Study of a functional linked to Bessel bridge
The authors investigate the behaviour of \(I_t (x,y):= E[\exp \int^t_0 l( X^{x,y}_{s,t} )ds]\), where (i) \((X^{x,y}_{s,t}\); \(0\leq s\leq t)\) denotes the \(n\)-dimensional Bessel bridge from \(x\) to \(y\) over time \(t\), \(n> 2\), (ii) \(l: \mathbb{R}_+\to \mathbb{R}_+\) is a bounded function verifying: \(\gamma_-< l(x)< \gamma_+\) (resp. \(c_1\leq (l(x)- {{k(p)} \over {x^2}}) x^{2+ q-p}\leq c_2)\) for any \(x\in [0,1 ]\) (resp. \([1, +\infty[)\), where \(\gamma_-\), \(\gamma_+\), \(c_1\) and \(c_2\) are explicit and fixed constants, \(k(p)= p(n-p- 2)/2\), \(0<p< (n-2)/ 2\), \(q>p\). The authors show that for any \(y_0>0\), there exists a positive constant \(C\) (depending only on \(l\), \(y_0\), \(n\), \(p)\) such that \(\forall t>0\), \(\forall x\geq 0\), \(0\leq y\leq y_0\), \[ C^{-1} \Biggl( {{1+x+t} \over {1+x}} \Biggr)^p\leq I_t (x,y)\leq C \Biggl( {{1+x+t} \over {1+x}} \Biggr)^p. \] This result allows the authors to give explicit lower and upper bounds of the density of hyperbolic diffusions on \(\mathbb{R}_+\), i.e. the generator is the differential operator \[ H_{\alpha, \beta} (f) (r)= (f'' (r)+ \gamma_{\alpha, \beta} (r) f' (r))/2, \qquad \gamma_{\alpha, \beta} (r)= \alpha \coth r+ 2\beta \coth 2r. \]
scientific article; zbMATH DE number 847541