We prove a scaling limit theorem for two-type Galton–Watson branching processes with interaction. The limit theorem gives rise to a class of mixed-state branching processes with interaction used to simulate evolution for cell division affected by parasites. Such processes can also be obtained by the pathwise-unique solution to a stochastic equation system. Moreover, we present sufficient conditions for extinction with probability 1 and the exponential ergodicity in the $L^1$-Wasserstein distance of such processes in some cases.

In this paper, we introduce a unified framework based on the pathwise expansion method to derive explicit recursive formulas for cumulative distribution functions, option prices, and transition densities in multivariate diffusion models. A key innovation of our approach is the introduction of the quasi-Lamperti transform, which normalizes the diffusion matrix at the initial time. This transformation facilitates expansions using uncorrelated Brownian motions, effectively reducing multivariate problems to one-dimensional computations. Consequently, both the analysis and the computation are significantly simplified. We also present two novel applications of the pathwise expansion method. Specifically, we employ the proposed framework to compute the value-at-risk for stock portfolios and to evaluate complex derivatives, such as forward-starting options. Our method has the flexibility to accommodate models with diverse features, including stochastic risk premiums, stochastic volatility, and nonaffine structures. Numerical experiments demonstrate the accuracy and computational efficiency of our approach. In addition, as a theoretical contribution, we establish an equivalence between the pathwise expansion method and the Hermite polynomial-based expansion method in the literature.

We consider a population consisting of two types of individuals, each of which can produce offspring on two different islands (in particular, the islands can be interpreted as active or dormant individuals). We model the evolution of the population of each type using a two-type Feller diffusion with immigration and study the frequency of one type on each island, when the total population size on each island is forced to be constant at a dense set of times. This leads to the solution of a stochastic differential equation, which we call the asymmetric two-island frequency process. We derive properties of this process and obtain a large population limit as the total size of each island tends to infinity. Additionally, we compute the fluctuations of the process around its deterministic limit. We establish conditions under which the asymmetric two-island frequency process has a moment dual. The dual is a continuous-time two-dimensional Markov chain that can be interpreted in terms of mutation, branching, pairwise branching, coalescence, and a novel mixed selection–migration term.

The hard-core model has as its configurations the independent sets of some graph instance $G$. The probability distribution on independent sets is controlled by a ‘fugacity’ $\lambda \gt 0$, with higher $\lambda$ leading to denser configurations. We investigate the mixing time of Glauber (single-site) dynamics for the hard-core model on restricted classes of bounded-degree graphs in which a particular graph $H$ is excluded as an induced subgraph. If $H$ is a subdivided claw then, for all $\lambda$, the mixing time is $O(n\log n)$, where $n$ is the order of $G$. This extends a result of Chen and Gu for claw-free graphs. When $H$ is a path, the set of possible instances is finite. For all other $H$, the mixing time is exponential in $n$ for sufficiently large $\lambda$, depending on $H$ and the maximum degree of $G$.

We study the local limit in distribution of Bienaymé–Galton–Watson trees conditioned on having large sub-populations. Assuming a generic and aperiodic condition on the offspring distribution, we prove the existence of a limit given by a Kesten’s tree associated with a certain critical offspring distribution.

Consider a subcritical branching Markov chain. Let $Z_n$ denote the counting measure of particles of generation n. Under some conditions, we give a probabilistic proof for the existence of the Yaglom limit of $(Z_n)_{n\in\mathbb{N}}$ by the moment method, based on the spinal decomposition and the many-to-few formula. As a result, we give explicit integral representations of all quasi-stationary distributions of $(Z_n)_{n\in\mathbb{N}}$, whose proofs are direct and probabilistic, and do not rely on Martin boundary theory.

We consider the random series–parallel graph introduced by Hambly and Jordan (2004 Adv. Appl. Probab. 36, 824–838), which is a hierarchical graph with a parameter $p\in [0, \, 1]$. The graph is built recursively: at each step, every edge in the graph is either replaced with probability p by a series of two edges, or with probability $1-p$ by two parallel edges, and the replacements are independent of each other and of everything up to then. At the nth step of the recursive procedure, the distance between the extremal points on the graph is denoted by $D_n (p)$. It is known that $D_n(p)$ possesses a phase transition at $p=p_c \;:\!=\;\frac{1}{2}$; more precisely, $\frac{1}{n}\log {{\mathbb{E}}}[D_n(p)] \to \alpha(p)$ when $n \to \infty$, with $\alpha(p) >0$ for $p>p_c$ and $\alpha(p)=0$ for $p\le p_c$. We study the exponent $\alpha(p)$ in the slightly supercritical regime $p=p_c+\varepsilon$. Our main result says that as $\varepsilon\to 0^+$, $\alpha(p_c+\varepsilon)$ behaves like $\sqrt{\zeta(2) \, \varepsilon}$, where $\zeta(2) \;:\!=\; \frac{\pi^2}{6}$.

We consider an optimal stopping problem of a linear diffusion under Poisson constraint where the agent can adjust the arrival rate of new stopping opportunities. We assume that the agent may switch the rate of the Poisson process between two values. Maintaining the lower rate incurs no cost, whereas the higher rate requires effort that is captured by a cost function c. We study a broad class of payoff functions, cost functions and diffusion dynamics, for which we explicitly characterize the solution to the constrained stopping problem. We also characterize the case where switching to the higher rate is always suboptimal. The results are illustrated with two examples.

We prove two-sided bounds on the expected values of several geometric functionals of the convex hull of Brownian motion in $\mathbb {R}^n$ and their inverse processes. This extends some recent results of McRedmond and Xu (2017), Jovalekić (2021), and Cygan, Šebek, and the first author (2023) from the plane to higher dimensions. Our main result shows that the average time required for the convex hull in $\mathbb {R}^n$ to attain unit volume is at most $n\sqrt [n]{n!}$. The proof relies on a novel procedure that embeds an n-simplex of prescribed volume within the convex hull of the Brownian path run up to a certain stopping time. All of our bounds capture the correct order of asymptotic growth or decay in the dimension n.

Let $\pi$ be a probability distribution in $\mathbb{R}^d$ and f a test function, and consider the problem of variance reduction in estimating $\mathbb{E}_\pi(f)$. We first construct a sequence of estimators for $\mathbb{E}_\pi (f)$, say $({1}/{k})\sum_{i=0}^{k-1} g_n(X_i)$, where the $X_i$ are samples from $\pi$ generated by the Metropolized Hamiltonian Monte Carlo algorithm and $g_n$ is the approximate solution of the Poisson equation through the weak approximate scheme recently invented by Mijatović and Vogrinc (2018). Then we prove under some regularity assumptions that the estimation error variance $\sigma_\pi^2(g_n)$ can be as arbitrarily small as the approximation order parameter $n\rightarrow\infty$. To illustrate, we confirm that the assumptions are satisfied by two typical concrete models, a Bayesian linear inverse problem and a two-component mixture of Gaussian distributions.

Recently, there has been much progress in understanding stationary measures for colored (also called multi-species or multi-type) interacting particle systems, motivated by asymptotic phenomena and rich underlying algebraic and combinatorial structures (such as nonsymmetric Macdonald polynomials). In this paper, we present a unified approach to constructing stationary measures for most of the known colored particle systems on the ring and the line, including (1) the Asymmetric Simple Exclusion Process (multi-species ASEP, or mASEP); (2) the $q$-deformed Totally Asymmetric Zero Range Process (TAZRP) also known as the $q$-Boson particle system; (3) the $q$-deformed Pushing Totally Asymmetric Simple Exclusion Process ($q$-PushTASEP). Our method is based on integrable stochastic vertex models and the Yang–Baxter equation. We express the stationary measures as partition functions of new ‘queue vertex models’ on the cylinder. The stationarity property is a direct consequence of the Yang–Baxter equation. For the mASEP on the ring, a particular case of our vertex model is equivalent to the multiline queues of Martin (Stationary distributions of the multi-type ASEP, Electron. J. Probab. 25 (2020), 1–41). For the colored $q$-Boson process and the $q$-PushTASEP on the ring, we recover and generalize known stationary measures constructed using multiline queues or other methods by Ayyer, Mandelshtam and Martin (Modified Macdonald polynomials and the multispecies zero range process: II, Algebr. Comb. 6 (2022), 243–284; Modified Macdonald polynomials and the multispecies zero-range process: I, Algebr. Comb. 6 (2023), 243–284) and Bukh and Cox (Periodic words, common subsequences and frogs, Ann. Appl. Probab. 32 (2022), 1295–1332). Our proofs of stationarity use the Yang–Baxter equation and bypass the Matrix Product Ansatz (used for the mASEP by Prolhac, Evans and Mallick (The matrix product solution of the multispecies partially asymmetric exclusion process, J. Phys. A. 42 (2009), 165004)). On the line and in a quadrant, we use the Yang–Baxter equation to establish a general colored Burke’s theorem, which implies that suitable specializations of our queue vertex models produce stationary measures for particle systems on the line. We also compute the colored particle currents in stationarity.

We show that on every affine Weyl group natural random walks are noise sensitive in total variation.

In this paper, we prove that the hitting probability of the Minkowski sum of fractal percolations can be characterised by capacity. Then we extend this result to Minkowski sums of general random sets in $\mathbb Z^d$, including ranges of random walks and critical branching random walks, whose hitting probabilities are described by Newtonian capacity individually.

The continuous random energy model (CREM) was introduced by Bovier and Kurkova in 2004 as a toy model of disordered systems. Among other things, their work indicates that there exists a critical point $\beta_\mathrm{c}$ such that the partition function exhibits a phase transition. The present work focuses on the high-temperature regime where $\beta<\beta_\mathrm{c}$. We show that, for all $\beta<\beta_\mathrm{c}$ and for all $s>0$, the negative s moment of the CREM partition function is comparable with the expectation of the CREM partition function to the power of $-s$, up to constants that are independent of N.

We study a two-dimensional discounted optimal stopping zero-sum (or Dynkin) game related to perpetual redeemable convertible bonds expressed as game (or Israeli) options in a model of financial markets in which the behaviour of the ex-dividend price of a dividend-paying asset follows a generalized geometric Brownian motion. It is assumed that the dynamics of the random dividend rate of the asset paid to shareholders are described by the mean-reverting filtering estimate of an unobservable continuous-time Markov chain with two states. It is shown that the optimal exercise (conversion) and withdrawal (redemption) times forming a Nash equilibrium are the first times at which the asset price hits either lower or upper stochastic boundaries being monotone functions of the running value of the filtering estimate of the state of the chain. We rigorously prove that the optimal stopping boundaries are regular for the stopping region relative to the resulting two-dimensional diffusion process and that the value function is continuously differentiable with respect to the both variables. It is verified by means of a change-of-variable formula with local time on surfaces that the optimal stopping boundaries are determined as a unique solution to the associated coupled system of nonlinear Fredholm integral equations among the couples of continuous functions of bounded variation satisfying certain conditions. We also give a closed-form solution to the appropriate optimal stopping zero-sum game in the corresponding model with an observable continuous-time Markov chain.

We consider the problem of sequential matching in a stochastic block model with several classes of nodes and generic compatibility constraints. When the probabilities of connections do not scale with the size of the graph, we show that under the Ncond condition, a simple max-weight type policy allows us to attain an asymptotically perfect matching while no sequential algorithm attains perfect matching otherwise. The proof relies on a specific Markovian representation of the dynamics associated with Lyapunov techniques.

The gambler’s ruin problem for correlated random walks (CRWs), both with and without delays, is addressed using the optional stopping theorem for martingales. We derive closed-form expressions for the ruin probabilities and the expected game duration for CRWs with increments $\{1,-1\}$ and for symmetric CRWs with increments $\{1,0,-1\}$ (CRWs with delays). Additionally, a martingale technique is developed for general CRWs with delays. The gambler’s ruin probability for a game involving bets on two arbitrary patterns is also examined.

We give a notion of boundary pair $(\mathcal{B}_-,\mathcal{B}_+)$ for measured groupoids which generalizes the one introduced by Bader and Furman [BF14] for locally compact groups. In the case of a semidirect groupoid $\mathcal{G}=\Gamma \ltimes X$ obtained by a probability measure preserving action $\Gamma \curvearrowright X$ of a locally compact group, we show that a boundary pair is exactly $(B_- \times X, B_+ \times X)$, where $(B_-,B_+)$ is a boundary pair for $\Gamma$. For any measured groupoid $(\mathcal{G},\nu )$, we prove that the Poisson boundaries associated to the Markov operators generated by a probability measure equivalent to $\nu$ provide other examples of our definition. Following Bader and Furman [BF], we define algebraic representability for an ergodic groupoid $(\mathcal{G},\nu )$. In this way, given any measurable representation $\rho \,:\,\mathcal{G} \rightarrow H$ into the $\kappa$-points of an algebraic $\kappa$-group $\mathbf{H}$, we obtain $\rho$-equivariant maps $\mathcal{B}_\pm \rightarrow H/L_\pm$, where $L_\pm =\mathbf{L}_\pm (\kappa )$ for some $\kappa$-subgroups $\mathbf{L}_\pm \lt \mathbf{H}$. In the particular case when $\kappa =\mathbb{R}$ and $\rho$ is Zariski dense, we show that $L_\pm$ must be minimal parabolic subgroups.

We consider a stochastic model, called the replicator coalescent, describing a system of blocks of k different types that undergo pairwise mergers at rates depending on the block types: with rate $C_{ij}\geq 0$ blocks of type i and j merge, resulting in a single block of type i. The replicator coalescent can be seen as a generalisation of Kingman’s coalescent death chain in a multi-type setting, although without an underpinning exchangeable partition structure. The name is derived from a remarkable connection between the instantaneous dynamics of this multi-type coalescent when issued from an arbitrarily large number of blocks, and the so-called replicator equations from evolutionary game theory. By dilating time arbitrarily close to zero, we see that initially, on coming down from infinity, the replicator coalescent behaves like the solution to a certain replicator equation. Thereafter, stochastic effects are felt and the process evolves more in the spirit of a multi-type death chain.

We prove an ergodic theorem for Markov chains indexed by the Ulam–Harris–Neveu tree over large subsets with arbitrary shape under two assumptions: (i) with high probability, two vertices in the large subset are far from each other, and (ii) with high probability, those two vertices have their common ancestor close to the root. The assumption on the common ancestor can be replaced by some regularity assumption on the Markov transition kernel. We verify that these assumptions are satisfied for some usual trees. Finally, with Markov chain Monte Carlo considerations in mind, we prove that when the underlying Markov chain is stationary and reversible, the Markov chain, that is the line graph, yields minimal variance for the empirical average estimator among trees with a given number of nodes. In doing so, we prove that the Hosoya–Wiener polynomial is minimized over $[{-}1,1]$ by the line graph among trees of a given size.