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

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 $$0$$.

### MSC:

 60J65 Brownian motion 60J55 Local time and additive functionals