The Annals of Probability

Geometric and Symmetry Properties of a Nondegenerate Diffusion Process

M. Cohen de Lara

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A diffusion process with smooth and nondegenerate elliptic infinitesimal generator on a manifold $M$ induces a Riemannian metric $g$ on $M$. This paper discusses in detail different symmetry properties of such a diffusion by geometric methods. Partial differential equations associated with the generator are studied likewise. With an eye to modelling and applications to filtering, relationships between symmetries of deterministic systems and symmetries of diffusion processes are delineated. The incidence of a stochastic framework on the properties of an original deterministic system are then illustrated in different examples. The construction of a diffusion process with given symmetries is also addressed and resulting geometric problems are raised.

Article information

Ann. Probab., Volume 23, Number 4 (1995), 1557-1604.

First available in Project Euclid: 19 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60J60: Diffusion processes [See also 58J65]
Secondary: 58G35

Diffusion processes Riemannian metric diffeomorphisms time changes invariance group symmetry group perturbation algebra


de Lara, M. Cohen. Geometric and Symmetry Properties of a Nondegenerate Diffusion Process. Ann. Probab. 23 (1995), no. 4, 1557--1604. doi:10.1214/aop/1176987794.

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