Réflexion entre deux diffusions conjuguées

We observed, in a previous work, that Brownian motion reflected on an independent time-reversed Brownian motion is again Brownian motion. We present the generalisation of this result to pairs of conjugate diffusions (which are also dual, in the sense of Siegmund). To cite this article: F. Soucaliuc, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 1119–1124.     

A d-dimensional Brownian motion as a weak limit from a one-dimensional Poisson process

We show how from a unique standard Poisson process we can build a family of processes that converges in law to a d-dimensional standard Brownian motion for any d $\geqslant$ 1.     

Density of Space-Time Distribution of Brownian First Hitting of a Disc and a Ball

We compute the joint distribution of the site and the time at which a d-dimensional standard Brownian motion ((B˙t)) hits the surface of the ball ((U(a) ={—x—<a})) for the first time. The asymptotic form of its density is obtained when either the hitting time or the starting site ((B˙0)) becomes large. Our results entail that if Brownian motion is started at ((x)) and conditioned to hit ((U(a))), at time t, the distribution of the hitting site approaches the uniform distribution or the point mass at ((ax/—x—)) according as ((—x—/t)) tends to zero or infinity; in each case we provide a precise asymptotic estimate of the density. In the case when ((—x—/t)) tends to a positive constant we show the convergence of the density and derive an analytic expression of the limit density.     

Une généralisation de l'intégrale stochastique de Wick–Itô

The covariance of the fractional Brownian motion belongs to a family of positive functions introduced by Schoenberg in the 1930s. We show that one can define a stochastic integral for a large sub-family of the corresponding Gaussian second order stochastic processes. To cite this article: D. Alpay et al., C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

On the convergence to the multiple Wiener-Itô integral

We study the convergence to the multiple Wiener-Itô integral from processes with absolutely continuous paths. More precisely, consider a family of processes, with paths in the Cameron-Martin space, that converges weakly to a standard Brownian motion in C₀([0,T]). Using these processes, we construct a family that converges weakly, in the sense of the finite dimensional distributions, to the multiple Wiener-Itô integral process of a function f∈L²(n[0,T]). We prove also the weak convergence in the space C₀([0,T]) to the second-order integral for two important families of processes that converge to a standard Brownian motion.     

SLEs as boundaries of clusters of Brownian loops

In this research announcement, we show that SLE curves can in fact be viewed as boundaries of certain clusters of Brownian loops (of the clusters in a Brownian loop soup). For small densities c of loops, we show that the outer boundaries of the clusters created by the Brownian loop soup are SLE_κ-type curves where κ∈(8/3,4] and c related by the usual relation c=(3κ?8)(6?κ)/2κ (i.e., c corresponds to the central charge of the model). This gives (for any Riemann surface) a simple construction of a natural countable family of random disjoint SLE_κ loops, that behaves “nicely” under perturbation of the surface and is related to various aspects of conformal field theory and representation theory. To cite this article: W. Werner, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

The existence and uniqueness of the solution of an integral equation driven by a p-semimartingale of special type

We consider the integral equation driven by a standard Brownian motion and fractional Brownian motion (fBm). Since fBm is not a semimartingale, we cannot use the semimartingale theory to define an integral with respect to the fBm. Furthermore, a well-developed theory of stochastic differential equations is not applicable to solve it. Existence and uniqueness conditions are obtained for a solution in the space of continuous functions with q-bounded variation, q>2.     

On the diffusive behavior of isotropic diffusions in a random environment

We present here results concerning the asymptotic behavior of isotropic diffusions in random environment that are small perturbations of Brownian motion. When the space dimension is three or more we prove an invariance principle as well as transience. Our methods also apply to questions of homogenization in random media. To cite this article: A.-S. Sznitman, O. Zeitouni, C. R. Acad. Sci. Paris, Ser. I 339 (2004).     

Théorème limite pour une équation différentielle à coefficient aléatoire à mémoire longue

We consider an ordinary differential equation depending on a small parameter and with a long-range random coefficient. We establish that the solution of this ordinary differential equation converges to the solution of a stochastic differential equation driven by a fractional Brownian motion. The index of the fractional Brownian motion depends on the asymptotic behavior of the covariance function of the random coefficient. The proof of the convergence uses the T. Lyons theory of “rough paths”. To cite this article: R. Marty, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Brownian measures on Jordan-Virasoro curves associated to the Weil-Petersson metric

In this paper existence of the Brownian measure on Jordan curves with respect to the Weil-Petersson metric is established. The step from Brownian motion on the diffeomorphism group of the circle to Brownian motion on Jordan curves in C requires probabilistic arguments well beyond the classical theory of conformal welding, due to the lacking quasi-symmetry of canonical Brownian motion on Diff(S¹). A new key step in our construction is the systematic use of a Kählerian diffusion on the space of Jordan curves for which the welding functional gives rise to conformal martingales, together with a Douady-Earle type conformal extension of vector fields on the circle to the disk.     

Quasi-free quantum stochastic integrals for the CAR and CCR

A theory of quantum martingales and quantum stochastic integrals in quasi-free representations of the CAR and CCR is presented. For the CAR, the results generalize some of those developed in Barnett, Streater, and Wilde (J. Funct. Anal.48 (1982), 172–212, J. London Math. Soc.27 (1983), 373–384) and for the CCR, the results contain the standard Itô theory of stochastic integration with respect to Brownian motion as a special case.     

Karhunen-Loeve expansions for the m-th order detrended Brownian motion

The m-th order detrended Brownian motion is defined as the orthogonal component of projection of the standard Brownian motion onto the subspace spanned by polynomials of degree up to m. We obtain the Karhunen-Loeve expansion for the process and establish a connection with the generalized (m-th order) Brownian bridge developed by MacNeill (1978) in the study of distributions of polynomial regression. The resulting distribution identity is also verified by a stochastic Fubini approach. As applications, large and small deviation asymptotic behaviors for the L ₂ norm are given.     

Strong approximations in a charged-polymer model

We study the large-time behavior of the charged-polymer Hamiltonian H _n of Kantor and Kardar [Bernoulli case] and Derrida, Griffiths, and Higgs [Gaussian case], using strong approximations to Brownian motion. Our results imply, among other things, that in one dimension the process {H _[nt]}_0≤t≤1 behaves like a Brownian motion, time-changed by the intersection local-time process of an independent Brownian motion. Chung-type LILs are also discussed.     

The rate of quasi sure convergence in the functional limit theorem for increments of a Brownian motion

We investigate the quasi sure convergence of the functional limit for increments of a Brownian motion. The rate of quasi sure convergence in the functional limit for increments of a d-dimensional Brownian motion is derived. The main tool in the proof is large deviation and small deviation for Brownian motion in terms of (r,p)-capacity.     

Harmonic functions of subordinate killed Brownian motion

In this paper we study harmonic functions of subordinate killed Brownian motion in a domain D. We first prove that, when the killed Brownian semigroup in D is intrinsic ultracontractive, all nonnegative harmonic functions of the subordinate killed Brownian motion in D are continuous and then we establish a Harnack inequality for these harmonic functions. We then show that, when D is a bounded Lipschitz domain, both the Martin boundary and the minimal Martin boundary of the subordinate killed Brownian motion in D coincide with the Euclidean boundary ∂D. We also show that, when D is a bounded Lipschitz domain, a boundary Harnack principle holds for positive harmonic functions of the subordinate killed Brownian motion in D.     

Limiting laws associated with Brownian motion perturbated by normalized exponential weights

We perturb Brownian motion on the time interval [0,t] by an exponential weight; we show that for a large class of these weights the corresponding probability laws converge weakly as t→∞. To cite this article: B. Roynette et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

A change of variable formula for the 2D fractional Brownian motion of Hurst index bigger or equal to 1/4

We prove a change of variable formula for the 2D fractional Brownian motion of index H bigger or equal to 1/4. For H strictly bigger than 1/4, our formula coincides with that obtained by using the rough paths theory. For H=1/4 (the more interesting case), there is an additional term that is a classical Wiener integral against an independent standard Brownian motion.     

Invariant Measures for Transient Reflected Brownian Motion in a Wedge: Existence and Uniqueness

Our main result is the existence and uniqueness of an invariant measure for reflected Brownian motion (RBM) in a wedge that is transient to ∞. We also consider this question for RBM that has been killed at the corner of the wedge.     

Boundary-Crossing Identities for Diffusions Having the Time-Inversion Property

We review and study a one-parameter family of functional transformations, denoted by (S ^(β)) _β∈ℝ, which, in the case β<0, provides a path realization of bridges associated to the family of diffusion processes enjoying the time-inversion property. This family includes Brownian motions, Bessel processes with a positive dimension and their conservative h-transforms. By means of these transformations, we derive an explicit and simple expression which relates the law of the boundary-crossing times for these diffusions over a given function f to those over the image of f by the mapping S ^(β), for some fixed β∈ℝ. We give some new examples of boundary-crossing problems for the Brownian motion and the family of Bessel processes. We also provide, in the Brownian case, an interpretation of the results obtained by the standard method of images and establish connections between the exact asymptotics for large time of the densities corresponding to various curves of each family.     

Ten penalisation results of Brownian motion involving its one-sided supremum until first and last passage times, VIII

We penalise Brownian motion by a function of its one-sided supremum considered up to the last zero before t, respectively first zero after t, of that Brownian motion. This study presents some analogy with penalisation by the longest length of Brownian excursions, up to time t.