Schr\"odinger Bridge with Quadratic State Cost is Exactly Solvable

Schr\"odinger bridge is a diffusion process that steers a given distribution to another in a prescribed time while minimizing the effort to do so. It can be seen as the stochastic dynamical version of the optimal mass transport, and has growing applications in generative diffusion models and stochastic optimal control. In this work, we propose a regularized variant of the Schr\"odinger bridge with a quadratic state cost-to-go that incentivizes the optimal sample paths to stay close to a nominal level. Unlike the conventional Schr\"odinger bridge, the regularization induces a state-dependent rate of killing and creation of probability mass, and its solution requires determining the Markov kernel of a reaction-diffusion partial differential equation. We derive this Markov kernel in closed form. Our solution recovers the heat kernel in the vanishing regularization (i.e., diffusion without reaction) limit, thereby recovering the solution of the conventional Schr\"odinger bridge. Our results enable the use of dynamic Sinkhorn recursion for computing the Schr\"odinger bridge with a quadratic state cost-to-go, which would otherwise be challenging to use in this setting. We deduce properties of the new kernel and explain its connections with certain exactly solvable models in quantum mechanics.