  In quantizing gravity based on stochastic quantization method, the stochastic
time plays a role of the proper time. We study 2D and 4D Euclidean quantum
gravity in this context. By applying stochastic quantization method to real
symmetric matrix models, it is shown that the stochastic process defined by the
Langevin equation in loop space describes the time evolution of the
non-orientable loops which defines non-orientable 2D surfaces. The
corresponding Fokker-Planck hamiltonian deduces a non-orientable string field
theory at the continuum limit. The strategy, which we have learned in the
example of 2D quantum gravity, is applied to 4D case. Especially, the Langevin
equation for the stochastic process of 3-geometries is proposed to describe the
(Euclidean) time evolution in 4D quantum gravity with Ashtekar's canonical
variables. We present it in both lattice regularized version and the naive
continuum limit.
