The purpose of this study is to present a subgeometric convergence formula for the stationary distribution of the finite-level M/G/1-type Markov chain when taking its infinite-level limit, where the upper boundary level goes to infinity. This study is carried out using the fundamental deviation matrix, which is a block-decomposition-friendly solution to the Poisson equation of the deviation matrix. The fundamental deviation matrix provides a difference formula for the respective stationary distributions of the finite-level chain and the corresponding infinite-level chain. The difference formula plays a crucial role in the derivation of the main result of this paper, and the main result is used, for example, to derive an asymptotic formula for the loss probability in the MAP/GI/1/N queue.

We explore a simple model of network dynamics which has previously been applied to the study of information flow in the context of epidemic spreading. A random rooted network is constructed that evolves according to the following rule: at a constant rate, pairs of nodes (i, j) are randomly chosen to interact, with an edge drawn from i to j (and any other out-edge from i deleted) if j is strictly closer to the root with respect to graph distance. We characterise the dynamics of this random network in the limit of large size, showing that it instantaneously forms a tree with long branches that immediately collapse to depth two, then it slowly rearranges itself to a star-like configuration. This curious behaviour has consequences for the study of the epidemic models in which this information network was first proposed.

We study the local convergence of critical Galton–Watson trees under various conditionings. We give a sufficient condition, which serves to cover all previous known results, for the convergence in distribution of a conditioned Galton–Watson tree to Kesten’s tree. We also propose a new proof to give the limit in distribution of a critical Galton–Watson tree, with finite support, conditioned on having a large width.

Processes of random tessellations of the Euclidean space $\mathbb{R}^d$, $d\geq 1$, are considered that are generated by subsequent division of their cells. Such processes are characterized by the laws of the life times of the cells until their division and by the laws for the random hyperplanes that divide the cells at the end of their life times. The STIT (STable with respect to ITerations) tessellation processes are a reference model. In the present paper a generalization concerning the life time distributions is introduced, a sufficient condition for the existence of such cell division tessellation processes is provided, and a construction is described. In particular, for the case that the random dividing hyperplanes have a Mondrian distribution—which means that all cells of the tessellations are cuboids—it is shown that the intrinsic volumes, except the Euler characteristic, can be used as the parameter for the exponential life time distribution of the cells.

In this paper we study the drift parameter estimation for reflected stochastic linear differential equations of a large signal. We discuss the consistency and asymptotic distributions of trajectory fitting estimator (TFE).

Birth–death processes form a natural class where ideas and results on large deviations can be tested. We derive a large-deviation principle under an assumption that the rate of jump down (death) grows asymptotically linearly with the population size, while the rate of jump up (birth) grows sublinearly. We establish a large-deviation principle under various forms of scaling of the underlying process and the corresponding normalization of the logarithm of the large-deviation probabilities. The results show interesting features of dependence of the rate functional upon the parameters of the process and the forms of scaling and normalization.

In this paper, we derive new differential Harnack estimates of Li–Yau type for positive smooth solutions to a class of nonlinear parabolic equations in the form

\[ {\mathscr L}_\phi^{\mathsf a} [w]:= \left[ \frac{\partial}{\partial t} - \mathsf{a}(x,t) - \Delta_\phi \right] w (x,t) = \mathscr G(t, x, w(x,t)), \quad t>0, \]

$({\mathsf k},\, m)$

For every $n\geq 2$, Bourgain’s constant $b_n$ is the largest number such that the (upper) Hausdorff dimension of harmonic measure is at most $n-b_n$ for every domain in $\mathbb {R}^n$ on which harmonic measure is defined. Jones and Wolff (1988, Acta Mathematica 161, 131–144) proved that $b_2=1$. When $n\geq 3$, Bourgain (1987, Inventiones Mathematicae 87, 477–483) proved that $b_n>0$ and Wolff (1995, Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991), Princeton University Press, Princeton, NJ, 321–384) produced examples showing $b_n<1$. Refining Bourgain’s original outline, we prove that

$$\begin{align*}b_n\geq c\,n^{-2n(n-1)}/\ln(n),\end{align*}$$

$n\geq 3$

$c>0$

$b_3\geq 1\times 10^{-15}$

$b_4\geq 2\times 10^{-26}$

We study in a general graph-theoretic formulation a long-range percolation model introduced by Lamperti [27]. For various underlying digraphs, we discuss connections between this model and random exchange processes. We clarify, for all $n \in \mathbb{N}$, under which conditions the lattices $\mathbb{N}_0^n$ and $\mathbb{Z}^n$ are essentially covered in this model. Moreover, for all $n \geq 2$, we establish that it is impossible to cover the directed n-ary tree in our model.

Inaccuracy and information measures based on cumulative residual entropy are quite useful and have received considerable attention in many fields, such as statistics, probability, and reliability theory. In particular, many authors have studied cumulative residual inaccuracy between coherent systems based on system lifetimes. In a previous paper (Bueno and Balakrishnan, Prob. Eng. Inf. Sci. 36, 2022), we discussed a cumulative residual inaccuracy measure for coherent systems at component level, that is, based on the common, stochastically dependent component lifetimes observed under a non-homogeneous Poisson process. In this paper, using a point process martingale approach, we extend this concept to a cumulative residual inaccuracy measure between non-explosive point processes and then specialize the results to Markov occurrence times. If the processes satisfy the proportional risk hazard process property, then the measure determines the Markov chain uniquely. Several examples are presented, including birth-and-death processes and pure birth process, and then the results are applied to coherent systems at component level subject to Markov failure and repair processes.

The concurrency of edges, quantified by the number of edges that share a common node at a given time point, may be an important determinant of epidemic processes in temporal networks. We propose theoretically tractable Markovian temporal network models in which each edge flips between the active and inactive states in continuous time. The different models have different amounts of concurrency while we can tune the models to share the same statistics of edge activation and deactivation (and hence the fraction of time for which each edge is active) and the structure of the aggregate (i.e., static) network. We analytically calculate the amount of concurrency of edges sharing a node for each model. We then numerically study effects of concurrency on epidemic spreading in the stochastic susceptible-infectious-susceptible and susceptible-infectious-recovered dynamics on the proposed temporal network models. We find that the concurrency enhances epidemic spreading near the epidemic threshold, while this effect is small in many cases. Furthermore, when the infection rate is substantially larger than the epidemic threshold, the concurrency suppresses epidemic spreading in a majority of cases. In sum, our numerical simulations suggest that the impact of concurrency on enhancing epidemic spreading within our model is consistently present near the epidemic threshold but modest. The proposed temporal network models are expected to be useful for investigating effects of concurrency on various collective dynamics on networks including both infectious and other dynamics.

This paper investigates tail asymptotics of stationary distributions and quasi-stationary distributions (QSDs) of continuous-time Markov chains on subsets of the non-negative integers. Based on the so-called flux-balance equation, we establish identities for stationary measures and QSDs, which we use to derive tail asymptotics. In particular, for continuous-time Markov chains with asymptotic power law transition rates, tail asymptotics for stationary distributions and QSDs are classified into three types using three easily computable parameters: (i) super-exponential distributions, (ii) exponential-tailed distributions, and (iii) sub-exponential distributions. Our approach to establish tail asymptotics of stationary distributions is different from the classical semimartingale approach, and we do not impose ergodicity or moment bound conditions. In particular, the results also hold for explosive Markov chains, for which multiple stationary distributions may exist. Furthermore, our results on tail asymptotics of QSDs seem new. We apply our results to biochemical reaction networks, a general single-cell stochastic gene expression model, an extended class of branching processes, and stochastic population processes with bursty reproduction, none of which are birth–death processes. Our approach, together with the identities, easily extends to discrete-time Markov chains.

For an n-element subset U of $\mathbb {Z}^2$, select x from U according to harmonic measure from infinity, remove x from U and start a random walk from x. If the walk leaves from y when it first enters the rest of U, add y to it. Iterating this procedure constitutes the process we call harmonic activation and transport (HAT).

HAT exhibits a phenomenon we refer to as collapse: Informally, the diameter shrinks to its logarithm over a number of steps which is comparable to this logarithm. Collapse implies the existence of the stationary distribution of HAT, where configurations are viewed up to translation, and the exponential tightness of diameter at stationarity. Additionally, collapse produces a renewal structure with which we establish that the center of mass process, properly rescaled, converges in distribution to two-dimensional Brownian motion.

To characterize the phenomenon of collapse, we address fundamental questions about the extremal behavior of harmonic measure and escape probabilities. Among n-element subsets of $\mathbb {Z}^2$, what is the least positive value of harmonic measure? What is the probability of escape from the set to a distance of, say, d? Concerning the former, examples abound for which the harmonic measure is exponentially small in n. We prove that it can be no smaller than exponential in $n \log n$. Regarding the latter, the escape probability is at most the reciprocal of $\log d$, up to a constant factor. We prove it is always at least this much, up to an n-dependent factor.

In this paper, we consider absorbing Markov chains $X_n$ admitting a quasi-stationary measure $\mu $ on M where the transition kernel ${\mathcal P}$ admits an eigenfunction $0\leq \eta \in L^1(M,\mu )$. We find conditions on the transition densities of ${\mathcal P}$ with respect to $\mu $ which ensure that $\eta (x) \mu (\mathrm {d} x)$ is a quasi-ergodic measure for $X_n$ and that the Yaglom limit converges to the quasi-stationary measure $\mu $-almost surely. We apply this result to the random logistic map $X_{n+1} = \omega _n X_n (1-X_n)$ absorbed at ${\mathbb R} \setminus [0,1],$ where $\omega _n$ is an independent and identically distributed sequence of random variables uniformly distributed in $[a,b],$ for $1\leq a <4$ and $b>4.$

We study the $R_\beta$-positivity and the existence of zero-temperature limits for a sequence of infinite-volume Gibbs measures $(\mu_{\beta}(\!\cdot\!))_{\beta \geq 0}$ at inverse temperature $\beta$ associated to a family of nearest-neighbor matrices $(Q_{\beta})_{\beta \geq 0}$ reflected at the origin. We use a probabilistic approach based on the continued fraction theory previously introduced in Ferrari and Martínez (1993) and sharpened in Littin and Martínez (2010). Some necessary and sufficient conditions are provided to ensure (i) the existence of a unique infinite-volume Gibbs measure for large but finite values of $\beta$, and (ii) the existence of weak limits as $\beta \to \infty$. Some application examples are revised to put in context the main results of this work.

A comparison theorem for state-dependent regime-switching diffusion processes is established, which enables us to pathwise-control the evolution of the state-dependent switching component simply by Markov chains. Moreover, a sharp estimate on the stability of Markovian regime-switching processes under the perturbation of transition rate matrices is provided. Our approach is based on elaborate constructions of switching processes in the spirit of Skorokhod’s representation theorem varying according to the problem being dealt with. In particular, this method can cope with switching processes in an infinite state space and not necessarily of birth–death type. As an application, some known results on the ergodicity and stability of state-dependent regime-switching processes can be improved.

We consider an SIR (susceptible $\to$ infective $\to$ recovered) epidemic in a closed population of size n, in which infection spreads via mixing events, comprising individuals chosen uniformly at random from the population, which occur at the points of a Poisson process. This contrasts sharply with most epidemic models, in which infection is spread purely by pairwise interaction. A sequence of epidemic processes, indexed by n, and an approximating branching process are constructed on a common probability space via embedded random walks. We show that under suitable conditions the process of infectives in the epidemic process converges almost surely to the branching process. This leads to a threshold theorem for the epidemic process, where a major outbreak is defined as one that infects at least $\log n$ individuals. We show further that there exists $\delta \gt 0$, depending on the model parameters, such that the probability that a major outbreak has size at least $\delta n$ tends to one as $n \to \infty$.

In this paper, we propose new Metropolis–Hastings and simulated annealing algorithms on a finite state space via modifying the energy landscape. The core idea of landscape modification rests on introducing a parameter c, such that the landscape is modified once the algorithm is above this threshold parameter to encourage exploration, while the original landscape is utilized when the algorithm is below the threshold for exploitation purposes. We illustrate the power and benefits of landscape modification by investigating its effect on the classical Curie–Weiss model with Glauber dynamics and external magnetic field in the subcritical regime. This leads to a landscape-modified mean-field equation, and with appropriate choice of c the free energy landscape can be transformed from a double-well into a single-well landscape, while the location of the global minimum is preserved on the modified landscape. Consequently, running algorithms on the modified landscape can improve the convergence to the ground state in the Curie–Weiss model. In the setting of simulated annealing, we demonstrate that landscape modification can yield improved or even subexponential mean tunnelling time between global minima in the low-temperature regime by appropriate choice of c, and we give a convergence guarantee using an improved logarithmic cooling schedule with reduced critical height. We also discuss connections between landscape modification and other acceleration techniques, such as Catoni’s energy transformation algorithm, preconditioning, importance sampling, and quantum annealing. The technique developed in this paper is not limited to simulated annealing, but is broadly applicable to any difference-based discrete optimization algorithm by a change of landscape.

We analyze the long-term stability of a stochastic model designed to illustrate the adaptation of a population to variation in its environment. A piecewise deterministic process modeling adaptation is coupled to a Feller logistic diffusion modeling population size. As the individual features in the population become further away from the optimal ones, the growth rate declines, making population extinction more likely. Assuming that the environment changes deterministically and steadily in a constant direction, we obtain the existence and uniqueness of the quasi-stationary distribution, the associated survival capacity, and the Q-process. Our approach also provides several exponential convergence results (in total variation for the measures). From this synthetic information, we can characterize the efficiency of internal adaptation (i.e. population turnover from mutant invasions). When the latter is lacking, there is still stability, but because of the high level of population extinction. Therefore, any characterization of internal adaptation should be based on specific features of this quasi-ergodic regime rather than the mere existence of the regime itself.

The Brownian bridge or Lévy–Ciesielski construction of Brownian paths almost surely converges uniformly to the true Brownian path. We focus on the uniform error. In particular, we show constructively that at level N, at which there are $d=2^N$ points evaluated on the Brownian path, the uniform error and its square, and the uniform error of geometric Brownian motion, have upper bounds of order $\mathcal {O}(\sqrt {\ln d/d})$, matching the known orders. We apply the results to an option pricing example.