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Geometric evolution under isotropic stochastic flow. (English) Zbl 0890.60048
Summary: Consider an embedded hypersurface $$M$$ in $$\mathbb R^3$$. For $$\Phi_t$$ a stochastic flow of diffeomorphisms on $$\mathbb R^3$$ and $$x \in M$$, set $$x_t = \Phi_t (x)$$ and $$M_t = \Phi_t (M)$$. We assume $$\Phi_t$$ is an isotropic measure preserving flow and give an explicit description 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 $$\mathbb R^n$$. In this case, there are $$n-1$$ principal curvatures $$\lambda_1 (t), \dots, \lambda_{n-1} (t)$$. If $$P_k, k=1,\dots,n-1$$, are the elementary symmetric polynomials in $$\lambda_1, \dots, \lambda_{n-1}$$, then the vector $$(P_1 (\lambda_1 (t), \dots, \lambda_{n-1} (t)), \dots, P_{n-1} (\lambda_1 (t), \dots, \lambda_{n-1} (t)))$$ 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).

##### MSC:
 60H10 Stochastic ordinary differential equations (aspects of stochastic analysis) 60J60 Diffusion processes
##### Keywords:
stochastic flows; Lyapunov exponents; principal curvatures
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