On the distribution of sojourn times for the Brownian motion process on a bundle of rays. (English. Ukrainian original) Zbl 0990.60080

Shevchenko, G. M.

Theory Probab. Math. Stat. 63, 159-170 (2001); translation from Teor. Jmovirn. Mat. Stat. 63, 145-155 (2000).

Let \(S=\bigcup_{i=1}^{k}S_{i}\) be a collection of rays \(S_1,\ldots, S_{k}\) with common endpoint \(0\) and let \(b_1,\ldots,b_{k}\) and \(q_1,\ldots,q_{k}\), \(\sum_{j=1}^{k}q_{j}=1\), \(b_{j}>0\), \(q_{j}>0\), \(j=1,\ldots,k\), be given real numbers. The author considers a Wiener process \(x(t)\) which has on the ray \(S_{j}\) the diffusion coefficient \(b_{j}\) and at the point \(0\) the process \(x(t)\) chooses the ray \(S_{j}\) with probability \(q_{j}\). If \(u(t,x,\varphi)=E_{x}\varphi(x(t))\), then \[ \begin{aligned} {\partial u_{j}\over\partial t}(t,x,\varphi)&={b_{j}\over 2}{\partial^2 u_{j}\over\partial x^2}(t,x,\varphi),\quad x\in S_{j}\setminus\{0\},\;t>0,\\ \displaystyle u_1(t,0+,\varphi)&=\ldots=u_{k}(t,0+,\varphi),\quad \sum_{j=1}^{k}q_{j}{\partial u_{j}\over\partial x}(t,0+,\varphi)=0.\end{aligned} \] The density of the transition probability for \(x(t)\) is obtained. Let us denote \(\zeta_{i}=\int_0^{t}{\mathbf 1}_{S_{i}}(x(s)) ds\), \(i=1,\ldots,k\). The author derives the joint distribution density of \(\zeta_1,\ldots,\zeta_{k-1}\) for the cases of delay at the point \(0\) or without delay at the point \(0\).

Reviewer: A.D.Borisenko (Kyïv)

MSC:

60J65	Brownian motion
60J55	Local time and additive functionals

Keywords:

distribution; occupation times; Brownian motion; bundle of rays; transition probability