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Abel transform and integrals of Bessel local times. (English) Zbl 0937.60080
Fix \(n\in]-1,0]\) and let \(R_t\) (resp. \(r_t)\) denote a Bessel process (resp. Bessel bridge) of dimension \(2(1+n)\) and started from 0. Denote by \(L_t(R)\) (resp. \(L_t(r))\) the corresponding local time at level 0, and by \({\mathcal L}\) the generator of \(R_t\). The first result here is a characterization of the laws of \(\int^s_0\varphi(v-s)dL_v(R)\) and \(\int^s_0\varphi(v-s)dL_v(r)\), for locally bounded positive \(\varphi\), given by means of some integral transform with respect to \(s\) of the Laplace transform of these laws. This is obtained by solving the parabolic equation \([{\partial\over\partial t}w={\mathcal L}w,w(0,\cdot)=w_0,w(\cdot,0)=f]\) in two ways, using the Dirichlet and the Fokker-Planck representations.
Using then the Abel integral transform \(f\mapsto A_f(t)=c_n\int^t_0(t-u)^nf'(u)du\), several consequences are deduced from this first result. The main one is as follows: for \(f\) belonging to some large class, and for a variable \(Z\) independent of \(R_t\) and having density \(A_f(1-t)1_{[0,1]}(t)\), then \(\int^Z_0{Af\over f}(1-Z+v)dL_v(R)\) is independent of \(R_Z\) and has an exponential law. The last section is a list of 48 formulae on integral transforms of local times.
Reviewer: J.Franchi (Paris)

60J65 Brownian motion
60J55 Local time and additive functionals
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
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