We show that the large-eddy motions in turbulent fluid flow obey a mod
ified hydrodynamic equation with a stochastic turbulent stress whose d
istribution is a causal Functional of the large-scale velocity field i
tself We do so by means of an exact procedure of ''statistical filteri
ng'' of the Navier-Stokes equations, which formally solves the closure
problem, and we discuss the relation of our analysis with the ''decim
ation theory'' of Kraichnan. We show that the statistical filtering pr
ocedure can be formulated using field-theoretic path-integral methods
within the Martin-Siggia-Rose (MSR) formalism for classical statistica
l dynamics. We also establish within the MSR formalism a ''least-effec
tive-action principle'' for mean turbulent velocity profiles, which ge
neralizes Onsager's principle of least dissipation. This minimum princ
iple is a consequence of a simple realizability inequality and therefo
re holds also in any realizable closure. Symanzik's theorem in field t
heory-which characterizes the static effective action as the minimum e
xpected value of the quantum Hamiltonian over all state vectors with p
rescribed expectations of fields-is extended to MSR theory with non-He
rmitian Hamiltonian, This allows stationary mean velocity profiles and
other turbulence statistics to be calculated variationally by a Rayle
igh-Ritz procedure. Finally, we develop approximations of the exact La
ngevin equations for large eddies, e.g., a random-coupling DIA model,
which yield new stochastic LES models. These are compared with stochas
tic subgrid modeling schemes proposed by Rose, Chasnov, Leith, and oth
ers; and various applications are discussed.