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- W2039772581 abstract "Consider an embedded hypersurface $M$ in $R^3$. For $Phi_t$ a stochastic flow of differomorphisms on $R^3$ and $x in M$, set $x_t = Phi_t (x)$ and $M_t = Phi_t (M)$. In this paper we will assume $Phi_t$ 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 $lambda_1 (t)$ 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 $R^n$. In this case, there are $n-1$ principal curvatures $lambda_1 (t), dotsc, lambda_{n-1} (t)$. If $P_k, k=1,dots,n-1$ are the elementary symmetric polynomials in $lambda_1, dotsc, lambda_{n-1}$, then the vector $(P_1 (lambda_1 (t), dotsc, lambda_{n-1} (t)), dotsc, P_{n-1} (lambda_1 (t), dotsc, lambda_{n-1} (t))$ 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)." @default.
- W2039772581 created "2016-06-24" @default.
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- W2039772581 date "1998-01-01" @default.
- W2039772581 modified "2023-10-01" @default.
- W2039772581 title "Geometric Evolution Under Isotropic Stochastic Flow" @default.
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- W2039772581 doi "https://doi.org/10.1214/ejp.v3-26" @default.
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