In this work, we present a theoretical and computational framework for constructing stochastic transport maps between probability distributions using diffusion processes. We begin by proving that the time-marginal distribution of the sum of two independent diffusion processes satisfies a Fokker-Planck equation. Building on this result and applying Ambrosio-Figalli-Trevisan's superposition principle, we establish the existence and uniqueness of solutions to the associated stochastic differential equation (SDE). Leveraging these theoretical foundations, we develop a method to construct (stochastic) transport maps between arbitrary probability distributions using dynamical ordinary differential equations (ODEs) and SDEs. Furthermore, we introduce a unified framework that generalizes and extends a broad class of diffusion-based generative models and sampling techniques. Finally, we analyze the convergence properties of particle approximations for the SDEs underlying our framework, providing theoretical guarantees for their practical implementation. This work bridges theoretical insights with practical applications, offering new tools for generative modeling and sampling in high-dimensional spaces.
Dynamical Measure Transport and Neural PDE Solvers for Sampling
California Institute of Technology（加州理工学院）；Zuse Institute Berlin（柏林祖斯研究所）；dida Datenschmiede GmbH（迪达数据 Schmiede 有限公司）；Simula Research Laboratory（西穆拉研究实验室）；University Hospital Freiburg（弗赖堡大学医院）；RWTH Aachen（亚琛工业大学）；NVIDIA（英伟达）
Posterior Sampling with Denoising Oracles via Tilted Transport
Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）；Center for Data Science, New York University（纽约大学数据科学中心）；Flatiron Institute（弗莱堡研究所）
Department of Management, Economics and Industrial Engineering, Politecnico di Milano（米兰理工大学管理、经济与工业工程系）；RFF-CMCC European Institute on Economics and the Environment (EIEE)（RFF-CMCC欧洲经济与环境研究所）；Euro-Mediterranean Center on Climate Change (CMCC)（欧洲-地中海气候变化中心）；Bocconi University, Department of Decision Sciences（博科尼大学决策科学系）；MIT, Department of Nuclear Science and Engineering（麻省理工学院核科学与工程系）
