Background
The expected displacement under the assumption of Brownian Motion for time step $\tau$ is given by $$\gamma(\tau) = D|\tau|$$ where $D$ is the diffusion coefficient.
If one assumes a 'pull back' mechanism to some point of attraction, the diffusion up to a given $\tau$ can be modeled with an Ornstein-Uhlenbeck process, with the expected displacement after $\tau$ time steps is given by $$\gamma(\tau) = \sigma \left(1 - \exp \left(-\frac{|\tau|}{\tau_m}\right) \right)$$ where $\sigma$ is the distance from 0 where the 'pull back' starts and $\tau_m$ the time lag at which the 'pull back' starts.
Further, if $\sigma \to \infty$ and $\tau_m \to \infty$ then an Ornstein-Uhlenbeck process is equivalent to Brownian motion with $D = \frac{\sigma}{\tau_m}$.
Question
Now my question: Is it possible to say for which $\sigma$ and $\tau_m$ there is no difference between an Ornstein-Uhlenbeck process and Brownian motion up to a given time lag $\tau$.