ON 2 MATHEMATICAL PROBLEMS OF CANONICAL QUANTIZATION .4.

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.