Geometric evolution under isotropic stochastic flow

Consider an embedded hypersurface M in ℝ³. For Φ_t a stochastic flow of differomorphisms on ℝ³ 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 λ₁(t) and λ₂(t) are the principal curvatures of M_t at x _t then the vector of mean curvature and Gauss curvature, (λ₁(t)+ λ₂(t), λ₁(t) λ₂(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 ℝⁿ. In this case, there are n-1 principal curvatures λ₁(t), . . . , λ_n-1(t). If P_k; k = 1, n-1 are the elementary symmetric polynomials in λ₁, . . . , λ_n-1, then the vector (P ₁(λ₁(t), . . . , λ_n-1(t)), . . . , P_n-1(λ₁(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.