Score-based diffusion models typically use Brownian perturbations, which provide tractable reverse-time dynamics but impose memoryless noising. We introduce Volterra generative models, a continuous-time score-based framework whose forward process injects path-dependent noise through fractional kernels. To handle the non-Markovian and non-semimartingale dynamics, we construct finite-dimensional Markovian lifts using Gaussian quadrature in both regimes and a hybrid finite-difference exponential approximation in the smooth regime. We prove squared error bounds, derive an augmented linear-Gaussian forward process, and show that the learning can remain data-dimensional by considering residual states and analytic auxiliary Gaussian scores. We also identify covariance and reverse-time degeneracies caused by shared Brownian factors and signed smooth-regime weights. The degeneracy motivates stabilized conditioning and, for stiff larger lifts, a Gaussian-bridge reconstruction sampler. Experiments on MNIST and CIFAR-10 show that persistent fractional perturbations with small Markovian lifts can improve score-based generation on MNIST and provide a promising extension to natural images, while the bridge sampler provides a stability mechanism for larger lifts.

翻译：基于分数的扩散模型通常采用布朗扰动，这类扰动虽能提供可逆反向时间动力学，但引入的是无记忆噪声。我们提出Volterra生成模型——一种连续时间框架下的分数驱动模型，其正向过程通过分数阶核注入路径依赖型噪声。为处理非马尔可夫与非半鞅动力学，我们分别采用高斯求积方法（适用于两类场景）和平滑场景下的混合有限差指数近似技术，构建有限维马尔可夫提升。我们证明了平方误差边界，推导出增广线性高斯正向过程，并通过考虑残差状态与分析辅助高斯分数，表明学习过程可维持数据维度。此外，我们识别出共享布朗因子与符号平滑权重导致的协方差及反向时间退化现象。这种退化促使我们引入稳定化条件处理，并针对刚性较大提升设计高斯桥重建采样器。在MNIST和CIFAR-10上的实验表明：采用小规模马尔可夫提升的持久分数扰动能提升MNIST上的分数驱动生成质量，为自然图像提供有前景的扩展方案，同时桥采样器为大提升提供了稳定性机制。