Wiener process in fine domain and perturbed Cauchy problem in the space of sequences (Q2777835)

The author proposes an estimate for the sojourn probability \(P_{\varepsilon}(T)=P\{w(\cdot)\in D(\varepsilon)\}\) as \(\varepsilon\to 0\), where \(D(\varepsilon)=\{\left[\varepsilon G_1(t),\varepsilon G_2(t),0\leq t\leq T\right]\}\), \(G_1(t),G_2(t)\) are two smooth functions, \(w(t)\) is a standard Wiener process. In the case \(\varepsilon=1\) we have \(P(T)=u(0,T)\), where \(u(x,t)\) is a solution of the boundary value heat conduction problem NEWLINE\[NEWLINE2u_t=u_{xx}, \quad t\in[0,T], \;x\in[G_1(T-t),G_2(T-t)];NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(G_i(T-t),t)=0, \quad u(x,0)=1,\;x\in[G_1(T),G_2(T)].NEWLINE\]NEWLINE Solutions to this equation are presented in the form \(u(x,t)=\sum_{k\geq 1}q_k(t)\varphi_k(x,t)\), where NEWLINE\[NEWLINE\varphi_k(x,t)=\sqrt{2/g(t)}\sin(k\pi(x-G_1(T-t))/g(t)),\quad g(t)=G_2(T-t)-G_1(T-t),NEWLINE\]NEWLINE and \(q_k(t)\) are determined from the corresponding Volterra equation. In the case where a small parameter \(\varepsilon\) is introduced in the boundary conditions of the heat conduction equation it can be transformed to the equation \(2u_t=\varepsilon^{-2}u_{zz}, \varepsilon z=x, (t,z)\in D(1)\). This approach gives an asymptotic estimate (as \(\varepsilon\to 0\)) for the considered sojourn probability.