Open Access
1998 Geometric Evolution Under Isotropic Stochastic Flow
Michael Cranston, Yves Le Jan
Author Affiliations +
Electron. J. Probab. 3: 1-36 (1998). DOI: 10.1214/EJP.v3-26


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).


Download Citation

Michael Cranston. Yves Le Jan. "Geometric Evolution Under Isotropic Stochastic Flow." Electron. J. Probab. 3 1 - 36, 1998.


Accepted: 12 February 1998; Published: 1998
First available in Project Euclid: 29 January 2016

zbMATH: 0890.60048
MathSciNet: MR1610230
Digital Object Identifier: 10.1214/EJP.v3-26

Primary: 60H10
Secondary: 60J60

Keywords: Lyapunov exponents , principal curvatures , Stochastic flows

Vol.3 • 1998
Back to Top