Reflecting Brownian motions: Quasimartingales and strong Caccioppoli sets

A notion ofstrong Caccioppoli set is defined for bounded Euclidean domains. It is shown that stationary (normally) reflecting Brownian motion on the closure of a bounded Euclidean domain is a quasimartingale on each compact time interval if and only if the domain is a strong Caccioppoli set. A similar result is shown to hold for symmetric reflecting diffusion processes.Research supported in part by NSF Grant DMS 91-01675.Research supported in part by NSF Grants DMS 86-57483 and 90-23335.     

Smoothness for the collision local time of two multidimensional bifractional Brownian motions

Let ${B^{{H_i},{K_i}}} = \{ B_t^{{H_i},{K_i}},t \ge 0\} ,{\rm{ }}i = 1,2$ be two independent, d-dimensional bifractional Brownian motions with respective indices H _i ∈ (0, 1) and K _i ∈ (0, 1]. Assume d ? 2. One of the main motivations of this paper is to investigate smoothness of the collision local time $${l_T} = \int_0^T {\delta (B_s^{{H_1},{K_1}} - B_s^{{H_2},{K_2}}} )ds,{\rm{ }}T > 0,$$ , where δ denotes the Dirac delta function. By an elementary method we show that l _T is smooth in the sense of Meyer-Watanabe if and only if min{H ₁ K ₁,H ₂ K ₂} < 1/(d + 2).     

Large deviations for the radial processes of the Brownian motions on hyperbolic spaces

Using the heat kernel estimates by Davies (1989) and Anker et al. (1996), we show large deviations for the radial processes of the Brownian motions on hyperbolic spaces.     

Perturbed Brownian motions

Summary. We study `perturbed Brownian motions', that can be, loosely speaking, described as follows: they behave exactly as linear Brownian motion except when they hit their past maximum or/and maximum where they get an extra `push'. We define with no restrictions on the perturbation parameters a process which has this property and show that its law is unique within a certain `natural class' of processes. In the case where both perturbations (at the maximum and at the minimum) are self-repelling, we show that in fact, more is true: Such a process can almost surely be constructed from Brownian paths by a one-to-one measurable transformation. This generalizes some results of Carmona-Petit-Yor and Davis. We also derive some fine properties of perturbed Brownian motions (Hausdorff dimension of points of monotonicity for example). Received: 17 May 1996 / In revised form: 21 January 1997     

On optimal stopping of multidimensional diffusions

This paper develops an approach for solving perpetual discounted optimal stopping problems for multidimensional diffusions, with special emphasis on the $d$-dimensional Wiener process. We first obtain some verification theorems for diffusions, based on the Green kernel representation of the value function. Specializing to the multidimensional Wiener process, we apply the Martin boundary theory to obtain a set of tractable integral equations involving only harmonic functions that characterize the stopping region of the problem in the bounded case. The approach is illustrated through the optimal stopping problem of a $d$-dimensional Wiener process with a positive definite quadratic form reward function.     

Two-state free Brownian motions

In a two-state free probability space (A,φ,ψ), we define an algebraic two-state free Brownian motion to be a process with two-state freely independent increments whose two-state free cumulant generating function Rφ,ψ(z) is quadratic. Note that a priori, the distribution of the process with respect to the second state ψ is arbitrary. We show, however, that if A is a von Neumann algebra, the states φ, ψ are normal, and φ is faithful, then there is only a one-parameter family of such processes. Moreover, with the exception of the actual free Brownian motion (corresponding to φ=ψ), these processes only exist for finite time.     

Intersections of Brownian motions

This article presents a survey of the theory of the intersections of Brownian motion paths. Among other things, we present a truly elementary proof of a classical theorem of A. Dvoretzky, P. Erdős and S. Kakutani. This proof is motivated by old ideas of P. Lévy that were originally used to investigate the curve of planar Brownian motion.     

Horo-tight spheres in hyperbolic space

We study horo-tight immersions of manifolds into hyperbolic spaces. The main result gives several characterizations of horo-tightness of spheres, answering a question proposed by Cecil and Ryan. For instance, we prove that a sphere is horo-tight if and only if it is tight in the hyperbolic sense. For codimension bigger than one, it follows that horo-tight spheres in hyperbolic space are metric spheres. We also prove that horo-tight hyperspheres are characterized by the property that both of its total absolute horospherical curvatures attend their minimum value. We also introduce the notion of weak horo-tightness: an immersion is weak horo-tight if only one of its total absolute curvature attends its minimum. We prove a characterization theorem for weak horo-tight hyperspheres.     

Reflecting Brownian motions and a deletion result for Sobolev spaces of order (1, 2)

Let D be an open set in ^d and E be a relatively closed subset of D having zero Lebesgue measure. A necessary and sufficient integral condition is given for the Sobolev spaces W ^1,2 (D) and W ^1,2(D\E) to be the same. The latter is equivalent to (normally) reflecting Brownian motion (RBM) on being indistinguishable (in distribution) from RBM on . This integral condition is satisfied, for example, when E has zero (d–1)-dimensional Hausdorff measure. Therefore it is possible to delete from D a relatively closed subset E having positive capacity but nevertheless the RBM on is indistinguishable from the RBM on , or equivalently, W ^1,2(D\E)=W^1,2(D). An example of such kind is: D=² and E is the Cantor set. In the proof of above mentioned results, a detailed study of RBMs on general open sets is given. In particular, a semimartingale decomposition and approximation result previously proved in [3] for RBMs on bounded open sets is extended to the case of unbounded open sets.Research supported in part by NSF Grant DMS 86-57483.     

Hyperbolic and fractional hyperbolic Brownian motion

We examine the hyperbolic, planar Brownian motion and its time-fractional version. The analogy between the hyperbolic Brownian motion and Brownian motion on the sphere is also analysed. We examine in detail the connection between the equations governing the distributions in the Cartesian and hyperbolic coordinates. We discuss the time-fractional generalization of hyperbolic Brownian motion and give a representation of it as composition of classical hyperbolic Brownian motion with a reflecting Brownian motion on the line.     

Concentration for multidimensional diffusions and their boundary local times

We prove that probability laws of certain multidimensional semimartingales which includes time-inhomogenous diffusions, under suitable assumptions, satisfy quadratic transportation cost inequality under the uniform metric. From this we derive concentration properties of Lipschitz functions of process paths that depend on the entire history. In particular, we estimate concentration of boundary local time of reflected Brownian motions on a polyhedral domain. We work out explicit applications of consequences of measure concentration for the case of Brownian motion with rank-based drifts.     

Pseudo Jordan domains and reflecting Brownian motions

Summary The manifold metric between two points in a planar domain is the minimum of the lengths of piecewiseC ¹ curves in the domain connecting these two points. We define a bounded simply connected planar region to be a pseudo Jordan domain if its boundary under the manifold metric is topologically homeomorphic to the unit circle. It is shown that reflecting Brownian motionX on a pseudo Jordan domain can be constructed starting at all points except those in a boundary subset of capacity zero.X has the expected Skorokhod decomposition under a condition which is satisfied when G has finite 1-dimensional lower Minkowski content.     

Rates of convergence of diffusions with drifted Brownian potentials

We are interested in the asymptotic behaviour of a diffusion process with drifted Brownian potential. The model is a continuous time analogue to the random walk in random environment studied in the classical paper of Kesten, Kozlov, and Spitzer. We not only recover the convergence of the diffusion process which was previously established by Kawazu and Tanaka, but also obtain all the possible convergence rates. An interesting feature of our approach is that it shows a clear relationship between drifted Brownian potentials and Bessel processes.

     

Uniqueness and stability in multidimensional hyperbolic inverse problems

Under a weak regularity assumption, we prove the uniqueness in multidimensional hyperbolic inverse problems with a single measurement. Moreover we show that our uniqueness results yield the best possible Lipschitz stability in L²-space in the inverse problems by means of the exact observability inequality.     

Markov-modulated Brownian motions perturbed by catastrophes

ABSTRACT

We consider a one-sided Markov-modulated Brownian motion perturbed by catastrophes that occur at some rates depending on the modulating process. When a catastrophe occurs, the level drops to zero for a random recovery period. Then the process evolves normally until the next catastrophe. We use a semi-regenerative approach to obtain the stationary distribution of this perturbed MMBM. Next, we determine the stationary distribution of two extensions: we consider the case of a temporary change of regime after each recovery period and the case where the catastrophes can only happen above a fixed threshold. We provide some simple numerical illustrations.     

Boundary trace of reflecting Brownian motions

We establish a uniform dimensional result for normally reflected Brownian motion (RBM) in a large class of non-smooth domains. Hausdorff dimensions for the boundary occupation time and the boundary trace of RBM are determined. Extensions to stable-like jump processes and to symmetric reflecting diffusions are also given.Mathematics Subject Classification (2000):Primary 60G17, 60J60, Secondary 28A80, 30C35, 60G52, 60J50