 |
|
|
| | | | | |
|
|
|
|
|
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).
|
Full text: PDF
Pages: 1-36
Published on: February 12, 1998
|
Bibliography
-
P. Baxendale, (1984), Brownian motions in the diffeomorphism
Group. I, Compositio Math. 53, 19--50.
Math. Review 86e:58086
-
P. Baxendale and T.Harris, (1986), Isotropic stochastic flows,
Annals of Prob., Vol 14, 1155--1179.
Math. Review 88c:60030
-
R. Bishop and S. Goldberg, (1980), Tensor Analysis on
Manifolds, Dover, Toronto.
Math. Review 82g:53001
-
Hardy, Littlewood, Polya, (1973), Inequalities, Cambridge
University Press, Cambridge.
Math. Review 89d:26016
-
T. Harris, (1981), Brownian motions on the homeomorphisms of
the plane, Annals of Prob. Vol. 9, 232--254.
Math. Review 82e:60130
-
N. Ikeda, S. Watanabe, (1989), Stochastic Differential Equations
and Diffusion Processes, North-Holland, New York.
Math. Review 90m:60069
-
Y. LeJan, (1985), On isotropic Brownian motions, Zeit. für
Wahr., 70, 609--620.
Math. Review 87a:60090
-
Y. LeJan, (1991), Asymototic properties of isotropic Brownian
flows, in Spatial Stochastic Processes, ed. K. Alexander, J. Watkins,
Birkh"auser, Boston, 219--232.
Math. Review 92k:60142
-
Y. LeJan, S. Watanabe (1984), Stochastic flows of
diffeomorphisms, Proc. of the Taniguchi Symposium, 307--332, North Holland,
New York.
Math. Review 86i:58140
-
M. Spivak, (1979), A comprehensive introduction to differential
geometry, Publish or Perish, Berkeley.
Math. Review 82g:53003e
|
|
|
|
|
|
|
|
| | | | |
Electronic Journal of Probability. ISSN: 1083-6489 |
|
Context