Let $H_B$ be the reproducing kernel Hilbert space (RKHS) of the Brownian Motion $(B_t)$ on $[0,1]$. It is well known that with probability 1 the paths of $(B_t)$ are *not* contained in $H_B$.
But is there another (potentially larger) RKHS $H$ that does contain the paths of $(B_t)$ with probability 1?

**Background information.** An RKHS $H$ on $[0,1]$ is a Hilbert space of functions $f:[0,1]\to \mathbb R$ such that the point evaluations
\begin{align*}
\delta_t :H &\to \mathbb R \\
f &\mapsto f(t)
\end{align*}
are continuous for all $t\in [0,1]$. In this case, $k:[0,1]^2\to \mathbb R$ defined by
\begin{align*}
k(s,t) := \langle \delta_s,\delta_t\rangle_{H'}
\end{align*}
is the reproducing kernel of $H$, where $H'$ denotes the dual of $H$. Reproducing kernels are exactly those functions that are symmetric and positive semi-definite.
It is well-known that there is a one-to-one relationship between reproducing kernels and RKHSs.

For a centered stochastic process $(X_t)$, the covariance function $(s,t) \mapsto \mathbb E X_s X_t$ is symmetric and positive semi-definite and thus a reproducing kernel.

In the case of the Brownian motion, we have $\mathbb E B_s B_t = \min\{s,t\}$. In my question, the corresponding, uniquely defined RKHS of this reproducing kernel is denoted by $H_B$.