Ornstein Uhlenbeck Process is Gaussian process $ X=\left\{X(t), t \in \mathbb{R}\right\} $ with expected value function $ \mu(t)=0 \quad \forall t \in \mathbb{R} $ and covariance function $ \gamma(s, t)=e^{-|t-s|} \forall t, s \in \mathbb{R} $
Let $ W=\left\{W(t), t \geqslant 0\right\} $ a stochastic process with $ W(0)=0 $ and $ W(t)=\sqrt{t} X\left(\frac{\log t}{2}\right), t>0 $.
(a) Calculate expected value & covariance function of $ W $.
(b) Show: $ W $ is a Wiener process.
For a.): The expected value function should also be 0 and the covariance function should be min(s,t).
For b.): There is a definition for the Wiener process via Gaussian processes, which would exactly fulfill the properties from a). One would only have to show that it is a Gaussian process, then it would already be a Wiener process or not?