A pathwise approach to semilinear SPDEs with Lévy drivers
Abstract
We extend the notion of robust viscosity solutions to semilinear rough partial differential equations (RPDEs) of the form ∂t u + ℒt u + f(t,x,u,σ⊤∇x u) + h(t,x,u) diamond dW = 0, 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.
Related
- Existence, Uniqueness, and Pathwise Regularity for Multidimensional Semilinear SPDEs with Locally Lipschitz Coefficients and Rough Initial Data
- Finite Explosion for One-dimensional Lévy Driven SDEs and its Application to SPDEs
- Sharp Error Estimates for a Fully Discrete Finite Element Method for Semilinear SPDEs with Multiplicative Noise and Nonsmooth Initial Data
- Stability of Solutions to Semilinear Exterior Dirichlet Problems under Convergence of Lévy Symbols
- Pathwise Shrinkage and Probabilistic Calibration of the Multiplicative Half-Cauchy Process
BibTeX
@misc{becherer2026pathwise,
title = {{A pathwise approach to semilinear SPDEs with Lévy drivers}},
author = {Dirk Becherer and Peter Friz and Yuchen Sun},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35176},
eprint = {2609.35176},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35176},
}
RIS
TY - UNPB
TI - A pathwise approach to semilinear SPDEs with Lévy drivers
AU - Dirk Becherer
AU - Peter Friz
AU - Yuchen Sun
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35176
UR - https://arxiv.org/abs/2609.35176
AB - We extend the notion of robust viscosity solutions to semilinear rough partial differential equations (RPDEs) of the form ∂_t u + ℒ_t u + f(t,x,u,σ^⊤∇_x u) + h(t,x,u) diamond dW = 0, 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.
ER -