## Abstract

Consider an embedded hypersurface M in ℝ^{3}. For Φ_{t} a stochastic flow of differomorphisms on ℝ^{3} and x ε M, set x_{t} = Φ_{t}(x) and M_{t} = Φ_{t}(M). In this paper we will assume Φ_{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 λ_{1}(t) and λ_{2}(t) are the principal curvatures of M_{t} at x _{t} then the vector of mean curvature and Gauss curvature, (λ_{1}(t)+ λ_{2}(t), λ_{1}(t) λ_{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 λ_{1}(t), . . . , λ_{n-1}(t). If P_{k}; k = 1, n-1 are the elementary symmetric polynomials in λ_{1}, . . . , λ_{n-1}, then the vector (P _{1}(λ_{1}(t), . . . , λ_{n-1}(t)), . . . , P_{n-1}(λ_{1}(t), . . . , λ_{n-1}(t)) is a diffusion and we compute the generator explicitly. Again no projection of this di usion 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). Stochastic flows, Lyapunov exponents, principal curvatures.

Original language | English (US) |
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Journal | Electronic Journal of Probability |

Volume | 3 |

State | Published - Feb 12 1998 |

## Keywords

- Lyapunov exponents
- Principal curvatures
- Stochastic flows

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty