The $$\frac{4}{3}$$ -Variation of the Derivative of the Self-intersection Brownian Local Time and Related Processes

Let [Formula: see text] denote the local time of Brownian motion. Our main result is to show that for each fixed t[Formula: see text] as h → 0, where η is a normal random variable with mean zero and variance one, that is independent of [Formula: see text]. This generalizes our previous result for the second moment. We also explain why our approach will not work for higher moments.

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Large deviations and Strassen's limit points of Brownian local time processes

Statistics & Probability Letters ◽

10.1016/s0167-7152(96)00206-4 ◽

1997 ◽

Vol 34 (4) ◽

pp. 385-395 ◽

Cited By ~ 1

Author(s):

Bin Chen

Keyword(s):

Large Deviations ◽

Local Time ◽

Brownian Local Time ◽

Limit Points

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Hilbert transform of G-Brownian local time

Stochastics and Dynamics ◽

10.1142/s0219493714500063 ◽

2014 ◽

Vol 14 (04) ◽

pp. 1450006 ◽

Cited By ~ 4

Author(s):

Litan Yan ◽

Qinghua Zhang ◽

Bo Gao

Keyword(s):

Brownian Motion ◽

Local Time ◽

Hilbert Transform ◽

Natural Extension ◽

Itô Integral ◽

Brownian Local Time ◽

Ito Integral ◽

Natural Result ◽

The Hilbert Transform

Let B be a G-Brownian motion with quadratic process 〈B〉 under the G-expectation. In this paper, we consider the integrals [Formula: see text] We show that the integral diverges and the convergence [Formula: see text] exists in 𝕃2 for all a ∈ ℝ, t > 0. This shows that [Formula: see text] coincides with the Hilbert transform of the local time [Formula: see text] of G-Brownian motion B for every t. The functional is a natural extension to classical cases. As a natural result we get a sublinear version of Yamada's formula [Formula: see text] where the integral is the Itô integral under the G-expectation.

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DIVERGENCE RESULTS FOR SELF-INTERSECTION LOCAL TIMES OF GAUSSIAN ${\mathscr S}' ({\mathbb R}^d)$-PROCESSES

Infinite Dimensional Analysis Quantum Probability and Related Topics ◽

10.1142/s0219025701000668 ◽

2001 ◽

Vol 04 (04) ◽

pp. 417-488 ◽

Cited By ~ 4

Author(s):

ANNA TALARCZYK

Keyword(s):

Local Time ◽

Necessary Conditions ◽

Particle Systems ◽

The Self ◽

Local Times ◽

Intersection Local Time ◽

Convergence In Law ◽

Stable Particle ◽

Gaussian Formula ◽

Intersection Local Times

For various types of Gaussian [Formula: see text]-processes we consider the case when the self-intersection local time (SILT) does not exist. We study the rate of divergence of the corresponding approximating processes obtaining, after suitable normalizations convergence in law to some [Formula: see text]-valued processes (not necessarily Gaussian). We also obtain some new necessary conditions for the existence of SILT. We give examples associated with fluctuation limits of α-stable particle systems.

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