On Striking Identities About the Exponential Functionals of the Brownian Bridge and Brownian Motion

The negative moment of order 1, resp. of order 1/2, for the integral on (0, 1) of the exponential of α times the Brownian bridge, resp. the Brownian motion, does not depend on α. We give a simple explanation and a reinforcement of this property in the case of the Brownian bridge. We then discuss how different the case of the Brownian motion is.     

First exit distribution and path continuity of Hunt processes

This paper gives a characterization of a Hunt process path by the first exit left limit distribution. It is also showed that if the first exit left limit distribution leaving any ball from the center is a uniform distribution on the sphere, then the Levy Processes are a scaled Brownian motion.     

A Functional Limit Theorem for Random Walk Conditioned to Stay Non-Negative

In this paper we consider an aperiodic integer-valued randomwalk S and a process S* that is a harmonic transform of S killedwhen it first enters the negative half; informally, S* is ‘Sconditioned to stay non-negative’. If S is in the domainof attraction of the standard normal law, without centring,a suitably normed and linearly interpolated version of S convergesweakly to standard Brownian motion, and our main result is thatunder the same assumptions a corresponding statement holds forS*, the limit of course being the three-dimensional Bessel process.As this process can be thought of as Brownian motion conditionedto stay non-negative, in essence our result shows that the interchangeof the two limit operations is valid. We also establish somerelated results, including a local limit theorem for S*, anda bivariate renewal theorem for the ladder time and height process,which may be of independent interest.     

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     

Pinning class of the Wiener measure by a functional: related martingales and invariance properties

For a given functional Y on the path space, we define the pinning class of the Wiener measure as the class of probabilities which admit the same conditioning given Y as the Wiener measure. Using stochastic analysis and the theory of initial enlargement of filtration, we study the transformations (not necessarily adapted) which preserve this class. We prove, in this non Markov setting, a stochastic Newton equation and a stochastic Noether theorem. We conclude the paper with some non canonical representations of Brownian motion, closely related to our study.Mathematics Subject Classification (2000): 60G44, 60H07, 60H20, 60H30     

Chung’s functional law of the iterated logarithm for the Brownian sheet

In this paper, we investigate functional limit problem for path of a Brownian sheet, Chung's functional law of the iterated logarithm for a Brownian sheet is obtained. The main tool in the proof is large deviation and small deviation for a Brownian sheet.     

Representations of the Brownian snake with drift

We consider a path-valued process which is a generalization of the classical Brownian snake introduced by Le Gall. More precisely we add a drift term b to the lifetime process, which may depends on the spatial process. Consequently, this introduces a coupling between the lifetime process and the spatial motion. This process can be obtained from the standard Brownian snake by Girsanov's theorem or by killing of the spatial motion. It can also be viewed as the limit of discrete snakes or, in some special cases, as conditioned Brownian snakes. We also use this process to describe the solutions of the non-linear partial differential equation j u =4 u 2 +4 bu .     

Sample path properties of stochastic integrals,and stochastic differentiation

Let B be a Brownian motion, and X = H.B be a stochastic integral of B. We give conditions on the smoothness of the process H which imply that if Ms a singular point of the sample path of B (ω) (such as a local maximum, a slow point, or a fast point) then t is also a singular point of X (ω). In the final section we give an application to stochastic differential equations     

Functional Non-Central and Central Limit Theorems for Bivariate Appell Polynomials

We consider quadratic forms in bivariate Appell polynomials involving strongly dependent time series. Both the spectral density of these time series and the Fourier transform of the kernel of the quadratic forms are regularly varying at the origin and hence may diverge, for example, like a power function. We obtain functional limit theorems for these quadratic forms by extending the recent results on the convergence of their finite-dimensional distributions. Some of these are functional central limit theorems where the limiting process is Brownian motion. Others are functional non-central limit theorems where the limiting processes are typically not Gaussian or, if they are Gaussian, then they are not Brownian motion.     

Small and Large Scale Behavior of the Poissonized Telecom Process

The stable Telecom process has infinite variance and appears as a limit of renormalized renewal reward processes. We study its Poissonized version where the infinite variance stable measure is replaced by a Poisson point measure. We show that this Poissonized version converges to the stable Telecom process at small scales and to the Gaussian fractional Brownian motion at large scales. This process is therefore locally as well as asymptotically self-similar. The value of the self-similarity parameter at large scales, namely the self-similarity parameter of the limit fractional Brownian motion, depends on the form the Poissonized Telecom process. The Poissonized Telecom process is a Poissonized mixed moving average. We investigate more general Poissonized mixed moving averages as well.     

Path Space Large Deviations of a Large Buffer with Gaussian Input Traffic

We consider a queue fed by Gaussian traffic and give conditions on the input process under which the path space large deviations of the queue are governed by the rate function of the fractional Brownian motion. As an example we consider input traffic that is composed of of independent streams, each of which is a fractional Brownian motion, having different Hurst indices.     

Characterization of the least concave majorant of brownian motion,conditional on a vertex point,with application to construction

The characterization of the least concave majorant of brownian motion by Pitman (1983,Seminar on Stochastic Processes, 1982 (eds. E. Cinlar, K. L. Chung and R. K. Getoor), 219–228, Birkhäuser, Boston) is tweaked, conditional on a vertex point. The joint distribution of this vertex point is derived and is shown to be generated with extreme ease. A procedure is then outlined by which one can construct the least concave majorant of a standard Brownian motion path over any finite, closed subinterval of (0, ∞). This construction is exact in distribution. One can also construct a linearly interpolated version of the Brownian motion path (i.e. we construct the Brownian motion path over a grid of points and linearly interpolate) corresponding to this least concave majorant over the same finite interval. A discussion of how to translate the aforementioned construction to the least concave majorant of a Brownian bridge is also presented.     

THE INDUCED H-STRUCTURE ON FUNCTION SPACES

Abstract

Let (Z,Γ) be an H-structure. Then, for each exponential object Y in TOP, an H-structure is induced on the topological space C_t(Y,Z) of continuous maps equipped with the appropriate function space topology t (e.g. t = T_is, where T_is is the Isbell topology on C(Y,Z)).

If (Z,Γ) is H-associative (resp.admits inversion), then the induced H-structure is also H-associative (resp. admits inversion).

If (Z,Γ) is H-associative and admits inversion (e.g. a topological group) then all path components of C_t(Y,Z) belong to the same homotopy type.

We also study the special case of (Z,Γ) being a topological group. Moreover, we prove that certain functions between function spaces are H-homomorphisms of the induced H-structures in the function spaces.     

Brownian Gibbs property for Airy line ensembles

Consider a collection of N Brownian bridges $B_{i}:[-N,N] \to \mathbb{R} $ , B _i (?N)=B _i (N)=0, 1≤i≤N, conditioned not to intersect. The edge-scaling limit of this system is obtained by taking a weak limit as N→∞ of the collection of curves scaled so that the point (0,2^1/2 N) is fixed and space is squeezed, horizontally by a factor of N ^2/3 and vertically by N ^1/3. If a parabola is added to each of the curves of this scaling limit, an x-translation invariant process sometimes called the multi-line Airy process is obtained. We prove the existence of a version of this process (which we call the Airy line ensemble) in which the curves are almost surely everywhere continuous and non-intersecting. This process naturally arises in the study of growth processes and random matrix ensembles, as do related processes with “wanderers” and “outliers”. We formulate our results to treat these relatives as well. Note that the law of the finite collection of Brownian bridges above has the property—called the Brownian Gibbs property—of being invariant under the following action. Select an index 1≤k≤N and erase B _k on a fixed time interval (a,b)?(?N,N); then replace this erased curve with a new curve on (a,b) according to the law of a Brownian bridge between the two existing endpoints (a,B _k (a)) and (b,B _k (b)), conditioned to intersect neither the curve above nor the one below. We show that this property is preserved under the edge-scaling limit and thus establish that the Airy line ensemble has the Brownian Gibbs property. An immediate consequence of the Brownian Gibbs property is a confirmation of the prediction of M. Prähofer and H. Spohn that each line of the Airy line ensemble is locally absolutely continuous with respect to Brownian motion. We also obtain a proof of the long-standing conjecture of K. Johansson that the top line of the Airy line ensemble minus a parabola attains its maximum at a unique point. This establishes the asymptotic law of the transversal fluctuation of last passage percolation with geometric weights. Our probabilistic approach complements the perspective of exactly solvable systems which is often taken in studying the multi-line Airy process, and readily yields several other interesting properties of this process.     

Finite dimensional characteristic functions of Brownian rough path

The Brownian rough path is the canonical lifting of Brownian motion to the free nilpotent Lie group of order 2: Equivalently, it is a process taking values in the algebra of Lie polynomials of degree 2, which is described explicitly by the Brownian motion coupled with its area process. The aim of this article is to compute the finite dimensional characteristic functions of the Brownian rough path in ?^d and obtain an explicit formula for the case when d = 2     

Brownian motion on the continuum tree

Summary We construct Brownian motion on a continuum tree, a structure introduced as an asymptotic limit to certain families of finite trees. We approximate the Dirichlet form of Brownian motion on the continuum tree by adjoining one-dimensional Brownian excursions. We study the local times of the resulting diffusion. Using time-change methods, we find explicit expressions for certain hitting probabilities and the mean occupation density of the process.     

Well-posedness and Large Deviations of the Stochastic Modified Camassa-Holm Equation

We derive the heat trace asymptotics of the generator of subordinate Brownian motion on Euclidean space for a class of Laplace exponents. The terms in the asymptotic expansion can be computed to arbitrary order and depend both on the geometry of Euclidean space and the short-time behaviour of the process. If the Blumenthal-Getoor index of the process is rational, then the asymptotics may contain logarithmic terms. The key assumption is the existence of a suitable density for the Lévy measure of the subordinator. The analysis is highly explicit.     

SPDE with generalized drift and fractional-type noise

We consider a stochastic partial differential equation involving a second order differential operator whose drift is discontinuous. The equation is driven by a Gaussian noise which behaves as a Wiener process in space and the time covariance generates a signed measure. This class includes the Brownian motion, fractional Brownian motion and other related processes. We give a necessary and sufficient condition for the existence of the solution and we study the path regularity of this solution.     

代数结构上的模糊拟伪$b$-度量

研究代数结构上的模糊(拟)伪$b$-度量. 主要结论有: (1)设$G$是一个抽象群, $\tau$是$G$上一个左不变模糊拟伪$b$-度量(伪$b$-度量)诱导的拓扑,如果$(G,\tau)$是右拓扑群,那么$(G,\tau)$是一个仿拓扑群(拓扑群); (2)设$S$是一个半群,如果$\tau$是$S$上一个不变模糊拟伪$b$-度量诱导的拓扑,那么$(S,\tau)$是一个拓扑半群.     

On the Weighted Local Time and the Tanaka Formula for the Multidimensional Fractional Brownian Motion

Abstract

We determine the weighted local time for the multidimensional fractional Brownian motion from the occupation time formula. We also discuss on the Itô and Tanaka formula for the multidimensional fractional Brownian motion. In these formulas the Skorohod integral is applicable if the Hurst parameter of fractional Brownian motion is greater than 1/2. If the Hurst parameter is less than 1/2, then we use the Skorohod type integral introduced by Nualart and Zakai for the stochastic integral and establish the Itô and Tanaka formulas.