Generalized bochner theorems and the spectrum of complete manifolds

Bochner's theorem that a compact Riemannian manifold with positive Ricci curvature has vanishing first cohomology group has various extensions to complete noncompact manifolds with Ricci possibly negative. One still has a vanishing theorem for L ² harmonic one-forms if the infimum of the spectrum of the Laplacian on functions is greater than minus the infimum of the Ricci curvature. This result and its analogues for p-forms yield vanishing results for certain infinite volume hyperbolic manifolds. This spectral condition also imposes topological restrictions on the ends of the manifold. More refined results are obtained by taking a certain Brownian motion average of the Ricci curvature; if this average is positive, one has a vanishing theorem for the first cohomology group with compact supports on the universal cover of a compact manifold. There are corresponding results for L ² harmonic spinors on spin manifolds.     

Sur la mesure de sortie du super mouvement brownien

Summary We study some properties of the exit measure of super Brownian motion from a smooth domainD inR ^d . In particular, we give precise estimates for the probability that the exit measure gives a positive mass to a small ball on the boundary. As an application, we compute the Hausdorff dimension of the support of the exit measure. In dimension 2, we prove that the exit measure is absolutely continuous with respect to the Lebesgue measure on the boundary. In connection with Dynkin's work, our results give some information on the behavior of solutions of u=u ² inD, and are related to the characterization of removable singularities at the boundary. As a consequence of our estimates, we give a sufficient condition for the uniqueness of the positive solution of u=u ² inD that tends to on an open subsetO of D and to 0 on the complement in D of the closure ofO. Our proofs use the path-valued process studied in [L2, L3].

     

Two renewal theorems for general random walks tending to infinity

Summary. Necessary and sufficient conditions for the existence of moments of the first passage time of a random walk S _n into [x, ∞) for fixed x≧ 0, and the last exit time of the walk from (−∞, x], are given under the condition that S _n →∞ a.s. The methods, which are quite different from those applied in the previously studied case of a positive mean for the increments of S _n , are further developed to obtain the “order of magnitude” as x→∞ of the moments of the first passage and last exit times, when these are finite. A number of other conditions of interest in renewal theory are also discussed, and some results for the first time for which the random walk remains above the level x on K consecutive occasions, which has applications in option pricing, are given. Received: 18 September 1995/In revised form: 28 February 1996     

Pathwise construction of additive H-transforms of super-Brownian motion

Summary The pathwise construction of additiveH-transforms of the super-Brownian motion is carried out as a modification of Le Gall's construction of superprocesses. It provides then the explicit conditioning of the super-Brownian motion on its exit behaviour at its Martin boundary, which yields an additiveH-transforms of the super-Brownian motion. The condition turns out to be that the space-time point of death of the super-Brownian motion converges in the Martin topology of the Brownian motion.Supported by an EC-Individual-Fellowship under Contract No. ERBCHBICT930682 and the SFB 256 of the University of Bonn, Germany     

Log-Sobolev inequality on non-convex Riemannian manifolds

Counterexamples are constructed to show that when the second fundamental form of the boundary is bounded below by a negative constant, any curvature lower bound is not enough to imply the log-Sobolev inequality. This indicates that in the study of functional inequalities on non-convex manifolds, the concavity of the boundary cannot be compensated by the positivity of the curvature. Next, when the boundary is merely concave on a bounded domain, a criterion on the log-Sobolev inequality known for convex manifolds is proved. Finally, when the concave part of the boundary is unbounded, a Sobolev inequality for a weighted volume measure is established, which implies an explicit sufficient condition for the log-Sobolev inequality to hold on non-convex manifolds.     

First exit time probability for multidimensional diffusions: A PDE-based approach

First exit time distributions for multidimensional processes are key quantities in many areas of risk management and option pricing. The aim of this paper is to provide a flexible, fast and accurate algorithm for computing the probability of the first exit time from a bounded domain for multidimensional diffusions. First, we show that the probability distribution of this stopping time is the unique (weak) solution of a parabolic initial and boundary value problem. Then, we describe the algorithm which is based on a combination of the sparse tensor product finite element spaces and an hp-discontinuous Galerkin method. We illustrate our approach with several examples. We also compare the numerical results to classical Monte Carlo methods.     

On the conditioned exit measures of super Brownian motion

In this paper we present a martingale related to the exit measures of super Brownian motion. By changing measure with this martingale in the canonical way we have a new process associated with the conditioned exit measure. This measure is shown to be identical to a measure generated by a non-homogeneous branching particle system with immigration of mass. An application is given to the problem of conditioning the exit measure to hit a number of specified points on the boundary of a domain. The results are similar in flavor to the “immortal particle” picture of conditioned super Brownian motion but more general, as the change of measure is given by a martingale which need not arise from a single harmonic function. Received: 27 August 1998 / Revised version: 8 January 1999     

The importance of strictly local martingales; applications to radial Ornstein–Uhlenbeck processes

For a wide class of local martingales (M _t ) there is a default function, which is not identically zero only when (M _t ) is strictly local, i.e. not a true martingale. This default in the martingale property allows us to characterize the integrability of functions of sup _s≤t M _s in terms of the integrability of the function itself. We describe some (paradoxical) mean-decreasing local sub-martingales, and the default functions for Bessel processes and radial Ornstein–Uhlenbeck processes in relation to their first hitting and last exit times. Received: 6 August 1996 / Revised version: 27 July 1998     

The structure of a Brownian bubble

Summary We examine local geometric properties of level sets of the Brownian sheet, and in particular, we identify the asymptotic distribution of the area of sets which correspond to excursions of the sheet high above a given level in the neighborhood of a particular random point. It is equal to the area of certain individual connected components of the random set {(s, t):B(t)>b(s)}, whereB is a standard Brownian motion andb is (essentially) a Bessel process of dimension 3. This limit distribution is studied and, in particular, explicit formulas are given for the probability that a point belongs to a specific connected component, and for the expected area of a component given the height of the excursion ofB(t)-b(s) in this component. These formulas are evaluated numerically and compared with the results from direct simulations ofB andb.The research of this author was partially supported by grants DMS-9103962 from the National Science Foundation and DAAL03-92-6-0323 from the Army Research Office     

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.     

Geography of the level sets of the Brownian sheet

Summary We describe geometric properties of {W>}, whereW is a standard real-valued Brownian sheet, in the neighborhood of the first hitP of the level set {W>} along a straight line or smooth monotone curveL. In such a neighborhood we use a decomposition of the formW(s, t)=–b(s)+B(t)+x(s, t), whereb(s) andB(t) are particular diffusion processes andx(s, t) is comparatively small, to show thatP is not on the boundary of any connected component of {W>}. Rather, components of this set form clusters nearP. An integral test for thorn-shaped neighborhoods ofL with tip atP that do not meet {W>} is given. We then analyse the position and size of clusters and individual connected components of {W>} near such a thorn, giving upper bounds on their height, width and the space between clusters. This provides a local picture of the level set. Our calculations are based on estimates of the length of excursions ofB andb and an accounting of the error termx.The research of this author was partially supported by NSF grant DMS-9103962, and, during the period of revision, by grant DAAL03-92-6-0323 from the Army Research Office     

Sharp estimates for intrinsic ultracontractivity on C 1,α-domains

Let be the first Dirichlet eigenfunction on a connected bounded C ^1,α-domain in and the corresponding Dirichlet heat kernel. It is proved that where λ₂ > λ₁ are the first two Dirichlet eigenvalues. This estimate is sharp for both short and long times. Bounded Lipschitz domains, elliptic operators on manifolds, and a general framework are also discussed. Supported in part by Creative Research Group Fund of the National Foundation of China (no. 10121101), the 973-Project in China and RFDP(20040027009).     

Extremal ranks and transformation of variables for extremes of functions of multivariate Gaussian processes

The exit rate from a ‘safe region’ plays an important role in dynamic reliability theory with multivariate random loads. For Gaussian processes the exit rate is simply calculated only for spherical or linear boundaries. However, many smooth boundaries, not of any of these types, are asymptotically spherical in variables of lower dimension, having a greater curvature in the remaining variables. As is shown in this paper, the asymptotic exit rate is then simply expressed as the exit rate from a sphere for a process of the lower dimensions, corrected by an explicit factor.The procedure circumvents the need to calculate complicated exit rate integrals for general boundaries, reducing the problem to a Gaussian probability integral for independent variables.A result of independent interest relates the tail distribution for a sum of a noncentral χ²-variable and a weighted sum of squares of noncentral normal variables, to the tail distribution of the χ²-variable only.     

Skorokhod embeddings, minimality and non-centred target distributions

In this paper we consider the Skorokhod embedding problem for target distributions with non-zero mean. In the zero-mean case, uniform integrability provides a natural restriction on the class of embeddings, but this is no longer suitable when the target distribution is not centred. Instead we restrict our class of stopping times to those which are minimal, and we find conditions on the stopping times which are equivalent to minimality. We then apply these results, firstly to the problem of embedding non-centred target distributions in Brownian motion, and secondly to embedding general target laws in a diffusion. We construct an embedding (which reduces to the Azema-Yor embedding in the zero-target mean case) which maximises the law of sup_s≤TB_s among the class of minimal embeddings of a general target distribution μ in Brownian motion. We then construct a minimal embedding of μ in a diffusion X which maximises the law of sup_s≤Th(X_s) for a general function h.     

A probabilistic approach to Martin boundaries for manifolds with ends

Summary We use martingale methods and coupling arguments to prove results of Li and Tam (1987) and Donnelly (1986) characterizing positive and bounded harmonic functions, respectively, on certain manifolds with ends.Research supported by a grant from NSA/NSF     

State space models on special manifolds

This paper concerns modeling time series observations in state space forms considered on the Stiefel and Grassmann manifolds. We develop a state space model relating the time series observations to a sequence of unobserved state or parameter matrices assuming the matrix Langevin noise processes on the Stiefel manifolds. We show a Bayes method for estimating the state matrices by the posterior modes. We consider a further extended state space model where two sequences of unobserved state matrices are involved. A simple state space model on the Grassmann manifolds with matrix Langevin noise processes is also investigated.     

Weak convergence to a Markov process: The martingale approach

Summary In this article, we obtain some sufficient conditions for weak convergence of a sequence of processes {X _n } toX, whenX arises as a solution to a well posed martingale problem. These conditions are tailored for application to the case when the state space for the processesX _n ,X is infinite dimensional. The usefulness of these conditions is illustrated by deriving Donsker's invariance principle for Hilbert space valued random variables. Also, continuous dependence of Hilbert space valued diffusions on diffusion and drift coefficients is proved.Research supported by National Board for Higher Mathematics, Bombay, IndiaPart of the work was done at University of California, Santa Barbara, USA     

Exit time moments, boundary value problems, and the geometry of domains in Euclidean space

Let X _t be a diffusion in Euclidean space. We initiate a study of the geometry of smoothly bounded domains in Euclidean space using the moments of the exit time for particles driven by X _t , as functionals on the space of smoothly bounded domains. We provide a characterization of critical points for each functional in terms of an overdetermined boundary value problem. For Brownian motion we prove that, for each functional, the boundary value problem which characterizes critical points admits solutions if and only if the critical point is a ball, and that all critical points are maxima. Received: 23 January 1997 / Revised version: 21 January 1998     

Diffusions on path and loop spaces: existence, finite dimensional approximation and Hölder continuity

Summary. We construct Ornstein–Uhlenbeck processes and more general diffusion processes on path and loop spaces of Riemannian manifolds by finite dimensional approximation. We also show H?lder continuity of the sample paths w.r.t. the supremum norm. The proofs are based on the Lyons–Zheng decomposition. Received: 6 September 1996 / In revised form: 1 April 1997