Inference and Collective Steering for Stochastic and Deterministic Systems

This book makes an original and substantial contribution to the mathematical theory of stochastic particle flows in settings where particles may drift, stop, be created, or be killed. The work builds on the classical Schrödinger bridge framework, introduced to reconcile observed statistical data with stochastic dynamics, and extends it in a new and physically meaningful direction. The thesis' central innovation is the incorporation of spatio-temporal creation and killing into the Schrödinger bridge paradigm. This allows one to model, for instance, particles transported by a flow that may be deposited along the way, or new particles entering the system, while still matching prescribed statistical observations. Beyond this physical intuition, the thesis develops a rigorous Markovian probabilistic framework in which drift, stopping, creation, and killing jointly determine probability laws on sample paths. A particularly important feature of the work is its dual interpretation: as an inference problem, it asks how stochastic dynamics should be updated in light of observed statistics; as a control problem, it asks how drift and creation/killing rates should be chosen to steer a population toward prescribed distributions. By placing this duality on a rigorous foundation and linking it to optimal mass transport, the thesis opens a novel and promising direction.