An Lp-Variational Formula on Wiener Space
Abstract
We establish a stochastic control representation for Lp-norms on Wiener space. For every p≥1 and every non-negative universally measurable functional φ, we show that |φ(W)|p = supa 𝔼[ e−1/2∫0T|at|2 dt φ( W+√(p−1)∫0· at dt ) ], where W is a d-dimensional Brownian motion and the supremum is taken over all progressively measurable controls a satisfying ∫0T|at|2 dt<∞ almost surely. This identity can be viewed as a multiplicative Lp-analogue of the Boué--Dupuis variational formula. We also provide an extension to strong solutions of stochastic differential equations with path-dependent coefficients. Building on the variational formula, we discuss finite-p estimates for small-noise limits of stochastic differential equations with path-dependent coefficients. These include quantitative Freidlin-Wentzell bounds, a functional Gaussian approximation and concentration estimates, and moderate-deviation bounds. Moreover, we discuss consequences for functionals determined up to a stopping time, obtaining in particular probability bounds that identify the sharp exponential decay rate after optimization over controls. The proof of the variational formula combines probabilistic arguments with convex duality methods, viscosity theory for Hamilton-Jacobi-Bellman equations, and a dynamic programming principle.
Related
- Stochastic Euler Products in the Paley-Wiener Space
- A Bootstrap Proof of the Abstract Wiener Space Theorem for Fréchet Spaces
- Stability of concentration in the Paley-Wiener space
- TAP variational formula for 2-spin models with general interaction matrix and marginality
- Renormalization Group on Wiener Space: Spectral Theory and Universality
BibTeX
@misc{criens2026lpvariational,
title = {{An $L^p$-Variational Formula on Wiener Space}},
author = {David Criens and Carsten Hartmann and Michael Kupper},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15828},
eprint = {2609.15828},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15828},
}
RIS
TY - UNPB
TI - An L^p-Variational Formula on Wiener Space
AU - David Criens
AU - Carsten Hartmann
AU - Michael Kupper
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15828
UR - https://arxiv.org/abs/2609.15828
AB - We establish a stochastic control representation for L^p-norms on Wiener space. For every p≥1 and every non-negative universally measurable functional φ, we show that |φ(W)|_p = sup_a 𝔼[ e^−1/2∫_0^T|a_t|^2 dt φ( W+√(p−1)∫_0^· a_t dt ) ], where W is a d-dimensional Brownian motion and the supremum is taken over all progressively measurable controls a satisfying ∫_0^T|a_t|^2 dt<∞ almost surely. This identity can be viewed as a multiplicative L^p-analogue of the Boué--Dupuis variational formula. We also provide an extension to strong solutions of stochastic differential equations with path-dependent coefficients. Building on the variational formula, we discuss finite-p estimates for small-noise limits of stochastic differential equations with path-dependent coefficients. These include quantitative Freidlin-Wentzell bounds, a functional Gaussian approximation and concentration estimates, and moderate-deviation bounds. Moreover, we discuss consequences for functionals determined up to a stopping time, obtaining in particular probability bounds that identify the sharp exponential decay rate after optimization over controls. The proof of the variational formula combines probabilistic arguments with convex duality methods, viscosity theory for Hamilton-Jacobi-Bellman equations, and a dynamic programming principle.
ER -