Motivated by several models introduced in the physics literature to study the nonequilibrium coarsening dynamics of one-dimensional systems, we consider a large class of “hierarchical coalescence processes” (HCP). An HCP consists of an infinite sequence of coalescence processes $\{\xi^{(n)}(\cdot)\}_{n\ge1}$: each process occurs in a different “epoch” (indexed by $n$) and evolves for an infinite time, while the evolution in subsequent epochs are linked in such a way that the initial distribution of $\xi^{(n+1)}$ coincides with the final distribution of $\xi^{(n)}$. Inside each epoch the process, described by a suitable simple point process representing the boundaries between adjacent intervals (domains), evolves as follows. Only intervals whose length belongs to a certain epoch-dependent finite range are active, that is, they can incorporate their left or right neighboring interval with quite general rates. Inactive intervals cannot incorporate their neighbors and can increase their length only if they are incorporated by active neighbors. The activity ranges are such that after a merging step the newly produced interval always becomes inactive for that epoch but active for some future epoch.

Without making any mean-field assumption we show that: (i) if the initial distribution describes a renewal process, then such a property is preserved at all later times and all future epochs; (ii) the distribution of certain rescaled variables, for example, the domain length, has a well-defined and universal limiting behavior as $n\to\infty$ independent of the details of the process (merging rates, activity ranges, …). This last result explains the universality in the limiting behavior of several very different physical systems (e.g., the East model of glassy dynamics or the Paste-all model) which was observed in several simulations and analyzed in many physics papers. The main idea to obtain the asymptotic result is to first write down a recursive set of nonlinear identities for the Laplace transforms of the relevant quantities on different epochs and then to solve it by means of a transformation which in some sense linearizes the system.

Let $B_{s}$ be a $d$-dimensional Brownian motion and $\omega(dx)$ be an independent Poisson field on $\mathbb{R}^{d}$. The almost sure asymptotics for the logarithmic moment generating function

\[\log\mathbb{E}_{0}\exp\left\{\pm\theta\int_{0}^{t}\overline{V}(B_{s})\,ds\right\}\qquad (t\to\infty)\]

are investigated in connection with the renormalized Poisson potential of the form

\[\overline{V}(x)=\int_{\mathbb{R}^{d}}{\frac{1}{\vert y-x\vert^{p}}}[\omega(dy)-dy],\qquad x\in\mathbb{R}^{d}.\]

The investigation is motivated by some practical problems arising from the models of Brownian motion in random media and from the parabolic Anderson models.

This paper examines the existence of the self-intersection local time for a superprocess over a stochastic flow in dimensions $d\leq3$, which through constructive methods, results in a Tanaka-like representation. The superprocess over a stochastic flow is a superprocess with dependent spatial motion, and thus Dynkin’s proof of existence, which requires multiplicity of the log-Laplace functional, no longer applies. Skoulakis and Adler’s method of calculating moments is extended to higher moments, from which existence follows.

For an ergodic Markov chain $\{X(t)\}$ on $\mathbb{N}$, with a stationary distribution $\pi$, let $T_{n}>0$ denote a hitting time for $[n]^{c}$, and let $X_{n}=X(T_{n})$. Around 2005 Guy Louchard popularized a conjecture that, for $n\to\infty$, $T_{n}$ is almost $\operatorname{Geometric}(p)$, $p=\pi([n]^{c})$, $X_{n}$ is almost stationarily distributed on $[n]^{c}$ and that $X_{n}$ and $T_{n}$ are almost independent, if $p(n):=\sup_{i}p(i,[n]^{c})\to0$ exponentially fast. For the chains with $p(n)\to0$, however slowly, and with $\sup_{i,j}\|p(i,\cdot)-p(j,\cdot)\|_{\mathrm{TV}}<1$, we show that Louchard’s conjecture is indeed true, even for the hits of an arbitrary $S_{n}\subset\mathbb{N}$ with $\pi(S_{n})\to0$. More precisely, a sequence of $k$ consecutive hit locations paired with the time elapsed since a previous hit (for the first hit, since the starting moment) is approximated, within a total variation distance of order $k\sup_{i}p(i,S_{n})$, by a $k$-long sequence of independent copies of $(\ell_{n},t_{n})$, where $t_{n}=\operatorname{Geometric}(\pi(S_{n}))$, $\ell_{n}$ is distributed stationarily on $S_{n}$ and $\ell_{n}$ is independent of $t_{n}$. The two conditions are easily met by the Markov chains that arose in Louchard’s studies as likely sharp approximations of two random compositions of a large integer $\nu$, a column-convex animal (cca) composition and a Carlitz (C) composition. We show that this approximation is indeed very sharp for each of the random compositions, read from left to right, for as long as the sum of the remaining parts stays above $\ln^{2}\nu$. Combining the two approximations, a composition—by its chain, and, for $S_{n}=[n]^{c}$, the sequence of hit locations paired each with a time elapsed from the previous hit—by the independent copies of $(\ell_{n},t_{n})$, enables us to determine the limiting distributions of $\mu=o(\ln\nu)$ and $\mu=o(\nu^{1/2})$ largest parts of the random cca-composition and the random C-composition, respectively. (Submitted to Annals of Probability in June 2009.)

We prove that a normalized sequence of multiple Wigner integrals (in a fixed order of free Wigner chaos) converges in law to the standard semicircular distribution if and only if the corresponding sequence of fourth moments converges to 2, the fourth moment of the semicircular law. This extends to the free probabilistic, setting some recent results by Nualart and Peccati on characterizations of central limit theorems in a fixed order of Gaussian Wiener chaos. Our proof is combinatorial, analyzing the relevant noncrossing partitions that control the moments of the integrals. We can also use these techniques to distinguish the first order of chaos from all others in terms of distributions; we then use tools from the free Malliavin calculus to give quantitative bounds on a distance between different orders of chaos. When applied to highly symmetric kernels, our results yield a new transfer principle, connecting central limit theorems in free Wigner chaos to those in Gaussian Wiener chaos. We use this to prove a new free version of an important classical theorem, the Breuer–Major theorem.

We offer a unified approach to the theory of convex minorants of Lévy processes with continuous distributions. New results include simple explicit constructions of the convex minorant of a Lévy process on both finite and infinite time intervals, and of a Poisson point process of excursions above the convex minorant up to an independent exponential time. The Poisson–Dirichlet distribution of parameter 1 is shown to be the universal law of ranked lengths of excursions of a Lévy process with continuous distributions above its convex minorant on the interval $[0,1]$.

We consider a class of stochastic PDEs of Burgers type in spatial dimension $1$, driven by space–time white noise. Even though it is well known that these equations are well posed, it turns out that if one performs a spatial discretization of the nonlinearity in the “wrong” way, then the sequence of approximate equations does converge to a limit, but this limit exhibits an additional correction term.

This correction term is proportional to the local quadratic cross-variation (in space) of the gradient of the conserved quantity with the solution itself. This can be understood as a consequence of the fact that for any fixed time, the law of the solution is locally equivalent to Wiener measure, where space plays the role of time. In this sense, the correction term is similar to the usual Itô–Stratonovich correction term that arises when one considers different temporal discretizations of stochastic ODEs.

Backward stochastic differential equations (BSDEs) in the sense of Pardoux–Peng [Lecture Notes in Control and Inform. Sci. 176 (1992) 200–217] provide a non-Markovian extension to certain classes of nonlinear partial differential equations; the nonlinearity is expressed in the so-called driver of the BSDE. Our aim is to deal with drivers which have very little regularity in time. To this end, we establish continuity of BSDE solutions with respect to rough path metrics in the sense of Lyons [Rev. Mat. Iberoam. 14 (1998) 215–310] and so obtain a notion of “BSDE with rough driver.” Existence, uniqueness and a version of Lyons’ limit theorem in this context are established. Our main tool, aside from rough path analysis, is the stability theory for quadratic BSDEs due to Kobylanski [Ann. Probab. 28 (2000) 558–602].

In this paper, we introduce a definition of BV functions in a Gelfand triple which is an extension of the definition of BV functions in [Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 21 (2010) 405–414] by using Dirichlet form theory. By this definition, we can consider the stochastic reflection problem associated with a self-adjoint operator $A$ and a cylindrical Wiener process on a convex set $\Gamma$ in a Hilbert space $H$. We prove the existence and uniqueness of a strong solution of this problem when $\Gamma$ is a regular convex set. The result is also extended to the nonsymmetric case. Finally, we extend our results to the case when $\Gamma=K_{\alpha}$, where $K_{\alpha}=\{f\in L^{2}(0,1)|f\geq-\alpha\}$, $\alpha\geq0$.

For a large class of absolutely continuous probabilities $P$ it is shown that, for $r>0$, for $n$-optimal $L^{r}(P)$-codebooks $\alpha_{n}$, and any Voronoi partition $V_{n,a}$ with respect to $\alpha_{n}$ the local probabilities $P(V_{n,a})$ satisfy $P(V_{a,n})\approx n^{-1}$ while the local $L^{r}$-quantization errors satisfy $\int_{V_{n,a}}\|x-a\|^{r}\,dP(x)\approx n^{-(1+r/d)}$ as long as the partition sets $V_{n,a}$ intersect a fixed compact set $K$ in the interior of the support of $P$.

We derive a decoupling formula for the Brownian sheet which has the following ready consequence: An $N$-parameter Brownian sheet in $\mathbf{R}^{d}$ has double points if and only if $d<4N$. In particular, in the critical case where $d=4N$, the Brownian sheet does not have double points. This answers an old problem in the folklore of the subject. We also discuss some of the geometric consequences of the mentioned decoupling, and establish a partial result concerning $k$-multiple points in the critical case $k(d-2N)=d$.