We define a new state-space for the coalescing Brownian flow, also known as the Brownian web, on the circle. The elements of this space are families of order-preserving maps of the circle, depending continuously on two time parameters and having a certain weak flow property. The space is equipped with a complete separable metric. A larger state-space, allowing jumps in time, is also introduced, and equipped with a Skorokhod-type metric, also complete and separable. We prove that the coalescing Brownian flow is the weak limit in this larger space of a family of flows which evolve by jumps, each jump arising from a small localized disturbance of the circle. A local version of this result is also obtained, in which the weak limit law is that of the coalescing Brownian flow on the line. Our set-up is well adapted to time-reversal and our weak limit result provides a new proof of time-reversibility of the coalescing Brownian flow. We also identify a martingale associated with the coalescing Brownian flow on the circle and use this to make a direct calculation of the Laplace transform of the time to complete coalescence.

We prove the first robust dimension free isoperimetric result for the standard Gaussian measure $\gamma_{n}$ and the corresponding boundary measure $\gamma_{n}^{+}$ in $\mathbb{R} ^{n}$. The main result in the theory of Gaussian isoperimetry (proven in the 1970s by Sudakov and Tsirelson, and independently by Borell) states that if $\gamma_{n}(A)=1/2$ then the surface area of $A$ is bounded by the surface area of a half-space with the same measure, $\gamma_{n}^{+}(A)\leq(2\pi)^{-1/2}$. Our results imply in particular that if $A\subset\mathbb{R} ^{n}$ satisfies $\gamma_{n}(A)=1/2$ and $\gamma_{n}^{+}(A)\leq(2\pi)^{-1/2}+\delta$ then there exists a half-space $B\subset\mathbb{R} ^{n}$ such that $\gamma_{n}(A\Delta B)\leq C\smash{\log^{-1/2}}(1/\delta)$ for an absolute constant $C$. Since the Gaussian isoperimetric result was established, only recently a robust version of the Gaussian isoperimetric result was obtained by Cianchi et al., who showed that $\gamma_{n}(A\Delta B)\le C(n)\sqrt{\delta}$ for some function $C(n)$ with no effective bounds. Compared to the results of Cianchi et al., our results have optimal (i.e., no) dependence on the dimension, but worse dependence on $\delta$.

We study the asymptotic behavior of a multidimensional random walk in a general cone. We find the tail asymptotics for the exit time and prove integral and local limit theorems for a random walk conditioned to stay in a cone. The main step in the proof consists in constructing a positive harmonic function for our random walk under minimal moment restrictions on the increments. For the proof of tail asymptotics and integral limit theorems, we use a strong approximation of random walks by Brownian motion. For the proof of local limit theorems, we suggest a rather simple approach, which combines integral theorems for random walks in cones with classical local theorems for unrestricted random walks. We also discuss some possible applications of our results to ordered random walks and lattice path enumeration.

We study random bipartite planar maps defined by assigning nonnegative weights to each face of a map. We prove that for certain choices of weights a unique large face, having degree proportional to the total number of edges in the maps, appears when the maps are large. It is furthermore shown that as the number of edges $n$ of the planar maps goes to infinity, the profile of distances to a marked vertex rescaled by $n^{-1/2}$ is described by a Brownian excursion. The planar maps, with the graph metric rescaled by $n^{-1/2}$, are then shown to converge in distribution toward Aldous’ Brownian tree in the Gromov–Hausdorff topology. In the proofs, we rely on the Bouttier–di Francesco–Guitter bijection between maps and labeled trees and recent results on simply generated trees where a unique vertex of a high degree appears when the trees are large.

We prove the existence and nontriviality of the $d$-dimensional 4 Minkowski content for the Schramm–Loewner evolution ($\mathrm{SLE} _{\kappa}$) with $\kappa<8$ and $d=1+\frac{\kappa}{8}$. We show that this is a multiple of the natural parameterization.

The large deviations principle for the empirical measure for both continuous and discrete time Markov processes is well known. Various expressions are available for the rate function, but these expressions are usually as the solution to a variational problem, and in this sense not explicit. An interesting class of continuous time, reversible processes was identified in the original work of Donsker and Varadhan for which an explicit expression is possible. While this class includes many (reversible) processes of interest, it excludes the case of continuous time pure jump processes, such as a reversible finite state Markov chain. In this paper, we study the large deviations principle for the empirical measure of pure jump Markov processes and provide an explicit formula of the rate function under reversibility.

We consider the random normal matrix ensemble associated with a potential in the plane of sufficient growth near infinity. It is known that asymptotically as the order of the random matrix increases indefinitely, the eigenvalues approach a certain equilibrium density, given in terms of Frostman’s solution to the minimum energy problem of weighted logarithmic potential theory. At a finer scale, we may consider fluctuations of eigenvalues about the equilibrium. In the present paper, we give the correction to the expectation of the fluctuations, and we show that the potential field of the corrected fluctuations converge on smooth test functions to a Gaussian free field with free boundary conditions on the droplet associated with the potential.

In this paper, we study the 2D stochastic quasi-geostrophic equation on $\mathbb{T}^{2}$ for general parameter $\alpha\in(0,1)$ and multiplicative noise. We prove the existence of weak solutions and Markov selections for multiplicative noise for all $\alpha\in(0,1)$. In the subcritical case $\alpha>1/2$, we prove existence and uniqueness of (probabilistically) strong solutions. Moreover, we prove ergodicity for the solution of the stochastic quasi-geostrophic equations in the subcritical case driven by possibly degenerate noise. The law of large numbers for the solution of the stochastic quasi-geostrophic equations in the subcritical case is also established. In the case of nondegenerate noise and $\alpha>2/3$ in addition exponential ergodicity is proved.

With $\mathcal{Q}_{q,n}$ the distribution of $n$ minus the rank of a matrix chosen uniformly from the collection of all $n\times(n+m)$ matrices over the finite field $\mathbb{F}_{q}$ of size $q\ge2$, and $\mathcal{Q}_{q}$ the distributional limit of $\mathcal{Q}_{q,n}$ as $n\rightarrow\infty$, we apply Stein’s method to prove the total variation bound

\[\frac{1}{8q^{n+m+1}}\leq\|\mathcal{Q}_{q,n}-\mathcal{Q}_{q}\|_{\mathrm{TV}}\leq\frac{3}{q^{n+m+1}}.\] In addition, we obtain similar sharp results for the rank distributions of symmetric, symmetric with zero diagonal, skew symmetric, skew centrosymmetric and Hermitian matrices.

We characterize all translation invariant half-planar maps satisfying a certain natural domain Markov property. For $p$-angulations with $p\ge3$ where all faces are simple, we show that these form a one-parameter family of measures $\mathbb{H}^{(p)}_{\alpha}$. For triangulations, we also establish existence of a phase transition which affects many properties of these maps. The critical maps are the well-known half-plane uniform infinite planar maps. The subcritical maps are identified as all possible limits of uniform measures on finite maps with given boundary and area.

We introduce a class of Lévy processes subject to specific regularity conditions, and consider their Feynman–Kac semigroups given under a Kato-class potential. Using new techniques, first we analyze the rate of decay of eigenfunctions at infinity. We prove bounds on $\lambda$-subaveraging functions, from which we derive two-sided sharp pointwise estimates on the ground state, and obtain upper bounds on all other eigenfunctions. Next, by using these results, we analyze intrinsic ultracontractivity and related properties of the semigroup refining them by the concept of ground state domination and asymptotic versions. We establish the relationships of these properties, derive sharp necessary and sufficient conditions for their validity in terms of the behavior of the Lévy density and the potential at infinity, define the concept of borderline potential for the asymptotic properties and give probabilistic and variational characterizations. These results are amply illustrated by key examples.

We consider a self-interacting process described in terms of a single-server system with service stations at each point of the real line. The customer arrivals are given by a Poisson point processes on the space–time half plane. The server adopts a greedy routing mechanism, traveling toward the nearest customer, and ignoring new arrivals while in transit. We study the trajectories of the server and show that its asymptotic position diverges logarithmically in time.

Consider a standard ${\Lambda }$-coalescent that comes down from infinity. Such a coalescent starts from a configuration consisting of infinitely many blocks at time $0$, but its number of blocks $N_{t}$ is a finite random variable at each positive time $t$. Berestycki et al. [Ann. Probab. 38 (2010) 207–233] found the first-order approximation $v$ for the process $N$ at small times. This is a deterministic function satisfying $N_{t}/v_{t}\to1$ as $t\to0$. The present paper reports on the first progress in the study of the second-order asymptotics for $N$ at small times. We show that, if the driving measure $\Lambda$ has a density near zero which behaves as $x^{-\beta}$ with $\beta\in(0,1)$, then the process $(\varepsilon^{-1/(1+\beta)}(N_{\varepsilon t}/v_{\varepsilon t}-1))_{t\ge0}$ converges in law as $\varepsilon\to0$ in the Skorokhod space to a totally skewed $(1+\beta)$-stable process. Moreover, this process is a unique solution of a related stochastic differential equation of Ornstein–Uhlenbeck type, with a completely asymmetric stable Lévy noise.

We consider stochastic dynamical systems on $\mathbb{R}$, that is, random processes defined by $X_{n}^{x}=\Psi_{n}(X_{n-1}^{x})$, $X_{0}^{x}=x$, where $\Psi_{n}$ are i.i.d. random continuous transformations of some unbounded closed subset of $\mathbb{R}$. We assume here that $\Psi_{n}$ behaves asymptotically like $A_{n}x$, for some random positive number $A_{n}$ [the main example is the affine stochastic recursion $\Psi_{n}(x)=A_{n}x+B_{n}$]. Our aim is to describe invariant Radon measures of the process $X_{n}^{x}$ in the critical case, when $\mathbb{E}\log A_{1}=0$. We prove that those measures behave at infinity like $\frac{dx}{x}$. We study also the problem of uniqueness of the invariant measure. We improve previous results known for the affine recursions and generalize them to a larger class of stochastic dynamical systems which include, for instance, reflected random walks, stochastic dynamical systems on the unit interval $[0,1]$, additive Markov processes and a variant of the Galton–Watson process.

This article is concerned with modulus of continuity of Brownian local times. Specifically, we focus on three closely related problems: (a) Limit theorem for a Brownian modulus of continuity involving Riesz potentials, where the limit law is an intricate Gaussian mixture. (b) Central limit theorems for the projections of $L^{2}$ modulus of continuity for a one-dimensional Brownian motion. (c) Extension of the second result to a two-dimensional Brownian motion. Our proofs rely on a combination of stochastic calculus and Malliavin calculus tools, plus a thorough analysis of singular integrals.

In this paper, we establish the existence of a stochastic flow of Sobolev diffeomorphisms

\[\mathbb{R}^{d}\ni x\quad\longmapsto\quad\phi_{s,t}(x)\in\mathbb{R}^{d},\qquad s,t\in\mathbb{R}\] for a stochastic differential equation (SDE) of the form

\[dX_{t}=b(t,X_{t})\,dt+dB_{t},\qquad s,t\in\mathbb{R},X_{s}=x\in\mathbb{R}^{d}.\] The above SDE is driven by a bounded measurable drift coefficient $b:\mathbb{R}\times\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ and a $d$-dimensional Brownian motion $B$. More specifically, we show that the stochastic flow $\phi_{s,t}(\cdot)$ of the SDE lives in the space $L^{2}(\Omega;W^{1,p}(\mathbb{R}^{d},w))$ for all $s,t$ and all $p\in(1,\infty)$, where $W^{1,p}(\mathbb{R}^{d},w)$ denotes a weighted Sobolev space with weight $w$ possessing a $p$th moment with respect to Lebesgue measure on $\mathbb{R}^{d}$. From the viewpoint of stochastic (and deterministic) dynamical systems, this is a striking result, since the dominant “culture” in these dynamical systems is that the flow “inherits” its spatial regularity from that of the driving vector fields.

The spatial regularity of the stochastic flow yields existence and uniqueness of a Sobolev differentiable weak solution of the (Stratonovich) stochastic transport equation

\[\cases{d_{t}u(t,x)+(b(t,x)\cdot Du(t,x))\,dt+\sum_{i=1}^{d}e_{i}\cdot Du(t,x)\circ dB_{t}^{i}=0,\cr u(0,x)=u_{0}(x),}\] where $b$ is bounded and measurable, $u_{0}$ is $C_{b}^{1}$ and $\{e_{i}\}_{i=1}^{d}$ a basis for $\mathbb{R}^{d}$. It is well known that the deterministic counterpart of the above equation does not in general have a solution.