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The statistical description of the scalar conservation law of the form ρ t = H (ρ) x with H : R → R a smooth convex function has been an object of interest when the initial profile ρ (· , 0) is random. The special case when H (ρ) = ρ 2 2 (Burgers equation) has in particular received extensive interest in the past and is now understood for various random initial conditions. We prove in this paper a conjecture on the profile of the solution at any time t > 0 for a general class of Hamiltonians H and show that it is a stationary piecewise-smooth Feller process. Along the way, we study the excursion process of the two-sided linear Brownian motion W below any strictly convex function ϕ with superlinear growth and derive a generalized Chernoff distribution of the random variable argmax z ∈ R (W (z) - ϕ (z)) . Finally, when ρ (· , 0) is a white noise derived from an abrupt Lévy process, we show that the structure of shocks of the solution is a.s discrete at any fixed time t > 0 under some mild assumptions on H. [ABSTRACT FROM AUTHOR]

One major open conjecture in the area of critical random graphs, formulated by statistical physicists, and supported by a large amount of numerical evidence over the last decade (Braunstein et al. in Phys Rev Lett 91(16):168701, 2003; Wu et al. in Phys Rev Lett 96(14):148702, 2006; Braunstein et al. Int J Bifurc Chaos 17(07):2215-2255, 2007; Chen et al. in Phys Rev Lett 96(6):068702, 2006) is as follows: for a wide array of random graph models with degree exponent $$\tau \in (3,4)$$ , distances between typical points both within maximal components in the critical regime as well as on the minimal spanning tree on the giant component in the supercritical regime scale like $$n^{(\tau -3)/(\tau -1)}$$ . In this paper we study the metric space structure of maximal components of the multiplicative coalescent, in the regime where the sizes converge to excursions of Lévy processes 'without replacement' (Aldous and Limic Electron in J Probab 3(3):59, 1998), yielding a completely new class of limiting random metric spaces. A by-product of the analysis yields the continuum scaling limit of one fundamental class of random graph models with degree exponent $$\tau \in (3,4)$$ where edges are rescaled by $$n^{-(\tau -3)/(\tau -1)}$$ yielding the first rigorous proof of the above conjecture. The limits in this case are compact 'tree-like' random fractals with a dense collection of hubs (infinite degree vertices), a finite number of which are identified with leaves to form shortcuts. In a special case, we show that the Minkowski dimension of the limiting spaces equal $$(\tau -2)/(\tau -3)$$ a.s., in stark contrast to the Erdős-Rényi scaling limit whose Minkowski dimension is 2 a.s. It is generally believed that dynamic versions of a number of fundamental random graph models, as one moves from the barely subcritical to the critical regime can be approximated by the multiplicative coalescent. In work in progress, the general theory developed in this paper is used to prove analogous limit results for other random graph models with degree exponent $$\tau \in (3,4)$$ . Our proof makes crucial use of inhomogeneous continuum random trees (ICRT), which have previously arisen in the study of the entrance boundary of the additive coalescent. We show that tilted versions of the same objects using the associated mass measure, describe connectivity properties of the multiplicative coalescent. Since convergence of height processes of corresponding approximating $$\mathbf {p}$$ -trees is not known, we use general methodology in Athreya et al. (2014) and develop novel techniques relying on first showing convergence in the Gromov-weak topology and then extending this to Gromov-Hausdorff-Prokhorov convergence by proving a global lower mass-bound. [ABSTRACT FROM AUTHOR]

This paper extends recent results on ageing in mean field spin glasses on short time scales, obtained by Ben Arous and Gün (Commun Pure Appl Math 65:77–127, 2012 ) in law with respect to the environment, to results that hold almost surely, respectively in probability, with respect to the environment. It is based on the methods put forward in (Gayrard in Aging in reversible dynamics of disordered systems. II. Emergence of the arcsine law in the random hopping time dynamics of the REM, 2010 ; Electron J Probab 17(58): 1–33, 2012 ) and naturally complements (Bovier and Gayrard in Ann Probab, 2012 ). [ABSTRACT FROM AUTHOR]

A classical result, due to Lamperti, establishes a one-to-one correspondence between a class of strictly positive Markov processes that are self-similar, and the class of one-dimensional Lévy processes. This correspondence is obtained by suitably time-changing the exponential of the Lévy process. In this paper we generalise Lamperti's result to processes in n dimensions. For the representation we obtain, it is essential that the same time-change be applied to all coordinates of the processes involved. Also for the statement of the main result we need the proper concept of self-similarity in higher dimensions, referred to as multi-self-similarity in the paper. The special case where the Lévy process ξ is standard Brownian motion in n dimensions is studied in detail. There are also specific comments on the case where ξ is an n-dimensional compound Poisson process with drift. Finally, we present some results concerning moment sequences, obtained by studying the multi-self-similar processes that correspond to n-dimensional subordinators. [ABSTRACT FROM AUTHOR]

We give an explicit construction of the increasing tree-valued process introduced by Abraham and Delmas using a random point process of trees and a grafting procedure. This random point process will be used in companion papers to study record processes on Lévy trees. We use the Poissonian structure of the jumps of the increasing tree-valued process to describe its behavior at the first time the tree grows higher than a given height, using a spinal decomposition of the tree, similar to the classical Bismut and Williams decompositions. We also give the joint distribution of this exit time and the ascension time which corresponds to the first infinite jump of the tree-valued process. [ABSTRACT FROM AUTHOR]

We apply the Dirichlet form theory to stochastic differential equations with jumps as extension of Malliavin calculus reasoning. As in the continuous case, this weakens significantly the assumptions on the coefficients of the SDE. Thanks to the flexibility of the Dirichlet forms language, this approach brings also an important simplification which was neither available nor visible previously: an explicit formula giving the carré du champ matrix, i.e., the Malliavin matrix. Following this formula a new procedure appears, called the lent particle method which shortens the computations both theoretically and in concrete examples. In this paper which uses the construction done in Bouleau and Denis (J. Funct. Analysis 257:1144–1174, 2009) we restrict ourselves to the existence of densities; smoothness will be studied separately. [ABSTRACT FROM AUTHOR]

A recent result of Bertoin, Doney and Maller (Ann. Prob., 2007) gives an integral condition to characterize the class of Lévy processes X( t) for which lim sup $$_{t\downarrow 0}|X(t)|/\sqrt{t} \in (0,\infty)$$ occurs almost surely (a.s.). For such processes we have a kind of almost sure “iterated logarithm” result, but without the logs. In the present paper we prove a functional version of this result, which then opens the way to various interesting applications obtained via a continuous mapping theorem. We set these out in a rigorous framework, including a characterisation of the existence of an a.s. cluster set for the interpolated process, appropriate to the continuous time situation. The applications relate to functional laws for the supremum, reflected and a variety of other processes, including a class of stochastic differential equations, where we aim to give as informative a description as we can of the functional limit sets. [ABSTRACT FROM AUTHOR]

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We consider deterministic fast–slow dynamical systems on R m × Y of the form x k + 1 (n) = x k (n) + n - 1 a (x k (n) ) + n - 1 / α b (x k (n) ) v (y k) , y k + 1 = f (y k) , where α ∈ (1 , 2) . Under certain assumptions we prove convergence of the m-dimensional process X n (t) = x ⌊ n t ⌋ (n) to the solution of the stochastic differential equation d X = a (X) d t + b (X) ⋄ d L α , where L α is an α -stable Lévy process and ⋄ indicates that the stochastic integral is in the Marcus sense. In addition, we show that our assumptions are satisfied for intermittent maps f of Pomeau–Manneville type. [ABSTRACT FROM AUTHOR]

We consider Metropolis dynamics of the Random Energy Model. We prove that the classical two-time correlation function that allows one to establish aging converges almost surely to the arcsine law distribution function, as predicted in the physics literature, in the optimal domain of the time-scale and temperature parameters where this result can be expected to hold. To do this we link the two-time correlation function to a certain continuous-time clock process which, after proper rescaling, is proven to converge to a stable subordinator almost surely in the random environment and in the fine J 1 -topology of Skorohod. This fine topology then enables us to deduce from the arcsine law for stable subordinators the asymptotic behavior of the two-time correlation function that characterizes aging. [ABSTRACT FROM AUTHOR]

The purpose of the present work is twofold. First, we develop the theory of general self-similar growth-fragmentation processes by focusing on martingales which appear naturally in this setting and by recasting classical results for branching random walks in this framework. In particular, we establish many-to-one formulas for growth-fragmentations and define the notion of intrinsic area of a growth-fragmentation. Second, we identify a distinguished family of growth-fragmentations closely related to stable Lévy processes, which are then shown to arise as the scaling limit of the perimeter process in Markovian explorations of certain random planar maps with large degrees (which are, roughly speaking, the dual maps of the stable maps of Le Gall and Miermont in Ann Probab 39:1-69, 2011). As a consequence of this result, we are able to identify the law of the intrinsic area of these distinguished growth-fragmentations. This generalizes a geometric connection between large Boltzmann triangulations and a certain growth-fragmentation process, which was established in Bertoin et al. (Ann Probab, accepted). [ABSTRACT FROM AUTHOR]

We introduce flows of branching processes with competition, which describe the evolution of general continuous state branching populations in which interactions between individuals give rise to a negative density dependence term. This generalizes the logistic branching processes studied by Lambert (Ann Appl Probab 15(2):1506-1535, 2005). Following the approach developed by Dawson and Li (Ann Probab 40(2):813-857, 2012), we first construct such processes as the solutions of certain flow of stochastic differential equations. We then propose a novel genealogical description for branching processes with competition based on interactive pruning of Lévy-trees, and establish a Ray-Knight representation result for these processes in terms of the local times of suitably pruned forests. [ABSTRACT FROM AUTHOR]

We consider random walks and Lévy processes in a homogeneous group G. For all p>0, we completely characterise (almost) all G-valued Lévy processes whose sample paths have finite p-variation, and give sufficient conditions under which a sequence of G-valued random walks converges in law to a Lévy process in p-variation topology. In the case that G is the free nilpotent Lie group over Rd, so that processes of finite p-variation are identified with rough paths, we demonstrate applications of our results to weak convergence of stochastic flows and provide a Lévy-Khintchine formula for the characteristic function of the signature of a Lévy process. At the heart of our analysis is a criterion for tightness of p-variation for a collection of càdlàg strong Markov processes. [ABSTRACT FROM AUTHOR]

We establish uniform sub-exponential tail bounds for the width, height and maximal outdegree of critical Bienaymé-Galton-Watson trees conditioned on having a large fixed size, whose offspring distribution belongs to the domain of attraction of a stable law. This extends results obtained for the height and width by Addario-Berry, Devroye and Janson in the finite variance case. [ABSTRACT FROM AUTHOR]

We consider branching Brownian motion on the real line with the following selection mechanism: every time the number of particles exceeds a (large) given number N, only the N right-most particles are kept and the others killed. After rescaling time by $$\log ^3N$$ , we show that the properly recentred position of the $$\lceil \alpha N\rceil $$ -th particle from the right, $$\alpha \in (0,1)$$ , converges in law to an explicitly given spectrally positive Lévy process. This behaviour has been predicted to hold for a large class of models falling into the universality class of the FKPP equation with weak multiplicative noise [Brunet et al. in Phys Rev E 73(5):056126 2006] and is proven here for the first time for such a model. [ABSTRACT FROM AUTHOR]

Donsker-type functional limit theorems are proved for empirical processes arising from discretely sampled increments of a univariate Lévy process. In the asymptotic regime the sampling frequencies increase to infinity and the limiting object is a Gaussian process that can be obtained from the composition of a Brownian motion with a covariance operator determined by the Lévy measure. The results are applied to derive the asymptotic distribution of natural estimators for the distribution function of the Lévy jump measure. As an application we deduce Kolmogorov-Smirnov type tests and confidence bands. [ABSTRACT FROM AUTHOR]

We compute the persistence exponent of the integral of a stable Lévy process in terms of its self-similarity and positivity parameters. This solves a problem raised by Shi (Lower tails of some integrated processes. In: Small deviations and related topics (problem panel ). Along the way, we investigate the law of the stable process $$L$$ evaluated at the first time its integral $$X$$ hits zero, when the bivariate process $$(X,L)$$ starts from a coordinate axis. This extends classical formulæ by McKean (J Math Kyoto Univ 2:227-235, ) and Gor'kov (Soviet. Math. Dokl. 16:904-908, ) for integrated Brownian motion. [ABSTRACT FROM AUTHOR]

We give superharmonic functions and derive sharp bounds for the expected exit time and probability of survival for isotropic unimodal Lévy processes in smooth domains. [ABSTRACT FROM AUTHOR]

For $$\alpha > 0$$, the $$\alpha $$-Lipschitz minorant of a function $$f\!:\!\mathbb{R }\rightarrow \mathbb{R }$$ is the greatest function $$m : \mathbb{R }\rightarrow \mathbb{R }$$ such that $$m \le f$$ and $$|m(s)-m(t)| \le \alpha |s-t|$$ for all $$s,t \in \mathbb{R }$$, should such a function exist. If $$X=(X_t)_{t \in \mathbb{R }}$$ is a real-valued Lévy process that is not pure linear drift with slope $$\pm \alpha $$, then the sample paths of $$X$$ have an $$\alpha $$-Lipschitz minorant almost surely if and only if $$| \mathbb{E }[X_1] | < \alpha $$. Denoting the minorant by $$M$$, we investigate properties of the random closed set $$\fancyscript{Z} := \{ t \in \mathbb{R }: M_t = X_t \wedge X_{t-} \}$$, which, since it is regenerative and stationary, has the distribution of the closed range of some subordinator 'made stationary' in a suitable sense. We give conditions for the contact set $$\fancyscript{Z}$$ to be countable or to have zero Lebesgue measure, and we obtain formulas that characterize the Lévy measure of the associated subordinator. We study the limit of $$\fancyscript{Z}$$ as $$\alpha \rightarrow \infty $$ and find for the so-called abrupt Lévy processes introduced by Vigon that this limit is the set of local infima of $$X$$. When $$X$$ is a Brownian motion with drift $$\beta $$ such that $$|\beta | < \alpha $$, we calculate explicitly the densities of various random variables related to the minorant. [ABSTRACT FROM AUTHOR]

We prove a new rearrangement inequality for multiple integrals, which partly generalizes a result of Friedberg and Luttinger (Arch Ration Mech 61:35-44, ) and can be interpreted as involving symmetric rearrangements of domains around $$\infty $$. As applications, we prove two comparison results for general Lévy processes and their symmetric rearrangements. The first application concerns the survival probability of a point particle in a Poisson field of moving traps following independent Lévy motions. We show that the survival probability can only increase if the point particle does not move, and the traps and the Lévy motions are symmetrically rearranged. This essentially generalizes an isoperimetric inequality of Peres and Sousi (Geom Funct Anal 22(4):1000-1014, ) for the Wiener sausage. In the second application, we show that the $$q$$-capacity of a Borel measurable set for a Lévy process can only decrease if the set and the Lévy process are symmetrically rearranged. This result generalizes an inequality obtained by Watanabe (Z Wahrsch Verw Gebiete 63:487-499, ) for symmetric Lévy processes. [ABSTRACT FROM AUTHOR]

Let $$X$$ be a real valued Lévy process that is in the domain of attraction of a stable law without centering with norming function $$c.$$ As an analogue of the random walk results in Vatutin and Wachtel (Probab Theory Relat Fields 143(1–2):177–217, 2009 ) and Doney (Probab Theory Relat Fields 152(3–4):559–588, 2012 ), we study the local behaviour of the distribution of the lifetime $$\zeta $$ under the characteristic measure $$\underline{n}$$ of excursions away from $$0$$ of the process $$X$$ reflected in its past infimum, and of the first passage time of $$X$$ below $$0,$$ $$T_{0}=\inf \{t>0:X_{t}<0\},$$ under $$\mathbb{P }_{x}(\cdot ),$$ for $$x>0,$$ in two different regimes for $$x,$$ viz. $$x=o(c(\cdot ))$$ and $$x>D c(\cdot ),$$ for some $$D>0.$$ We sharpen our estimates by distinguishing between two types of path behaviour, viz. continuous passage at $$T_{0}$$ and discontinuous passage. In order to prove our main results we establish some sharp local estimates for the entrance law of the excursion process associated to $$X$$ reflected in its past infimum. [ABSTRACT FROM AUTHOR]

We consider a process given by a n-dimensional fractional Brownian motion with Hurst parameter $${\frac{1}{4} < H < \frac{1}{2}}$$, along with an associated Lévy area-like process, and prove the smoothness of the density for this process with respect to Lebesgue measure. [ABSTRACT FROM AUTHOR]

We study the pointwise regularity properties of the Lévy fields introduced by T. Mori; these fields are the most natural generalization of Lévy processes to the multivariate setting. We determine their spectrum of singularities, and we show that their Hölder singularity sets satisfy a large intersection property in the sense of K. Falconer. [ABSTRACT FROM AUTHOR]

Let X be a n-dimensional Ornstein-Uhlenbeck process, solution of the S.D.E. where A is a real n × n matrix and B a Lévy process without Gaussian part. We show that when A is non-singular, the law of X1 is absolutely continuous in if and only if the jumping measure of B fulfils a certain geometric condition with respect to A, which we call the exhaustion property. This optimal criterion is much weaker than for the background driving Lévy process B, which might be singular and sometimes even have a one-dimensional discrete jumping measure. This improves on a result by Priola and Zabczyk. [ABSTRACT FROM AUTHOR]

Scale functions play a central role in the fluctuation theory of spectrally negative Lévy processes and often appear in the context of martingale relations. These relations are often require excursion theory rather than Itô calculus. The reason for the latter is that standard Itô calculus is only applicable to functions with a sufficient degree of smoothness and knowledge of the precise degree of smoothness of scale functions is seemingly incomplete. The aim of this article is to offer new results concerning properties of scale functions in relation to the smoothness of the underlying Lévy measure. We place particular emphasis on spectrally negative Lévy processes with a Gaussian component and processes of bounded variation. An additional motivation is the very intimate relation of scale functions to renewal functions of subordinators. The results obtained for scale functions have direct implications offering new results concerning the smoothness of such renewal functions for which there seems to be very little existing literature on this topic. [ABSTRACT FROM AUTHOR]

We give a 'small time' functional version of Chung's 'other' law of the iterated logarithm for Lévy processes with non-vanishing Brownian component. This is an analogue of the 'other' law of the iterated logarithm at 'large times' for Lévy processes and random walks with finite variance, as extended to a functional version by Wichura. As one of many possible applications, we mention a functional law for a two-sided passage time process. [ABSTRACT FROM AUTHOR]

We wish to characterize when a Lévy process X t crosses boundaries b( t), in a two-sided sense, for small times t, where b( t) satisfies very mild conditions. An integral test is furnished for computing the value of sup t→0| X t|/ b( t) = c. In some cases, we also specify a function b( t) in terms of the Lévy triplet, such that sup t→0 | X t|/ b( t) = 1. [ABSTRACT FROM AUTHOR]

We completely describe the size and large intersection properties of the Hölder singularity sets of Lévy processes. We also study the set of times at which a given function cannot be a modulus of continuity of a Lévy process. The Hölder singularity sets of the sample paths of certain random wavelet series are investigated as well. [ABSTRACT FROM AUTHOR]

We prove a theorem on the Lebesgue measure of the range of additive Lévy Processes and then use this theorem to remove Condition (1.3) of Theorem 1.5 of Khoshnevisan et al. (Ann Probab 31:1097–1141, 2003). [ABSTRACT FROM AUTHOR]

We consider the height process of a Lévy process with no negative jumps, and its associated continuous tree representation. Using Lévy snake tools developed by Le Gall-Le Jan and Duquesne-Le Gall, with an underlying Poisson process, we construct a fragmentation process, which in the stable case corresponds to the self-similar fragmentation described by Miermont. For the general fragmentation process we compute a family of dislocation measures as well as the law of the size of a tagged fragment. We also give a special Markov property for the snake which is of its own interest. [ABSTRACT FROM AUTHOR]

Let X 1, ..., X N denote N independent, symmetric Lévy processes on R d . The corresponding additive Lévy process is defined as the following N-parameter random field on R d : $${\mathfrak{X}(t) :=X_1(t_1) + \cdots + X_N(t_N)\quad(t\in {\bf R}^N_+).}$$ Khoshnevisan and Xiao (Ann Probab 30(1):62–100, 2002) have found a necessary and sufficient condition for the zero-set $${\mathfrak{X}^{-1}(\{0\})}$$ of $${\mathfrak{X}}$$ to be non-trivial with positive probability. They also provide bounds for the Hausdorff dimension of $${\mathfrak{X}^{-1}(\{0\})}$$ which hold with positive probability in the case that $${\mathfrak{X}^{-1}(\{0\})}$$ can be non-void. Here we prove that the Hausdorff dimension of $${\mathfrak{X}^{-1}(\{0\})}$$ is a constant almost surely on the event $${\{\mathfrak{X}^{-1}(\{0\})\neq\varnothing\}}$$ . Moreover, we derive a formula for the said constant. This portion of our work extends the well known formulas of Horowitz (Israel J Math 6:176–182, 1968) and Hawkes (J Lond Math Soc 8:517–525, 1974) both of which hold for one-parameter Lévy processes. More generally, we prove that for every nonrandom Borel set F in (0,∞) N , the Hausdorff dimension of $${\mathfrak{X}^{-1}(\{0\})\cap F}$$ is a constant almost surely on the event $${\{\mathfrak{X}^{-1}(\{0\})\cap F\neq\varnothing\}}$$ . This constant is computed explicitly in many cases. [ABSTRACT FROM AUTHOR]

We find an analytical condition characterising when the probability that a Lévy Process leaves a symmetric interval upwards goes to one as the size of the interval is shrunk to zero. We show that this is also equivalent to the probability that the process is positive at time t going to one as t goes to zero and prove some related sequential results. For each α > 0 we find an analytical condition equivalent to $${X_{T_r} T_r^{-1/\alpha}\stackrel{p}{\longrightarrow} \infty}$$ and $${X_t t^{-1/\alpha}\stackrel {p} {\longrightarrow} \infty}$$ as $${r,t \to 0}$$ where X is a Lévy Process and T r the time it first leaves an interval of radius r [ABSTRACT FROM AUTHOR]

We consider last-passage percolation models in two dimensions, in which the underlying weight distribution has a heavy tail of index α < 2. We prove scaling laws and asymptotic distributions, both for the passage times and for the shape of optimal paths; these are expressed in terms of a family (indexed by α) of “continuous last-passage percolation” models in the unit square. In the extreme case α = 0 (corresponding to a distribution with slowly varying tail) the asymptotic distribution of the optimal path can be represented by a random self-similar measure on [0,1], whose multifractal spectrum we compute. By extending the continuous last-passage percolation model to $$\mathbb{R}^2$$ we obtain a heavy-tailed analogue of the Airy process, representing the limit of appropriately scaled vectors of passage times to different points in the plane. We give corresponding results for a directed percolation problem based on α-stable Lévy processes, and indicate extensions of the results to higher dimensions. [ABSTRACT FROM AUTHOR]

We investigate the random continuous trees called Lévy trees, which are obtained as scaling limits of discrete Galton-Watson trees. We give a mathematically precise definition of these random trees as random variables taking values in the set of equivalence classes of compact rooted R-trees, which is equipped with the Gromov-Hausdorff distance. To construct Lévy trees, we make use of the coding by the height process which was studied in detail in previous work. We then investigate various probabilistic properties of Lévy trees. In particular we establish a branching property analogous to the well-known property for Galton-Watson trees: Conditionally given the tree below levela, the subtrees originating from that level are distributed as the atoms of a Poisson point measure whose intensity involves a local time measure supported on the vertices at distanceafrom the root. We study regularity properties of local times in the space variable, and prove that the support of local time is the full level set, except for certain exceptional values ofacorresponding to local extinctions. We also compute several fractal dimensions of Lévy trees, including Hausdorff and packing dimensions, in terms of lower and upper indices for the branching mechanism function?which characterizes the distribution of the tree. We finally discuss some applications to super-Brownian motion with a general branching mechanism. [ABSTRACT FROM AUTHOR]

We show that stochastic processes with linear conditional expectations and quadratic conditional variances are Markov, and their transition probabilities are related to a three-parameter family of orthogonal polynomials which generalize the Meixner polynomials. Special cases of these processes are known to arise from the non-commutative generalizations of the Lévy processes. [ABSTRACT FROM AUTHOR]

We study a natural fragmentation process of the so-called stable tree introduced by Duquesne and Le Gall, which consists in removing the nodes of the tree according to a certain procedure that makes the fragmentation self-similar with positive index. Explicit formulas for the semigroup are given, and we provide asymptotic results. We also give an alternative construction of this fragmentation, using paths of Lévy processes, hence echoing the two alternative constructions of the standard additive coalescent by fragmenting the Brownian continuum random tree or using Brownian paths, respectively due to Aldous-Pitman and Bertoin. [ABSTRACT FROM AUTHOR]

If X is a symmetric Lévy process on the line, then there exists a non-decreasing, càdlàg process H such that X(H(x)) = x for all x >= 0 if and only if X is recurrent and has a non-trivial Gaussian component. The minimal such H is a subordinator K. The law of K is identified and shown to be the same as that of a linear time change of the inverse local time at 0 of X. When X is Brownian motion, K is just the usual ladder times process and this result extends the classical result of Lévy that the maximum process has the same law as the local time at 0. Write G[sub t] for last point in the range of K prior to t. In a parallel with classical fluctuation theory, the process Z := (X[sub t] - X[sub G[sub t] ] )[sub t] [sub >=0] is Markov with local time at 0 given by (X[sub G[sub t] ] )[sub t] [sub >=0] . The transition kernel and excursion measure of Z are identified. A similar programme is outlined for Lévy processes on the circle. This leads to the construction of a stopping time such that the stopped local times constitute a stationary process indexed by the circle. [ABSTRACT FROM AUTHOR]

Given (μ[sub t] )[sub t] [sub >=0] a transient convolution semigroup over R[sup d] , we characterize the transient semi-groups which are subordinate in the wide sense to the latter, via some quotient relation between the two Lévy generators. This is also equivalent to an inclusion between excessive cones. Thanks to a theorem of Hirsch, this allows us in big dimensions to give a probabilistic description of some measures satisfying the classical maximum principle. [ABSTRACT FROM AUTHOR]

Abstract. We show that the sample paths of most Levy processes are multifractal functions and we determine their spectrum of singularities. [ABSTRACT FROM AUTHOR]