Characterization of Brownian motion on manifolds through integration by parts

The theory of generalized white noise functionals (white noise calculus) initiated in [2] has been considerably developed in recent years, in particular, toward applications to quantum physics, see e.g. [5], [7] and references cited therein. On the other hand, since H. Yoshizawa [4], [23] discussed an infinite dimensional rotation group to broaden the scope of an investigation of Brownian motion, there have been some attempts to introduce an idea of group theory into the white noise calculus. For example, conformal invariance of Brownian motion with multidimensional parameter space [6], variational calculus of white noise functionals [14], characterization of the Levy Laplacian [17] and so on.

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Analysis of the geomagnetic activity of the <i>D<sub>st</sub></i> index and self-affine fractals using wavelet transforms

Nonlinear Processes in Geophysics ◽

10.5194/npg-11-303-2004 ◽

2004 ◽

Vol 11 (3) ◽

pp. 303-312 ◽

Cited By ~ 19

Author(s):

H. L. Wei ◽

S. A. Billings ◽

M. Balikhin

Keyword(s):

Brownian Motion ◽

Geomagnetic Activity ◽

Power Law ◽

Wavelet Transforms ◽

Effective Measure ◽

Dst Index ◽

Power Spectral ◽

Power Law Exponent ◽

Brownian Motion Model

Abstract. The geomagnetic activity of the Dst index is analyzed using wavelet transforms and it is shown that the Dst index possesses properties associated with self-affine fractals. For example, the power spectral density obeys a power-law dependence on frequency, and therefore the Dst index can be viewed as a self-affine fractal dynamic process. In fact, the behaviour of the Dst index, with a Hurst exponent H≈0.5 (power-law exponent β≈2) at high frequency, is similar to that of Brownian motion. Therefore, the dynamical invariants of the Dst index may be described by a potential Brownian motion model. Characterization of the geomagnetic activity has been studied by analysing the geomagnetic field using a wavelet covariance technique. The wavelet covariance exponent provides a direct effective measure of the strength of persistence of the Dst index. One of the advantages of wavelet analysis is that many inherent problems encountered in Fourier transform methods, such as windowing and detrending, are not necessary.

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R&D projects analyzed by semimartingale methods

Journal of Applied Probability ◽

10.1017/s0021900200037761 ◽

1985 ◽

Vol 22 (02) ◽

pp. 288-299 ◽

Cited By ~ 3

Author(s):

Knut K. Aase

Keyword(s):

Brownian Motion ◽

Stochastic Equation ◽

Compound Poisson Process ◽

General Nature ◽

Decision Variable ◽

Non Linear Equation ◽

Special Cases ◽

Dynamic Stochastic ◽

Present Setting

In this article we examine R&D projects where the project status changes according to a general dynamic stochastic equation. This allows for both continuous and jump behavior of the project status. The time parameter is continuous. The decision variable includes a non-stationary resource expenditure strategy and a stopping policy which determines when the project should be terminated. Characterization of stationary policies becomes straightforward in the present setting. A non-linear equation is determined for the expected discounted return from the project. This equation, which is of a very general nature, has been considered in certain special cases, where it becomes manageable. The examples include situations where the project status changes according to a compound Poisson process, a geometric Brownian motion, and a Brownian motion with drift. In those cases we demonstrate how the exact solution can be obtained and the optimal policy found.

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Deterministic characterization of viability for stochastic differential equation driven by fractional Brownian motion

ESAIM Control Optimisation and Calculus of Variations ◽

10.1051/cocv/2011188 ◽

2011 ◽

Vol 18 (4) ◽

pp. 915-929 ◽

Cited By ~ 5

Author(s):

Tianyang Nie ◽

Aurel Răşcanu

Keyword(s):

Differential Equation ◽

Brownian Motion ◽

Stochastic Differential Equation ◽

Fractional Brownian Motion

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Texture characterization of atmospheric fine particles by fractional Brownian motion analysis

Atmospheric Environment ◽

10.1016/j.atmosenv.2003.10.031 ◽

2004 ◽

Vol 38 (6) ◽

pp. 935-940 ◽

Cited By ~ 7

Author(s):

Chin-Hsiang Luo ◽

Che-Yen Wen ◽

Jiun-Jian Liaw ◽

Shih-Hsuan Chiu ◽

Whei-May Grace Lee

Keyword(s):

Brownian Motion ◽

Fractional Brownian Motion ◽

Motion Analysis ◽

Fine Particles ◽

Texture Characterization

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CHARACTERIZATION OF LASER PROPAGATION THROUGH TURBULENT MEDIA BY QUANTIFIERS BASED ON THE WAVELET TRANSFORM

Fractals ◽

10.1142/s0218348x04002471 ◽

2004 ◽

Vol 12 (02) ◽

pp. 223-233 ◽

Cited By ~ 11

Author(s):

LUCIANO ZUNINO ◽

DARÍO G. PÉREZ ◽

MARIO GARAVAGLIA ◽

OSVALDO A. ROSSO

Keyword(s):

Time Series ◽

Brownian Motion ◽

Hurst Exponent ◽

Wavelet Theory ◽

Complementary Information ◽

Laser Propagation ◽

Entropy Formula ◽

Center Position ◽

The Mean

The propagation of a laser beam through turbulent media is modeled as a fractional Brownian motion (fBm). Time series corresponding to the center position of the laser spot (coordinates x and y) after traveling across air in turbulent motion, with different strength, are analyzed by the wavelet theory. Two quantifiers are calculated, the Hurst exponent, H, and the mean Normalized Total Wavelet Entropy, [Formula: see text]. It is verified that both quantifiers give complementary information about the turbulence.

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