We generalize Lyons’ rough paths theory in order to give a pathwise meaning to some nonlinear infinite-dimensional evolution equation associated to an analytic semigroup and driven by an irregular noise. As an illustration, we discuss a class of linear and nonlinear 1d SPDEs driven by a space–time Gaussian noise with singular space covariance and Brownian time dependence.

We consider decompositions of processes of the form Y=f(t, X_t) where X is a semimartingale. The function f is not required to be differentiable, so Itô’s lemma does not apply.

In the case where f(t, x) is independent of t, it is shown that requiring f to be locally Lipschitz continuous in x is enough for an Itô-style decomposition to exist. In particular, Y will be a Dirichlet process. We also look at the case where f(t, x) can depend on t, possibly discontinuously. It is shown, under some additional mild constraints on f, that the same decomposition still holds. Both these results follow as special cases of a more general decomposition which we prove, and which applies to nondifferentiable functions of Dirichlet processes.

Possible applications of these results to the theory of one-dimensional diffusions are briefly discussed.

We study the behavior of the two-dimensional Ising model in a finite box at temperatures that are below, but very close to, the critical temperature. In a regime where the temperature approaches the critical point and, simultaneously, the size of the box grows fast enough, we establish a large deviation principle that proves the appearance of a round Wulff crystal.

In this paper we determine the percolation threshold for an arbitrary sequence of dense graphs (G_n). Let λ_n be the largest eigenvalue of the adjacency matrix of G_n, and let G_n(p_n) be the random subgraph of G_n obtained by keeping each edge independently with probability p_n. We show that the appearance of a giant component in G_n(p_n) has a sharp threshold at p_n=1/λ_n. In fact, we prove much more: if (G_n) converges to an irreducible limit, then the density of the largest component of G_n(c/n) tends to the survival probability of a multi-type branching process defined in terms of this limit. Here the notions of convergence and limit are those of Borgs, Chayes, Lovász, Sós and Vesztergombi.

In addition to using basic properties of convergence, we make heavy use of the methods of Bollobás, Janson and Riordan, who used multi-type branching processes to study the emergence of a giant component in a very broad family of sparse inhomogeneous random graphs.

We consider the Ising model on a general tree under various boundary conditions: all plus, free and spin-glass. In each case, we determine when the root is influenced by the boundary values in the limit as the boundary recedes to infinity. We obtain exact capacity criteria that govern behavior at critical temperatures. For plus boundary conditions, an L³ capacity arises. In particular, on a spherically symmetric tree that has n^αbⁿ vertices at level n (up to bounded factors), we prove that there is a unique Gibbs measure for the ferromagnetic Ising model at the relevant critical temperature if and only if α≤1/2. Our proofs are based on a new link between nonlinear recursions on trees and L^p capacities.

Consider a Λ-coalescent that comes down from infinity (meaning that it starts from a configuration containing infinitely many blocks at time 0, yet it has a finite number N_t of blocks at any positive time t>0). We exhibit a deterministic function v:(0, ∞)→(0, ∞) such that N_t/v(t)→1, almost surely, and in L^p for any p≥1, as t→0. Our approach relies on a novel martingale technique.

We study a continuous time random walk X in an environment of i.i.d. random conductances μ_e∈[1, ∞). We obtain heat kernel bounds and prove a quenched invariance principle for X. This holds even when ${\mathbb{E}}\mu_{e}=\infty$.

We prove a sharp rate of convergence to stationarity for a natural random walk on a compact Riemannian manifold (M, g). The proof includes a detailed study of the spectral theory of the associated operator.

If X is a stable process of index α∈(0, 2) whose Lévy measure has density cx^−α−1 on (0, ∞), and S₁=sup_0<t≤1X_t, it is known that P(S₁>x)∽Aα⁻¹x^−α as x→∞ and P(S₁≤x)∽Bα⁻¹ρ⁻¹x^αρ as x↓0. [Here ρ=P(X₁>0) and A and B are known constants.] It is also known that S₁ has a continuous density, m say. The main point of this note is to show that m(x)∽Ax^−(α+1) as x→∞ and m(x)∽Bx^αρ−1 as x↓0. Similar results are obtained for related densities.

We consider a symmetric positive definite weakly exchangeable infinite random matrix and show that, under the technical condition that its elements take a finite number of values, the Ghirlanda–Guerra identities imply ultrametricity.

Splitting trees are those random trees where individuals give birth at a constant rate during a lifetime with general distribution, to i.i.d. copies of themselves. The width process of a splitting tree is then a binary, homogeneous Crump–Mode–Jagers (CMJ) process, and is not Markovian unless the lifetime distribution is exponential (or a Dirac mass at {∞}). Here, we allow the birth rate to be infinite, that is, pairs of birth times and life spans of newborns form a Poisson point process along the lifetime of their mother, with possibly infinite intensity measure.

A splitting tree is a random (so-called) chronological tree. Each element of a chronological tree is a (so-called) existence point (v, τ) of some individual v (vertex) in a discrete tree where τ is a nonnegative real number called chronological level (time). We introduce a total order on existence points, called linear order, and a mapping φ from the tree into the real line which preserves this order. The inverse of φ is called the exploration process, and the projection of this inverse on chronological levels the contour process.

For splitting trees truncated up to level τ, we prove that a thus defined contour process is a Lévy process reflected below τ and killed upon hitting 0. This allows one to derive properties of (i) splitting trees: conceptual proof of Le Gall–Le Jan’s theorem in the finite variation case, exceptional points, coalescent point process and age distribution; (ii) CMJ processes: one-dimensional marginals, conditionings, limit theorems and asymptotic numbers of individuals with infinite versus finite descendances.

Let $\{L_t^x; (x, t)∈R^1×R_+^1\}$ denote the local time of Brownian motion, and $$α_t:=∫_{−∞}^∞(L_t^x)^2 dx.$$ Let $η=N(0, 1)$ be independent of $α_t$. For each fixed $t$, $$\frac{\int_{-\infty}^{\infty}(L_{t}^{x+h}-L_{t}^{x})^{2}\,dx-4ht}{h^{3/2}}\stackrel{\mathcal{L}}{\rightarrow}\biggl(\frac{64}{3}\biggr)^{1/2}\sqrt{\alpha_{t}}\eta$$ as $h→0$. Equivalently, $$\frac{\int_{-\infty}^{\infty}(L^{x+1}_{t}-L^{x}_{t})^{2}\,dx-4t}{t^{3/4}}\stackrel{\mathcal{L}}{\rightarrow}\biggl(\frac{64}{3}\biggr)^{1/2}\sqrt{\alpha_{1}}\eta$$ as $t→∞$.

We correct an error in the paper identified in the title. The error affects the factor before the square root of entropy in Theorem 1; the same factor appears in Theorem 2.