A method for solving the problem of reconstructing a measure beginning
with its logarithmic derivative is Presented. The method completes th
at of solving the stochastic differential equation via Dirichlet forms
proposed by S. Albeverio and M. Rockner. As a result one obtains the
mathematical apparatus for the stochastic quantization. The apparatus
is applied to prove the existence of the Feynman-Kac measure of the si
ne-Gordon and lambdaphi2n/(1 + kappa2phi2n)-models. A synthesis of bot
h mathematical problems of canonical quantization is obtained in the f
orm of a second-order martingale problem for vacuum noise. It is shown
that in stochastic mechanics the martingale problem is an analog of N
ewton's second law and enables us to find the Nelson's stochastic traj
ectories without determining the wave functions.