Consider the following degenerate Fokker-Planck equation in $\mathbb{R}^d$ $$\partial_t u = \nabla\cdot(D\,\nabla u + Cxu),\quad u(t=0) = u_0 \label{1}\tag{1}$$ where $D \in \mathbb{R}^{d \times d}$ is constant, symmetric, and positive semi-definite and $C \in \mathbb{R}^{d \times d}$ satisfies certain technical conditions (see for instance Definition 2.1. in this paper) to ensure the global well-posedness and smoothness of the solution to \eqref{1}. It is shown in the aforementioned paper (via a rather sophisticated Fourier transform based computations) that the explicit solution of \eqref{1} is given by $$u(x,t) = \frac{1}{(2\pi)^{\frac{d}{2}}\sqrt{\det W(t)}} \int_{y \in \mathbb{R}^d} \mathrm{e}^{-\frac 12 (x-\mathrm{e}^{-Ct}y)^\intercal W(t)^{-1} (x-\mathrm{e}^{-Ct}y)} u_0(y) \, \mathrm{d} y \label{2}\tag{2}$$ with the kernel $W(t)$ being defined by $$W(t) = 2\int_0^t \mathrm{e}^{-Cs} D \mathrm{e}^{-C^\intercal s} \, \mathrm{d} s.$$ May I know if it is possible to directly verify that \eqref{2} solves \eqref{1}? I have to admit that taking derivative of a determinant of a time-dependent matrix (which is $W(t)$ in this case) is not a pleasant computation to carry out...
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