The Markov transition kernel is defined as $q_{\gamma}(s, x, t, y) = (2 \pi \gamma (t - s))^{-\frac{n}{2}} \exp \left( -\frac{1}{2} \frac{ \left\lVert x - y \right\rVert^{2}}{ \gamma (t - s)} \right)$
Given two probability distributions $\rho_{0}, \rho_{1}$ on $\mathbb{R}^{n}$ and the Markov transition kernel $q_{\gamma}(s,x,t,y)$, the Schrodinger Bridge between $\rho_{0}$ and $\rho_{1}$ has a density at time $t$ that factors as: \begin{align*} \rho(t, x) = g(t, x) f(t, x) \end{align*} with $f, g$ solving the Schrodinger system and satisfying the boundary conditions: \begin{align*} g(t, x) = \int_{\mathbb{R}^{n}} q_\gamma(t,x,1,y) g(1, y) \mathrm{d} y, &\quad \rho(0, x) = g(0, x) f(0, x)\\ f(t, x) = \int_{\mathbb{R}^{n}} q_\gamma(0,y,t,x) f(0, y) \mathrm{d} y, &\quad \rho(1, x) = g(1, x) f(1, x) \end{align*}
References: https://www.ima.umn.edu/materials/2015-2016/W1.25-29.16/24533/IMA_workshop.pdf
I am working on this problem, and I found very little resources on this topic. Can anyone point me to properties of the above system of equations? And also properties of the entropic interpolation. Thank you!