On the Probability of a Markov Point Falling into a Plane Region with Small Diameter
М. С. Никольский · Theory of Probability and Its Applications · 1964
The following problem arises in the field of optimum control (see [1]). A point in a plane with a probability density $p(\sigma ,x,\tau ,y)$, that satisfies Kolmogorov’s equation \[ \frac{{\partial p}} {{\partial \sigma }} + \sum\limits_{i,j = 1}^2 {a^{ij} } (\sigma ,x)\frac{{\partial ^2 p}} {{\partial x^i \partial x^j }} + \sum\limits_{i = 1}^2 {b^i } (\sigma ,x)\frac{{\partial p}} {{\partial x^i }} = 0. \] A second point z moves in the same plane in accordance with the equation $z = z(t)$. A closed curve $S_t = z(t) + \varepsilon S$ moves together with z. It is similar to a stationary curve S with a small similarity coefficient $\varepsilon $. It is required to calculate the probability $\varphi (\sigma ,x,\tau )$ that a random point will intersect curve $S_A $ during the time interval $\sigma \leqq t \leqq \tau $ if at time $\sigma $ the point z is at $z(\sigma )$ and the random point is at x. It is shown in the paper that with some restrictions imposed on the coefficients in Kolmogorov’s equation for $|x - z(\sigma )| > r_0 $, where $r_0 $ is any non-zero constant, the following is true: \[ \varphi (\sigma ,x,\tau ) = \frac{{2\pi }}{{|\log \varepsilon |}}\int_\sigma ^\tau {p(\sigma ,x,s,z(s))} ds + o\left( {\frac{1} {{|\log \varepsilon |}}} \right). \]