  The proximity effect in superfluid $^3$He partly filled with high porosity
aerogel is discussed. This system can be regarded as a dirty Fermi
liquid/spin-triplet p-wave superfluid junction. Our attention is mainly paid to
the case when the dirty layer is in the normal state owing to the impurity
pair-breaking effect by the aerogel. We use the quasiclassical Green's function
to determine self-consistently the spatial variations of the p-wave order
parameter and the impurity self-energy. On the basis of the fully
self-consistent calculation, we analyze the spatial dependence of the pair
function (anomalous Green's function). The spin-triplet pair function has in
general even-frequency odd-parity and odd-frequency even-parity components. We
show that the admixture of the even- and odd-frequency pairs occurs near the
aerogel/superfluid $^3$He-B interface. Among those Cooper pairs, only the
odd-frequency s-wave pair can penetrate deep into the aerogel layer. As a
result, the proximity-induced superfluidity in a thick aerogel layer is
dominated by the Cooper pair with the odd-frequency s-wave symmetry. We also
analyze the local density of states and show that it has a characteristic
zero-energy peak reflecting the existence of the odd-frequency s-wave pair, in
agreement with previous works using the Usadel equation.
