scholarly journals On the Asymptotic Behavior of the Hyperbolic Brownian Motion

2014 ◽  
Vol 154 (6) ◽  
pp. 1550-1568 ◽  
Author(s):  
Valentina Cammarota ◽  
Alessandro De Gregorio ◽  
Claudio Macci
Keyword(s):  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Hyperbolic Brownian Motion

scholarly journals ASYMPTOTIC BEHAVIOR OF THE PRINCIPAL EIGENVALUE FOR A CLASS OF NON-LOCAL ELLIPTIC OPERATORS RELATED TO BROWNIAN MOTION WITH SPATIALLY DEPENDENT RANDOM JUMPS

2011 ◽  
Vol 13 (06) ◽  
pp. 1077-1093
Author(s):  
NITAY ARCUSIN ◽  
ROSS G. PINSKY
Keyword(s):  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Large Time ◽  
Dirichlet Boundary ◽  
Principal Eigenvalue ◽  
Probability Measures ◽  
Rate Of Decay ◽  
Non Local ◽  
Random Jumps ◽  
Explicit Constant

Let D ⊂ Rd be a bounded domain and let [Formula: see text] denote the space of probability measures on D. Consider a Brownian motion in D which is killed at the boundary and which, while alive, jumps instantaneously according to a spatially dependent exponential clock with intensity γV to a new point, according to a distribution [Formula: see text]. From its new position after the jump, the process repeats the above behavior independently of what has transpired previously. The generator of this process is an extension of the operator -Lγ,μ, defined by [Formula: see text] with the Dirichlet boundary condition, where Cμ is the "μ-centering" operator defined by [Formula: see text] The principal eigenvalue, λ0(γ, μ), of Lγ, μ governs the exponential rate of decay of the probability of not exiting D for large time. We study the asymptotic behavior of λ0(γ, μ) as γ → ∞. In particular, if μ possesses a density in a neighborhood of the boundary, which we call μ, then [Formula: see text] If μ and all its derivatives up to order k - 1 vanish on the boundary, but the kth derivative does not vanish identically on the boundary, then λ0(γ, μ) behaves asymptotically like [Formula: see text], for an explicit constant ck.

Hyperbolic and fractional hyperbolic Brownian motion

Stochastics ◽  
2007 ◽  
Vol 79 (6) ◽  
pp. 505-522 ◽  
Author(s):  
Lanjun Lao ◽  
Enzo Orsingher
Keyword(s):  
Brownian Motion ◽  
Hyperbolic Brownian Motion

scholarly journals Nonparametric Regression with Subfractional Brownian Motion via Malliavin Calculus

2014 ◽  
Vol 2014 ◽  
pp. 1-14
Author(s):  
Yuquan Cang ◽  
Junfeng Liu ◽  
Yan Zhang
Keyword(s):  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Nonparametric Regression ◽  
Local Time ◽  
Kernel Function ◽  
Malliavin Calculus ◽  
Bandwidth Parameter ◽  
Subfractional Brownian Motion ◽  
Normal Law

We study the asymptotic behavior of the sequenceSn=∑i=0n-1K(nαSiH1)(Si+1H2-SiH2),asntends to infinity, whereSH1andSH2are two independent subfractional Brownian motions with indicesH1andH2, respectively.Kis a kernel function and the bandwidth parameterαsatisfies some hypotheses in terms ofH1andH2. Its limiting distribution is a mixed normal law involving the local time of the sub-fractional Brownian motionSH1. We mainly use the techniques of Malliavin calculus with respect to sub-fractional Brownian motion.

scholarly journals Asymptotic Behavior of the Stochastic Rayleigh-van der Pol Equations with Jumps

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
ZaiTang Huang ◽  
ChunTao Chen
Keyword(s):  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Stochastic Bifurcation ◽  
Brownian Motion Process ◽  
Van Der Pol ◽  
Limit Circle ◽  
Random Attractors ◽  
Motion Process ◽  
Van Der Pol Equations ◽  
The Stability

We study the stability, attractors, and bifurcation of stochastic Rayleigh-van der Pol equations with jumps. We first established the stochastic stability and the large deviations results for the stochastic Rayleigh-van der Pol equations. We then examine the existence limit circle and obtain some new random attractors. We further establish stochastic bifurcation of random attractors. Interestingly, this shows the effect of the Poisson noise which can stabilize or unstabilize the system which is significantly different from the classical Brownian motion process.

scholarly journals A DYNAMICAL APPROACH TO FRACTIONAL BROWNIAN MOTION

Fractals ◽  
1994 ◽  
Vol 02 (01) ◽  
pp. 81-94 ◽  
Author(s):  
RICCARDO MANNELLA ◽  
PAOLO GRIGOLINI ◽  
BRUCE J. WEST
Keyword(s):  
Dynamical System ◽  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Correlation Function ◽  
Fractional Brownian Motion ◽  
Gaussian Processes ◽  
The Other ◽  
Hurst Coefficient ◽  
And Diffusion ◽  
Clear Relation

Herein we develop a dynamical foundation for fractional Brownian motion. A clear relation is established between the asymptotic behavior of the correlation function and diffusion in a dynamical system. Then, assuming that scaling is applicable, we establish a connection between diffusion (either standard or anomalous) and the dynamical indicator known as the Hurst coefficient. We argue on the basis of numerical simulations that although we have been able to prove scaling only for "Gaussian" processes, our conclusions may well apply to a wider class of systems. On the other hand, systems exist for which scaling might not hold, so we speculate on the possible consequences of the various relations derived in the paper on such systems.

Winding angle and maximum winding angle of the two-dimensional random walk

1991 ◽  
Vol 28 (4) ◽  
pp. 717-726 ◽  
Author(s):  
Claude Bélisle ◽  
Julian Faraway
Keyword(s):  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Random Walk ◽  
Random Walks ◽  
Computer Simulations ◽  
Asymptotic Distributions ◽  
Two Dimensional ◽  
Exact Distributions ◽  
The Difference ◽  
Winding Angle

Recent results on the winding angle of the ordinary two-dimensional random walk on the integer lattice are reviewed. The difference between the Brownian motion winding angle and the random walk winding angle is discussed. Other functionals of the random walk, such as the maximum winding angle, are also considered and new results on their asymptotic behavior, as the number of steps increases, are presented. Results of computer simulations are presented, indicating how well the asymptotic distributions fit the exact distributions for random walks with 10m steps, for m = 2, 3, 4, 5, 6, 7.

scholarly journals Sharp estimates of the Green function of hyperbolic Brownian motion

2015 ◽  
Vol 228 (3) ◽  
pp. 197-221 ◽  
Author(s):  
Kamil Bogus ◽  
Tomasz Byczkowski ◽  
Jacek Małecki
Keyword(s):  
Brownian Motion ◽  
Green Function ◽  
Sharp Estimates ◽  
The Green Function ◽  
Hyperbolic Brownian Motion

scholarly journals Hyperbolic Harmonic Functions and Hyperbolic Brownian Motion

2020 ◽  
Vol 30 (5) ◽  
Author(s):  
Sirkka-Liisa Eriksson ◽  
Terhi Kaarakka
Keyword(s):  
Brownian Motion ◽  
Half Space ◽  
Laplace Operator ◽  
Harmonic Functions ◽  
Riemannian Metric ◽  
Hyperbolic Metric ◽  
Hyperbolic Functions ◽  
The Hyperbolic Metric ◽  
Hyperbolic Brownian Motion ◽  
Hyperbolic Harmonic Functions

Abstract We study harmonic functions with respect to the Riemannian metric $$\begin{aligned} ds^{2}=\frac{dx_{1}^{2}+\cdots +dx_{n}^{2}}{x_{n}^{\frac{2\alpha }{n-2}}} \end{aligned}$$ d s 2 = d x 1 2 + ⋯ + d x n 2 x n 2 α n - 2 in the upper half space $$\mathbb {R}_{+}^{n}=\{\left( x_{1},\ldots ,x_{n}\right) \in \mathbb {R}^{n}:x_{n}>0\}$$ R + n = { x 1 , … , x n ∈ R n : x n > 0 } . They are called $$\alpha $$ α -hyperbolic harmonic. An important result is that a function f is $$\alpha $$ α -hyperbolic harmonic íf and only if the function $$g\left( x\right) =x_{n}^{-\frac{ 2-n+\alpha }{2}}f\left( x\right) $$ g x = x n - 2 - n + α 2 f x is the eigenfunction of the hyperbolic Laplace operator $$\bigtriangleup _{h}=x_{n}^{2}\triangle -\left( n-2\right) x_{n}\frac{\partial }{\partial x_{n}}$$ △ h = x n 2 ▵ - n - 2 x n ∂ ∂ x n corresponding to the eigenvalue $$\ \frac{1}{4}\left( \left( \alpha +1\right) ^{2}-\left( n-1\right) ^{2}\right) =0$$ 1 4 α + 1 2 - n - 1 2 = 0 . This means that in case $$\alpha =n-2$$ α = n - 2 , the $$n-2$$ n - 2 -hyperbolic harmonic functions are harmonic with respect to the hyperbolic metric of the Poincaré upper half-space. We are presenting some connections of $$\alpha $$ α -hyperbolic functions to the generalized hyperbolic Brownian motion. These results are similar as in case of harmonic functions with respect to usual Laplace and Brownian motion.

scholarly journals On Bernstein–Kantorovich invariance principle in Hölder spaces and weighted scan statistics

2020 ◽  
Vol 24 ◽  
pp. 186-206
Author(s):  
Alfredas Račkauskas ◽  
Charles Suquet
Keyword(s):  
Asymptotic Behavior ◽  
Brownian Motion ◽  
Weak Convergence ◽  
Invariance Principle ◽  
Brownian Bridge ◽  
Polygonal Line ◽  
Scan Statistics ◽  
Partial Sums ◽  
Kantorovich Theorem ◽  
The One

Let ξn be the polygonal line partial sums process built on i.i.d. centered random variables Xi, i ≥ 1. The Bernstein-Kantorovich theorem states the equivalence between the finiteness of E|X1|max(2,r) and the joint weak convergence in C[0, 1] of n−1∕2ξn to a Brownian motion W with the moments convergence of E∥n−1/2ξn∥∞r to E∥W∥∞r. For 0 < α < 1∕2 and p (α) = (1 ∕ 2 - α) -1, we prove that the joint convergence in the separable Hölder space Hαo of n−1∕2ξn to W jointly with the one of E∥n−1∕2ξn∥αr to E∥W∥αr holds if and only if P(|X1| > t) = o(t−p(α)) when r < p(α) or E|X1|r < ∞ when r ≥ p(α). As an application we show that for every α < 1∕2, all the α-Hölderian moments of the polygonal uniform quantile process converge to the corresponding ones of a Brownian bridge. We also obtain the asymptotic behavior of the rth moments of some α-Hölderian weighted scan statistics where the natural border for α is 1∕2 − 1∕p when E|X1|p < ∞. In the case where the Xi’s are p regularly varying, we can complete these results for α > 1∕2 − 1∕p with an appropriate normalization.

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