On the law of the minimum of the solutions to a class of unidimensional SDEs

Abstract

We prove that the law of the minimum $m:={\min }_{t\in [0,1]}\xi (t)$ of the solution $\xi$ to a one-dimensional ODE with good nonlinearity has continuous density with respect to the Lebesgue measure. As a byproduct of the procedure, we show that the sets $\{x\in C([0,1]):\min x>r\}$ have finite perimeter with respect to the law $\nu$ of the solution $\xi (\cdot )$ in $L^2(0,1)$.