A pathwise approach to semilinear SPDEs with Lévy drivers

Source abstract

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.