% \newpage
\pagenumbering{gobble}
\section*{Ornstein--Uhlenbeck integrator}

An additional example of a randomised integrator is an {\em integrated Ornstein--Uhlenbeck process},
derived as follows. Define,
on the interval $s \in [t_k,t_{k+1})$, the pair of equations
\begin{subequations}
\label{eq:OU2}
\begin{align}
dU&=Vdt, \quad U(t_k)=U_k,\\
dV&=-\Lambda Vdt+\sqrt{2\Sigma} \, dW, \quad V(t_k)=f(U_k).
\end{align}
\end{subequations}
Here $W$ is a standard Brownian motion and
$\Lambda$ and $\Sigma$ are invertible matrices, possibly depending on $h$.
The approximating function $\gh(s)$ is thus defined by $V(s)$,
an Ornstein--Uhlenbeck process. 

Integrating (\ref{eq:OU2}b) we obtain
\begin{equation}
\label{eq:6}
V(s)=\exp\bigl(-\Lambda(s-t_k)\bigr)f(U_{k})+\chi_k(s-t_k), 
\end{equation}
where $\quad s \in [t_k,t_{k+1})$ and the $\{\chi_k\}$ form an i.i.d. sequence of Gaussian random
functions defined on $[0,h]$ with 
$$\chi_k(s)=\sqrt{2\Sigma}\int_0^{s}\exp\bigl(\Lambda(\tau-s)\bigr) \, dW(\tau).$$
Note that the $h$-dependence of $\hC$ comes through
the time-interval on which $\chi_k$ is defined, and through $\Lambda$
and $\Sigma$.

Integrating (\ref{eq:OU2}a), using \eqref{eq:6}, we obtain
\begin{equation}
\label{eq:probgex}
U(s)=U_k+\Lambda^{-1}\Bigl(I-\exp\bigl(-\Lambda (s-t_k)\bigr)\Bigr)f(U_{k})+\xi_k(s-t_k),
\end{equation}
where $\quad s \in [t_k,t_{k+1}]$, and, for $t \in [0,h]$,
\begin{equation}
\label{eq:chi2}
\xi_k(t)=\int_{0}^{t}\chi_{k}(\tau)d\tau.
\end{equation}
The numerical method \eqref{eq:probgex} 
may be written in the form \eqref{eq:probg2}, and hence
\eqref{eq:probg} at the grid-points, with the definition
$$\psh(u)=u+\Lambda^{-1}\Bigl(I-\exp\bigl(-\Lambda h\bigr)\Bigr)f(u).$$
This integrator is first order accurate and satisfies
Assumptions \ref{a:2} with $p=1$. Choosing to scale $\Sigma$ with
$h$ so that $q \ge 1$ in Assumptions \ref{a:1} leads to convergence
of the numerical method with order $1$.


Had we carried out the above analysis in the case $\Lambda=0$ we would
have obtained the probabilistic Euler method \eqref{eq:prob2},
and hence \eqref{eq:prob} at grid points, used as our canonical
example in the earlier developments.


\section*{Convergence rate of $L_2$ convergence}

When considering the Poisson problem in two dimensions, as discussed in section \ref{sec:pde}, we expect an improved
rate of convergence in the function space $H$. We now show that such a result 
also holds in our random setting.

Note that, under Assumptions \ref{assumption:pde1}, 
if we introduce $\gamma_{jk}$ that is $1$ when two basis functions
have overlapping support, and $0$ otherwise, then
$\gamma_{jk}$ is symmetric and there is constant $C$, independent of 
$j$ and $J$, such that $\sum_{k=1}^J \gamma_{jk} \le C.$
Now let $\varphi$ solve the equation
$a(\varphi,v)=\langle e,v \rangle, \quad \forall v \in \cV.$
Then $\|\varphi\|_{H^2} \le C|e|.$
%\begin{equation}
%\label{eq:vp}
%\|\varphi\|_{H^2} \le C|e|.
%\end{equation}
We define $\vars$ and $\varr$ in analogy with the definitions of
$\vs$ and $\vr$. Following the usual arguments for application of the
Aubin--Nitsche trick \cite{jo92}, we have
$
|e|^2=a(e,\varphi)=a(e,\varphi-\vars-\varr).
$
Thus
\begin{equation}
|e|^2  \le  \|e\|_a\|\varphi-\vars-\varr\|_a
 \le  \sqrt{2}\|e\|_a\Bigl(\|\varphi-\vars\|_a^2+\|\varr\|_a^2\Bigl)^{\frac12}. \label{eq:dnb2}
\end{equation}
We note that
$\varr(x)=\sum_{j=1}^{J} \varphi(x_j)\phr_j(x)=\|\varphi\|_{H^2}
\sum_{j=1}^{J} a_j \phr_j(x)$
where, by Sobolev embedding ($d=2$ here),
$a_j:=\varphi(x_j)/\|\varphi\|_{H^2}$
satisfies $\max_{1 \le j \le J}|a_j| \le C.$
Note, however, that the $a_j$ are random and correlated with all of
the random basis functions.
Using this, together with \eqref{eq:bnd}, in \eqref{eq:dnb2}, we obtain
$$|e|^2  \le C\|e\|_a\Bigl(h^2+\bigl\|\sum_{j=1}^{J} a_j \phr_j(x)\bigr\|_a^2\Bigr)^{\frac12}\|\varphi\|_{H^2}.$$
We see that
$$|e| \le C\|e\|_a\Bigl(h^2+
\sum_{j=1}^J \sum_{k=1}^J a_ja_k \,a(\phr_j,\phr_k)\Bigr)^{\frac12}.$$
From this and the symmetry of $\gamma_{jk}$, we obtain
\begin{align*}
|e| &\le C\|e\|_a\Bigl(h^2+\sum_{j=1}^J \sum_{k=1}^J \gamma_{jk}
\bigl(\|\phr_j\|_a^2+\|\phr_k\|_a^2\bigr)\Bigr)^{\frac12}\\
&\le C\|e\|_a\Bigl(h^2+2C\sum_{j=1}^J \|\phr_j\|_a^2\Bigr)^{\frac12}.
\end{align*}
Taking expectations, using that $p=q=1$, we find, using Assumptions \ref{assumption:pde1}, that
$\E |e| \le Ch \bigl(\E\bigl\|e\|_a^2\bigr)^{\frac12} \le Ch^2$
as desired.  Thus we recover the extra
order of convergence over the rate $1$ in the $\|\cdot\|_a$ norm
(although the improved rate is in $L^1(\Omega;H)$ whilst the lower rate
of convergence is in $L^2(\Omega;\cV).$).
