We present detailed discussions on the stochastic Hamiltonians for
non-critical string field theories on the basis of matrix models. Beginning
from the simplest c=0 case, we derive the explicit forms of the Hamiltonians
for the higher critical case k=3 (which corresponds to c=−22/5) and for the
case c=1/2, directly from the double-scaled matrix models. In particular, for
the two-matrix case, we do not put any restrictions on the spin configurations
of the string fields. The properties of the resulting infinite algebras of
Schwinger-Dyson operators associated with the Hamiltonians and the derivation
of the Virasoro and W3​ algebras therefrom are also investigated. Our results
suggest certain universal structure of the stochastic Hamiltonians, which might
be useful for an attempt towards a background independent string field theory.Comment: 70 pages, LaTeX, typographical errors are corrected, to be published
in Phys. Rev.