\"Uber die Ausgangsmomentverteilung

I. S. Zhitomirskii

Teoriâ veroâtnostej i ee primeneniâ, Tome 8 (1963) no. 2, pp. 156-166

Résumé

Assume that in a topological space $E$ a Markov process $x(t,\omega),t >0,\omega\in\Omega$, is given which is continuous from the right and let $\Gamma$ be a closed set of the space $E$. We consider the problem of the distribution $P(t)$ of the time of the first exit from $\Gamma$, for $x(0,\omega)\in\Gamma$, or the equivalent problem of the probability $Q(t)$ of "no-exit" from $\Gamma$ during the time interval $(0,t]$. Let $\mathcal{B}$ be the $\sigma$-algebra generated by the open sets of the space $E$. We set $$\begin{gathered} A_t (M)=\{\omega:x(t,\omega)\in M\},\qquad t\geq0,\qquad M\in\mathcal{B},\hfill\\D_t^{t'}(\omega)=\left\{{x:x(u,\omega)=x,u\in\left[{t,t'}\right]}\right\},\hfill\\B_t^{t'}=\left\{{\omega:D_t^{t'}\subseteq\Gamma}\right\},\hfill\\\end{gathered}$$ and denote by $\mathcal{P}(t,x,t',M)=\{\mathbf{P}_{t,x}\{A_{t'}(M)\},0\leq t\leq t',M\in\mathcal{B}$, the transition function and by $\mu _0(M)$ a probability measure which satisfies the condition $\mu _0(\Gamma )=1$, and $$\begin{gathered}P(A)=\int_\Gamma\mathbf{P}_{0,x}(A)\mu _0(dx),\hfill\\G(t,M)=P\left\{{B_0^t A_t (M)}\right\},\qquad t>0,\quad M \in\mathcal{B}\hfill\\\end{gathered}.$$ Obviously, $Q(t)=P\left({B_0^t}\right)=G(t,\Gamma).$ We assume that $M\in\mathcal{C}_\mu,\mathcal{C}_\mu\subseteq\mathcal{B}$, if the finite limit $$K_t [u]=\mathop{\lim}\limits_{t'\to t+0}\frac{1}{{t'-t}}\left[{\int_{E}{\mathcal{P}\left({t,x;t',M}\right)\mu(dx)-\mu}(M)}\right]$$ exists. Then the following theorem holds: Theorem. If the condition $P\left\{B_0^t B_0^{t'}A_{t'}(\Gamma)\right\}=o(t'-t)$ for $t'\to t+0$ is satisfied, then the function $G(t,M)$ satisfies the equation $$\frac{{\partial^*}}{{\partial t}}G(t,M)=K_t [G],\qquad t\geq0,\quad M\subseteq\Gamma,\quad M \in\mathcal{C}_G,$$ and the condition $$\begin{gathered} G(0,M)=\mu _0 (M),\qquad M\in\mathcal{B},\hfill\\G(t,M)=0,\qquad M\subseteq \overline\Gamma,\quad t\geq0.\hfill\\\end{gathered}$$ Here ${\partial^*{{G(t,M)}/{\partial t}}}$ is a partial right derivative. As an example we consider the application of the method to the shot effect.