  We formulate a new model for transport in stochastic media with long-range
spatial correlations where exponential attenuation (controlling the propagation
part of the transport) becomes power law. Direct transmission over optical
distance $\tau(s)$, for fixed physical distance $s$, thus becomes
$(1+\tau(s)/a)^{-a}$, with standard exponential decay recovered when
$a\to\infty$. Atmospheric turbulence phenomenology for fluctuating optical
properties rationalizes this switch. Foundational equations for this
generalized transport model are stated in integral form for $d=1,2,3$ spatial
dimensions. A deterministic numerical solution is developed in $d=1$ using
Markov Chain formalism, verified with Monte Carlo, and used to investigate
internal radiation fields. Standard two-stream theory, where diffusion is
exact, is recovered when $a=\infty$. Differential diffusion equations are not
presently known when $a<\infty$, nor is the integro-differential form of the
generalized transport equation. Monte Carlo simulations are performed in $d=2$,
as a model for transport on random surfaces, to explore scaling behavior of
transmittance $T$ when transport optical thickness $\tau_\text{t} \gg 1$.
Random walk theory correctly predicts $T \propto
\tau_\text{t}^{-\min\{1,a/2\}}$ in the absence of absorption. Finally, single
scattering theory in $d=3$ highlights the model's violation of angular
reciprocity when $a<\infty$, a desirable property at least in atmospheric
applications. This violation is traced back to a key trait of generalized
transport theory, namely, that we must distinguish more carefully between two
kinds of propagation: one that ends in a virtual or actual detection, the other
in a transition from one position to another in the medium.
