ABSTRACT
Stochastic integration in Banach space and Banach space geometry
Jan van Neerven,June 18, 2003
Let $E$ be a real Banach space.
In this talk we construct stochastic integrals for $E$-valued functions
(with respect to a real Brownian motion $\{B(t)\}_{t\ge 0}$) and ${\mathcal
L}(E)$-valued functions (with respect to $E$-valued an Brownian motion
$\{B_E(t)\}_{t\ge 0}$), where ${\mathcal L}(E)$ denotes the space of bounded
linear operators on $E$.
We address the following problem: if $\Phi:(0,T)\to {\mathcal L}(E)$ is
stochastically integrable with respect to $\{B_E(t)\}_{t\ge 0}$, does it
follows that the orbits $\Phi x:(0,T)\to E$ (with $x\in E$) are
stochastically integrable with respect to $\{B(t)\}_{t\ge 0}$ and vice versa?
It turns out that the answer to these seemingly innocent questions
depends on the geometry of the underlying Banach space $E$. More precisely,
we show that an affirmative answer can be given precisely for the class
of Banach spaces with Pisier's property $(\alpha)$.
This requires a reformulation of property $(\alpha)$ in terms of a
natural isomorphism between certain spaces of radonifying operators.
Back to the history of the seminar
or the Colloquium Stochastiek homepage.
Igor Grubisic
(grubisic@math.uu.nl)