This paper introduces a rigorous framework for defining generative diffusion models in infinite dimensions via Doob's h-transform. Rather than relying on time reversal of a noising process, a reference diffusion is forced towards the target distribution by an exponential change of measure. Compared to existing methodology, this approach readily generalises to the infinite-dimensional setting, hence offering greater flexibility in the diffusion model. The construction is derived rigorously under verifiable conditions, and bounds with respect to the target measure are established. We show that the forced process under the changed measure can be approximated by minimising a score-matching objective and validate our method on both synthetic and real data.

翻译：本文通过Doob的h变换，提出了一个在无限维空间中定义生成扩散模型的严格框架。该方法不依赖于噪声过程的时间反转，而是通过指数测度变换将参考扩散过程推向目标分布。与现有方法相比，该方案能自然地推广到无限维场景，从而为扩散模型提供了更大的灵活性。该构建过程在可验证条件下严格推导，并建立了关于目标测度的误差界。我们证明，变换测度下的强制过程可通过最小化分数匹配目标来近似，并在合成数据与真实数据上验证了所提方法的有效性。