Random function covariance
You have two different meanings for ${γ}_{xx}$ when you write stuff such as
$${γ}_{xx}(τ) = {γ}_{xx}(t+τ,t)$$
because on the LHS, the function ${γ}_{xx}$ has one argument while on the RHS it has two arguments. Be that as it may, a very important property of what you call the covariance but what most everyone else on dsp.SE would call the autocovariance function is that
$$\text{For all values of }\tau, |{γ}_{xx}(τ)| \leq {γ}_{xx}(0).\tag{1}$$
It is easy to verify that $\exp\left(-(\tau-t_0)^2\right)$ has value $\exp(-t_0^2)&lt;1$ at $\tau=0$, and larger value $1$ at $\tau=t_0$, in contradiction of $(1)$, and so $\exp\left(-(\tau-t_0)^2\right)$ cannot possibly be a valid autocovariance (except, of course, for the special case of $t_0=0$).