In this paper we study the quasi-stationary behavior of absorbed one-dimensional diffusions. We obtain necessary and sufficient conditions for the exponential convergence to a unique quasi-stationary distribution in total variation, uniformly with respect to the initial distribution. An important tool is provided by one-dimensional strict local martingale diffusions coming down from infinity. We prove, under mild assumptions, that their expectation at any positive time is uniformly bounded with respect to the initial position. We provide several examples and extensions, including the sticky Brownian motion and some one-dimensional processes with jumps.

We formulate and prove a shape theorem for a continuous-time continuous-space stochastic growth model under certain general conditions. Similar to the classical lattice growth models, the proof makes use of the subadditive ergodic theorem. A precise expression for the speed of propagation is given in the case of a truncated free-branching birth rate.

In this paper we investigate gradient estimation for a class of contracting stochastic systems on a continuous state space. We find conditions on the one-step transitions, namely differentiability and contraction in a Wasserstein distance, that guarantee differentiability of stationary costs. Then we show how to estimate the derivatives, deriving an estimator that can be seen as a generalization of the forward sensitivity analysis method used in deterministic systems. We apply the results to examples, including a neural network model.

Given a sequence (M k , Q k )k ≥ 1 of independent and identically distributed random vectors with nonnegative components, we consider the recursive Markov chain (X n )n ≥ 0, defined by the random difference equation X n = M n X n - 1 + Q n for n ≥ 1, where X 0 is independent of (M k , Q k )k ≥ 1. Criteria for the null recurrence/transience are provided in the situation where (X n )n ≥ 0 is contractive in the sense that M 1 ⋯ M n → 0 almost surely, yet occasional large values of the Q n overcompensate the contractive behavior so that positive recurrence fails to hold. We also investigate the attractor set of (X n )n ≥ 0 under the sole assumption that this chain is locally contractive and recurrent.

In this paper we consider random distance graphs motivated by applications in neurobiology. These models can be viewed as examples of inhomogeneous random graphs, notably outside of the so-called rank-1 case. Treating these models in the context of the general theory of inhomogeneous graphs helps us to derive the asymptotics for the size of the largest connected component. In particular, we show that certain random distance graphs behave exactly as the classical Erdős–Rényi model, not only in the supercritical phase (as already known) but in the subcritical case as well.

We consider the discounted continuous-time Markov decision process (CTMDP), where the negative part of each cost rate is bounded by a drift function, say w, whereas the positive part is allowed to be arbitrarily unbounded. Our focus is on the existence of a stationary optimal policy for the discounted CTMDP problems out of the more general class. Both constrained and unconstrained problems are considered. Our investigations are based on the continuous-time version of the Veinott transformation. This technique has not been widely employed in the previous literature on CTMDPs, but it clarifies the roles of the imposed conditions in a rather transparent way.

In this paper we study the random walk on the hypercube (ℤ / 2ℤ)n which at each step flips k randomly chosen coordinates. We prove that the mixing time for this walk is of the order (n / k)logn. We also prove that if k = o(n) then the walk exhibits cutoff at (n / 2k)logn with window n / 2k.

We study an affine two-factor model introduced by Barczy et al. (2014). One component of this two-dimensional model is the so-called α-root process, which generalizes the well-known Cox–Ingersoll–Ross process. In the α = 2 case, this two-factor model was used by Chen and Joslin (2012) to price defaultable bonds with stochastic recovery rates. In this paper we prove exponential ergodicity of this two-factor model when α ∈ (1, 2). As a possible application, our result can be used to study the parameter estimation problem of the model.

Stochastic encounter-mating (SEM) models describe monogamous permanent pair formation in finite zoological populations of multitype females and males. In this paper we study SEM models with Poisson firing times. First, we prove that the model enjoys a fluid limit as the population size diverges, that is, the stochastic dynamics converges to a deterministic system governed by coupled ordinary differential equations (ODEs). Then we convert these ODEs to the well-known Lotka–Volterra and replicator equations from population dynamics. Next, under the so-called fine balance condition which characterizes panmixia, we solve the corresponding replicator equations and give an exact expression for the fluid limit. Finally, we consider the case with two types of female and male. Without the fine balance assumption, but under certain symmetry conditions, we give an explicit formula for the limiting mating pattern, and then use it to characterize assortative mating.

We prove that the probability substitution matrices obtained from a continuous-time Markov chain form a multiplicatively closed set if and only if the rate matrices associated with the chain form a linear space spanning a Lie algebra. The key original contribution we make is to overcome an obstruction, due to the presence of inequalities that are unavoidable in the probabilistic application, which prevents free manipulation of terms in the Baker–Campbell–Haursdorff formula.

Conformal loop ensembles (CLEs) are random collections of loops in a simply connected domain, whose laws are characterized by a natural conformal invariance property. The set of points not surrounded by any loop is a canonical random connected fractal set — a random and conformally invariant analog of the Sierpinski carpet or gasket.

In the present paper, we derive a direct relationship between the CLEs with simple loops ( $\text{CLE}_{\unicode[STIX]{x1D705}}$ for $\unicode[STIX]{x1D705}\in (8/3,4)$ , whose loops are Schramm’s $\text{SLE}_{\unicode[STIX]{x1D705}}$ -type curves) and the corresponding CLEs with nonsimple loops ( $\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$ with $\unicode[STIX]{x1D705}^{\prime }:=16/\unicode[STIX]{x1D705}\in (4,6)$ , whose loops are $\text{SLE}_{\unicode[STIX]{x1D705}^{\prime }}$ -type curves). This correspondence is the continuum analog of the Edwards–Sokal coupling between the $q$ -state Potts model and the associated FK random cluster model, and its generalization to noninteger  $q$ .

Like its discrete analog, our continuum correspondence has two directions. First, we show that for each $\unicode[STIX]{x1D705}\in (8/3,4)$ , one can construct a variant of $\text{CLE}_{\unicode[STIX]{x1D705}}$ as follows: start with an instance of $\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$ , then use a biased coin to independently color each $\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$ loop in one of two colors, and then consider the outer boundaries of the clusters of loops of a given color. Second, we show how to interpret $\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$ loops as interfaces of a continuum analog of critical Bernoulli percolation within $\text{CLE}_{\unicode[STIX]{x1D705}}$ carpets — this is the first construction of continuum percolation on a fractal planar domain. It extends and generalizes the continuum percolation on open domains defined by $\text{SLE}_{6}$ and $\text{CLE}_{6}$ .

These constructions allow us to prove several conjectures made by the second author and provide new and perhaps surprising interpretations of the relationship between CLEs and the Gaussian free field. Along the way, we obtain new results about generalized $\text{SLE}_{\unicode[STIX]{x1D705}}(\unicode[STIX]{x1D70C})$ curves for $\unicode[STIX]{x1D70C}<-2$ , such as their decomposition into collections of $\text{SLE}_{\unicode[STIX]{x1D705}}$ -type ‘loops’ hanging off of $\text{SLE}_{\unicode[STIX]{x1D705}^{\prime }}$ -type ‘trunks’, and vice versa (exchanging $\unicode[STIX]{x1D705}$ and $\unicode[STIX]{x1D705}^{\prime }$ ). We also define a continuous family of natural $\text{CLE}$ variants called boundary conformal loop ensembles (BCLEs) that share some (but not all) of the conformal symmetries that characterize $\text{CLE}$ s, and that should be scaling limits of critical models with special boundary conditions. We extend the $\text{CLE}_{\unicode[STIX]{x1D705}}$ / $\text{CLE}_{\unicode[STIX]{x1D705}^{\prime }}$ correspondence to a $\text{BCLE}_{\unicode[STIX]{x1D705}}$ / $\text{BCLE}_{\unicode[STIX]{x1D705}^{\prime }}$ correspondence that makes sense for the wider range $\unicode[STIX]{x1D705}\in (2,4]$ and $\unicode[STIX]{x1D705}^{\prime }\in [4,8)$ .

Uniformization transforms a pseudo-Poisson process with unequal intensities (leaving rates) into one with uniform intensity. Self-transitions is the price to pay. Two intensities arise when one considers an absorbing barrier of a Markov process as a body in its own right: a pair of Markov processes intertwined by an extended Chapman–Kolmogorov equation naturally arises. We show that Sauer's two-state space empathy theory handles such intertwined processes. The price of self-transitions is also avoided.

The classical SIR epidemic model is generalized to incorporate a detection process of infectives in the course of time. Our purpose is to determine the exact distribution of the population state at the first detection instant and the following ones. An extension is also discussed that allows the parameters to change with the number of detected cases. The followed approach relies on simple martingale arguments and uses a special family of Abel–Gontcharoff polynomials.

In this paper we treat a pure death process coming down from infinity as a natural generalization of the death process associated with the Kingman coalescent. We establish a number of limit theorems including a strong law of large numbers and a large deviation theorem.

Quasi-stationary distributions (QSDs) arise from stochastic processes that exhibit transient equilibrium behaviour on the way to absorption. QSDs are often mathematically intractable and even drawing samples from them is not straightforward. In this paper the framework of sequential Monte Carlo samplers is utilised to simulate QSDs and several novel resampling techniques are proposed to accommodate models with reducible state spaces, with particular focus on preserving particle diversity on discrete spaces. Finally, an approach is considered to estimate eigenvalues associated with QSDs, such as the decay parameter.

Markov chain Monte Carlo (MCMC) methods provide an essential tool in statistics for sampling from complex probability distributions. While the standard approach to MCMC involves constructing discrete-time reversible Markov chains whose transition kernel is obtained via the Metropolis–Hastings algorithm, there has been recent interest in alternative schemes based on piecewise deterministic Markov processes (PDMPs). One such approach is based on the zig-zag process, introduced in Bierkens and Roberts (2016), which proved to provide a highly scalable sampling scheme for sampling in the big data regime; see Bierkens et al. (2016). In this paper we study the performance of the zig-zag sampler, focusing on the one-dimensional case. In particular, we identify conditions under which a central limit theorem holds and characterise the asymptotic variance. Moreover, we study the influence of the switching rate on the diffusivity of the zig-zag process by identifying a diffusion limit as the switching rate tends to ∞. Based on our results we compare the performance of the zig-zag sampler to existing Monte Carlo methods, both analytically and through simulations.

We introduce a new parameter to discuss the behavior of a genetic algorithm. This parameter is the mean number of exact copies of the best-fit chromosomes from one generation to the next. We believe that the genetic algorithm operates best when this parameter is slightly larger than 1 and we prove two results supporting this belief. We consider the case of the simple genetic algorithm with the roulette wheel selection mechanism. We denote by ℓ the length of the chromosomes, m the population size, p C the crossover probability, and p M the mutation probability. Our results suggest that the mutation and crossover probabilities should be tuned so that, at each generation, the maximal fitness multiplied by (1 - p C)(1 - p M)ℓ is greater than the mean fitness.

We consider a general class of epidemic models obtained by applying the random time changes of Ethier and Kurtz (2005) to a collection of Poisson processes and we show the large deviation principle for such models. We generalise the approach followed by Dolgoarshinnykh (2009) in the case of the SIR epidemic model. Thanks to an additional assumption which is satisfied in many examples, we simplify the recent work of Kratz and Pardoux (2017).

We consider a Yule process until the total population reaches size n ≫ 1, and assume that neutral mutations occur with high probability 1 - p (in the sense that each child is a new mutant with probability 1 - p, independently of the other children), where p = p n ≪ 1. We establish a general strategy for obtaining Poisson limit laws and a weak law of large numbers for the number of subpopulations exceeding a given size and apply this to some mutation regimes of particular interest. Finally, we give an application to subcritical Bernoulli bond percolation on random recursive trees with percolation parameter p n tending to 0.

In this paper we compute the absorbing time T n of an n-dimensional discrete-time Markov chain comprising n components, each with an absorbing state and evolving in mutual exclusion. We show that the random absorbing time T n is well approximated by a deterministic time t n that is the first time when a fluid approximation of the chain approaches the absorbing state at a distance 1 / n. We provide an asymptotic expansion of t n that uses the spectral decomposition of the kernel of the chain as well as the asymptotic distribution of T n , relying on extreme values theory. We show the applicability of this approach with three different problems: the coupon collector, the erasure channel lifetime, and the coupling times of random walks in high-dimensional spaces.