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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)

MSC:
 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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