On Optimal Stochastic Ballistic Transports

For a given Lagrangian $L:[0,T]\times M\times M^\ast\rightarrow \mathbb{R}_+$ and probability measures $\mu\in\mathcal{P}(M^\ast)$, $\nu\in \mathcal{P}(M)$, we introduce the stochastic ballistic transportation problems \begin{align}\tag{$\star$} \underline{B}(\mu,\nu):=\inf\left\{\mathbb{E}\left[\langle V,X_0\rangle +\int_0^T L(t,X,\beta(t,X))\,dt\right]\middle\rvert V\sim\mu,X_T\sim \nu\right\}\\\tag{$\star\star$} \overline{B}(\nu,\mu):=\sup\left\{\mathbb{E}\left[\langle V,X_T\rangle -\int_0^T L(t,X,\beta(t,X))\,dt\right]\middle\rvert V\sim\mu,X_0\sim \nu\right\} \end{align} where $X$ is a diffusion process with drift $\beta$. This cost is based on the stochastic optimal transport problem presented by Mikami and the deterministic ballistic transport introduced by Ghoussoub. We obtain a Kantorovich-style duality result that reformulates this problem in terms of solutions to the Hamilton-Jacobi-Bellman equation \begin{equation*} \frac{\partial\phi}{\partial t}+\frac{1}{2}\Delta \phi+H(t,x,\nabla\phi)=0, \end{equation*} and show how optimal processes may be thereby attained.