Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.35176

A pathwise approach to semilinear SPDEs with Lévy drivers

Dirk Becherer·Peter Friz·Yuchen Sun
View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Dirk Becherer, Peter Friz, Yuchen Sun. 2026-09-28. A pathwise approach to semilinear SPDEs with Lévy drivers. https://arxiv.org/abs/2609.35176

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The discrete periodic Pitman transform: invariances, braid relations, and Burke properties

We develop the theory of the discrete periodic Pitman transform, first introduced by Corwin, Gu, and the fifth author. We prove that the discrete periodic Pitman transform satisfies the same braid relations that are satisfied for the full-line Pitman transform shown by Biane, Bougerol, and O'Connell. This defines a group action of the infinite symmetric group on sequences of vectors in $\mathbb R^{\mathbb Z_N}$. We prove that, for polymers in a periodic environment, single-path and multi-path partition functions are preserved under the action of this transform on the weights in the polymer model. Combined with a new inhomogeneous Burke property for the periodic Pitman transform, we prove a multi-path invariance result for the periodic inverse-gamma polymer under permutations of the column parameters. In the limit to the full-line case, we obtain a multi-path extension of a recent invariance result of Bates, Emrah, Martin, Seppäläinen, and the fifth author, in both positive and zero-temperature.

math.PR↗
arXiv2026

The Berry-Esseen Constant Conjecture is Eventually True

We prove that the Esseen's constant $0.4097...$ is valid in the iid Berry--Esseen inequality as long as the sample sizes $n\ge N$. The threshold $N$ is universal over all real summand distributions with positive variance and a finite third absolute moment. In particular, one may take $N=2\left\lceil\exp(10^{17})\right\rceil. $

math.PR↗
arXiv2026

Local decisions, diffusive influence, and lower bounds for graphical balanced allocation

In graphical two-choice allocation, each arriving ball is assigned to one endpoint of a random edge. We study rules whose decision is a monotone function of the two endpoint loads, allowing edge-dependent thresholds and fresh randomization. Such a rule has an exact unit-discrepancy coupling: adding one ball to the initial state produces one tagged discrepancy at every later time. We represent the tag by conditional-expectation projections on the marked edge space and obtain diffusive displacement bounds. A transport-volume inequality then converts slow propagation of influence into lower bounds for the load gap. On the cycle with $n$ vertices, from every initial distribution and at every physical time $t\ge 1/n$, the expected gap is at least a constant times $\min\{\sqrt n,t^{1/4}\}$, and the gap exceeds this scale with probability at least $1/8$. After exactly $k\ge1$ allocations, the corresponding scale is $\min\{\sqrt n,(k/n)^{1/4}\}$. No stationarity, symmetry, recurrence, or moment assumption is used. The general inequality also yields a lower bound of order $\sqrt{L/K+\log K}$ on the $L\times K$ discrete torus $C_L\square C_K$. These results separate endpoint-local rules from global-information strategies that achieve polylogarithmic gaps on cycles.

math.PR↗