Abstract: Let $M$ be a compact 1-manifold. Given a continuous function $g:M\to \R_+$ we consider the following ordinary differential equation: \begin{center}$\|\dot{f}(t)\|=g(t)$, where $f:M\to \R^2$.\end{center} We construct a probability measure on the space of almost everywhere differentiable solutions of this differential equation and study this measure. A solution of this equation can be viewed as an isometric immersion of a compact 1-manifold into $\R^2$. Nash's convergence technique in the proof of isometric $C^1$-immersion theorem plays an important role in the construction.

This is a joint work with Mahuya Datta.