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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
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