Let $\Omega=C([0,1],\mathbb{R})$, $(\Pi_t)_{t\in [0,1]}$ the canonical process with $\Pi_t(\omega)=\omega_t$, $\mathcal{F}=\sigma(\Pi)$ and $\mathbb{F}$ the filtration generated by $\Pi$. Let $F:[0,1] \rightarrow \mathbb{R}$ be a continuous nondecreasing function.
I am currently reading a paper in which the following assertion is used without any reference.
There exists a unique probability measure $\mu$ on $(\Omega,\mathcal{F})$ such that $\Pi$ is a centered Gaussian process on $(\Omega,\mathcal{F},\mathbb{F},\mu)$ with $\mathrm{Cov}[\Pi_s,\Pi_t]=F(\min(s,t))$.
I can not find this claim anywhere. I would be very grateful for a reference to this assertion.
For $F(x)=x$ we know that the Gaussian process $\Pi_t$ is a Brownian motion. A general nondecreasing $F$ is differentiable almost everywhere on $[0,1]$. Then the process
$$ \int_0^t \sqrt{F'(s)}\,d\Pi_s=G(t)\Pi_t-G(0)\Pi_0-\int_0^t\Pi_s\,dG(s)\,,\quad\text{ where }G(s):=\sqrt{F'(s)} $$ has variance $\int_0^t F'(s)\,ds=F(t)$ and covariance function $F(\min(s,t))$.
The measure $\mu$ you are looking for is then the image (push forward) of the Wiener measure under the map $$ \Pi_{\,\textstyle.}\mapsto G\,\Pi_{\,\textstyle.}-G(0)\Pi_0-\int_0^{\,\textstyle\cdot}\Pi_s\,dG(s) $$ from $C([0,1],\mathbb R)$ to itself.