  The capacity of additive Gaussian noise (AGN) channels, $Y_t=X_t+V_t, t=1,
\ldots, n$, $\frac{1}{n} {\bf E}\big\{\sum_{t=1}^n |X_t|^2 \big\}\leq \kappa,
\kappa \in [0,\infty)$, with time-invariant channel input feedback strategies,
is characterized and conditions are identified for entropy rates, and limit of
average power to exist, when the noise is described by {\it stable and
unstable} autoregressive models, AR$(c)$, $V_t=cV_{t-1}+ W_t, V_0=v_0, t=1,
\ldots, n$, where $c\in (-\infty,\infty)$, $W_t, t=1,\ldots, n$, is a zero
mean, variance $K_W$, independent Gaussian sequence, independent of $V_0$. For
stable AR$(c), c\in (-1,1)$ the conditions are necessary and sufficient for
asymptotic stationarity of the processes $(X_t, Y_t), t=1, 2, \ldots$. New
closed form capacity formulas and lower bounds are derived, for the AR$(c),
c\in (-\infty,\infty)$ noise, which are fundamentally different from existing
formulas in the literature, and illustrate multiple regimes of capacity, as a
function of the parameters $(c,K_W,\kappa)$, as follows.\\ 1) feedback
increases capacity for the regime, $c^2 \in (1, \infty),$ for $\kappa >
\frac{K_W\big(1+\sqrt{4c^2-3}\big)}{2\big(c^2-1\big)^2}$, \\ 2) feedback does
not increase capacity for the regime $c^2 \in (1, \infty)$, for $\kappa \leq
\frac{K_W\big(1+\sqrt{4c^2-3}\big)}{2\big(c^2-1\big)^2}$, and \\ 3) feedback
does not increase capacity for the regime $c \in [-1,1]$, for $ \kappa \in
[0,\infty)$.
