Stochastic control formulations of the Schrödinger bridge problem between $\mu,\nu$ are well known (e.g Chen et al Eq. 4.23)
$$\inf \limits_{p_t, v_t} \int_0^T \int \frac{1}{2}\lvert v_t\rvert^2 p_t dx dt \\ \text{subject to }\frac{\partial p_t}{\partial t} = - \operatorname{div}(p_t v_t) + \frac{1}{2}\Delta p_t, \text{ and} p_0 = \mu, p_T = \nu $$
which can be related to the static Shannon entropy-regularized optimal transport
$$ \inf \limits_{p(x_0,x_T)} \int \lvert x_T - x_0\rvert^2 p(x_0, x_T) dx - \epsilon ~ H[p_{0,T}] = \inf \limits_{p(x_0,x_1)} \epsilon KL[p(x_0, x_T) : e^{-\frac{1}{\epsilon}\lvert x_T - x_0\rvert^2}] \\ \text{subject to } p_0 = \mu, p_T = \nu. $$
I understand this result as the transition probability $q(x_T|x_0)$ of the Brownian motion $dX_t = dW_t$ or $\frac{\partial q_t}{\partial t}=\frac{1}{2}\Delta q_t$ being Gaussian, with $\epsilon = 2T$. However, most references restrict attention to the Euclidean cost.
I am interested in how this picture changes for strictly convex difference costs $c(x_T - x_0)$, in which the unregularized dynamical OT formulation would be (Villani Ex. 7.2)
$$\inf \limits_{p_t, v_t} \int_0^T \int c(v_t) p_t dx dt \\ \text{subject to } \frac{\partial p_t}{\partial t} = - \operatorname{div}(p_t v_t) \text{ and} p_0 = \mu, p_T = \nu $$
However, I am not sure how to relate a stochastic version of this to the static problem, where we would like to obtain
$$ \inf \limits_{p(x_0,x_T)} \epsilon KL[p(x_0, x_T) : e^{-\frac{1}{\epsilon} c(x_T-x_0)}]. $$
Is it possible to modify the Fokker–Planck equation (i.e. constraints in the original dynamical problem) or to induce the desired transition probability $q(x_T|x_0) \propto e^{-\frac{1}{\epsilon} c(x_T-x_0)}$ ?
