An isometric embedding of the g(t)-Brownian motion with application in stability and homotopy group

Abstract

In this work, we construct a [Formula: see text]-Brownian motion via an isometric embedding, whose approach permit to define the Laplace operator associated with parametrized metric [Formula: see text], for every [Formula: see text]. We present an Itô formula for stochastic flow acting on time-dependent tensor fields, in particular to the metric [Formula: see text] and consequently to the norm of a stochastic process in [Formula: see text]. We use this approach to study stability by [Formula: see text]th moment exponent of [Formula: see text]-Brownian motion and its applications on homotopy groups.