D-dimensional Brownian Motion Research Articles

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.

Favourite sites of transient Brownian motion

We present an accurate description for the location of maximum of d-dimensional Brownian motion. In case d = 1, this is a well-known theorem of Csáki et al. (1987a). We also deduce, as application, a version of the iterated logarithm law for the favourite site of transient Brownian motion.

Small ball problem via wavelets for Gaussian processes

The recently developed theory of wavelets is used to estimate small ball probability for a class of Gaussian processes including d-dimensional fractional Brownian motion.

A Note on the Long Time Asymptotics of the Brownian Motion with Application to the Theory of Quantum Measurement

Estimates of growth of the standard d-dimensional Brownian motion W(t) and its integral V(t) = ∫t0 W (s) ds are obtained, as t → ∞, and an application is discussed to the long time asymptotics of the solutions of the nonlinear stochastic equation of the quantum filtering theory.

Limits of Spatial Branching Populations

In critical branching and migrating populations, mobility of the individuals counteracts, and the clumping effect caused by the branching favours local extinction of the population in the large time limit. For example, d-dimensional critical binary branching Brownian motion (d>1) with a spatially inhomogeneous branching rate V(x), when started off with a homogeneous Poisson population, persists if V(x) ~ ||x||d-2(log||x||)-(1+e), and suffers local extinction if V(x) ~ ||x||d-2(log||x||)-1 as ||x||→∞; this can be derived from a probabilistic persistence criterion (Theorem 2). Besides presenting this result, the paper reviews conditions on the parameters of various other models which are necessary and sufficient for persistence, and discusses related results for superprocesses. Common to the proofs is the method of analysing the backward tree of an individual encountered in an old population, originally due to Kallenberg (1977) and Liemant (1981) in discrete time settings.

The Einstein relation for the displacement of a test particle in a random environment

Consider a stochastic system evolving in time, in which one observes the displacement of a tagged particle, X( t). Assume that this displacement process converges weakly to d-dimensional centered Brownian motion with covariance D, when space and time are appropriately scaled: X ε ( t) = εX( ε −2 t), ε→0. Now perturb the process by putting a small “force” ε a on the test particle. We prove on three different examples that under previous scaling the perturbed process converges to Brownian motion having the same covariance D, but an additional drift of the form M · a. We show that M, the “mobility” of the test particle, and D are related to each other by the Einstein formula M=( 1 2 β)D where β = 1/ kT( T being temperature and k Boltzmann's constant) is defined in such a way that the reversible state for the modified dynamics gets the correct Boltzmann factor. The method used to verify (1) is the calculus of Radon-Nikodym derivatives of measures in the space of trajectories (Girsanov's formula). Scaling simultaneously force and displacement has also a technical advantage: there is no need to show existence, under the perturbed evolution, of an invariant measure for the process “environment seen from the test particle” such that it is equivalent to the invariant measure under the unperturbed evolution.

Moments of the Lifetime of Conditioned Brownian Motion in Cones

Let $\tau$ be the time it takes standard d-dimensional Brownian motion, started at a point inside a cone $\Gamma$ in ${\mathbb {R}^d}$ which has aperture angle $\theta$, to leave the cone. Burkholder has determined the smallest p, denoted $p(\theta ,d)$, such that $E{\tau ^p} = \infty$. We show that if $y \in \partial \Gamma$ then the smallest p, such that $E({\tau ^p}|{B_\tau } = y) = \infty$, is $p = 2p(\theta ,d) + (d - 2)/2$.

Moments of the lifetime of conditioned Brownian motion in cones

Let τ \tau be the time it takes standard d-dimensional Brownian motion, started at a point inside a cone Γ \Gamma in R d {\mathbb {R}^d} which has aperture angle θ \theta , to leave the cone. Burkholder has determined the smallest p, denoted p ( θ , d ) p(\theta ,d) , such that E τ p = ∞ E{\tau ^p} = \infty . We show that if y ∈ ∂ Γ y \in \partial \Gamma then the smallest p, such that E ( τ p | B τ = y ) = ∞ E({\tau ^p}|{B_\tau } = y) = \infty , is p = 2 p ( θ , d ) + ( d − 2 ) / 2 p = 2p(\theta ,d) + (d - 2)/2 .

Conditional expectations of brownian functionals and their applications

This work studies the analytical expressions of the expectations of the forms E g ∫ 0 T ƒ(t, w(t)) dt w(T) ∈ B and E g ∫ 0 T ƒ(t, w(t)) dt ; w(T) ∈ B with { w( t), t ≥ 0} being a d-dimensional Brownian motion. Its application in obtaining solutions of Fokker-Planck equations is studied. Finally, a generalization from Brownian motion to diffusion processes is given in a one-dimensional setting.

An extension of a result of Komatsu and Takashima

If A is any time set of dimension α and {B(t):t≥0} a d-dimensional Brownian motion then the image of A by {B(t):t≥0} has Hansdorff dimension equal to the minimum of 2α and d quasi-everywhere. This result extends a theorem of Kahane

Sur une conjecture de M. Kac

We consider the following heat conduction problem. Let K be a compact set in Euclidean space ℝ3. Suppose that K is held at the temperature 1, while the surrounding medium is at the temperature 0 at time 0. Following Spitzer we investigate the asymptotic behaviour of the integral E K (t) which represents the total energy flow in time t from the set K to the surrounding medium ℝ3−K. An asymptotic expansion is given for E K (t) which refines a theorem due to Spitzer. This expansion also verifies and improves a formal calculation of Kac. Similar results are proved in higher dimensions. Up to the constant m(K), the quantity E K (t) can be interpreted as the expected value of the volume of the Wiener sausage associated with K and a d-dimensional Brownian motion. This point of view both plays a major role in the proofs and leads to a probabilistic interpretation of the different terms of the expansion.

Kahane using brownian local times of intersection

We show that if a set E in the positive real line has Hausdorff dimension greater than d/2 m, then the m-fold algebraic sum of the image of E by d-dimensional Brownian motion has an interior point. This extends a result of Kahane. The proof uses techniques found in Rosen (1983) and Geman, Horowitz and Rosen. We then show that the results do not hold for random sets and demonstrate that the above condition on the Hausdorff dimension of E is not close to being necessary

Reflected Brownian motion with skew symmetric data in a polyhedral domain

This paper is concerned with the characterization and invariant measures of certain reflected Brownian motions (RBM's) in polyhedral domains. The kind of RBM studied here behaves like d-dimensional Brownian motion with constant drift μ in the interior of a simple polyhedron and is instantaneously reflected at the boundary in directions that depend on the face that is hit. Under the assumption that the directions of reflection satisfy a certain skew symmetry condition first introduced in Harrison-Williams [9], it is shown that such an RBM can be characterized in terms of a family of submartingales and that it reaches non-smooth parts of the boundary with probability zero. In [9], a purely analytic problem associated with such an RBM was solved. Here the exponential form solution obtained in [9] is shown to be the density of an invariant measure for the RBM. Furthermore, if the density is integrable over the polyhedral state space, then it yields the unique stationary distribution for the RBM. In the proofs of these results, a key role is played by a dual process for the RBM and by results in [9] for reflected Brownian motions on smooth approximating domains.

Joint continuity and representations of additive functionals of d-dimensional Brownian motion

Conditions are given on a family of measures { μ a , 0⩽ a⩽1} so that the corresponding family { A a t , 0⩽ a⩽1} of additive functionals of d-dimensional Brownian motion will be jointly continuous in a and t, a.s. This is then used to give a d-dimensional analogue to the representation A t = ∝ L y t μ( dy) that is valid for one-dimensional Brownian motion, where L y t is local time at y. In place of local times at points, local times at hyperplanes are used.

On the asymptotic behaviour of the empirical random field of the brownian motion

Let ξ t , t ⩾ 0, be a d-dimensional Brownian motion. The asymptotic behaviour of the random field ƒ↦∫ t 0ƒ(ξ s ) d s is investigated, where ƒ belongs to a Sobolev space of periodic functions. Particularly a central limit theorem and a law of iterated logarithm are proved leading to a so-called universal law of iterated logarithm.

Levy's stochastic area formula in higher dimensions

Let ( W t ) = ( W 1 t , W 2 t ,…, W d t ), d ⩾ 2, be a d-dimensional standard Brownian motion and let A( t) be a bounded measurable function from R + into the space of d × d skew-symmetric matrices and x( t) such a function into R d . A class of stochastic processes (L t A,x), a particular example of which is Levy's “stochastic area” L t = 1 2 ∝ 0 −t (W 1 s,dW 2 s − W 2 s,dW 1 s) , is dealt with. The joint characteristic function of W t and L 1 A, x is calculated and based on this result a formula for fundamental solutions for the hypoelliptic operators which generate the diffusions ( W t , L t A, x ) is given.

Functionals of diffusion processes as stochastic integrals

Let be a standard d-dimensional Brownian motion and L a smooth functional on the space C = Cd[0, 1] of continuous functions from [0,1] to Rd, such that Denote Thenis a square integrable martingale which can be represented as a stochastic integral; consequently the random variable has the representationFor Fréchet differentiable functions L, Clark(1) gave the following formula for the integrand e: