We consider the 3D Landau equation for moderately soft potentials [$\gamma\in(-2,0)$ with the usual notation] as well as a stochastic system of $N$ particles approximating it. We first establish some strong/weak stability estimates for the Landau equation, which are fully satisfactory only when $\gamma\in[-1,0)$. We next prove, under some appropriate conditions on the initial data, the so-called propagation of molecular chaos, that is, that the empirical measure of the particle system converges to the unique solution of the Landau equation. The main difficulty is the presence of a singularity in the equation. When $\gamma\in(-1,0)$, the strong-weak uniqueness estimate allows us to use a coupling argument and to obtain a rate of convergence. When $\gamma\in(-2,-1]$, we use the classical martingale method introduced by McKean. To control the singularity, we have to take advantage of the regularity provided by the entropy dissipation. Unfortunately, this dissipation is too weak for some (very rare) aligned configurations. We thus introduce a perturbed system with an additional noise, show the propagation of chaos for this system and finally prove that the additional noise is almost never used in the limit $N\to\infty$.

By establishing a characterization for Sobolev differentiability of random fields, we prove the weak differentiability of solutions to stochastic differential equations with local Sobolev and super-linear growth coefficients with respect to the starting point. Moreover, we also study the strong Feller property and the irreducibility to the associated diffusion semigroup.

In a variety of problems in pure and applied probability, it is relevant to study the large exceedance probabilities of the perpetuity sequence $Y_{n}:=B_{1}+A_{1}B_{2}+\cdots+(A_{1}\cdots A_{n-1})B_{n}$, where $(A_{i},B_{i})\subset(0,\infty)\times\mathbb{R}$. Estimates for the stationary tail distribution of $\{Y_{n}\}$ have been developed in the seminal papers of Kesten [Acta Math. 131 (1973) 207–248] and Goldie [Ann. Appl. Probab. 1 (1991) 126–166]. Specifically, it is well known that if $M:=\sup_{n}Y_{n}$, then $\mathbb{P}\{M>u\}\sim\mathcal{C}_{M}u^{-\xi}$ as $u\to\infty$. While much attention has been focused on extending such estimates to more general settings, little work has been devoted to understanding the path behavior of these processes. In this paper, we derive sharp asymptotic estimates for the normalized first passage time $T_{u}:=(\log u)^{-1}\inf\{n:Y_{n}>u\}$. We begin by showing that, conditional on $\{T_{u}<\infty\}$, $T_{u}\to\rho$ as $u\to\infty$ for a certain positive constant $\rho$. We then provide a conditional central limit theorem for $\{T_{u}\}$, and study $\mathbb{P}\{T_{u}\in G\}$ as $u\to\infty$ for sets $G\subset[0,\infty)$. If $G\subset[0,\rho)$, then we show that $\mathbb{P}\{T_{u}\in G\}u^{I(G)}\to C(G)$ as $u\to\infty$ for a certain large deviation rate function $I$ and constant $C(G)$. On the other hand, if $G\subset(\rho,\infty)$, then we show that the tail behavior is actually quite complex and different asymptotic regimes are possible. We conclude by extending our results to the corresponding forward process, understood in the sense of Letac [In Random Matrices and Their Applications (Brunswick, Maine, 1984) (1986) 263–273 Amer. Math. Soc.], namely to the reflected process $M_{n}^{\ast}:=\max\{A_{n}M_{n-1}^{\ast}+B_{n},0\}$, $n\in\mathbb{Z}_{+}$. Using Siegmund duality, we relate the first passage times of $\{Y_{n}\}$ to the finite-time exceedance probabilities of $\{M_{n}^{\ast}\}$, yielding a new result concerning the convergence of $\{M_{n}^{\ast}\}$ to its stationary distribution.

A theory of existence and uniqueness is developed for general stochastic differential mean field games with common noise. The concepts of strong and weak solutions are introduced in analogy with the theory of stochastic differential equations, and existence of weak solutions for mean field games is shown to hold under very general assumptions. Examples and counter-examples are provided to enlighten the underpinnings of the existence theory. Finally, an analog of the famous result of Yamada and Watanabe is derived, and it is used to prove existence and uniqueness of a strong solution under additional assumptions.

We begin by studying inventory accumulation at a LIFO (last-in-first-out) retailer with two products. In the simplest version, the following occur with equal probability at each time step: first product ordered, first product produced, second product ordered, second product produced. The inventory thus evolves as a simple random walk on $\mathbb{Z}^{2}$. In more interesting versions, a $p$ fraction of customers orders the “freshest available” product regardless of type. We show that the corresponding random walks scale to Brownian motions with diffusion matrices depending on $p$.

We then turn our attention to the critical Fortuin–Kastelyn random planar map model, which gives, for each $q>0$, a probability measure on random (discretized) two-dimensional surfaces decorated by loops, related to the $q$-state Potts model. A longstanding open problem is to show that as the discretization gets finer, the surfaces converge in law to a limiting (loop-decorated) random surface. The limit is expected to be a Liouville quantum gravity surface decorated by a conformal loop ensemble, with parameters depending on $q$. Thanks to a bijection between decorated planar maps and inventory trajectories (closely related to bijections of Bernardi and Mullin), our results about the latter imply convergence of the former in a particular topology. A phase transition occurs at $p=1/2$, $q=4$.

We study the Euler characteristic of an excursion set of a stationary isotropic Gaussian random field $X:\Omega\times\mathbb{R}^{d}\to\mathbb{R}$. Let us fix a level $u\in\mathbb{R}$ and let us consider the excursion set above $u$, $A(T,u)=\{t\in T:X(t)\ge u\}$ where $T$ is a bounded cube $\subset\mathbb{R}^{d}$. The aim of this paper is to establish a central limit theorem for the Euler characteristic of $A(T,u)$ as $T$ grows to $\mathbb{R}^{d}$, as conjectured by R. Adler more than ten years ago [Ann. Appl. Probab. 10 (2000) 1–74].

The required assumption on $X$ is $C^{3}$ regularity of the trajectories, non degeneracy of the Gaussian vector $X(t)$ and derivatives at any fixed point $t\in\mathbb{R}^{d}$ as well as integrability on $\mathbb{R}^{d}$ of the covariance function and its derivatives. The fact that $X$ is $C^{3}$ is stronger than Geman’s assumption traditionally used in dimension one. Nevertheless, our result extends what is known in dimension one to higher dimension. In that case, the Euler characteristic of $A(T,u)$ equals the number of up-crossings of $X$ at level $u$, plus eventually one if $X$ is above $u$ at the left bound of the interval $T$.

We prove that every component of the wired uniform spanning forest (${\mathsf{WUSF}}$) is one-ended almost surely in every transient reversible random graph, removing the bounded degree hypothesis required by earlier results. We deduce that every component of the ${\mathsf{WUSF}}$ is one-ended almost surely in every supercritical Galton–Watson tree, answering a question of Benjamini, Lyons, Peres and Schramm [Ann. Probab. 29 (2001) 1–65].

Our proof introduces and exploits a family of Markov chains under which the oriented ${\mathsf{WUSF}}$ is stationary, which we call the wired cycle-breaking dynamics.

We are interested in the randomly biased random walk on the supercritical Galton–Watson tree. Our attention is focused on a slow regime when the biased random walk $(X_{n})$ is null recurrent, making a maximal displacement of order of magnitude $(\log n)^{3}$ in the first $n$ steps. We study the localization problem of $X_{n}$ and prove that the quenched law of $X_{n}$ can be approximated by a certain invariant probability depending on $n$ and the random environment. As a consequence, we establish that upon the survival of the system, $\frac{|X_{n}|}{(\log n)^{2}}$ converges in law to some non-degenerate limit on $(0,\infty)$ whose law is explicitly computed.

In proving large deviation estimates, the lower bound for open sets and upper bound for compact sets are essentially local estimates. On the other hand, the upper bound for closed sets is global and compactness of space or an exponential tightness estimate is needed to establish it. In dealing with the occupation measure $L_{t}(A)=\frac{1}{t}\int_{0}^{t}{\mathbf{1}}_{A}(W_{s})\,\mathrm{d}s$ of the $d$-dimensional Brownian motion, which is not positive recurrent, there is no possibility of exponential tightness. The space of probability distributions $\mathcal{M}_{1}(\mathbb{R}^{d})$ can be compactified by replacing the usual topology of weak convergence by the vague toplogy, where the space is treated as the dual of continuous functions with compact support. This is essentially the one point compactification of $R^{d}$ by adding a point at $\infty$ that results in the compactification of $\mathcal{M}_{1}(\mathbb{R}^{d})$ by allowing some mass to escape to the point at $\infty$. If one were to use only test functions that are continuous and vanish at $\infty$, then the compactification results in the space of sub-probability distributions $\mathcal{M}_{\le1}(\mathbb{R}^{d})$ by ignoring the mass at $\infty$.

The main drawback of this compactification is that it ignores the underlying translation invariance. More explicitly, we may be interested in the space of equivalence classes of orbits $\tilde{\mathcal{M}}_{1}=\tilde{\mathcal{M}}_{1}(\mathbb{R}^{d})$ under the action of the translation group $\mathbb{R}^{d}$ on $\mathcal{M}_{1}(\mathbb{R}^{d})$. There are problems for which it is natural to compactify this space of orbits. We will provide such a compactification, prove a large deviation principle there and give an application to a relevant problem.

In the present paper, we study the chaotic representation property for certain families ${\mathscr{X}}$ of square integrable martingales on a finite time interval $[0,T]$. For this purpose, we introduce the notion of compensated-covariation stability of such families. The chaotic representation property will be defined using iterated integrals with respect to a given family ${\mathscr{X}}$ of square integrable martingales having deterministic mutual predictable covariation $\langle X,Y\rangle$ for all $X,Y\in{\mathscr{X}}$. The main result of the present paper is stated in Theorem 5.8 below: If ${\mathscr{X}}$ is a compensated-covariation stable family of square integrable martingales such that $\langle X,Y\rangle$ is deterministic for all $X,Y\in{\mathscr{X}}$ and, furthermore, the system of monomials generated by ${\mathscr{X}}$ is total in $L^{2}(\Omega,\mathscr{F}^{\mathscr{X}}_{T},\mathbb{P})$, then ${\mathscr{X}}$ possesses the chaotic representation property with respect to the $\sigma$-field $\mathscr{F}^{\mathscr{X}}_{T}$. We shall apply this result to the case of Lévy processes. Relative to the filtration $\mathbb{F}^{L}$ generated by a Lévy process $L$, we construct families of martingales which possess the chaotic representation property. As an illustration of the general results, we will also discuss applications to continuous Gaussian families of martingales and independent families of compensated Poisson processes. We conclude the paper by giving, for the case of Lévy processes, several examples of concrete families ${\mathscr{X}}$ of martingales including Teugels martingales.

The directed polymer model at intermediate disorder regime was introduced by Alberts–Khanin–Quastel [Ann. Probab. 42 (2014) 1212–1256]. It was proved that at inverse temperature $\beta n^{-\gamma}$ with $\gamma=1/4$ the partition function, centered appropriately, converges in distribution and the limit is given in terms of the solution of the stochastic heat equation. This result was obtained under the assumption that the disorder variables posses exponential moments, but its universality was also conjectured under the assumption of six moments. We show that this conjecture is valid and we further extend it by exhibiting classes of different universal limiting behaviors in the case of less than six moments. We also explain the behavior of the scaling exponent for the log-partition function under different moment assumptions and values of $\gamma$.

We define a characteristic function for probability measures on the signatures of geometric rough paths. We determine sufficient conditions under which a random variable is uniquely determined by its expected signature, thus partially solving the analogue of the moment problem. We furthermore study analyticity properties of the characteristic function and prove a method of moments for weak convergence of random variables. We apply our results to signature arising from Lévy, Gaussian and Markovian rough paths.

In this paper, we study the behavior of Brownian motion conditioned on the event that its local time at zero stays below a given increasing function $f$ up to time $T$. For a class of nonincreasing functions $f$, we show that the conditioned process converges, as $T\rightarrow\infty$, to a limit process and we derive necessary and sufficient conditions for the limit to be transient. In the transient case, the limit process is described explicitly, and in the recurrent case we quantify the entropic repulsion phenomenon by describing the repulsion envelope, stating how much slower than $f$ the local time of the process grows as a result of the conditioning. The methodology is based on a fine analysis of the subordinator given by the inverse local time of the Brownian motion. We describe the probability of general subordinator to stay above a given curve up to time $T$ via the solution of a general ordinary linear differential equation. For the specific case of the inverse local time of the Brownian motion, we explicitly and precisely compute the asymptotics of this probability for a large class of functions.

Let $G$ be a finitely generated group equipped with a finite symmetric generating set and the associated word length function $\vert \cdot\vert $. We study the behavior of the probability of return for random walks driven by symmetric measures $\mu$ that are such that $\sum\rho(\vert x\vert )\mu(x)<\infty$ for increasing regularly varying or slowly varying functions $\rho$, for instance, $s\mapsto(1+s)^{\alpha}$, $\alpha\in(0,2]$, or $s\mapsto(1+\log(1+s))^{\varepsilon}$, $\varepsilon>0$. For this purpose, we develop new relations between the isoperimetric profiles associated with different symmetric probability measures. These techniques allow us to obtain a sharp $L^{2}$-version of Erschler’s inequality concerning the Følner functions of wreath products. Examples and assorted applications are included.

Consider a discrete-time martingale $\{X_{t}\}$ taking values in a Hilbert space $\mathcal{H}$. We show that if for some $L\geq1$, the bounds $\mathbb{E}[\|X_{t+1}-X_{t}\|_{\mathcal{H}}^{2}\vert X_{t}]=1$ and $\|X_{t+1}-X_{t}\|_{\mathcal{H}}\leq L$ are satisfied for all times $t\geq0$, then there is a constant $c=c(L)$ such that for $1\leq R\leq\sqrt{t}$,

\[\mathbb{P}(\|X_{t}-X_{0}\|_{\mathcal{H}}\leq R)\leq c\frac{R}{\sqrt{t}}.\] Following Lee and Peres [Ann. Probab. 41 (2013) 3392–3419], this estimate has applications to small-ball estimates for random walks on vertex-transitive graphs: We show that for every infinite, connected, vertex-transitive graph $G$ with bounded degree, there is a constant $C_{G}>0$ such that if $\{Z_{t}\}$ is the simple random walk on $G$, then for every $\varepsilon>0$ and $t\geq1/\varepsilon^{2}$,

\[\mathbb{P}(\mathsf{dist}_{G}(Z_{t},Z_{0})\leq \varepsilon \sqrt{t})\leq C_{G}\varepsilon,\] where $\mathsf{dist}_{G}$ denotes the graph distance in $G$.