The Shortest Path Around an Island Outside the Shallows

#### I.V. Evstigneev

1995, v.1, №3, 407-418

ABSTRACT

Let $\xi = (\xi_x(\omega))_{x\in R^2}$ be a random field on the plane $R^2$. Let $t_0$ be a domain in $R^2$ and $c$ a real number. We analyse the variational problem of finding the shortest path $\lambda(\omega)$ around the domain $t_0$ outside the random set $\{x\in R^2:\xi_x(\omega) < c\}$. We consider the random domain $\tau(\omega)$ bounded by the curve $\lambda(\omega)$. We prove that $\tau(\omega)$ is a splitting random domain for the field $\xi$. This yields the following result. If the field $\xi$ is Markovian, then $\xi$ possesses the strong Markov property with respect to $\tau$.

Keywords: variational problem,random fields,Markov property,Hausdorff measures