Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

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Strong Approximations in Probability and Statistics.

Let {W(t): t ∈ ∝} be two-sided Brownian motion, originating from zero, and let V(a) be defined by V(a)=sup}t ∈ ∝: W(t)−(t−a)2 is maximal}. Then {V(a): a ∈ ℝ} is a Markovian jump process, running through the locations of maxima of two-sided Brownian motion with respect to the parabolas fa(t)=(t−a)2. We give an analytic expression for the infinitesimal generators of the processes a ∈ ℝ, in terms of Airy functions in Theorem 4.1. This makes it possible to develop asymptotics for the global behavior of a large class of isotonic estimators (i.e. estimators derived under order restrictions). An example of this is given in Groeneboom (1985), where the asymptotic distribution of the (standardized) L1-distance between a decreasing density and the Grenander maximum likelihood estimator of this density is determined. On our way to Theorem 4.1 we derive some other results. For example, we give an analytic expression for the joint density of the maximum and the location of the maximum of the process {W(t)−ct2: t ∈ ℝ}, where c is an aribrary positive constant. We also determine the Laplace transform of the integral over a Brownian excursion. These last results also have recently been derived by several other authors, using a variety of methods.

Brownian motion with a parabolic drift and Airy functions

In this article we study a one dimensional model for a polymer in a poor sol-vent: the random walk on Z penalized by its range. More precisely, we consider a Gibbs transformation of the law of the simple symmetric random walk by a weight exp( − h n |R n | ) , with |R n | the number of visited sites and h n a size-dependent positive parameter. We use gambler’s ruin estimates to obtain exact asymptotics for the partition function, that enables us to obtain a precise description of trajectories, in particular scaling limits for the center and the amplitude of the range. A phase transition for the ﬂuctuations around an optimal amplitude is identiﬁed at h n ≍ n 1 / 4 , inherent to the underlying lattice structure.

Scaling limits for the random walk penalized by its range in dimension one

In this article we study a \emph{non-directed} polymer model in dimension $d\ge 2$: we consider a simple symmetric random walk on $\mathbb{Z}^d$ which interacts with a random environment, represented by i.i.d. random variables $(\omega_x)_{x\in \mathbb{Z}^d}$. The model consists in modifying the law of the random walk up to time (or length) $N$ by the exponential of $\sum_{x\in \mathcal{R}_N}\beta (\omega_x-h)$ where $\mathcal{R}_N$ is the range of the walk, \textit{i.e.} the set of visited sites up to time $N$, and $\beta\geq 0,\, h\in \mathbb{R}$ are two parameters. We study the behavior of the model in a weak-coupling regime, that is taking $\beta:=\beta_N$ vanishing as the length $N$ goes to infinity, and in the case where the random variables $\omega$ have a heavy tail with exponent $\alpha\in (0,d)$. We are able to obtain precisely the behavior of polymer trajectories under all possible weak-coupling regimes $\beta_N = \hat \beta N^{-\gamma}$ with $\gamma \geq 0$: we find the correct transversal fluctuation exponent $\xi$ for the polymer (it depends on $\alpha$ and $\gamma$) and we give the limiting distribution of the rescaled log-partition function. This extends existing works to the non-directed case and to higher dimensions.

Non-directed polymers in heavy-tail random environment in dimension $d\geq 2$

In this article we study a \emph{non-directed polymer model} on $\mathbb Z$, that is a one-dimensional simple random walk placed in a random environment. More precisely, the law of the random walk is modified by the exponential of the sum of ``rewards'' (or penalities) $\beta \omega_x -h$ sitting on the range of the random walk, where $(\omega_x)_{x\in \mathbb Z}$ are i.i.d.\ random variables (the disorder), and where $\beta\geq 0$ (disorder strength) and $h\in \mathbb{R}$ (external field) are two parameters. When $\beta=0,h>0$, this corresponds to a random walk penalized by its range; when $\beta>0, h=0$, this corresponds to the ``standard'' polymer model in random environment, except that it is non-directed. In this work, we allow the parameters $\beta,h$ to vary according to the length of the random walk, and we study in detail the competition between the \emph{stretching effect} of the disorder, the \emph{folding effect} of the external field (if $h\ge 0$), and the \emph{entropy cost} of atypical trajectories. We prove a complete description of the (rich) phase diagram. For instance, in the case $\beta>0, h=0$ of the non-directed polymer, if $\go_x$ ha a finite second moment, we find a transversal fluctuation exponent $\xi=2/3$, and we identify the limiting distribution of the rescaled log-partition function.

Ran Wei

Papers

One-dimensional polymers in random environments: stretching vs. folding

Non-directed polymers in heavy-tail random environment in dimension $d\geq 2$