We study one-dimensional iterated function systems in which the choice of map is governed by a finite-state Markov chain rather than by independent sampling. Let X⊂ℝ be a compact interval, let f_1,… ,f_m:X→ X be C^1 maps, and let (J_n)_n≥ 0 be a finite-state Markov chain on {1,… ,m} with transition matrix P=(P_ij) . The associated random dynamical system is X_n+1=f_J_n+1(X_n), and its natural Markov state is the lifted process (X_n,J_n)∈ X×{1,… ,m}. We introduce the vector-valued derivative transfer operator (𝕃Φ )_i(x) = ∑ _j=1^m P_ij|f_j'(x)| ϕ _j(f_j(x)), Φ =(ϕ _1,… ,ϕ _m), where each component φ _i∈ C(X) corresponds to a switching mode. The operator acts on positive continuous vector fields on X . Our main result proves the Collatz–Wielandt type formula ρ (𝕃) = inf _Φ >0max _1≤ i≤ msup _x∈ X(𝕃Φ )_i(x)/ϕ _i(x). Equivalently, ρ (𝕃) is the optimal contraction threshold over all continuous mode-dependent weights w_i:X→ (0,∞ ) . In particular, ρ (𝕃)<1 is equivalent to the existence of such weights for which the lifted Markov operator is a strict contraction along fibres with fixed mode marginal, in the associated mode-dependent weighted Wasserstein geometry. When the switching chain is irreducible and aperiodic, this fibrewise contraction combines with finite-state mode mixing to yield a unique invariant probability measure on X×{1,… ,m} and exponential convergence to it in a full weighted Wasserstein metric. Conversely, if ρ (𝕃)≥ 1 , then no continuous choice of mode-dependent weights can yield fibrewise Wasserstein contractivity of this type with contraction factor strictly smaller than one. The result gives a sharp spectral criterion for contractive geometry in Markov-switched random dynamical systems and shows explicitly how temporal correlations in the switching process affect ergodicity beyond the averaged i.i.d. theory.
In this paper, we are concerned with the averaging principle of slow-fast stochastic differential equations as the time scale parameter tends to zero, where both the slow and fast processes include path-dependent coefficients. We first extend the mixed Itô formula established in [39] to a mixed Itô product formula. Then, we obtain the well-posedness and some uniform estimates of solutions to slow-fast stochastic differential equations. Furthermore, we investigate the regularity of solutions for the Poisson equation on ℝ^d with a parameter in D([0,T],ℝ^d) . Based on this, we prove the strong convergence of the averaging principle for the underlying system, and obtain the optimal 1/2-order convergence rate, which does not depend on the regularity of coefficients in equations about the fast variable.
We consider the recently introduced coupon collector problem with reset button, which incorporates the possibility of a total reset. This means that at any time t all coupons collected up to that point are lost (removed from the collection) with a given probability. We further investigate the exact and asymptotic properties, as the number of coupons tends to infinity, of the expected waiting time until either a small subcollection of coupons or the full collection is sampled despite the resets.
We study inequalities that compare the expected cost of the sum with that of the difference of two observations from a stationary reversible Markov process. For symmetric conditionally negative definite cost functions, the associated sum–difference cost gap admits a positive spectral representation. In discrete time, the gap is represented as a moment sequence of a finite positive measure supported on the spectrum of the transition operator. In continuous time, it is represented as the Laplace transform of a finite positive measure associated with the generator. These representations yield inequalities at every even lag in discrete time and at every positive time in continuous time, together with complete monotonicity and spectral gap bounds for nonlinear relaxation. As applications, we recover absolute moment inequalities under a quadratic-form positivity condition and extend them to a broader class of costs generated by Bernstein functions.
In this paper, we introduce drifted versions of the generalized counting process (GCP) with a deterministic drift and a random drift. The composition of a stable subordinator with an independent inverse stable subordinator is taken as the random drift. We derive the probability laws for these drifted versions, and obtain their governing fractional differential equations. Also, we study the GCP time-changed with random times related to Brownian motion, for example, the Brownian first passage time with or without drift and Brownian sojourn time on the positive half-line. For these time-changed processes, we obtain the governing system of differential equations of their state probabilities, probability generating function, etc. A similar study is done for the composition of GCP with independent elastic Brownian motion and the squared Bessel process. Further, we consider a time-changed GCP where the time-change is done by subordinators linked to the incomplete gamma function. Later, we study the fractional integral of GCP and its time-changed variant.
