Geometric Evolution Under Isotropic Stochastic Flow
M. Cranston, University of Rochester
Y. Le Jan, Université de Paris, Sud
Abstract
Consider an embedded hypersurface $M$ in $RR^3$. For $Phi_t$ a
stochastic flow of differomorphisms on $RR^3$ 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 $RR^n$. In this case, there are $n-1$ principal curvatures
$lambda_1 (t), dotsc, lambda_{n-1} (t)$. If $P_k, k=1$, $n-1$ 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).
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