  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 $W_3$ 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.
