Concentration of additive functionals of Stratonovich-type

Abstract

Additive functionals $\overline{J}_t=\frac{1}{t}\int_0^tU(X_s)\circ dX_s$ 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 $(X_s)_{0\le s\le t}$, whereby $X_0$ is initiated from some general measure. Concentration results on $\overline{J}_t$, albeit desirable, are virtually nonexistent. They turn out to be significantly more challenging to prove than for classical Lebesgue-type functionals $\overline{\rho}_t=\frac{1}{t}\int_0^t V(X_s)ds$ 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 $\overline{J}_t$ 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 ${\rm Var}(\overline{J}_t)$. Strikingly, under detailed balance the concentration of $\overline{J}_t$ 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 $\overline{\rho}_t$.