Consider two vector fields $b_0,b_1\in C^2([0,1]\times\mathbb{R}^d;\mathbb{R}^d)$ and the solutions $\rho_0,\rho_1\in AC([0,1];\mathcal{P}_2(\mathbb{R}^d))$ to the associated Fokker-Planck equations $$ \partial_t \rho_i=\Delta \rho_i+div(\rho_ib_i).$$ Suppose that $\rho_0(0,\cdot)=\rho_1(0,\cdot)=\rho_{in}$ and $\rho_0(1,\cdot)=\rho_1(1,\cdot)=\rho_{fin}$.
The question is: does there exist a family $\{\rho_\lambda\}_{\lambda\in[0,1]}\subset AC([0,1];\mathcal{P}_2(\mathbb{R}^d))$ such that
for every $t\in[0,1]$, $\lambda\rightarrow\rho_\lambda$ is a curve of probability measures induced by some coupling $m_t\in \mathcal{P}(\mathbb{R}^d\times \mathbb{R}^d)$ with marginals $\rho_0(t,\cdot)$ and $\rho_1(t,\cdot)$
for every $\lambda\in[0,1]$, $\rho_\lambda(0,\cdot)=\rho_{in}$ and $\rho_\lambda(1,\cdot)=\rho_{fin}$.
So we have two different absolutely continuous curves of probability measures connecting two given distibution (e.g. $\rho_{in},\rho_{fin}\in C^1(\mathbb{R}^d)\cap\mathcal{P}_2(\mathbb{R}^d)$) and I would like to have an homotopy between the two which keeps fixed the extrema $\rho_{in}$ and $\rho_{fin}$ in the transition.
I could use a linear interpolation ($\rho_\lambda=(1-\lambda)\rho_0+\lambda\rho_1$) to be sure that the extrema in time are fixed, but that would produce a "transition" with non absolutely continuous curves (I need that continuity because I want to use a displacement convexity condition in $\lambda$, not a flat convexity one). The most difficult condition to satisfy for me is keeping fixed the marginal at time $t=1$ because of the forward nature of the Fokker-Planck equation.
In other words, by the Ambrosio superposition principle, there are two path measures $P_0,P_1\in\mathcal{P}(C([0,T];\mathbb{R}^d))$ induced respectively by $\rho_0$ and $\rho_1$, does there exist a coupling $\overline{P}$ between $P_0$ and $P_1$ such that $$ e(0)_\#\overline{P}=\rho_{in}\quad \text{and}\quad e(1)_\#\overline{P}=\rho_{fin}$$ where $e(t)_\#$ stands for the push-forward along the evaluation of curves at time $t\in[0,1]$.
