I am interested in the support of the fractional stationary Ornstein Uhlenbeck process (first kind). In particular, I want to know whether smooth functions (or even only smooth functions starting at 0) are in the support. To put it into some context:
Problem:
We consider the differential equation
\begin{align} dY = \alpha Y dt + \sigma d B^H_t \qquad \alpha<0, \sigma > 0 \end{align} where $B^H$ is a fractional Brownian motion with Hurst parameter $H \in (\frac{1}{3},\frac{1}{2}]$.
The solution is given (provided that $Y$ is started at time $t_0$ with value $Y_0$) by
$$Y_t = Y_0e^{\alpha(t-t_0)} + \sigma e^{\alpha t} \int_{t_0}^t e^{-\alpha s} dB^H$$
Now upon taking the pullback limit $t_0 \to -\infty$ we end up with a stationary Gaussian process $O_t=\sigma \int_{-\infty}^t e^{\alpha(t-s)} dB^H $ with covariance function
$$ R(s,t) := Cov(O_s, O_t) = \sigma^2 \frac{\Gamma(2H+1)\sin(\pi H)}{2\pi} \int_{-\infty}^{\infty}e^{ix(t-s)} \frac{|{x}|^{1-2H}}{\alpha^2 + x^2} dx$$
All this has already been discussed (for example here). With that being said I am now interested in the support of this process. In particular, I want to know whether $\forall f \in C_0^\infty([0,T])$ (Smooth functions starting in $0$) and $\forall \varepsilon >0$ we have
$$ \mathbb{P}\left( ||O - f|| < \varepsilon \right) > 0$$
where $||\cdot||$ either denotes the uniform norm or the $\beta$-Hoelder norm ($\beta<H$) on $[0,T]$.
My Thoughts:
I have 2.5 thoughts on this.
(1): If $H=1/2$, i.e. we have standard Brownian motion, then we can use that $O_0$ is normal distributed, since it is Gaussian, and therefore $\mathbb{P}(O_0 \in (-\eta, \eta)) >0$ (for any $\eta >0$). Moreover, since the increments are independent and the support of the OU process started at time $t_0=0$ and initial value $\xi = 0$ is well known, we get some result along the lines of
$$ \mathbb{P}(||O-f|| < \eta + \varepsilon) > 0 \qquad f \in C_0^\infty([0,T]) $$
However, this obviously does not work anymore as soon as $H\neq 1/2$ as increments are then no longer independent, or is there some way to salvage that?
(2): The second idea is via the Cameron-Martin space. It is well known that the support of a Gaussian measure on $E$ is the closure of the CM in $E$. Moreover, the CM is the closure of
$$ H^0 := span\left\{R(t,\cdot) : t \in [0,T] \right\} $$
in the scalar product
$$ (R(t,\cdot), R(s, \cdot))_{H^0} := R(s,t)$$
(see for example "Selected Topics in Malliavin Calculus" by L. Decreusefond). However, here I run into the problem that the covariance is far too ugly to really derive any reasonable result here. Even though I think that this might lead to a solution once massaged with enough functional analysis.
(3):
This one is less an idea and more an observation. Namely that the covariance $R(s,t)$ seems to be the Fourier transform of $$f(x) = C[\sigma, H] \frac{|x|^{1-2H}}{\alpha^2 + x^2}$$ at $t-s$. However, I am have no idea whether there are any results on supports of Gaussian measure with covariances given by fourier transforms. I tried to google it but found very few things and what I found did not really seem helpful.
Any help is appreciated :)
