In recent works on the theory of machine learning, it has been observed that heavy tail properties of stochastic gradient descent (SGD) can be studied in the probabilistic framework of stochastic recursions. In particular, Gürbüzbalaban et al. (2021) considered a setup corresponding to linear regression for which iterations of SGD can be modelled by a multivariate affine stochastic recursion $X_n=A_nX_{n-1}+B_n$ for independent and identically distributed pairs $(A_n,B_n)$, where $A_n$ is a random symmetric matrix and $B_n$ is a random vector. However, their approach is not completely correct and, in the present paper, the problem is put into the right framework by applying the theory of irreducible-proximal matrices.

We investigate the EM approximation for $\mathbb{R}^d$-valued ergodic stochastic differential equations (SDEs) driven by rotationally invariant $\alpha$-stable processes ($\alpha\in(1,2)$) with Markovian switching. The coefficient g violates the dissipative condition for certain states of the switching process. Using the Lindeberg principle, we establish quantitative error bounds between the original process $(X_t,R_t)_{t\geqslant 0}$ and its Euler–Maruyama (EM) scheme under a specially designed metric. Furthermore, we derive both a central limit theorem and a moderate derivation principle for the empirical measures of both the SDE and its EM scheme. The theoretical results are subsequently validated through a concrete example.

For Markov chains and Markov processes exhibiting a form of stochastic monotonicity (higher states have higher transition probabilities in terms of stochastic dominance), stability and ergodicity results can be obtained with the use of order-theoretic mixing conditions. We complement these results by providing quantitative bounds on deviations between distributions. We also show that well-known total variation bounds can be recovered as a special case.

We prove new results about comparing the efficiency of general state space Markov chain Monte Carlo algorithms that randomly select a possibly different reversible method at each step (previously known only for finite state spaces). We also provide new, simpler, more accessible proofs of key results, and analyse numerous examples. We provide a full proof of the formula for the asymptotic variance for real-valued functionals on $\varphi$-irreducible reversible Markov chains, first introduced by Kipnis and Varadhan (1986, Commun. Math. Phys. 104, 1–19). Given two Markov kernels P and Q with stationary measure $\pi$, we say that the Markov kernel P efficiency-dominates the Markov kernel Q if the asymptotic variance with respect to P is at most the asymptotic variance with respect to Q for every real-valued functional $f\in L^2(\pi)$. Assuming only a basic background in functional analysis, we prove that for two reversible Markov kernels P and Q, P efficiency-dominates Q if and only if the operator $\mathcal{Q}-\mathcal{P}$, where $\mathcal{P}$ is the operator on $L^2(\pi)$ that maps $f\mapsto\int f(y)P(\cdot,\mathrm{d}y)$ and similarly for $\mathcal{Q}$, is positive on $L^2(\pi)$, i.e. $\langle f,\left(\mathcal{Q}-\mathcal{P}\right)f\rangle\geq0$ for every $f\in L^2(\pi)$ (previous proofs for general state spaces use technical results from monotone operator function theory). We use this result to show that under mild conditions, sandwich variants of data augmentation algorithms efficiency-dominate the original algorithm. We also provide other easy-to-check sufficient conditions for efficiency dominance, some of which are generalized from the finite state space case. We also provide a proof based on that of Tierney (1998, Ann. Appl. Prob. 8, 1–9) that Peskun dominance is a sufficient condition for efficiency dominance for reversible kernels. Using these results, we show that Markov kernels formed by random selection of other ‘component’ Markov kernels will always efficiency-dominate another Markov kernel formed in this way, as long as the component kernels of the former efficiency-dominate those of the latter. These results on the efficiency dominance of combining component kernels generalizes the results on the efficiency dominance of combined chains introduced by Neal and Rosenthal (2024, J. Appl. Prob. 62, 188–208) from finite state spaces to general state spaces.

In this paper we study degree-penalized contact processes on Galton-Watson (GW) trees and the configuration model. The model we consider is a modification of the usual contact process on a graph. In particular, each vertex can be either infected or healthy. When infected, each vertex heals at rate one. Also, when infected, a vertex u with degree $d_u$ infects its neighboring vertex v with degree $d_v$ with rate $\lambda / f(d_u, d_v)$ for some positive function f. In the case $f(d_u, d_v)=\max (d_u, d_v)^\mu $ for some $\mu \ge 0$, the infection is slowed down to and from high-degree vertices. This is in line with arguments used in social network science: people with many contacts do not have the time to infect their neighbors at the same rate as people with fewer contacts.

We show that new phase transitions occur in terms of the parameter $\mu $ (at $1/2$) and the degree distribution D of the GW tree.

• • When $\mu \ge 1$, the process goes extinct for all distributions D for all sufficiently small $\lambda>0$;

• • When $\mu \in [1/2, 1)$, and the tail of D weakly follows a power law with tail-exponent less than $1-\mu $, the process survives globally but not locally for all $\lambda $ small enough;

• • When $\mu \in [1/2, 1)$, and $\mathbb {E}[D^{1-\mu }]<\infty $, the process goes extinct almost surely, for all $\lambda $ small enough;

• • When $\mu <1/2$, and D is heavier than stretched exponential with stretch-exponent $1-2\mu $, the process survives (locally) with positive probability for all $\lambda>0$.

We also study the product case, where $f(d_u,d_v)=(d_u d_v)^\mu $. In that case, the situation for $\mu < 1/2$ is the same as the one described above, but $\mu \ge 1/2$ always leads to a subcritical contact process for small enough $\lambda>0$ on all graphs. Furthermore, for finite random graphs with prescribed degree sequences, we establish the corresponding phase transitions in terms of the length of survival.

We consider steady-state diffusion in a bounded planar domain with multiple small targets on a smooth boundary. Using the method of matched asymptotic expansions, we investigate the competition of these targets for a diffusing particle and the crucial role of surface reactions on the targets. We start from the classical problem of splitting probabilities for perfectly reactive targets with Dirichlet boundary conditions and improve some earlier results. We discuss how this approach can be generalised to partially reactive targets characterised by a Robin boundary condition. In particular, we show how partial reactivity reduces the effective size of the target. In addition, we consider more intricate surface reactions modelled by mixed Steklov-Neumann or Steklov-Neumann-Dirichlet problems. We provide the first derivation of the asymptotic behaviour of the eigenvalues and eigenfunctions for these spectral problems in the small-target limit. Finally, we show how our asymptotic approach can be extended to interior targets in the bulk and to exterior problems where diffusion occurs in an unbounded planar domain outside a compact set. Direct applications of these results to diffusion-controlled reactions are discussed.

Following the pivotal work of Sevastyanov (1957), who considered branching processes with homogeneous Poisson immigration, much has been done to understand the behaviour of such processes under different types of branching and immigration mechanisms. Recently, the case where the times of immigration are generated by a non-homogeneous Poisson process has been considered in depth. In this work, we demonstrate how we can use the framework of point processes in order to go beyond the Poisson process. As an illustration, we show how to transfer techniques from the case of Poisson immigration to the case where it is spanned by a determinantal point process.

We study quasi-stationary distributions and quasi-limiting behaviour of Markov chains in general reducible state spaces with absorption. First, we consider state spaces that can be decomposed into two successive subsets (with communication possible in a single direction), differentiating between three situations, and characterize the exponential order of magnitude and the exact polynomial correction, called the polynomial convergence parameter, for the leading-order term of the semigroup for large time. Second, we consider general Markov chains with finitely or countably many communication classes by applying the first results iteratively over the communication classes of the chain. We conclude with an application of these results to the case of denumerable state spaces, where we prove existence for a quasi-stationary distribution without assuming irreducibility before absorption, but only aperiodicity, existence of a Lyapunov function, and existence of a point with almost surely finite return time.

Designing efficient and rigorous numerical methods for sequential decision-making under uncertainty is a difficult problem that arises in many applications frameworks. In this paper we focus on the numerical solution of a subclass of impulse control problems for the piecewise deterministic Markov process (PDMP) when the jump times are hidden. We first state the problem as a partially observed Markov decision process (POMDP) on a continuous state space and with controlled transition kernels corresponding to some specific skeleton chains of the PDMP. We then proceed to build a numerically tractable approximation of the POMDP by tailor-made discretizations of the state spaces. The main difficulty in evaluating the discretization error comes from the possible random jumps of the PDMP between consecutive epochs of the POMDP and requires special care. Finally, we discuss the practical construction of discretization grids and illustrate our method on simulations.

A general way to represent stochastic differential equations (SDEs) on smooth manifolds is based on the Schwartz morphism. In this manuscript, we are interested in SDEs on a smooth manifold $M$ that are driven by p-dimensional Wiener process $W_t \in \mathbb{R}^p$ and time $t$. In terms of the Schwartz morphism, such an SDE is represented by a Schwartz morphism that morphs the semimartingale $(t,W_t)\in\mathbb{R}^{p+1}$ into a semimartingale on the manifold $M$. We show that it is possible to construct such Schwartz morphisms using special maps that we call diffusion generators. We show that one of the ways to construct a diffusion generator is by considering the flow of differential equations. One particular case is the construction of diffusion generators using Lagrangian vector fields. Using the diffusion generator approach, we also give the extended Itô formula (also known as generalized Itô formula or Itô–Wentzell formula) for SDEs on manifolds.

We study a continuous-time mutually catalytic branching model on the $\mathbb{Z}^{d}$. The model describes the behavior of two different populations of particles, performing random walk on the lattice in the presence of branching, that is, each particle dies at a certain rate and is replaced by a random number of offspring. The branching rate of a particle in one population is proportional to the number of particles of another population at the same site. We study the long time behavior for this model, in particular, coexistence and noncoexistence of two populations in the long run. Finally, we construct a sequence of renormalized processes and use duality techniques to investigate its limiting behavior.

We study stationary distributions in the context of stochastic reaction networks. In particular, we are interested in complex balanced reaction networks and the reduction of such networks by assuming that a set of species (called non-interacting species) are degraded fast (and therefore essentially absent from the network), implying that some reaction rates are large relative to others. Technically, we assume that these reaction rates are scaled by a common parameter N and let $N\to\infty$. The limiting stationary distribution as $N\to\infty$ is compared with the stationary distribution of the reduced reaction network obtained by elimination of the non-interacting species. In general, the limiting stationary distribution could differ from the stationary distribution of the reduced reaction network. We identify various sufficient conditions under which these two distributions are the same, including when the reaction network is detailed balanced and when the set of non-interacting species consists of intermediate species. In the latter case, the limiting stationary distribution essentially retains the form of the complex balanced distribution. This finding is particularly surprising given that the reduced reaction network could be non-weakly reversible and might exhibit unconventional kinetics.

In this paper, we introduce a new technique to study the distribution in residue classes of sets of integers with digit and sum-of-digits restrictions. From our main theorem, we derive a necessary and sufficient condition for integers with missing digits to be uniformly distributed in arithmetic progressions, extending previous results going back to the work of Erdős, Mauduit and Sárközy. Our approach uses Markov chains and does not rely on Fourier analysis as many results of this nature do. Our results apply more generally to the class of multiplicatively invariant sets of integers. This class, defined by Glasscock, Moreira and Richter using symbolic dynamics, is an integer analogue to fractal sets and includes all missing digits sets. We address uniform distribution in this setting, partially answering an open question posed by the same authors.

We investigate a specific class of irreducible, level-dependent, discrete-time, GI/M/1-type Markov chains. The transition matrices possess a block lower-Hessenberg structure, which shows asymptotic convergence along the rows as the level approaches infinity. Criteria are presented for recurrence, transience, positive recurrence, geometric ergodicity, and geometric transience in terms of elements of the transition matrices. These criteria are established by employing drift functions and matrix-generating functions. Furthermore, we discuss the extension of the main results to the continuous-time case.

We show that the Potts model on a graph can be approximated by a sequence of independent and identically distributed spins in terms of Wasserstein distance at high temperatures. We prove a similar result for the Curie–Weiss–Potts model on the complete graph, conditioned on being close enough to any of its equilibrium macrostates, in the low-temperature regime. Our proof technique is based on Stein’s method for comparing the stationary distributions of two Glauber dynamics with similar updates, one of which is rapid mixing and contracting on a subset of the state space. Along the way, we prove a new upper bound on the mixing time of the Glauber dynamics for the conditional measure of the Curie–Weiss–Potts model near an equilibrium macrostate.

We study sequential optimal stopping with partial reversibility. The optimal stopping problem is subject to implementation delay, which is random and exponentially distributed. Once the stopping decision is made, the decision maker can, by incurring a cost, call the decision off and restart the stopping problem. The optimization criterion is to maximize the expected present value of the total payoff. We characterize the value function in terms of a Bellman principle for a wide class of payoff functions and potentially multidimensional strong Markov dynamics. We also analyse the case of linear diffusion dynamics and characterize the value function and the optimal decision rule for a wide class of payoff functions.

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 $G$.