Stochastic modelling of closed curves in the plane

The most versatile simulation method is stochastic simulation. Initially, Enrico Fermi in the 1930s in Italy, and then John von Neumann and Stanislav Ulam in the 1940s at Los Alamos, proposed using a stochastic approach to approximate multidimensional integrals in transport equations arising in connection with the problem of the motion of a neutron in an isotropic medium. After the start of the use of computers, there was a big breakthrough, and this method began to be applied in a wide variety of problems, for which the stochastic approach proved to be more effective than other mathematical methods. In this paper, we study the shape of a random convex oval in a plane and a more general problem, the shape of a random closed curve in a plane, investigate the isoperimetric ratio – the ratio of the squared length of a curve to the area of bounded curve. The value of this ratio, due to the isoperimetric inequality, is limited and characterizes the deviation of the curve from the circle. A finite-dimensional manifold of closed regular curves in the plane and its infinite- dimensional analog are defined. The probability distributions of the isoperimetric ratio on them are studied. The main result is to establish an analytical law for the probability distribution of the ratio – as Frechet distributions, which are a particular case of the generalized distribution of extreme values. The main method used is the Fourier expansion of the support set function on the plane and the use of mathematical packages Mathematica and Matlab for stochastic modeling.