We combine Stein’s method with a version of Malliavin calculus on the Poisson space. As a result, we obtain explicit Berry–Esséen bounds in Central limit theorems (CLTs) involving multiple Wiener–Itô integrals with respect to a general Poisson measure. We provide several applications to CLTs related to Ornstein–Uhlenbeck Lévy processes.

Dawson and Perkins [Ann. Probab. 26 (1988) 1088–1138] constructed a stochastic model of an interacting two-type population indexed by a countable site space which locally undergoes a mutually catalytic branching mechanism. In Klenke and Mytnik [Preprint (2008), arXiv:0901.0623], it is shown that as the branching rate approaches infinity, the process converges to a process that is called the infinite rate mutually catalytic branching process (IMUB). It is most conveniently characterized as the solution of a certain martingale problem. While in the latter reference, a noise equation approach is used in order to construct a solution to this martingale problem, the aim of this paper is to provide a Trotter-type construction.

The construction presented here will be used in a forthcoming paper, Klenke and Mytnik [Preprint (2009)], to investigate the long-time behavior of IMUB (coexistence versus segregation of types).

This paper is partly based on the Ph.D. thesis of the second author (2008), where the Trotter approach was first introduced.

Given a bounded $\mathcal{C}^{2}$ domain G⊂ℝ^m, functions $g\in\mathcal{C}(\partial G,{\mathbb{R}})$ and $h\in\mathcal {C}(\overline{G},{\mathbb{R}}\setminus\{0\})$, let u denote the unique viscosity solution to the equation −2Δ_∞u=h in G with boundary data g. We provide a representation for u as the value of a two-player zero-sum stochastic differential game.

The solution of a parabolic stochastic partial differential equation (SPDE) driven by an infinite-dimensional Brownian motion is in general not a semi-martingale anymore and does in general not satisfy an Itô formula like the solution of a finite-dimensional stochastic ordinary differential equation (SODE). In particular, it is not possible to derive stochastic Taylor expansions as for the solution of a SODE using an iterated application of the Itô formula. Consequently, until recently, only low order numerical approximation results for such a SPDE have been available. Here, the fact that the solution of a SPDE driven by additive noise can be interpreted in the mild sense with integrals involving the exponential of the dominant linear operator in the SPDE provides an alternative approach for deriving stochastic Taylor expansions for the solution of such a SPDE. Essentially, the exponential factor has a mollifying effect and ensures that all integrals take values in the Hilbert space under consideration. The iteration of such integrals allows us to derive stochastic Taylor expansions of arbitrarily high order, which are robust in the sense that they also hold for other types of driving noise processes such as fractional Brownian motion. Combinatorial concepts of trees and woods provide a compact formulation of the Taylor expansions.

This paper has two agendas. Firstly, we exhibit new results for coverage processes. Let p(n, m, α) be the probability that n spherical caps of angular radius α in S^m do not cover the whole sphere S^m. We give an exact formula for p(n, m, α) in the case α∈[π/2, π] and an upper bound for p(n, m, α) in the case α∈[0, π/2] which tends to p(n, m, π/2) when α→π/2. In the case α∈[0, π/2] this yields upper bounds for the expected number of spherical caps of radius α that are needed to cover S^m.

Secondly, we study the condition number ${\mathscr{C}}(A)$ of the linear programming feasibility problem ∃x∈ℝ^m+1Ax≤0, x≠0 where A∈ℝ^n×(m+1) is randomly chosen according to the standard normal distribution. We exactly determine the distribution of ${\mathscr{C}}(A)$ conditioned to A being feasible and provide an upper bound on the distribution function in the infeasible case. Using these results, we show that $\mathbf{E}(\ln{\mathscr{C}}(A))\le2\ln(m+1)+3.31$ for all n>m, the sharpest bound for this expectancy as of today. Both agendas are related through a result which translates between coverage and condition.

In this article, we extend the theory of multiplicative chaos for positive definite functions in ℝ^d of the form f(x)=λ²ln⁺ R/|x|+g(x), where g is a continuous and bounded function. The construction is simpler and more general than the one defined by Kahane in [Ann. Sci. Math. Québec 9 (1985) 105–150]. As a main application, we provide a rigorous mathematical meaning to the Kolmogorov–Obukhov model of energy dissipation in a turbulent flow.

We generalize the asymptotic shape theorem in first passage percolation on ℤ^d to cover the case of general semimetrics. We prove a structure theorem for equivariant semimetrics on topological groups and an extended version of the maximal inequality for ℤ^d-cocycles of Boivin and Derriennic in the vector-valued case. This inequality will imply a very general form of Kingman’s subadditive ergodic theorem. For certain classes of generalized first passage percolation, we prove further structure theorems and provide rates of convergence for the asymptotic shape theorem. We also establish a general form of the multiplicative ergodic theorem of Karlsson and Ledrappier for cocycles with values in separable Banach spaces with the Radon–Nikodym property.

In this paper, we study metastability in large volumes at low temperatures. We consider both Ising spins subject to Glauber spin-flip dynamics and lattice gas particles subject to Kawasaki hopping dynamics. Let β denote the inverse temperature and let Λ_β⊂ℤ² be a square box with periodic boundary conditions such that lim_β→∞|Λ_β|=∞. We run the dynamics on Λ_β, starting from a random initial configuration where all of the droplets (clusters of plus-spins and clusters of particles, respectively) are small. For large β and for interaction parameters that correspond to the metastable regime, we investigate how the transition from the metastable state (with only small droplets) to the stable state (with one or more large droplets) takes place under the dynamics. This transition is triggered by the appearance of a single critical droplet somewhere in Λ_β. Using potential-theoretic methods, we compute the average nucleation time (the first time a critical droplet appears and starts growing) up to a multiplicative factor that tends to 1 as β→∞. It turns out that this time grows as Ke^Γβ/|Λ_β| for Glauber dynamics and as Kβe^Γβ/|Λ_β| for Kawasaki dynamics, where Γ is the local canonical (resp. grand-canonical) energy, to create a critical droplet and K is a constant reflecting the geometry of the critical droplet, provided these times tend to infinity (which puts a growth restriction on |Λ_β|). The fact that the average nucleation time is inversely proportional to |Λ_β| is referred to as homogeneous nucleation because it says that the critical droplet for the transition appears essentially independently in small boxes that partition Λ_β.

Consider n+m nonintersecting Brownian bridges, with n of them leaving from 0 at time t=−1 and returning to 0 at time t=1, while the m remaining ones (wanderers) go from m points a_i to m points b_i. First, we keep m fixed and we scale a_i, b_i appropriately with n. In the large-n limit, we obtain a new Airy process with wanderers, in the neighborhood of $\sqrt{2n}$, the approximate location of the rightmost particle in the absence of wanderers. This new process is governed by an Airy-type kernel, with a rational perturbation.

Letting the number m of wanderers tend to infinity as well, leads to two Pearcey processes about two cusps, a closing and an opening cusp, the location of the tips being related by an elliptic curve. Upon tuning the starting and target points, one can let the two tips of the cusps grow very close; this leads to a new process, which might be governed by a kernel, represented as a double integral involving the exponential of a quintic polynomial in the integration variables.

We consider a point process sequence induced by a stationary symmetric α-stable (0<α<2) discrete parameter random field. It is easy to prove, following the arguments in the one-dimensional case in [Stochastic Process. Appl. 114 (2004) 191–210], that if the random field is generated by a dissipative group action then the point process sequence converges weakly to a cluster Poisson process. For the conservative case, no general result is known even in the one-dimensional case. We look at a specific class of stable random fields generated by conservative actions whose effective dimensions can be computed using the structure theorem of finitely generated Abelian groups. The corresponding point processes sequence is not tight, and hence needs to be properly normalized in order to ensure weak convergence. This weak limit is computed using extreme value theory and some counting techniques.

We consider a class of backward stochastic differential equations (BSDEs) driven by Brownian motion and Poisson random measure, and subject to constraints on the jump component. We prove the existence and uniqueness of the minimal solution for the BSDEs by using a penalization approach. Moreover, we show that under mild conditions the minimal solutions to these constrained BSDEs can be characterized as the unique viscosity solution of quasi-variational inequalities (QVIs), which leads to a probabilistic representation for solutions to QVIs. Such a representation in particular gives a new stochastic formula for value functions of a class of impulse control problems. As a direct consequence, this suggests a numerical scheme for the solution of such QVIs via the simulation of the penalized BSDEs.

Following the recent work of Sznitman [Probab. Theory Related Fields 145 (2009) 143–174], we investigate the microscopic picture induced by a random walk trajectory on a cylinder of the form G_N×ℤ, where G_N is a large finite connected weighted graph, and relate it to the model of random interlacements on infinite transient weighted graphs. Under suitable assumptions, the set of points not visited by the random walk until a time of order |G_N|² in a neighborhood of a point with ℤ-component of order |G_N| converges in distribution to the law of the vacant set of a random interlacement on a certain limit model describing the structure of the graph in the neighborhood of the point. The level of the random interlacement depends on the local time of a Brownian motion. The result also describes the limit behavior of the joint distribution of the local pictures in the neighborhood of several distant points with possibly different limit models. As examples of G_N, we treat the d-dimensional box of side length N, the Sierpinski graph of depth N and the d-ary tree of depth N, where d≥2.

Let U⊆C be a bounded domain with smooth boundary and let F be an instance of the continuum Gaussian free field on U with respect to the Dirichlet inner product ∫_U∇f(x)⋅∇g(x) dx. The set T(a; U) of a-thick points of F consists of those z∈U such that the average of F on a disk of radius r centered at z has growth $\sqrt{a/\pi}\log\frac{1}{r}$ as r→0. We show that for each 0≤a≤2 the Hausdorff dimension of T(a; U) is almost surely 2−a, that ν_2−a(T(a; U))=∞ when 0<a≤2 and ν₂(T(0; U))=ν₂(U) almost surely, where ν_α is the Hausdorff-α measure, and that T(a; U) is almost surely empty when a>2. Furthermore, we prove that T(a; U) is invariant under conformal transformations in an appropriate sense. The notion of a thick point is connected to the Liouville quantum gravity measure with parameter γ given formally by $\Gamma(dz)=e^{\sqrt{2\pi}\gamma F(z)}\,dz$ considered by Duplantier and Sheffield.

Let (X_t, t≥0) be a continuous time simple random walk on ℤ^d (d≥3), and let l_T(x) be the time spent by (X_t, t≥0) on the site x up to time T. We prove a large deviations principle for the q-fold self-intersection local time I_T=∑_x∈ℤ^dl_T(x)^q in the critical case q=d/(d−2). When q is integer, we obtain similar results for the intersection local times of q independent simple random walks.