分野別セミナー

講演要旨：
Some parts of this talk are based on joint works
with Dr. M. Hayashi and with Prof. H. Kunita.

Asymptotic expansion of a Wiener-Poisson functional
Let $F$ be a Wiener-Poisson functional.
Suppose $F=F(e)$ depends on a small parameter $e>0$.
We would like to define
$$
\Phi \circ F(e)
$$
for $\Phi \in {\cal S}'$, and make an asymptotic
expansion
$$
\Phi \circ F(e) \sim \Phi_0 + e \Phi_1
+ e^2 \Phi_2 + ...
$$
in some norm with respect to $e$. Here $ {\cal S}'$
dnotes the space of tempered distributions. Examples
are $\Phi = d_{\{x_0\}}$, $\Phi = H(x) =1_{\{x>0\}}$,
and $\Phi = (x-x_0)_+$.
Main topics are :
Wiener-Poisson space
Sobolev space over the Wiener-Poisson space
Composition with the (Schwartz) distributions (2 ways)
The density function of $F$ at $x \in R^d$ can formally
be defined by $E[d_x (F)] = \langle d_x (F), 1 \rangle$,
where Dirac's delta function $d_x$ is an element of the
tempered distributions ${\cal S}'$.