We extend the spatial $\Lambda$-Fleming–Viot process introduced in [Electron. J. Probab. 15 (2010) 162–216] to incorporate recombination. The process models allele frequencies in a population which is distributed over the two-dimensional torus $\mathbb{T} (L)$ of sidelength $L$ and is subject to two kinds of reproduction events: small events of radius $\mathcal{O} (1)$ and much rarer large events of radius $\mathcal{O} (L^{\alpha})$ for some $\alpha\in(0,1]$. We investigate the correlation between the times to the most recent common ancestor of alleles at two linked loci for a sample of size two from the population. These individuals are initially sampled from “far apart” on the torus. As $L$ tends to infinity, depending on the frequency of the large events, the recombination rate and the initial distance between the two individuals sampled, we obtain either a complete decorrelation of the coalescence times at the two loci, or a sharp transition between a first period of complete correlation and a subsequent period during which the remaining times needed to reach the most recent common ancestor at each locus are independent. We use our computations to derive approximate probabilities of identity by descent as a function of the separation at which the two individuals are sampled.

We prove that the mixing time of the Glauber dynamics for random $k$-colorings of the complete tree with branching factor $b$ undergoes a phase transition at $k=b(1+o_{b}(1))/\ln{b}$. Our main result shows nearly sharp bounds on the mixing time of the dynamics on the complete tree with $n$ vertices for $k=Cb/\ln{b}$ colors with constant $C$. For $C\geq1$ we prove the mixing time is $O(n^{1+o_{b}(1)}\ln{n})$. On the other side, for $C<1$ the mixing time experiences a slowing down; in particular, we prove it is $O(n^{1/C+o_{b}(1)}\ln{n})$ and $\Omega(n^{1/C-o_{b}(1)})$. The critical point $C=1$ is interesting since it coincides (at least up to first order) with the so-called reconstruction threshold which was recently established by Sly. The reconstruction threshold has been of considerable interest recently since it appears to have close connections to the efficiency of certain local algorithms, and this work was inspired by our attempt to understand these connections in this particular setting.

In this paper, we consider smooth shot noise processes and their expected number of level crossings. When the kernel response function is sufficiently smooth, the mean number of crossings function is obtained through an integral formula. Moreover, as the intensity increases, or equivalently, as the number of shots becomes larger, a normal convergence to the classical Rice’s formula for Gaussian processes is obtained. The Gaussian kernel function, that corresponds to many applications in physics, is studied in detail and two different regimes are exhibited.

We derive a large deviation principle for the empirical currents of lattice gas dynamics which combine a fast stirring mechanism (Symmetric Simple Exclusion Process) and creation/annihilation mechanisms (Glauber dynamics). Previous results on the density large deviations can be recovered from this general large deviation principle. The contribution of external driving forces due to reservoirs at the boundary of the system is also taken into account.

The Metropolis-adjusted Langevin (MALA) algorithm is a sampling algorithm which makes local moves by incorporating information about the gradient of the logarithm of the target density. In this paper we study the efficiency of MALA on a natural class of target measures supported on an infinite dimensional Hilbert space. These natural measures have density with respect to a Gaussian random field measure and arise in many applications such as Bayesian nonparametric statistics and the theory of conditioned diffusions. We prove that, started in stationarity, a suitably interpolated and scaled version of the Markov chain corresponding to MALA converges to an infinite dimensional diffusion process. Our results imply that, in stationarity, the MALA algorithm applied to an $N$-dimensional approximation of the target will take $\mathcal{O}(N^{1/3})$ steps to explore the invariant measure, comparing favorably with the Random Walk Metropolis which was recently shown to require $\mathcal{O}(N)$ steps when applied to the same class of problems. As a by-product of the diffusion limit, it also follows that the MALA algorithm is optimized at an average acceptance probability of $0.574$. Previous results were proved only for targets which are products of one-dimensional distributions, or for variants of this situation, limiting their applicability. The correlation in our target means that the rescaled MALA algorithm converges weakly to an infinite dimensional Hilbert space valued diffusion, and the limit cannot be described through analysis of scalar diffusions. The limit theorem is proved by showing that a drift-martingale decomposition of the Markov chain, suitably scaled, closely resembles a weak Euler–Maruyama discretization of the putative limit. An invariance principle is proved for the martingale, and a continuous mapping argument is used to complete the proof.

We present a new class of multifractal process on $\mathbb{R}$, constructed using an embedded branching process. The construction makes use of known results on multitype branching random walks, and along the way constructs cascade measures on the boundaries of multitype Galton–Watson trees. Our class of processes includes Brownian motion subjected to a continuous multifractal time-change.

In addition, if we observe our process at a fixed spatial resolution, then we can obtain a finite Markov representation of it, which we can use for on-line simulation. That is, given only the Markov representation at step $n$, we can generate step $n+1$ in $O(\log n)$ operations. Detailed pseudo-code for this algorithm is provided.

We solve the problem of mean-variance hedging for general semimartingale models via stochastic control methods. After proving that the value process of the associated stochastic control problem has a quadratic structure, we characterize its three coefficient processes as solutions of semimartingale backward stochastic differential equations and show how they can be used to describe the optimal trading strategy for each conditional mean-variance hedging problem. For comparison with the existing literature, we provide alternative equivalent versions of the BSDEs and present a number of simple examples.

The reconstruction of phylogenies from DNA or protein sequences is a major task of computational evolutionary biology. Common phenomena, notably variations in mutation rates across genomes and incongruences between gene lineage histories, often make it necessary to model molecular data as originating from a mixture of phylogenies. Such mixed models play an increasingly important role in practice.

Using concentration of measure techniques, we show that mixtures of large trees are typically identifiable. We also derive sequence-length requirements for high-probability reconstruction.

In this article we propose a generalization of the theory of diffusion approximation for random ODE to a nonlinear system of random Schrödinger equations. This system arises in the study of pulse propagation in randomly birefringent optical fibers. We first show existence and uniqueness of solutions for the random PDE and the limiting equation. We follow the work of Garnier and Marty [Wave Motion 43 (2006) 544–560], Marty [Problèmes d’évolution en milieux aléatoires: Théorèmes limites, schémas numériques et applications en optique (2005) Univ. Paul Sabatier], where a linear electric field is considered, and we get an asymptotic dynamic for the nonlinear electric field.

In this article we study a class of stochastic functional differential equations driven by Lévy processes (in particular, $\alpha$-stable processes), and obtain the existence and uniqueness of Markov solutions in small time intervals. This corresponds to the local solvability to a class of quasi-linear partial integro-differential equations. Moreover, in the constant diffusion coefficient case, without any assumptions on the Lévy generator, we also show the existence of a unique maximal weak solution for a class of semi-linear partial integro-differential equation systems under bounded Lipschitz assumptions on the coefficients. Meanwhile, in the nondegenerate case (corresponding to $\Delta^{\alpha/2}$ with $\alpha\in(1,2]$), based upon some gradient estimates, the existence of global solutions is established too. In particular, this provides a probabilistic treatment for the nonlinear partial integro-differential equations, such as the multi-dimensional fractal Burgers equations and the fractal scalar conservation law equations.

The rate of convergence of the distribution of the length of the longest increasing subsequence, toward the maximal eigenvalue of certain matrix ensembles, is investigated. For finite-alphabet uniform and nonuniform i.i.d. sources, a rate of $\log n/\sqrt{n}$ is obtained. The uniform binary case is further explored, and an improved $1/\sqrt{n}$ rate obtained.

The Fleming–Viot measure-valued diffusion is a Markov process describing the evolution of (allelic) types under mutation, selection and random reproduction. We enrich this process by genealogical relations of individuals so that the random type distribution as well as the genealogical distances in the population evolve stochastically. The state space of this tree-valued enrichment of the Fleming–Viot dynamics with mutation and selection (TFVMS) consists of marked ultrametric measure spaces, equipped with the marked Gromov-weak topology and a suitable notion of polynomials as a separating algebra of test functions.

The construction and study of the TFVMS is based on a well-posed martingale problem. For existence, we use approximating finite population models, the tree-valued Moran models, while uniqueness follows from duality to a function-valued process. Path properties of the resulting process carry over from the neutral case due to absolute continuity, given by a new Girsanov-type theorem on marked metric measure spaces.

To study the long-time behavior of the process, we use a duality based on ideas from Dawson and Greven [On the effects of migration in spatial Fleming–Viot models with selection and mutation (2011c) Unpublished manuscript] and prove ergodicity of the TFVMS if the Fleming–Viot measure-valued diffusion is ergodic. As a further application, we consider the case of two allelic types and additive selection. For small selection strength, we give an expansion of the Laplace transform of genealogical distances in equilibrium, which is a first step in showing that distances are shorter in the selective case.