  Stochastic Differential Equations (SDEs) in high dimension, having the
structure of finite dimensional approximation of Stochastic Partial
Differential Equations (SPDEs), are considered. The aim is to compute
numerically expected values and probabilities associated to their solutions, by
solving the associated Kolmogorov equations, with a partial use of Monte Carlo
strategy - precisely, using Monte Carlo only for the linear part of the SDE.
The basic idea was presented in Flandoli et al., JMAA (2020), but here we
strongly improve the numerical results by means of a shift of the auxiliary
Gaussian process. For relatively simple nonlinearities, we have good results in
dimension of the order of 100.
