Existence and decay estimates for time dependent parabolic equation with application to Duncan–Mortensen–Zakai equation

where fo and V are possibly time dependent functions. Given an initial function at time zero, we would like to know existence of a positive solution of this equation. Furthermore if the initial function decay fast in spatial direction, we would like to know the spatial decay property of the solution for later time. In fact, in order for numerical calculation to be carried out effectively, we need to know quantitively this decay property. In this paper, we provide precise estimates of such an equation under reasonable assumptions on / and V. In applications /; and V may not be smooth in time. We have therefore avoided any argument involving differentiation of fc and V in time. A typical equation that can be treated are those arised in nonlinear filtering problem where the robust Duncan-Mortensen-Zakai equation has our form. We demonstrate existence and give decay estimate of this equation. D. Strook pointed out that his paper with Norris (Heat flows with uniformly elliptic coefficients, Proceedings LMS (3), Vol.62, #2, (1991), 373-402) is closely related to section 1 of this paper where they treated the case with bounded coefficients. We were also informed that Fleming-Mitter, Sussmann, Baras-Blankenship-Hopkins have obtained important estimates on the DMZ equation. However the latter two papers are focused on one spatial dimension, while the former paper needs the boundedness of/ and V/.