The parametrix method for SPDEs and conditional transition densities

We extend the classic parametrix method in the context of evolution SPDEs. Our method is based on a It\^o-Wentzell reduction of the SPDE to a PDE with random coefficients to which we apply a revised parametrix technique to construct a fundamental solution. This approach avoids the use of the Duhamel's principle for SPDEs and the related measurability issues that appear in the stochastic framework. We first apply our method to a parabolic SPDE with random and measurable coefficients; then we expand our study to a class degenerate SPDEs of Kolmogorov type, we discuss the filtering problem for the associated partially observable degenerate diffusion and show existence, regularity and Gaussian-type estimates of a conditional transition density. In the second part of the work, we consider non degenerate Brownian SDEs with H\"older continuous in space diffusion coefficient and unbounded drift with linear growth. We derive two sided bounds for the associated density and pointwise controls of its derivatives up to order two under some additional spatial H\"older continuity assumptions on the drift. Importantly, the estimates reflect the transport of the initial condition by the unbounded drift through an auxiliary, possibly regularized, flow.