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Geometric Evolution Under Isotropic Stochastic Flow

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Geometric Evolution Under Isotropic Stochastic Flow
2.	Creator	Author's name, affiliation, country	Michael Cranston; University of Rochester
2.	Creator	Author's name, affiliation, country	Yves Le Jan; Université de Paris, Sud
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	Stochastic flows, Lyapunov exponents, principal curvatures
3.	Subject	Subject classification	60H10, 60J60
4.	Description	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).
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	1998-02-12
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ejp.ejpecp.org/article/view/26
10.	Identifier	Digital Object Identifier	10.1214/EJP.v3-26
11.	Source	Journal/conference title; vol., no. (year)	Electronic Journal of Probability; Vol 3
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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