Concentration for sub-geometrically ergodic diffusions

Establish concentration inequalities for Stratonovich additive functionals of sub-geometrically ergodic diffusions, for which the Poincaré inequality does not apply and a different analytical approach is required.

Background

The paper proves concentration bounds for Stratonovich additive functionals of geometrically ergodic diffusions, relying substantially on a Poincaré inequality for the symmetric part of the generator. Sub-geometric ergodicity falls outside this framework because the requisite Poincaré inequality is unavailable. The authors identify the derivation of corresponding concentration results as unresolved.

References

Corresponding results for sub-geometrically ergodic diffusion, however, remain elusive and will require a somewhat different approach as the Poincaré inequality does not apply.

— Concentration of additive functionals of Stratonovich-type  (2609.01581 - Bebon et al., 1 Sep 2026) in Section 'Open questions'

Similarly, extensions to the time-inhomogeneous case , multiplicative noise diffusions, or results on the concentration around the transient, time-dependent mean remain to be established, which will likely require dedicated two-sided bounds and is thus likely to pose a substantially greater challenge.

— Concentration of additive functionals of Stratonovich-type  (2609.01581 - Bebon et al., 1 Sep 2026) in Section 'Open questions'

Similarly, extensions to the time-inhomogeneous case , multiplicative noise diffusions, or results on the concentration around the transient, time-dependent mean remain to be established, which will likely require dedicated two-sided bounds and is thus likely to pose a substantially greater challenge.

— Concentration of additive functionals of Stratonovich-type  (2609.01581 - Bebon et al., 1 Sep 2026) in Section 'Open questions'