We give an explicit construction of the weak local limit of a class of preferential attachment graphs. This limit contains all local information and allows several computations that are otherwise hard, for example, joint degree distributions and, more generally, the limiting distribution of subgraphs in balls of any given radius $k$ around a random vertex in the preferential attachment graph. We also establish the finite-volume corrections which give the approach to the limit.

A family of continuous-state branching processes with immigration are constructed as the solution flow of a stochastic equation system driven by time–space noises. The family can be regarded as an inhomogeneous increasing path-valued branching process with immigration. Two nonlocal branching immigration superprocesses can be defined from the flow. We identify explicitly the branching and immigration mechanisms of those processes. The results provide new perspectives into the tree-valued Markov processes of Aldous and Pitman [Ann. Inst. Henri Poincaré Probab. Stat. 34 (1998) 637–686] and Abraham and Delmas [Ann. Probab. 40 (2012) 1167–1211].

We establish the local existence of pathwise solutions for the stochastic Euler equations in a three-dimensional bounded domain with slip boundary conditions and a suitable nonlinear multiplicative noise. In the two-dimensional case we obtain the global existence of these solutions with additive or linear-multiplicative noise. Finally, we show that, in the three-dimensional case, the addition of linear multiplicative noise provides a regularizing effect; the global existence of solutions occurs with high probability if the initial data is sufficiently small, or if the noise coefficient is sufficiently large.

Let $A_{n}$ be an $n$ by $n$ random matrix whose entries are independent real random variables with mean zero, variance one and with subexponential tail. We show that the logarithm of $|\det A_{n}|$ satisfies a central limit theorem. More precisely,

\begin{eqnarray*}&&\sup_{x\in{\mathbf {R}}}\biggl|{\mathbf{P} }\biggl(\frac{\log(|\det A_{n}|)-({1}/{2})\log(n-1)!}{\sqrt{({1}/{2})\log n}}\le x\biggr)-{\mathbf{P} }\bigl(\mathbf{N} (0,1)\le x\bigr)\biggr|\\&&\qquad\le\log^{-{1}/{3}+o(1)}n.\end{eqnarray*}

We prove a central limit theorem for an additive functional of the $d$-dimensional fractional Brownian motion with Hurst index $H\in(\frac{1}{1+d},\frac{1}{d})$, using the method of moments, extending the result by Papanicolaou, Stroock and Varadhan in the case of the standard Brownian motion.

In this paper we propose a notion of viscosity solutions for path dependent semi-linear parabolic PDEs. This can also be viewed as viscosity solutions of non-Markovian backward SDEs, and thus extends the well-known nonlinear Feynman–Kac formula to non-Markovian case. We shall prove the existence, uniqueness, stability and comparison principle for the viscosity solutions. The key ingredient of our approach is a functional Itô calculus recently introduced by Dupire [Functional Itô calculus (2009) Preprint].

We consider measure-valued processes $X=(X_{t})$ that solve the following martingale problem: for a given initial measure $X_{0}$, and for all smooth, compactly supported test functions $\varphi$,

\begin{eqnarray*}X_{t}(\varphi )&=&X_{0}(\varphi)+\frac{1}{2}\int_{0}^{t}X_{s}(\Delta\varphi )\,ds+\theta \int_{0}^{t}X_{s}(\varphi )\,ds\\&&{}-\int_{0}^{t}X_{s}(L_{s}\varphi )\,ds+M_{t}(\varphi ).\end{eqnarray*}

Here $L_{s}(x)$ is the local time density process associated with $X$, and $M_{t}(\varphi )$ is a martingale with quadratic variation $[M(\varphi )]_{t}=\int_{0}^{t}X_{s}(\varphi ^{2})\,ds$. Such processes arise as scaling limits of SIR epidemic models. We show that there exist critical values $\theta_{c}(d)\in(0,\infty)$ for dimensions $d=2,3$ such that if $\theta>\theta_{c}(d)$, then the solution survives forever with positive probability, but if $\theta<\theta_{c}(d)$, then the solution dies out in finite time with probability 1. For $d=1$ we prove that the solution dies out almost surely for all values of $\theta$. We also show that in dimensions $d=2,3$ the process dies out locally almost surely for any value of $\theta$; that is, for any compact set $K$, the process $X_{t}(K)=0$ eventually.

A noise is a kind of homomorphism from a Boolean algebra of domains to the lattice of $\sigma$-fields. Leaving aside the homomorphism we examine its image, a Boolean algebra of $\sigma$-fields. The largest extension of such Boolean algebra of $\sigma$-fields, being well-defined always, is a complete Boolean algebra if and only if the noise is classical, which answers an old question of J. Feldman.

Let $X,X_{1},X_{2},\ldots$ be i.i.d. ${\mathbb{R}}^{d}$-valued real random vectors. Assume that ${\mathbf{E}X=0}$, $\operatorname{cov} X=\mathbb{C}$, $\mathbf{E}\Vert X\Vert^{2}=\sigma ^{2}$ and that $X$ is not concentrated in a proper subspace of $\mathbb{R}^{d}$. Let $G$ be a mean zero Gaussian random vector with the same covariance operator as that of $X$. We study the distributions of nondegenerate quadratic forms $\mathbb{Q}[S_{N}]$ of the normalized sums ${S_{N}=N^{-1/2}(X_{1}+\cdots+X_{N})}$ and show that, without any additional conditions,

\[\Delta_{N}\stackrel{\mathrm{def}}{=}\sup_{x}\bigl|\mathbf{P}\bigl\{\mathbb{Q}[S_{N}]\leq x\bigr\}-\mathbf{P}\bigl\{\mathbb{Q}[G]\leq x\bigr\}\bigr|={\mathcal{O}}\bigl(N^{-1}\bigr),\]

provided that $d\geq5$ and the fourth moment of $X$ exists. Furthermore, we provide explicit bounds of order ${\mathcal{O}}(N^{-1})$ for $\Delta_{N}$ for the rate of approximation by short asymptotic expansions and for the concentration functions of the random variables $\mathbb{Q}[S_{N}+a]$, $a\in{\mathbb{R}}^{d}$. The order of the bound is optimal. It extends previous results of Bentkus and Götze [Probab. Theory Related Fields 109 (1997a) 367–416] (for ${d\ge9}$) to the case $d\ge5$, which is the smallest possible dimension for such a bound. Moreover, we show that, in the finite dimensional case and for isometric $\mathbb{Q}$, the implied constant in ${\mathcal{O}}(N^{-1})$ has the form $c_{d}\sigma ^{d}(\det\mathbb{C})^{-1/2}\mathbf{E} \|\mathbb{C}^{-1/2}X\|^{4}$ with some $c_{d}$ depending on $d$ only. This answers a long standing question about optimal rates in the central limit theorem for quadratic forms starting with a seminal paper by Esséen [Acta Math. 77 (1945) 1–125].

We consider two first passage problems for stable processes, not necessarily symmetric, in one dimension. We make use of a novel method of path censoring in order to deduce explicit formulas for hitting probabilities, hitting distributions and a killed potential measure. To do this, we describe in full detail the Wiener–Hopf factorization of a new Lamperti-stable-type Lévy process obtained via the Lamperti transform, in the style of recent work in this area.