I'll begin this question with the finite-dimensional case, as a
warmup.
Let me say a continuous path $\omega : [0,1] \to \mathbb{R}^d$ is
hyperplanar if there exists a nonzero $x \in \mathbb{R}^d$ such
that $\omega(t) \cdot x = 0$ for all $t \in [0,1]$.
It is not hard to show:
Proposition. Let $B_t$ be a standard Brownian motion in $\mathbb{R}^d$. Almost surely, $(B_t : 0 \le t \le 1)$ is not hyperplanar.
To avoid possible confusion, let me emphasize that the hyperplane is
allowed to be random. That is, I am claiming:
$$\mathbb{P}(\exists x \in \mathbb{R}^d\\, \forall t \in [0,1] : B_t \cdot
x = 0) = 0$$
which should not be confused with the weaker statement
$$\forall x \in \mathbb{R}^d :\mathbb{P}(\forall t \in [0,1] : B_t \cdot
x = 0) = 0.$$
One possible proof of the proposition is to choose $0 < t_1 < \dots <
t_d < 1$, and show by induction that, almost surely, $B_{t_1}, \dots,
B_{t_d}$ are linearly independent in $\mathbb{R}^d$. (By the
induction hypothesis, $B_{t_1}, \dots, B_{t_{k-1}}$ span a
$k-1$-dimensional subspace of $\mathbb{R}^d$; by the Markov property
and the absolute continuity of the Gaussian distribution, $B_{t_k}$ is
almost surely not in this subspace.)
I am interested in the analogous statement for infinite dimensions.
Let $W$ be a real separable Banach space, and say a continuous path $\omega :
[0,1] \to W$ is hyperplanar if there exists a nonzero continuous
linear functional $f \in W^*$ such that $f(\omega(t)) = 0$ for all $t
\in [0,1]$. Let $\mu$ be a non-degenerate Gaussian measure on $W$
(so that $(W,\mu)$ is an abstract Wiener space), and let $B_t$ be a standard Brownian motion in $W$. That is, the process
$B_t$ has continuous sample paths and independent increments, starts
at 0, and the increments are distributed such that $(t-s)^{-1/2}(B_t –
B_s) \sim \mu$.
For infinite-dimensional $W$, what is the probability that $(B_t : 0 \le t \le 1)$ is hyperplanar?
As a start, I can show that the set of hyperplanar paths is analytic
in $C([0,1], W)$, and hence universally measurable, so the question
actually makes sense. We can also use a scaling argument and the
Blumenthal zero-one law to see that the probability must be either 0
or 1.
I am not sure which way my intuition points here. On the one hand, we
expect a Brownian motion to be pretty irregular and unconstrained,
suggesting the answer is 0 as in finite dimensions. On the other
hand, an infinite-dimensional space has a lot of hyperplanes. In principle, the answer could depend on the abstract Wiener space $(W,\mu)$.
Thanks!
Best Answer
As suggested in my comment, here's a simple fact which applies to any probability measure $\mu$ on (the Borel σ-algebra of) a second countable topological space $X$. There is a unique minimal closed subset $S$ of $X$ with $\mu(S)=1$ -- the support of $\mu$ -- and, if $X_1,X_2,\ldots$ is an IID sequence of random variables, each with distribution $\mu$, then $\lbrace X_1,X_2,\ldots\rbrace$ is almost surely a dense subset of $S$.
Now, in the situation described in the question, choose a sequence $0\le t_0 < t_1 < t_2 < \cdots \le1$. Then, $X_k\equiv(t_k-t_{k-1})^{-1/2}(B_{t_k}-B_{t_{k-1}})$ is an IID sequence of random variables each with distribution $\mu$. Also, as it is non-degenerate, the support of $\mu$ is not contained in a closed proper subspace of the Banach space $W$.