scholarly journals Particle picture approach to the self-intersection local time of density processes in S′(Rd)

2005 ◽  
Vol 115 (3) ◽  
pp. 449-479 ◽  
Author(s):  
Tomasz Bojdecki ◽  
Anna Talarczyk
Keyword(s):  
Local Time ◽  
The Self ◽  
Intersection Local Time

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

2001 ◽  
Vol 04 (04) ◽  
pp. 417-488 ◽  
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.

scholarly journals On the Self-Intersection Local Time of Subfractional Brownian Motion

2012 ◽  
Vol 2012 ◽  
pp. 1-27 ◽  
Author(s):  
Junfeng Liu ◽  
Zhihang Peng ◽  
Donglei Tang ◽  
Yuquan Cang
Keyword(s):  
Brownian Motion ◽  
White Noise ◽  
Local Time ◽  
Noise Analysis ◽  
The Self ◽  
White Noise Analysis ◽  
Intersection Local Time ◽  
Subfractional Brownian Motion

We study the problem of self-intersection local time ofd-dimensional subfractional Brownian motion based on the property of chaotic representation and the white noise analysis.

PARTICLE PICTURE APPROACH TO THE SELF-INTERSECTION LOCAL TIME OF BRANCHING DENSITY PROCESSES IN ${\mathcal S}' ({\mathbb R}^d)$

2007 ◽  
Vol 10 (03) ◽  
pp. 439-464 ◽  
Author(s):  
ANNA TALARCZYK
Keyword(s):  
Local Time ◽  
Particle System ◽  
Sufficient Conditions ◽  
Random Measure ◽  
Particle Systems ◽  
The Self ◽  
Intersection Local Time ◽  
Branching Particle System ◽  
Independent Particle ◽  
Density Process

The problems studied in this paper are associated with a critical branching particle system in [Formula: see text], where the particle motion is described by a Lévy process. We define the intersection local time (ILT) of two independent trees, i.e. two independent particle systems, each starting from a single particle and we give sufficient conditions for its existence. The [Formula: see text]-valued density process arises as the high density limit of a "charged" particle system, where the initial positions of particles are given by a Poisson random measure. We express the self-intersection local time of this density process by means of ILTs of pairs of trees.

scholarly journals Optimal Bounds for the Variance of Self-Intersection Local Times

2016 ◽  
Vol 2016 ◽  
pp. 1-10
Author(s):  
George Deligiannidis ◽  
Sergey Utev
Keyword(s):  
Random Walk ◽  
Local Time ◽  
Simple Random Walk ◽  
The Self ◽  
The Other ◽  
Local Times ◽  
Intersection Local Time ◽  
Moment Conditions ◽  
Symmetric Random Walk ◽  
Optimal Bounds

For a Zd-valued random walk (Sn)n∈N0, let l(n,x) be its local time at the site x∈Zd. For α∈N, define the α-fold self-intersection local time as Ln(α)≔∑xl(n,x)α. Also let LnSRW(α) be the corresponding quantities for the simple random walk in Zd. Without imposing any moment conditions, we show that the variance of the self-intersection local time of any genuinely d-dimensional random walk is bounded above by the corresponding quantity for the simple symmetric random walk; that is, var(Ln(α))=O(var⁡(LnSRW(α))). In particular, for any genuinely d-dimensional random walk, with d≥4, we have var⁡(Ln(α))=O(n). On the other hand, in dimensions d≤3 we show that if the behaviour resembles that of simple random walk, in the sense that lim infn→∞var⁡Lnα/var⁡(LnSRW(α))>0, then the increments of the random walk must have zero mean and finite second moment.

Derivative of intersection local time of independent symmetric stable motions

2017 ◽  
Vol 121 ◽  
pp. 18-28 ◽  
Author(s):  
Litan Yan ◽  
Xianye Yu ◽  
Ruqing Chen
Keyword(s):  
Local Time ◽  
Intersection Local Time

scholarly journals Condition for intersection occupation measure to be absolutely continuous

2020 ◽  
Vol 72 (9) ◽  
pp. 1304-1312
Author(s):  
X. Chen
Keyword(s):  
Stochastic Processes ◽  
Lebesgue Measure ◽  
Local Time ◽  
Minimal Condition ◽  
Occupation Measure ◽  
Absolutely Continuous ◽  
Intersection Local Time ◽  
Stationary Increments

UDC 519.21 Given the i.i.d. -valued stochastic processes with the stationary increments, a minimal condition is provided for the occupation measure to be absolutely continuous with respect to the Lebesgue measure on An isometry identity related to the resulting density (known as intersection local time) is also established.

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