[Paper Review] On some ergodicity properties for time inhomogeneous Markov processes with $T$-periodic semigroup

Reinhard Hoepfner, Eva Loecherbach|arXiv (Cornell University)|Dec 22, 2010

Markov Chains and Monte Carlo Methods16 references4 citations

TL;DR

This paper establishes verifiable sufficient conditions for positive Harris recurrence of time inhomogeneous Markov processes with T-periodic semigroups, focusing on d-dimensional diffusions. By combining Lyapunov function techniques with non-classical lower bounds on transition densities (from Bally), it provides explicit, checkable criteria in terms of drift and diffusion coefficients—applicable even to degenerate or unbounded coefficients—ensuring ergodicity and strong laws of large numbers for additive functionals.

ABSTRACT

We consider a time inhomogeneous strong Markov process $(ξ_t)_{t\ge 0}$ taking values in a Polish state space whose semigroup has a $T$-periodic structure. We give simple conditions which imply ergodicity of the grid chain $(ξ_{kT})_{k\in \mathbb{N}_0}$ In case of $d$-dimensional possibly degenerate diffusions, the conditions are stated in terms of drift and diffusion coefficient of the process.

Motivation & Objective

• To establish sufficient conditions for positive Harris recurrence of the T-grid chain in time inhomogeneous Markov processes with T-periodic semigroups.
• To extend classical ergodicity results to diffusions with possibly degenerate or unbounded coefficients.
• To provide explicit, verifiable criteria based on drift and diffusion coefficients that ensure long-term stability and ergodic behavior.
• To enable the application of strong laws of large numbers for additive functionals in non-stationary, periodically time-dependent diffusion processes.

Proposed method

• Utilizes the framework of Meyn and Tweedie’s Lyapunov function approach to establish positive Harris recurrence.
• Applies Bally’s non-classical lower bounds on transition densities for possibly degenerate diffusions to ensure uniform positivity of densities on compact sets.
• Imposes smoothness and bounded derivative conditions (up to order d+2) on drift and diffusion coefficients to ensure regularity and control of transition densities.
• Derives a sufficient condition involving a compact set K and a decay rate ε on the drift-trace term 2xᵀb(s,x) + tr(a(x)) outside K.
• Uses time-periodicity of the semigroup to reduce the problem to analyzing a time-homogeneous Markov chain on the T-grid.
• Constructs a verification inequality involving R, C₀, T, and dimension-dependent constants to ensure the existence of a Lyapunov function.

Experimental results

Research questions

• RQ1 Under what conditions on the drift and diffusion coefficients is the T-grid chain of a time inhomogeneous diffusion with T-periodic semigroup positive Harris recurrent?
• RQ2 Can ergodicity be established for diffusions with degenerate or unbounded coefficients using explicit, checkable criteria?
• RQ3 How can lower bounds on transition densities be leveraged to prove recurrence in time-inhomogeneous, periodic diffusion processes?
• RQ4 What role does the periodicity of the semigroup play in reducing the analysis of the full path process to a time-homogeneous Markov chain on the T-grid?
• RQ5 Can strong laws of large numbers be established for additive functionals in such non-stationary, periodically time-dependent diffusions?

Key findings

• The paper establishes a sufficient condition for positive Harris recurrence (H) of the T-grid chain based on a compact set K and a uniform negative drift-trace term 2xᵀb(s,x) + tr(a(x)) < -ε outside K.
• A quantitative verification inequality involving R, C₀, T, d, and m ensures the existence of a Lyapunov function, with explicit dependence on dimension and noise structure.
• For d-dimensional diffusions with T-periodic drift and bounded C^{d+2} derivatives, the condition (6) and the inequality (7) together guarantee ergodicity of the grid chain.
• The result applies even when the diffusion matrix is degenerate (i.e., has zero eigenvalues), extending beyond classical uniform ellipticity assumptions.
• The method enables strong laws of large numbers for additive functionals of the form ∫₀ᵗ F(s,ξₛ)Λ_T(ds) under periodicity and ergodicity.
• An example with geometric Brownian motion shows that the condition fails when the drift is too weak near zero, confirming the necessity of the compact set condition.