On Nummelin splitting for continuous time Harris recurrent Markov processes and application to kernel estimation for multi-dimensional diffusions

We introduce a sequence of stopping times that allow us to study an analogue of a life-cycle decomposition for a continuous time Markov process, which is an extension of the well-known splitting technique of Nummelin to the continuous time case. As a consequence, we are able to give deterministic equivalents of additive functionals of the process and to state a generalisation of Chen’s inequality. We apply our results to the problem of non-parametric kernel estimation of the drift of multi-dimensional recurrent, but not necessarily ergodic, diffusion processes.     

Potential analysis for positive recurrent Markov chains with asymptotically zero drift: Power-type asymptotics

We consider a positive recurrent Markov chain on R⁺${\mathbb{R}}^{+}$ with asymptotically zero drift which behaves like −_c1/x$-c_1/x$ at infinity; this model was first considered by Lamperti. We are interested in tail asymptotics for the stationary measure. Our analysis is based on construction of a harmonic function which turns out to be regularly varying at infinity. This harmonic function allows us to perform non-exponential change of measure. Under this new measure Markov chain is transient with drift like _c2/x$c_2/x$ at infinity and we compute the asymptotics for its Green function. Applying further the inverse transform of measure we deduce a power-like asymptotic behaviour of the stationary tail distribution. Such a heavy-tailed stationary measure happens even if the jumps of the chain are bounded. This model provides an example where possibly bounded input distributions produce non-exponential output.     

On growth,recurrence and the Choquet-Deny Theorem for <Emphasis Type="Italic">p</Emphasis>-adic Lie groups

We first study the growth properties of p-adic Lie groups and its connection with p-adic Lie groups of type R and prove that a non-type R p-adic Lie group has compact neighbourhoods of identity having exponential growth. This is applied to prove the growth dichotomy for a large class of p-adic Lie groups which includes p-adic algebraic groups. We next study p-adic Lie groups that admit recurrent random walks and prove the natural growth conjecture connecting growth and the existence of recurrent random walks, precisely we show that a p-adic Lie group admits a recurrent random walk if and only if it has polynomial growth of degree at most two. We prove this conjecture for some other classes of groups also. We also prove the Choquet-Deny Theorem for compactly generated p-adic Lie groups of polynomial growth and also show that polynomial growth is necessary and sufficient for the validity of the Choquet-Deny for all spread-out probabilities on Zariski-connected p-adic algebraic groups. Counter example is also given to show that certain assumptions made in the main results can not be relaxed.     

On the existence of smooth densities for jump processes

Summary We consider a Lévy processX _t and the solutionY _t of a stochastic differential equation driven byX _t; we suppose thatX _t has infinitely many small jumps, but its Lévy measure may be very singular (for instance it may have a countable support). We obtain sufficient conditions ensuring the existence of a smooth density forY _t: these conditions are similar to those of the classical Malliavin calculus for continuous diffusions. More generally, we study the smoothness of the law of variablesF defined on a Poisson probability space; the basic tool is a duality formula from which we estimate the characteristic function ofF.     

Valleys and the Maximum Local Time for Random Walk in Random Environment

Let ξ (n, x) be the local time at x for a recurrent one-dimensional random walk in random environment after n steps, and consider the maximum ξ^*(n) = max _x ξ(n, x). It is known that lim sup is a positive constant a.s. We prove that lim inf is a positive constant a.s. this answers a question of P. Révész [5]. The proof is based on an analysis of the valleys in the environment, defined as the potential wells of record depth. In particular, we show that almost surely, at any time n large enough, the random walker has spent almost all of its lifetime in the two deepest valleys of the environment it has encountered. We also prove a uniform exponential tail bound for the ratio of the expected total occupation time of a valley and the expected local time at its bottom.     

The law of iterated logarithm for additive functionals and martingale additive functionals of Harris recurrent Markov processes

We use Nummelin splitting in continuous time in order to prove laws of iterated logarithm for additive functionals of a Harris recurrent Markov process, with deterministic or random renormalization.     

A nontrivial solution of mountain-pass type for a hemivariational inequality

We consider a class of noncoercive hemivariational inequalities involving the p-Laplacian. Our goal is to obtain the existence of a nontrivial solution. Using the mountain-pass theorem for locally Lipschitz functionals we obtain the desired result.     

Extreme value theory for stochastic integrals of Legendre polynomials

We study in this paper the extremal behavior of stochastic integrals of Legendre polynomial transforms with respect to Brownian motion. As the main results, we obtain the exact tail behavior of the supremum of these integrals taken over intervals [0,h] with h>0 fixed, and the limiting distribution of the supremum on intervals [0,T] as T→∞. We show further how this limit distribution is connected to the asymptotic of the maximally selected quasi-likelihood procedure that is used to detect changes at an unknown time in polynomial regression models. In an application to global near-surface temperatures, we demonstrate that the limit results presented in this paper perform well for real data sets.     

Special conformal groups of a Riemannian manifold and Lie point symmetries of the nonlinear Poisson equation

We obtain a complete group classification of the Lie point symmetries of nonlinear Poisson equations on generic (pseudo) Riemannian manifolds M. Using this result we study their Noether symmetries and establish the respective conservation laws. It is shown that the projection of the Lie point symmetries on M are special subgroups of the conformal group of M. In particular, if the scalar curvature of M vanishes, the projection on M of the Lie point symmetry group of the Poisson equation with critical nonlinearity is the conformal group of the manifold. We illustrate our results by applying them to the Thurston geometries.     

Comparison of interacting diffusions and an application to their ergodic theory

Summary A general comparison argument for expectations of certain multitime functionals of infinite systems of linearly interacting diffusions differing in the diffusion coefficient is derived. As an application we prove clustering occurs in the case when the symmetrized interaction kernel is recurrent, and the components take values in an interval bounded on one side. The technique also gives an alternative proof of clustering in the case of compact intervals.     

Exit times for a class of piecewise exponential Markov processes with two-sided jumps

We consider first passage times for piecewise exponential Markov processes that may be viewed as Ornstein–Uhlenbeck processes driven by compound Poisson processes. We allow for two-sided jumps and as a main result we derive the joint Laplace transform of the first passage time of a lower level and the resulting undershoot when passage happens as a consequence of a downward (negative) jump. The Laplace transform is determined using complex contour integrals and we illustrate how the choice of contours depends in a crucial manner on the particular form of the negative jump part, which is allowed to belong to a dense class of probabilities. We give extensions of the main result to two-sided exit problems where the negative jumps are as before but now it is also required that the positive jumps have a distribution of the same type. Further, extensions are given for the case where the driving Lévy process is the sum of a compound Poisson process and an independent Brownian motion. Examples are used to illustrate the theoretical results and include the numerical evaluation of some concrete exit probabilities. Also, some of the examples show that for specific values of the model parameters it is possible to obtain closed form expressions for the Laplace transform, as is the case when residue calculus may be used for evaluating the relevant contour integrals.     

On strong solutions for positive definite jump diffusions

We show the existence of unique global strong solutions of a class of stochastic differential equations on the cone of symmetric positive definite matrices. Our result includes affine diffusion processes and therefore extends considerably the known statements concerning Wishart processes, which have recently been extensively employed in financial mathematics.Moreover, we consider stochastic differential equations where the diffusion coefficient is given by the αth positive semidefinite power of the process itself with 0.5<α<1 and obtain existence conditions for them. In the case of a diffusion coefficient which is linear in the process we likewise get a positive definite analogue of the univariate GARCH diffusions.     

Tail estimates for exponential functionals and applications to SDEs

In this paper, based on techniques of Malliavin calculus, we obtain an explicit bound for tail probabilities of a general class of exponential functionals. We apply the obtained results to derive asymptotic behaviors for the tail of the exponential functional of stochastic differential equations.     

Large deviations for increasing sequences on the plane

We prove a large deviation principle with explicit rate functions for the length of the longest increasing sequence among Poisson points on the plane. The rate function for lower tail deviations is derived from a 1977 result of Logan and Shepp about Young diagrams of random permutations. For the upper tail we use a coupling with Hammersley's particle process and convex-analytic techniques. Along the way we obtain the rate function for the lower tail of a tagged particle in a totally asymmetric Hammersley's process. Received: 22 July 1997 / Revised version: 23 March 1998     

The forest associated with the record process on a Lévy tree

We perform a pruning procedure on a Lévy tree and instead of throwing away the removed sub-tree, we regraft it on a given branch (not related to the Lévy tree). We prove that the tree constructed by regrafting is distributed as the original Lévy tree, generalizing a result of Addario-Berry, Broutin and Holmgren where only Aldous’s tree is considered. As a consequence, we obtain that the “average pruning time” of a leaf is distributed as the height of a leaf picked at random in the Lévy tree.     

Continuous dependence results for non-linear Neumann type boundary value problems

We obtain estimates on the continuous dependence on the coefficient for second-order non-linear degenerate Neumann type boundary value problems. Our results extend previous work of Cockburn et al., Jakobsen and Karlsen, and Gripenberg to problems with more general boundary conditions and domains. A new feature here is that we account for the dependence on the boundary conditions. As one application of our continuous dependence results, we derive for the first time the rate of convergence for the vanishing viscosity method for such problems. We also derive new explicit continuous dependence on the coefficients results for problems involving Bellman-Isaacs equations and certain quasilinear equation.     

Brownian motion on a homogeneous random fractal

Summary We introduce a simple random fractal based on the Sierpinski gasket and construct a Brownian motion upon the fractal. The properties of the process on the Sierpinski gasket are modified by the random environment. A sample path construction of the process via time truncation is used, which is a direct construction of the process on the fractal from the associated Dirichlet forms. We obtain estimates on the resolvent and transition density for the process and hence a value for the spectral dimension which satisfiesd _s=2d _f/d_w. A branching process in a random environment can be used to deduce some of the sample path properties of the process.     

Approximations of non-smooth integral type functionals of one dimensional diffusion processes

In this article, we obtain the weak and strong rates of convergence of time integrals of non-smooth functions of a one dimensional diffusion process. We propose the use of the exact simulation scheme to simulate the process at discretization points. In particular, we also present the rates of convergence for the weak and strong errors of approximation for the local time of a one dimensional diffusion process as an application of our method.     

The problem of the most visited site in random environment

We prove that the process of the most visited site of Sinai's simple random walk in random environment is transient. The rate of escape is characterized via an integral criterion. Our method also applies to a class of recurrent diffusion processes with random potentials. It is interesting to note that the corresponding problem for the usual symmetric Bernoulli walk or for Brownian motion remains open. Received: 17 April 1998