Geometric evolution under isotropic stochastic flow

M. Cranston, Y. LeJan

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Consider an embedded hypersurface M in ℝ3. For Φt a stochastic flow of differomorphisms on ℝ3 and x ε M, set xt = Φt(x) and Mt = Φt(M). In this paper we will assume Φt 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 xt. If λ1(t) and λ2(t) are the principal curvatures of Mt 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), . . . , λn-1(t). If Pk; k = 1, n-1 are the elementary symmetric polynomials in λ1, . . . , λn-1, then the vector (P 11(t), . . . , λn-1(t)), . . . , Pn-11(t), . . . , λn-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 languageEnglish (US)
JournalElectronic Journal of Probability
StatePublished - Feb 12 1998


  • Lyapunov exponents
  • Principal curvatures
  • Stochastic flows

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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