Concentration of additive functionals of Stratonovich-type
Abstract
Additive functionals J̅t=1/t∫0tU(Xs)∘ dXs of Stratonovich-type recently attracted much attention in the context of inference of thermodynamic properties of complex systems from observations U of individual fluctuating paths (Xs)0≤ s≤ t, whereby X0 is initiated from some general measure. Concentration results on J̅t, albeit desirable, are virtually nonexistent. They turn out to be significantly more challenging to prove than for classical Lebesgue-type functionals ρ̅t=1/t∫0t V(Xs)ds because the tilt deforms the full second-order structure of the Feynman-Kac generator instead of contributing an additive potential. This renders the generator generally non-self-adjoint even under detailed balance. We overcome this by working with a symmetrized Dirichlet form with a new effective potential that now couples the observable to the non-equilibrium character of the dynamics. We prove concentration inequalities for J̅t for any bounded, sufficiently smooth vector-valued function U of a general geometrically ergodic diffusion process Xs, including explicit sub-gamma and Bernstein-type inequalities, and we obtain explicit upper bounds on Var(J̅t). Strikingly, under detailed balance the concentration of J̅t is distinctively sub-Gaussian at all times and all deviations, with a variance proxy fixed by the noise alone and independent of the spectral gap, which has no analog for ρ̅t.
Related
- Large deviations for additive functionals of one-dimensional zero-range processes
- Thermodynamics of the space of trajectories governed by a combination of two additive boundary functionals
- CLT for quadratic additive functionals of three-dimensional and four-dimensional voter models
- Quantitative averaging of Markov-modulated additive functionals and Wentzell boundary homogenization
- Lyapunov-Krasovsky Functionals and the Existence of Unbounded Solutions of Functional-Differential Equations of Retarded Type
BibTeX
@misc{bebon2026concentration,
title = {{Concentration of additive functionals of Stratonovich-type}},
author = {Rick Bebon and Aljaž Godec and Angelika Rohde},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01581},
eprint = {2609.01581},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01581},
}
RIS
TY - UNPB
TI - Concentration of additive functionals of Stratonovich-type
AU - Rick Bebon
AU - Aljaž Godec
AU - Angelika Rohde
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01581
UR - https://arxiv.org/abs/2609.01581
AB - Additive functionals J̅_t=1/t∫_0^tU(X_s)∘ dX_s of Stratonovich-type recently attracted much attention in the context of inference of thermodynamic properties of complex systems from observations U of individual fluctuating paths (X_s)_0≤ s≤ t, whereby X_0 is initiated from some general measure. Concentration results on J̅_t, albeit desirable, are virtually nonexistent. They turn out to be significantly more challenging to prove than for classical Lebesgue-type functionals ρ̅_t=1/t∫_0^t V(X_s)ds because the tilt deforms the full second-order structure of the Feynman-Kac generator instead of contributing an additive potential. This renders the generator generally non-self-adjoint even under detailed balance. We overcome this by working with a symmetrized Dirichlet form with a new effective potential that now couples the observable to the non-equilibrium character of the dynamics. We prove concentration inequalities for J̅_t for any bounded, sufficiently smooth vector-valued function U of a general geometrically ergodic diffusion process X_s, including explicit sub-gamma and Bernstein-type inequalities, and we obtain explicit upper bounds on Var(J̅_t). Strikingly, under detailed balance the concentration of J̅_t is distinctively sub-Gaussian at all times and all deviations, with a variance proxy fixed by the noise alone and independent of the spectral gap, which has no analog for ρ̅_t.
ER -