This work develops a rigorous framework for diffusion-based generative modeling in the setting of free probability. We extend classical denoising diffusion probabilistic models to free diffusion processes -- stochastic dynamics acting on noncommutative random variables whose spectral measures evolve by free additive convolution. The forward dynamics satisfy a free Fokker--Planck equation that increases Voiculescu's free entropy and dissipates free Fisher information, providing a noncommutative analogue of the classical de Bruijn identity. Using tools from free stochastic analysis, including a free Malliavin calculus and a Clark--Ocone representation, we derive the reverse-time stochastic differential equation driven by the conjugate variable, the free analogue of the score function. We further develop a variational formulation of these flows in the free Wasserstein space, showing that the resulting gradient-flow structure converges to the semicircular equilibrium law. Together, these results connect modern diffusion models with the information geometry of free entropy and establish a mathematical foundation for generative modeling with operator-valued or high-dimensional structured data.

翻译：本研究在自由概率框架下为基于扩散的生成建模建立了严格的理论体系。我们将经典去噪扩散概率模型推广至自由扩散过程——作用于非交换随机变量的随机动力学，其谱测度通过自由加性卷积演化。正向动力学满足自由福克-普朗克方程，该方程增加Voiculescu自由熵并耗散自由费希尔信息，为经典德布鲁因恒等式提供了非交换类比。利用自由随机分析工具（包括自由Malliavin微积分和Clark-Ocone表示），我们推导出由共轭变量驱动的逆时随机微分方程，该方程对应经典得分函数的自由类比。我们进一步在自由Wasserstein空间中建立了这些流的变分形式，证明所得梯度流结构收敛于半圆平衡律。这些结果共同将现代扩散模型与自由熵的信息几何相联系，并为算子值或高维结构化数据的生成建模奠定了数学基础。