We describe a new general connection between $\Lambda$-coalescents and genealogies of continuous-state branching processes. This connection is based on the construction of an explicit coupling using a particle representation inspired by the lookdown process of Donnelly and Kurtz. This coupling has the property that the coalescent comes down from infinity if and only if the branching process becomes extinct, thereby answering a question of Bertoin and Le Gall. The coupling also offers new perspective on the speed of coming down from infinity and allows us to relate power-law behavior for $N^{\Lambda}(t)$ to the classical upper and lower indices arising in the study of pathwise properties of Lévy processes.

We consider one-dimensional hierarchical coalescence processes (in short HCPs) where two or three neighboring domains can merge. An HCP consists of an infinite sequence of stochastic coalescence processes: each process occurs in a different “epoch” and evolves for an infinite time, while the evolutions in subsequent epochs are linked in such a way that the initial distribution of epoch $n+1$ coincides with the final distribution of epoch $n$. Inside each epoch a domain can incorporate one of its neighboring domains or both of them if its length belongs to a certain epoch-dependent finite range.

Assuming that the distribution at the beginning of the first epoch is described by a renewal simple point process, we prove limit theorems for the domain length and for the position of the leftmost point (if any). Our analysis extends the results obtained in [Ann. Probab. 40 (2012) 1377–1435] to a larger family of models, including relevant examples from the physics literature [Europhys. Lett. 27 (1994) 175–180, Phys. Rev. E (3) 68 (2003) 031504]. It reveals the presence of a common abstract structure behind models which are apparently very different, thus leading to very similar limit theorems. Finally, we give here a full characterization of the infinitesimal generator for the dynamics inside each epoch, thus allowing us to describe the time evolution of the expected value of regular observables in terms of an ordinary differential equation.

We establish subgeometric bounds on convergence rate of general Markov processes in the Wasserstein metric. In the discrete time setting we prove that the Lyapunov drift condition and the existence of a “good” $d$-small set imply subgeometric convergence to the invariant measure. In the continuous time setting we obtain the same convergence rate provided that there exists a “good” $d$-small set and the Douc–Fort–Guillin supermartingale condition holds. As an application of our results, we prove that the Veretennikov–Khasminskii condition is sufficient for subexponential convergence of strong solutions of stochastic delay differential equations.

We analyze quasi-stationary distributions $\{\mu^{\varepsilon}\}_{\varepsilon>0}$ of a family of Markov chains $\{X^{\varepsilon}\}_{\varepsilon>0}$ that are random perturbations of a bounded, continuous map $F:M\to M$, where $M$ is a closed subset of $\mathbb{R}^{k}$. Consistent with many models in biology, these Markov chains have a closed absorbing set $M_{0}\subset M$ such that $F(M_{0})=M_{0}$ and $F(M\setminus M_{0})=M\setminus M_{0}$. Under some large deviations assumptions on the random perturbations, we show that, if there exists a positive attractor for $F$ (i.e., an attractor for $F$ in $M\setminus M_{0}$), then the weak* limit points of $\mu_{\varepsilon}$ are supported by the positive attractors of $F$. To illustrate the broad applicability of these results, we apply them to nonlinear branching process models of metapopulations, competing species, host-parasitoid interactions and evolutionary games.

We study the path properties of a random polymer attracted to a defect line by a potential with disorder, and we prove that in the delocalized regime, at any temperature, the number of contacts with the defect line remains in a certain sense “tight in probability” as the polymer length varies. On the other hand we show that at sufficiently low temperature, there exists a.s. a subsequence where the number of contacts grows like the log of the length of the polymer.

We use the inverse scattering transform and a diffusion approximation limit theorem to study the stability of soliton components of the solution of the nonlinear Schrödinger and Korteweg–de Vries equations under random perturbations of the initial conditions: for a wide class of rapidly oscillating random perturbations this problem reduces to the study of a canonical system of stochastic differential equations which depends only on the integrated covariance of the perturbation. We finally study the problem when the perturbation is weak, which allows us to analyze the stability of solitons quantitatively.

We propose a second order discretization for backward stochastic differential equations (BSDEs) with possibly nonsmooth boundary data. When implemented, the discretization method requires essentially the same computational effort with the Euler scheme for BSDEs of Bouchard and Touzi [Stochastic Process. Appl. 111 (2004) 175–206] and Zhang [Ann. Appl. Probab. 14 (2004) 459–488]. However, it enjoys a second order asymptotic rate of convergence, provided that the coefficients of the equation are sufficiently smooth. In the second part of the paper, we combine this discretization with higher order cubature formulas on Wiener space to produce a fully implementable second order scheme.

We study the convergence of a class of Runge–Kutta type schemes for backward stochastic differential equations (BSDEs) in a Markovian framework. The schemes belonging to the class under consideration benefit from a certain stability property. As a consequence, the overall rate of the convergence of these schemes is controlled by their local truncation error. The schemes are categorized by the number of intermediate stages implemented between consecutive partition time instances. We show that the order of the schemes matches the number $p$ of intermediate stages for $p\le3$. Moreover, we show that the so-called order barrier occurs at $p=3$, that is, that it is not possible to construct schemes of order $p$ with $p$ stages, when $p>3$. The analysis is done under sufficient regularity on the final condition and on the coefficients of the BSDE.

Ordinary differential equations obtained as limits of Markov processes appear in many settings. They may arise by scaling large systems, or by averaging rapidly fluctuating systems, or in systems involving multiple time-scales, by a combination of the two. Motivated by models with multiple time-scales arising in systems biology, we present a general approach to proving a central limit theorem capturing the fluctuations of the original model around the deterministic limit. The central limit theorem provides a method for deriving an appropriate diffusion (Langevin) approximation.

We consider the problem of minimizing queue-length costs in a system with heterogenous parallel servers, operating in a many-server heavy-traffic regime with nondegenerate slowdown. This regime is distinct from the well-studied heavy traffic diffusion regimes, namely the (single server) conventional regime and the (many-server) Halfin–Whitt regime. It has the distinguishing property that waiting times and service times are of comparable magnitudes. We establish an asymptotic lower bound on the cost and devise a sequence of policies that asymptotically attain this bound. As in the conventional regime, the asymptotics can be described by means of a Brownian control problem, the solution of which exhibits a state space collapse.

The present paper introduces a jump-diffusion extension of the classical diffusion default intensity model by means of subordination in the sense of Bochner. We start from the bi-variate process $(X,D)$ of a diffusion state variable $X$ driving default intensity and a default indicator process $D$ and time change it with a Lévy subordinator $\mathcal{T}$. We characterize the time-changed process $(X^{\phi}_{t},D^{\phi}_{t})=(X(\mathcal{T}_{t}),D(\mathcal{T}_{t}))$ as a Markovian–Itô semimartingale and show from the Doob–Meyer decomposition of $D^{\phi}$ that the default time in the time-changed model has a jump-diffusion or a pure jump intensity. When $X$ is a CIR diffusion with mean-reverting drift, the default intensity of the subordinate model (SubCIR) is a jump-diffusion or a pure jump process with mean-reverting jumps in both directions that stays nonnegative. The SubCIR default intensity model is analytically tractable by means of explicitly computed eigenfunction expansions of relevant semigroups, yielding closed-form pricing of credit-sensitive securities.

We introduce trap models on a finite volume $k$-level tree as a class of Markov jump processes with state space the leaves of that tree. They serve to describe the GREM-like trap model of Sasaki and Nemoto. Under suitable conditions on the parameters of the trap model, we establish its infinite volume limit, given by what we call a $K$-process in an infinite $k$-level tree. From this we deduce that the $K$-process also is the scaling limit of the GREM-like trap model on extreme time scales under a fine tuning assumption on the volumes.