Uniform upper bound on expected attraction time across representations when no κ>1 representation exists

Determine whether, for a fixed irreducible and positive recurrent Markov chain that admits no random dynamical system representation with attractor cardinality κ greater than 1, there exists a representation-independent upper bound on the expected time to hit the attractor, in particular a uniform bound on the quantity ∑_{x∈X} π(x) E[T_A(ω, x)] over all random dynamical system representations.

Background

The authors construct representations where the expected time to hit the attractor can be made arbitrarily large, even when the underlying Markov chain is ergodic of degree 2. This shows representation choice can dramatically affect hitting times.

They pose the question whether such blow-ups are still possible when the chain cannot be represented by any RDS with κ>1 (i.e., when every representation has κ=1), asking for a uniform upper bound across all representations in this restricted setting.

References

An interesting open question is whether there is an upper bound on the expected hitting time of the attractor, in particular of the quantity in eq:time_attraction_finite, which is uniform over all RDS representations, in the special case that the Markov chain has no RDS representation with $\kappa >1$.

eq:time_attraction_finite:

∑x∈Xπ(x)E[TA(ω,x)]<∞.\sum_{x \in X} \pi(x) E[T_A(\omega, x)] < \infty.

— Random attractors on countable state spaces  (2405.19898 - Chemnitz et al., 2024) in Example “Arbitrarily high mean attraction times” (Section 5, Examples)