-autonomous stochastic Cauchy problem on a real Banach space E, $dU(t) = A(t)U(t)dt + B(t)dW_\{H\}(t)$, t ∈ [0,T], U(0) = u₀. Here, $W_\{H\}$ is a cylindrical Brownian motion on a real separable Hilbert space H, $(B(t))_\{t∈[0,T]\}$ are closed and densely defined operators from a constant domain (B) ⊂ H into E, $(A(t))_\{t∈[0,T]\}$ denotes the generator of an evolution family on E, and u₀ ∈ E. In the first part, we study existence of weak and mild solutions by methods of van Neerven and Weis. Then we use a well-known factorisation method in the setting of evolution families to obtain time regularity of the solution. In the second part, we consider the parabolic case in the setting of Acquistapace and Terreni. By means of a factorisation method in the spirit of Da Prato, Kwapień, and Zabczyk we obtain space-time regularity results for parabolic evolution families on Banach spaces. We apply this theory to several examples. In the last part, relying on recent results of Dettweiler, van Neerven, and Weis, we prove a maximal regularity result where the A(t) are as in the setting of Kato and Tanabe.},
author = {Mark Veraar, Jan Zimmerschied},
journal = {Studia Mathematica},
keywords = {stochastic Cauchy problem; non-autonomous equation; parabolic evolution equation; evolution family; mild solution; weak solution; cylindrical Brownian motion; factorisation method; space-time regularity; maximal regularity; -calculus; -boundedness},
language = {eng},
number = {1},
pages = {1-34},
title = {Non-autonomous stochastic Cauchy problems in Banach spaces},
url = {http://eudml.org/doc/285107},
volume = {185},
year = {2008},
}

TY - JOUR
AU - Mark Veraar
AU - Jan Zimmerschied
TI - Non-autonomous stochastic Cauchy problems in Banach spaces
JO - Studia Mathematica
PY - 2008
VL - 185
IS - 1
SP - 1
EP - 34
AB - We study the non-autonomous stochastic Cauchy problem on a real Banach space E, $dU(t) = A(t)U(t)dt + B(t)dW_{H}(t)$, t ∈ [0,T], U(0) = u₀. Here, $W_{H}$ is a cylindrical Brownian motion on a real separable Hilbert space H, $(B(t))_{t∈[0,T]}$ are closed and densely defined operators from a constant domain (B) ⊂ H into E, $(A(t))_{t∈[0,T]}$ denotes the generator of an evolution family on E, and u₀ ∈ E. In the first part, we study existence of weak and mild solutions by methods of van Neerven and Weis. Then we use a well-known factorisation method in the setting of evolution families to obtain time regularity of the solution. In the second part, we consider the parabolic case in the setting of Acquistapace and Terreni. By means of a factorisation method in the spirit of Da Prato, Kwapień, and Zabczyk we obtain space-time regularity results for parabolic evolution families on Banach spaces. We apply this theory to several examples. In the last part, relying on recent results of Dettweiler, van Neerven, and Weis, we prove a maximal regularity result where the A(t) are as in the setting of Kato and Tanabe.
LA - eng
KW - stochastic Cauchy problem; non-autonomous equation; parabolic evolution equation; evolution family; mild solution; weak solution; cylindrical Brownian motion; factorisation method; space-time regularity; maximal regularity; -calculus; -boundedness
UR - http://eudml.org/doc/285107
ER -