Singular jump processes as generalized gradient flows

Abstract: We extend the generalized gradient-flow framework of Peletier, Rossi, Savar\'e, and Tse to singular jump processes on abstract metric spaces, moving beyond the translation-invariant kernels considered in $\mathbb{R}^d$ and $\mathbb{T}^d$ in previous contributions. To address the analytical challenges posed by singularities, we introduce reflecting solutions, a new solution concept inspired by reflected Dirichlet forms, which ensures the validity of a chain rule and restores uniqueness. We establish existence, stability, and compactness results for these solutions by approximating singular kernels with regularized ones, and we show their robustness under such approximations. The framework encompasses dissipative and balanced solutions, clarifies their relations, and highlights the role of density properties of Lipschitz functions in upgrading weak formulations to reflecting solutions. As an application, we demonstrate the versatility of our theory to nonlocal stochastic evolutions on configuration spaces.