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Le mouvement Brownien de Lévy indexé par \({\mathbb{R}}^ 3\) comme limite centrale de temps locaux d’intersection. (The Lévy Brownian motion in \({\mathbb{R}}^ 3\) as central limit of intersection local times). (French) Zbl 0653.60074

Séminaire de probabilités XXII, Strasbourg/France, Lect. Notes Math. 1321, 225-248 (1988).
[For the entire collection see Zbl 0635.00013.]
Let (\(\beta\),\(\beta\) ’) be two 3-dimensional, independent Brownian motions. The aim of this work is to precise the continuity of their intersection local times with respect to the space variable. These jointly continuous processes \(\alpha\) (x,t) can be defined by the following density of occupation formula: \[ \forall t>0,\quad \forall \quad Borel\quad functions\quad f:{\mathbb{R}}^ 3\to {\mathbb{R}}_+,\int^{t}_{0}du\int^{k}_{0}ds f(\beta_ u-\beta_ s')=\int dx f(x)\alpha (x,t). \] Our main result is that \((\beta_ t,\beta_ t',\sqrt{n}(\alpha (y/n,t)-\alpha (0,t))_{t,y}\) is weakly converging to \((\beta_ t,\beta_ t',(2/\sqrt{\pi}){\mathbb{B}}_{\alpha (0,t)}(y))_{t,y}\) when n tends to infinity where \(({\mathbb{B}}_ u(y))_{u>0,y\in {\mathbb{R}}^ 3}\) is a Gaussian process with mean zero, independent of (\(\beta\),\(\beta\) ’) and whose covariance is: \[ E[{\mathbb{B}}_ u(x){\mathbb{B}}_ v(y)]=((u\wedge v)/2)[\| x\| +\| y\| -\| x-y\|]. \] We also obtain extensions of this result for a class of additive functionals of (\(\beta\),\(\beta\) ’) involving in the limit the family of 3-dimensional fractional Brownian motions.
Reviewer: S.Weinryb

MSC:

60J65 Brownian motion
60J55 Local time and additive functionals
60F05 Central limit and other weak theorems
60G15 Gaussian processes

Citations:

Zbl 0635.00013
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