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

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