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1.
Multiple fractional integrals 总被引:2,自引:0,他引:2
Multiple integrals with respect to fractional Brownian motion (with H > 1/2) are constructed for a large class of functions. The first and second moments of the multiple integrals are explicitly
identified.
Received: 23 February 1998 / Revised version: 31 July 1998 相似文献
2.
M. Zähle 《Probability Theory and Related Fields》1998,111(3):333-374
The classical Lebesgue–Stieltjes integral ∫
b
a
fdg of real or complex-valued functions on a finite interval (a,b) is extended to a large class of integrands f and integrators g of unbounded variation. The key is to use composition formulas and integration-by-part rules for fractional integrals and
Weyl derivatives. In the special case of H?lder continuous functions f and g of summed order greater than 1 convergence of the corresponding Riemann–Stieltjes sums is proved.
The results are applied to stochastic integrals where g is replaced by the Wiener process and f by adapted as well as anticipating random functions. In the anticipating case we work within Slobodeckij spaces and introduce
a stochastic integral for which the classical It? formula remains valid. Moreover, this approach enables us to derive calculation
rules for pathwise defined stochastic integrals with respect to fractional Brownian motion.
Received: 14 January 1998 / Revised version: 9 April 1998 相似文献
3.
Any solution of the functional equation
where B is a Brownian motion, behaves like a reflected Brownian motion, except when it attains a new maximum: we call it an α-perturbed
reflected Brownian motion. Similarly any solution of
behaves like a Brownian motion except when it attains a new maximum or minimum: we call it an α,β-doubly perturbed Brownian
motion. We complete some recent investigations by showing that for all permissible values of the parameters α, α and β respectively,
these equations have pathwise unique solutions, and these are adapted to the filtration of B.
Received: 7 November 1997 / Revised version: 13 July 1998 相似文献
4.
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 相似文献
5.
Arun Kumar Palaniappan Vellaisamy 《Methodology and Computing in Applied Probability》2012,14(2):263-283
Normal inverse Gaussian (NIG) process was introduced by Barndorff-Nielsen (Scand J Statist 24:1–13, 1997) by subordinating Brownian motion with drift to an inverse Gaussian process. Increments of NIG process are independent and
are stationary. In this paper, we introduce dependence between the increments of NIG process, by subordinating fractional
Brownian motion to an inverse Gaussian process and call it fractional normal inverse Gaussian (FNIG) process. The basic properties
of this process are discussed. Its marginal distributions are scale mixtures of normal laws, infinitely divisible for the
Hurst parameter 1/2 ≤ H < 1 and are heavy tailed. First order increments of the process are stationary and possess long-range dependence (LRD) property.
It is shown that they have persistence of signs LRD property also. A generalization to an n-FNIG process is also discussed,
which allows Hurst parameter H in the interval (n − 1, n). Possible applications to mathematical finance and hydraulics are also pointed out. 相似文献
6.
Jean-Stéphane Dhersin Jean-François Le Gall 《Probability Theory and Related Fields》1997,108(1):103-129
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 相似文献
7.
Endre Csáki Miklós Csörgő Antónia Földes Zhan Shi 《Probability Theory and Related Fields》2000,117(4):515-531
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 相似文献
8.
Jean-François Delmas 《Probability Theory and Related Fields》1999,114(4):505-547
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 ℝd, 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 ℝd, d≥ 5.
Received: 7 April 1998 / Revised version: 2 October 1998 相似文献
9.
Jean-Claude Gruet 《Probability Theory and Related Fields》1998,111(4):489-516
We consider the word associated to the homotopic class of the Brownian path (properly closed) in the thrice punctured sphere.
We prove that its length has almost surely the same behaviour as a totally asymmetric Cauchy process on the line. More precisely,
the liminf has the same normalization in t
log(t) and the limsup can be described by the same integral test. They are the Brownian motion counterparts of some Lévy and Khintchine
results on continued fraction expansions.
Received: 17 December 1996 / Revised version: 23 February 1998 相似文献
10.
Mario V. Wüthrich 《Probability Theory and Related Fields》1998,112(3):299-319
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 相似文献
11.
Consider a d-dimensional Brownian motion X = (X
1,…,X
d
) and a function F which belongs locally to the Sobolev space W
1,2. We prove an extension of It? s formula where the usual second order terms are replaced by the quadratic covariations [f
k
(X), X
k
] involving the weak first partial derivatives f
k
of F. In particular we show that for any locally square-integrable function f the quadratic covariations [f(X), X
k
] exist as limits in probability for any starting point, except for some polar set. The proof is based on new approximation
results for forward and backward stochastic integrals.
Received: 16 March 1998 / Revised version: 4 April 1999 相似文献
12.
13.
In this paper we present a martingale related to the exit measures of super Brownian motion. By changing measure with this
martingale in the canonical way we have a new process associated with the conditioned exit measure. This measure is shown
to be identical to a measure generated by a non-homogeneous branching particle system with immigration of mass. An application
is given to the problem of conditioning the exit measure to hit a number of specified points on the boundary of a domain.
The results are similar in flavor to the “immortal particle” picture of conditioned super Brownian motion but more general,
as the change of measure is given by a martingale which need not arise from a single harmonic function.
Received: 27 August 1998 / Revised version: 8 January 1999 相似文献
14.
Let H be a Hilbert space and E a Banach space. We set up a theory of stochastic integration of ℒ(H,E)-valued functions with respect to H-cylindrical Liouville fractional Brownian motion with arbitrary Hurst parameter 0 < β < 1. For 0 < β < ? we show that a function Φ: (0, T) → ℒ(H,E) is stochastically integrable with respect to an H-cylindrical Liouville fractional Brownian motion if and only if it is stochastically integrable with respect to an H-cylindrical fractional Brownian motion. 相似文献
15.
Hirofumi Osada 《Probability Theory and Related Fields》1998,112(1):53-90
We prove the positivity of the self-diffusion matrix of interacting Brownian particles with hard core when the dimension of
the space is greater than or equal to 2. Here the self-diffusion matrix is a coefficient matrix of the diffusive limit of
a tagged particle. We will do this for all activities, z>0, of Gibbs measures; in particular, for large z– the case of high density particles. A typical example of such a particle system is an infinite amount of hard core Brownian
balls.
Received: 22 September 1997 / Revised version: 15 January 1998 相似文献
16.
Maria Jolis 《Journal of Theoretical Probability》2006,19(1):121-133
We construct a multiple Stratonovich-type integral with respect to Gaussian processes with covariance function of bounded variation. This construction is based on the previous definition of the multiple Itô-type integral given by Huang and Cambanis [Ann. Propab. 6(4), 585–614] and on a Hu–Meyer formula (that is, an expression of the multiple Stratonovich integral as a sum of Itô-type integrals of inferior or equal order) for the elementary functions. We also apply our results to the fractional Brownian motion with Hurst parameter $H > \frac{1}{2}We construct a multiple Stratonovich-type integral with respect to Gaussian processes with covariance function of bounded variation. This construction is based on the previous definition of the multiple It?-type integral given by Huang and Cambanis [Ann. Propab. 6(4), 585–614] and on a Hu–Meyer formula (that is, an expression of the multiple Stratonovich integral as a sum of It?-type integrals of inferior or equal order) for the elementary functions. We also apply our results to the fractional Brownian motion with Hurst parameter
. 相似文献
17.
Eric M. Rains 《Probability Theory and Related Fields》1998,112(3):411-423
Using the machinery of zonal polynomials, we examine the limiting behavior of random symmetric matrices invariant under conjugation
by orthogonal matrices as the dimension tends to infinity. In particular, we give sufficient conditions for the distribution
of a fixed submatrix to tend to a normal distribution. We also consider the problem of when the sequence of partial sums of
the diagonal elements tends to a Brownian motion. Using these results, we show that if O
n
is a uniform random n×n orthogonal matrix, then for any fixed k>0, the sequence of partial sums of the diagonal of O
k
n
tends to a Brownian motion as n→∞.
Received: 3 February 1998 / Revised version: 11 June 1998 相似文献
18.
Hirofumi Osada 《Probability Theory and Related Fields》2001,119(2):275-310
We construct a family of diffusions P
α = {P
x} on the d-dimensional Sierpinski carpet F^. The parameter α ranges over d
H
< α < ∞, where d
H
= log(3
d
− 1)/log 3 is the Hausdorff dimension of the d-dimensional Sierpinski carpet F^. These diffusions P
α are reversible with invariant measures μ = μ[α]. Here, μ are Radon measures whose topological supports are equal to F^ and satisfy self-similarity in the sense that μ(3A) = 3α·μ(A) for all A∈ℬ(F^). In addition, the diffusion is self-similar and invariant under local weak translations (cell translations) of the
Sierpinski carpet. The transition density p = p(t, x, y) is locally uniformly positive and satisfies a global Gaussian upper bound. In spite of these well-behaved properties, the
diffusions are different from Barlow-Bass' Brownian motions on the Sierpinski carpet.
Received: 30 September 1999 / Revised version: 15 June 2000 / Published online: 24 January 2000 相似文献
19.
How fast are the particles of super-Brownian motion? 总被引:5,自引:1,他引:4
Peter Mörters 《Probability Theory and Related Fields》2001,121(2):171-197
In this paper we investigate fast particles in the range and support ofsuper-Brownian motion in the historical setting. In
this setting eachparticle of super-Brownian motion alive at time t is represented by apath w:[0,t]→ℝ
d
and the state of historical super-Brownian motionis a measure on the set of paths. Typical particles have Brownian paths,however
in the uncountable collection of particles in the range of asuper-Brownian motion there are some which at exceptional times
movefaster than Brownian motion. We determine the maximal speed of allparticles during a given time period E, which turns out to be afunction of the packing dimension of E. A path w in the support ofhistorical super-Brownian motion at time t is called a-fast if . Wecalculate the Hausdorff dimension of the set of a-fast paths in thesupport and the range of historical super-Brownian motion. A valuabletool in the proofs is a uniform dimension
formula for the Browniansnake, which reduces dimension problems in the space of stopped paths to dimension problems on the
line.
Received: 27 January 2000 / Revised version: 28 August 2000 / Published online: 24 July 2001 相似文献
20.
Wensheng Wang 《Probability Theory and Related Fields》2003,126(2):203-220
Kesten and Spitzer have shown that certain random walks in random sceneries converge to stable processes in random sceneries.
In this paper, we consider certain random walks in sceneries defined using stationary Gaussian sequence, and show their convergence
towards a certain self-similar process that we call fractional Brownian motion in Brownian scenery.
Received: 17 April 2002 / Revised version: 11 October 2002 /
Published online: 15 April 2003
Research supported by NSFC (10131040).
Mathematics Subject Classification (2002): 60J55, 60J15, 60J65
Key words or phrases: Weak convergence – Random walk in random scenery – Local time – Fractional Brownian motion in Brownian scenery 相似文献