Geometric Evolution Under Isotropic Stochastic Flow

Abstract

Consider an embedded hypersurface $M$ in $R^3$. For $\Phi_t$ a stochastic flow of differomorphisms on $R^3$ and $x \in M$, set $x_t = \Phi_t (x)$ and $M_t = \Phi_t (M)$. In this paper we will assume $\Phi_t$ is an isotropic (to be defined below) measure preserving flow and give an explicit descripton by SDE's of the evolution of the Gauss and mean curvatures, of $M_t$ at $x_t$. If $\lambda_1 (t)$ and $\lambda_2 (t)$ are the principal curvatures of $M_t$ at $x_t$ then the vector of mean curvature and Gauss curvature, $(\lambda_1 (t) + \lambda_2 (t)$, $\lambda_1 (t) \lambda_2 (t))$, is a recurrent diffusion. Neither curvature by itself is a diffusion. In a separate addendum we treat the case of $M$ an embedded codimension one submanifold of $R^n$. In this case, there are $n-1$ principal curvatures $\lambda_1 (t), \dotsc, \lambda_{n-1} (t)$. If $P_k, k=1,\dots,n-1$ are the elementary symmetric polynomials in $\lambda_1, \dotsc, \lambda_{n-1}$, then the vector $(P_1 (\lambda_1 (t), \dotsc, \lambda_{n-1} (t)), \dotsc, P_{n-1} (\lambda_1 (t), \dotsc, \lambda_{n-1} (t))$ is a diffusion and we compute the generator explicitly. Again no projection of this diffusion onto lower dimensions is a diffusion. Our geometric study of isotropic stochastic flows is a natural offshoot of earlier works by Baxendale and Harris (1986), LeJan (1985, 1991) and Harris (1981).