We use the recently-developed multiparameter theory of additive Lévy processes to establish novel connections between an arbitrary Lévy process X in R^d, and a new class of energy forms and their corresponding capacities. We then apply these connections to solve two long-standing problems in the folklore of the theory of Lévy processes.

First, we compute the Hausdorff dimension of the image X(G) of a nonrandom linear Borel set G⊂R₊, where X is an arbitrary Lévy process in R^d. Our work completes the various earlier efforts of Taylor [Proc. Cambridge Phil. Soc. 49 (1953) 31–39], McKean [Duke Math. J. 22 (1955) 229–234], Blumenthal and Getoor [Illinois J. Math. 4 (1960) 370–375, J. Math. Mech. 10 (1961) 493–516], Millar [Z. Wahrsch. verw. Gebiete 17 (1971) 53–73], Pruitt [J. Math. Mech. 19 (1969) 371–378], Pruitt and Taylor [Z. Wahrsch. Verw. Gebiete 12 (1969) 267–289], Hawkes [Z. Wahrsch. verw. Gebiete 19 (1971) 90–102, J. London Math. Soc. (2) 17 (1978) 567–576, Probab. Theory Related Fields 112 (1998) 1–11], Hendricks [Ann. Math. Stat. 43 (1972) 690–694, Ann. Probab. 1 (1973) 849–853], Kahane [Publ. Math. Orsay (83-02) (1983) 74–105, Recent Progress in Fourier Analysis (1985b) 65–121], Becker-Kern, Meerschaert and Scheffler [Monatsh. Math. 14 (2003) 91–101] and Khoshnevisan, Xiao and Zhong [Ann. Probab. 31 (2003a) 1097–1141], where dim X(G) is computed under various conditions on G, X or both.

We next solve the following problem [Kahane (1983) Publ. Math. Orsay (83-02) 74–105]: When X is an isotropic stable process, what is a necessary and sufficient analytic condition on any two disjoint Borel sets F,G⊂R₊ such that with positive probability, X(F)∩X(G) is nonempty? Prior to this article, this was understood only in the case that X is a Brownian motion [Khoshnevisan (1999) Trans. Amer. Math. Soc. 351 2607–2622]. Here, we present a solution to Kahane’s problem for an arbitrary Lévy process X, provided the distribution of X(t) is mutually absolutely continuous with respect to the Lebesgue measure on R^d for all t>0.

As a third application of these methods, we compute the Hausdorff dimension and capacity of the preimage X⁻¹(F) of a nonrandom Borel set F⊂R^d under very mild conditions on the process X. This completes the work of Hawkes [Probab. Theory Related Fields 112 (1998) 1–11] that covers the special case where X is a subordinator.

We show that, for a stationary version of Hammersley’s process, with Poisson “sources” on the positive x-axis, and Poisson “sinks” on the positive y-axis, an isolated second-class particle, located at the origin at time zero, moves asymptotically, with probability 1, along the characteristic of a conservation equation for Hammersley’s process. This allows us to show that Hammersley’s process without sinks or sources, as defined by Aldous and Diaconis [Probab. Theory Related Fields 10 (1995) 199–213] converges locally in distribution to a Poisson process, a result first proved in Aldous and Diaconis (1995) by using the ergodic decomposition theorem and a construction of Hammersley’s process as a one-dimensional point process, developing as a function of (continuous) time on the whole real line. As a corollary we get the result that EL(t,t)/t converges to 2, as t→∞, where L(t,t) is the length of a longest North-East path from (0,0) to (t,t). The proofs of these facts need neither the ergodic decomposition theorem nor the subadditive ergodic theorem. We also prove a version of Burke’s theorem for the stationary process with sources and sinks and briefly discuss the relation of these results with the theory of longest increasing subsequences of random permutations.

We show that a sequence of stochastic spatial Lotka–Volterra models, suitably rescaled in space and time, converges weakly to super-Brownian motion with drift. The result includes both long range and nearest neighbor models, the latter for dimensions three and above. These theorems are special cases of a general convergence theorem for perturbations of the voter model.

Let B_t^H be a d-dimensional fractional Brownian motion with Hurst parameter H∈(0,1). Assume d≥2. We prove that the renormalized self-intersection local time $$\ell=\int_{0}^{T}\int_{0}^{t}\delta(B_{t}^{H}-B_{s}^{H})\,ds\,dt-\mathbb{E}\biggl(\int_{0}^{T}\int_{0}^{t}\delta (B_{t}^{H}-B_{s}^{H})\,ds\,dt\biggr)$$ exists in L² if and only if H<3/(2d), which generalizes the Varadhan renormalization theorem to any dimension and with any Hurst parameter. Motivated by a result of Yor, we show that in the case $3/4>H\geq\frac{3}{2d}$, r(ɛ)ℓ_ɛ converges in distribution to a normal law N(0,Tσ²), as ɛ tends to zero, where ℓ_ɛ is an approximation of ℓ, defined through (2), and r(ɛ)=|logɛ|⁻¹ if H=3/(2d), and r(ɛ)=ɛ^d−3/(2H) if 3/(2d)<H.

We study large deviations for the renormalized self-intersection local time of d-dimensional stable processes of index β∈(2d/3,d]. We find a difference between the upper and lower tail. In addition, we find that the behavior of the lower tail depends critically on whether β<d or β=d.

Let $S_1(n),…,S_p(n)$ be independent symmetric random walks in $\mathbb Z^d$. We establish moderate deviations and law of the iterated logarithm for the intersection of the ranges $$\#\{S_1[0,n]∩⋯∩S_p[0,n]\}$$ in the case $d=2, p≥2$ and the case $d=3, p=2.$

In this paper we give optimal constants in Talagrand’s concentration inequalities for maxima of empirical processes associated to independent and eventually nonidentically distributed random variables. Our approach is based on the entropy method introduced by Ledoux.

We study the influence of a multiplicative Gaussian noise, white in time and correlated in space, on the blow-up phenomenon in the supercritical nonlinear Schrödinger equation. We prove that any sufficiently regular and localized deterministic initial data gives rise to a solution which blows up in arbitrarily small time with a positive probability.

We study the Poisson equation Lu+f=0 in ℝ^d, where L is the infinitesimal generator of a diffusion process. In this paper, we allow the second-order part of the generator L to be degenerate, provided a local condition of Doeblin type is satisfied, so that, if we also assume a condition on the drift which implies recurrence, the diffusion process is ergodic. The equation is understood in a weak sense. Our results are then applied to diffusion approximation.

We construct a random matrix model for the bijection Ψ between clas- sical and free infinitely divisible distributions: for every d≥1, we associate in a quite natural way to each *-infinitely divisible distribution μ a distribution ℙ_d^μ on the space of d×d Hermitian matrices such that ℙ_d^μ*ℙ_d^ν=ℙ_d^μ*ν. The spectral distribution of a random matrix with distribution ℙ_d^μ converges in probability to Ψ(μ) when d tends to +∞. It gives, among other things, a new proof of the almost sure convergence of the spectral distribution of a matrix of the GUE and a projection model for the Marchenko–Pastur distribution. In an analogous way, for every d≥1, we associate to each *-infinitely divisible distribution μ, a distribution $\mathbb{L}_{d}^{\mu}$ on the space of complex (non-Hermitian) d×d random matrices. If μ is symmetric, the symmetrization of the spectral distribution of |M_d|, when M_d is $\mathbb{L}_{d}^{\mu}$-distributed, converges in probability to Ψ(μ).

We consider a totally monotone capacity on a Polish space and a sequence of bounded p.i.i.d. random variables. We show that, on a full set, any cluster point of empirical averages lies between the lower and the upper Choquet integrals of the random variables, provided either the random variables or the capacity are continuous.

In this paper we consider two ways to generalize the mathematical expectation of a random variable, the Choquet expectation and Peng’s g-expectation. An open question has been, after making suitable restrictions to the class of random variables acted on by the Choquet expectation, for what class of expectation do these two definitions coincide? In this paper we provide a necessary and sufficient condition which proves that the only expectation which lies in both classes is the traditional linear expectation. This settles another open question about whether Choquet expectation may be used to obtain Monte Carlo-like solution of nonlinear PDE: It cannot, except for some very special cases.

This article describes a purely analytic approach to urn models of the generalized or extended Pólya–Eggenberger type, in the case of two types of balls and constant “balance,” that is, constant row sum. The treatment starts from a quasilinear first-order partial differential equation associated with a combinatorial renormalization of the model and bases itself on elementary conformal mapping arguments coupled with singularity analysis techniques. Probabilistic consequences in the case of “subtractive” urns are new representations for the probability distribution of the urn’s composition at any time n, structural information on the shape of moments of all orders, estimates of the speed of convergence to the Gaussian limit and an explicit determination of the associated large deviation function. In the general case, analytic solutions involve Abelian integrals over the Fermat curve x^h+y^h=1. Several urn models, including a classical one associated with balanced trees (2–3 trees and fringe-balanced search trees) and related to a previous study of Panholzer and Prodinger, as well as all urns of balance 1 or 2 and a sporadic urn of balance 3, are shown to admit of explicit representations in terms of Weierstraß elliptic functions: these elliptic models appear precisely to correspond to regular tessellations of the Euclidean plane.