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Subexponential lower bounds for f-ergodic Markov processes
Miha Brešar1, Aleksandar Mijatović1
1Department of Statistics, University of Warwick, Coventry, UK.
Probability Theory and Related Fields
|February 26, 2026
Summary
Researchers developed a new criterion to establish lower bounds on Markov process convergence rates. This method uses novel martingale conditions and path-wise arguments for analyzing invariant measure tails and convergence speed.
Area of Science:
- Probability Theory
- Stochastic Processes
- Ergodic Theory
Background:
- Markov processes are fundamental in modeling dynamic systems.
- Understanding convergence rates to invariant measures is crucial for analyzing system long-term behavior.
- Existing methods primarily focus on upper bounds, leaving lower bounds less explored.
Purpose of the Study:
- To introduce a novel criterion for establishing lower bounds on the rate of convergence in f-variation for continuous-time ergodic Markov processes.
- To develop a general approach for proving lower bounds on invariant measure tails and convergence rates.
- To provide a method analogous to Lyapunov drift conditions for upper bounds.
Main Methods:
- Development of super- and submartingale conditions for specific Markov process functionals.
- Utilizing path-wise arguments to derive lower bounds on excursion heights and durations from bounded sets.
- Applying the criterion to elliptic diffusions and Lévy-driven stochastic differential equations.
Main Results:
- A new criterion for lower bounds on convergence rates in f-variation is established.
- The method successfully derives lower bounds on invariant measure tails.
- Applied to specific models, the derived lower bounds asymptotically match known upper bounds, confirming convergence rates.
Conclusions:
- The developed criterion offers a general approach for proving lower bounds on Markov process convergence.
- The methodology, using path-wise arguments and martingale conditions, is broadly applicable.
- This work complements existing methods for analyzing convergence rates and is expected to find wide application.
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