  The static optimal transport $(\mathrm{OT})$ problem between Gaussians seeks
to recover an optimal map, or more generally a coupling, to morph a Gaussian
into another. It has been well studied and applied to a wide variety of tasks.
Here we focus on the dynamic formulation of OT, also known as the Schr\"odinger
bridge (SB) problem, which has recently seen a surge of interest in machine
learning due to its connections with diffusion-based generative models. In
contrast to the static setting, much less is known about the dynamic setting,
even for Gaussian distributions. In this paper, we provide closed-form
expressions for SBs between Gaussian measures. In contrast to the static
Gaussian OT problem, which can be simply reduced to studying convex programs,
our framework for solving SBs requires significantly more involved tools such
as Riemannian geometry and generator theory. Notably, we establish that the
solutions of SBs between Gaussian measures are themselves Gaussian processes
with explicit mean and covariance kernels, and thus are readily amenable for
many downstream applications such as generative modeling or interpolation. To
demonstrate the utility, we devise a new method for modeling the evolution of
single-cell genomics data and report significantly improved numerical stability
compared to existing SB-based approaches.
