Brownian motion and random walk perturbed at extrema

Let b _t be Brownian motion. We show there is a unique adapted process x _t which satisfies dx _t = db _t except when x _t is at a maximum or a minimum, when it receives a push, the magnitudes and directions of the pushes being the parameters of the process. For some ranges of the parameters this is already known. We show that if a random walk close to b _t is perturbed properly, its paths are close to those of x _t . Received: 15 October 1997 / Revised version: 18 May 1998     

Hyperbolic branching Brownian motion

Summary. Hyperbolic branching Brownian motion is a branching diffusion process in which individual particles follow independent Brownian paths in the hyperbolic plane ? ² , and undergo binary fission(s) at rate λ > 0. It is shown that there is a phase transition in λ: For λ≦ 1/8 the number of particles in any compact region of ? ² is eventually 0, w.p.1, but for λ > 1/8 the number of particles in any open set grows to ∞ w.p.1. In the subcritical case (λ≦ 1/8) the set Λ of all limit points in ∂? ² (the boundary circle at ∞) of particle trails is a Cantor set, while in the supercritical case (λ > 1/8) the set Λ has full Lebesgue measure. For λ≦ 1/8 it is shown that w.p.1 the Hausdorff dimension of Λ is δ = (1−√1−8 λ)/2. Received: 2 November 1995 / In revised form: 22 October 1996     

Brownian motion in cones

Summary. We study the asymptotic behavior of Brownian motion and its conditioned process in cones using an infinite series representation of its transition density. A concise probabilistic interpretation of this series in terms of the skew product decomposition of Brownian motion is derived and used to show properties of the transition density. Received: 2 April 1996 / In revised form: 21 December 1996     

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     

Scaling identity for crossing Brownian motion in a Poissonian potential

We consider d-dimensional Brownian motion in a truncated Poissonian potential (d≥ 2). If Brownian motion starts at the origin and ends in the closed ball with center y and radius 1, then the transverse fluctuation of the path is expected to be of order |y|^ξ, whereas the distance fluctuation is of order |y|^χ. Physics literature tells us that ξ and χ should satisfy a scaling identity 2ξ− 1 = χ. We give here rigorous results for this conjecture. Received: 31 December 1997 / Revised version: 14 April 1998     

Increment sizes of the principal value of Brownian local time

Let W be a standard Brownian motion, and define Y(t)= ∫₀ ^t ds/W(s) as Cauchy's principal value related to local time. We determine: (a) the modulus of continuity of Y in the sense of P. Lévy; (b) the large increments of Y. Received: 1 April 1999 / Revised version: 27 September 1999 / Published online: 14 June 2000     

Intersection of independent regenerative sets

We investigate the nature of the intersection of two independent regenerative sets. The approach combines Bochners subordination and potential theory for a pair of Markov processes in duality. Received: 21 November 1997 / Revised version: 31 August 1998     

Cone paths for the planar Brownian snake

Summary. For the Brownian path-valued process of Le Gall (or Brownian snake) in , the times at which the process is a cone path are considered as a function of the size of the cone and the terminal position of the path. The results show that the paths for the path-valued process have local properties unlike those of a standard Brownian motion. Received: 29 January 1996 / In revised form: 21 June 1996     

Limiting angle of Brownian motion on certain manifolds

Summary. Suppose that M is a complete, simply connected Riemannian manifold of non-positive sectional curvature with dimension m ≧ 3. If, outside a fixed compact set, the sectional curvatures are bounded above by a negative constant multiple of the inverse of the square of the geodesic distance from a fixed point and below by another negative constant multiple of the square of the geodesic distance, then the angular part of Brownian motion on M tends to a limit as time tends to infinity, and the closure of the support of the distribution of this limit is the entire S ^m−1 . This improves a result of Hsu and March. Received: 7 December 1994/In revised form: 2 September 1995     

Wiener’s test for super-Brownian motion and the Brownian snake

We prove a Wiener-type criterion for super-Brownian motion and the Brownian snake.If F is a Borel subset of ℝ ^d and x ∈ ℝ ^d , we provide a necessary and sufficientcondition for super-Brownian motion started at δ _x to immediately hit the set F. Equivalently, this condition is necessary and sufficient for the hitting time of F by theBrownian snake with initial point x to be 0. A key ingredient of the proof isan estimate showing that the hitting probability of F is comparable, up to multiplicative constants,to the relevant capacity of F. This estimate, which is of independent interest, refines previous results due to Perkins and Dynkin. An important role is played by additivefunctionals of the Brownian snake, which are investigated here via the potentialtheory of symmetric Markov processes. As a direct application of our probabilisticresults, we obtain a necessary and sufficient condition for the existence in a domain D of a positivesolution of the equation Δ; u = u ² which explodes at a given point of ∂ D. Received: 5 January 1996 / In revised form: 30 October 1996     

Existence, uniqueness and qualitative properties of minima to radially symmetric non-coercive non-convex variational problems

We are concerned with the problem of existence, uniqueness and qualitative properties of solutions to the radially symmetric variational problem where is the ball of centered at the origin and with radius , the map is a normal integrand, and is a convex function of the second variable. This kind of problems, with non-convex lagrangians with respect to , arise in various fields of applied sciences, such as optimal design and nonlinear elasticity. Received June 18, 1998; in final form August 26, 1999 / Published online September 14, 2000     

Brownian motion, harmonic functions and hyperbolicity for Euclidean complexes

A Euclidean complex X is a simplicial complex whose simplices are (flat) Euclidean simplices. We construct a natural Brownian motion on X and show that if X has nonpositive curvature and satisfies Gromov's hyperbolicity condition, then, with probability one, Brownian motion tends to a random limit on the Gromov boundary. Applying a combination of geometric and probabilistic techniques we describe spaces of harmonic functions on X. Received November 18, 1999; in final form January 18, 2000 / Published online April 12, 2001     

Some properties of the range of super-Brownian motion

We consider a super-Brownian motion X. Its canonical measures can be studied through the path-valued process called the Brownian snake. We obtain the limiting behavior of the volume of the ɛ-neighborhood for the range of the Brownian snake, and as a consequence we derive the analogous result for the range of super-Brownian motion and for the support of the integrated super-Brownian excursion. Then we prove the support of X _t is capacity-equivalent to [0, 1]² in ℝ^d, d≥ 3, and the range of X, as well as the support of the integrated super-Brownian excursion are capacity-equivalent to [0, 1]⁴ in ℝ^d, d≥ 5. Received: 7 April 1998 / Revised version: 2 October 1998     

Exact capacity results for stable processes

Exact results are proved for the capacity of pullbacks of analytic sets by stable processes. Received: 25 May 1988 / Revised version: 15 September 1997     

Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously

. Consider site or bond percolation with retention parameter p on an infinite Cayley graph. In response to questions raised by Grimmett and Newman (1990) and Benjamini and Schramm (1996), we show that the property of having (almost surely) a unique infinite open cluster is increasing in p. Moreover, in the standard coupling of the percolation models for all parameters, a.s. for all p ₂>p ₁>p _c , each infinite p ₂-cluster contains an infinite p ₁-cluster; this yields an extension of Alexander's (1995) “simultaneous uniqueness” theorem. As a corollary, we obtain that the probability θ _v (p) that a given vertex v belongs to an infinite cluster is depends continuously on p throughout the supercritical phase p>p _c . All our results extend to quasi-transitive infinite graphs with a unimodular automorphism group. Received: 22 December 1997 / Revised version: 1 July 1998     

Brownian bridge on hyperbolic spaces and on homogeneous trees

Let B be the Brownian motion on a noncompact non Euclidean rank one symmetric space H. A typical examples is an hyperbolic space H _n , n > 2. For ν > 0, the Brownian bridge B ^(ν) of length ν on H is the process B _t , 0 ≤t≤ν, conditioned by B ₀ = B _ν = o, where o is an origin in H. It is proved that the process converges weakly to the Brownian excursion when ν→ + ∞ (the Brownian excursion is the radial part of the Brownian Bridge on ℝ³). The same result holds for the simple random walk on an homogeneous tree. Received: 4 December 1998 / Revised version: 22 January 1999     

On the discrepancy principle for iterative and parametric methods to solve linear ill-posed equations

Summary. For the numerical solution of (non-necessarily well-posed) linear equations in Banach spaces we consider a class of iterative methods which contains well-known methods like the Richardson iteration, if the associated resolvent operator fulfils a condition with respect to a sector. It is the purpose of this paper to show that for given noisy right-hand side the discrepancy principle (being a stopping rule for the iteration methods belonging to the mentioned class) defines a regularization method, and convergence rates are proved under additional smoothness conditions on the initial error. This extends similar results obtained for positive semidefinite problems in Hilbert spaces. Then we consider a class of parametric methods which under the same resolvent condition contains the method of the abstract Cauchy problem, and (under a weaker resolvent condition) the iterated method of Lavrentiev. A modified discrepancy principle is formulated for them, and finally numerical illustrations are presented. Received August 29, 1994 / Revised version received September 19, 1995     

Asymptotic behaviour of disconnection and non-intersection exponents

Summary. We study the asymptotic behaviour of disconnection and non-intersection exponents for planar Brownian motionwhen the number of considered paths tends to infinity. In particular, if η _n (respectively ξ (n, p)) denotes the disconnection exponent for n paths (respectively the non-intersection exponent for n paths versus p paths), then we show that lim _{n →∞} η _n /n = 1 2 and that for a > 0 and b > 0,lim _{n →∞} ξ ([na],[nb])/n = (√ a + √ b) ² /2. Received: 28 February 1996 / In revised form: 3 September 1996