发表机构
Weierstrass Institute（韦伊斯特拉斯研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种路径方法，将鲁棒粘性解推广到 Lévy 驱动的半线性粗糙偏微分方程，建立适定性与流性质，并给出基于粗糙 BSDE 的随机表示，为相应 SPDE 提供路径解释。
AI 中文摘要
我们将鲁棒粘性解的概念推广到如下形式的半线性粗糙偏微分方程（RPDE）：\\[ \partial_t u + \mathcal{L}_t u + f(t,x,u,\sigma^{\top}\nabla_x u) + h(t,x,u)\\,\diamond dW = 0, \quad u(T,x)=\xi(x), \\] 该方程由一条不连续路径 $W$ 驱动，其有限 $q$-变差满足 $q<2$。这尤其包括一大类 Lévy 过程的典型路径，例如 $\alpha$-稳定 Lévy 过程（$\alpha<2$）。我们的解的概念通过一个解映射来表述，该映射在驱动为光滑路径时与经典粘性解一致，并且在终端条件和驱动方面是连续的，其中驱动和解路径均在适当的 Skorokhod 型装饰路径度量中考虑（见文献 \cite{chevyrev_superdiffusive_2024}）。这些要求自然意味着鲁棒粘性解必须采用 Marcus 型跳跃，并且我们建立了适定性、流性质以及基于具有不连续 Young 驱动的粗糙倒向随机微分方程（BSDE）的随机表示（见文献 \cite{becherer_rough_2026}）。对于 $q<2$ 的有限 $q$-变差 Lévy 过程，我们的 RPDE 解概念为相应的 SPDE 提供了路径解释，这些 SPDE 被视为无限维函数空间中的马尔可夫过程。
英文摘要
We extend the notion of robust viscosity solutions to semilinear rough partial differential equations (RPDEs) of the form \[ \partial_t u + \mathcal{L}_t u + f(t,x,u,σ^{\top}\nabla_x u) + h(t,x,u)\,\diamond dW = 0, \quad u(T,x)=ξ(x), \] driven by a discontinuous path $W$ of finite $q$-variation for some $q<2$. This includes, in particular, typical paths from a broad class of Lévy processes, for instance from $α$-stable Lévy processes with $α<2$. Our notion of solution is formulated through a solution map that agrees with the classical viscosity solution for smooth drivers and is continuous in the terminal condition and the driver, with both the driver and the solution paths viewed in a suitable Skorokhod-type decorated-path metric \cite{chevyrev_superdiffusive_2024}. These requirements naturally imply that robust viscosity solutions must employ Marcus-type jumps, and we establish well-posedness, a flow property, and a stochastic representation in terms of the rough BSDEs with discontinuous Young drivers \cite{becherer_rough_2026}. For Lévy processes of finite $q$-variation with $q<2$, our notion of RPDE solutions provides a pathwise interpretation for corresponding SPDEs, which are seen to be Markov processes in an infinite-dimensional function space.
