Projection of the infinitesimal generator of a diffusion. (English) Zbl 0683.60055

Airault, Hélène

J. Funct. Anal. 85, No. 2, 353-391 (1989).

The author first considers the Wiener space X with the Ornstein-Uhlenbeck process. From a non-degenerated map g from X into a finite dimensional differentiable manifold V, the following invariants are constructed: (1) a Riemannian metric on V; (2) a vector field Z on V.
The image \(g_*\mu\) of the Wiener measure \(\mu\) through the map g has a density k with respect to the Riemannian volume constructed on V. The relation \(Z=\nabla \log k\) holds; the infinitesimal operator projected through the map g is equal to \(\Delta +Z\nabla\) where \(\Delta\) is the Laplacian associated to the Riemannian metric constructed on V. The vector field Z is exactly computed and is found to be a projection of an infinite version of the Laplacian introduced by J. Eells jun. and J. H. Sampson [Am. J. Math. 86, 109-160 (1964; Zbl 0122.401)].
Then the author considers the heat process on X: in that case a non- autonomous parabolic equation gives the law of \(g_*\mu\). The second order operator involved in this parabolic equation is strictly elliptic. The method is applied to the Heisenberg diffusion: in that case, the map g from X to \({\mathbb{R}}^ 3\) is constructed via the solution of the stochastic differential equation defining the Heisenberg diffusion.
In the first appendix, the author develops the method for a Markov chain; the map g lumps together two consecutive states into a single state. As example, the Bernoulli-Laplace model of diffusion [see W. Feller, An introduction to probability theory and its applications, Vol. 1 (1968; Zbl 0155.231)] is given. In the very particular case where \(r=4\), it happens that the image \(Y_{2n}=g(X_{2n})\) is Markov, though \(g(X_ n)\) is not Markov.
In the second appendix, the author using results of P. Malliavin and herself [Bull. Sci. Math., II. Ser. 112, No.1, 3-52 (1988; Zbl 0656.60046)] considers an Ornstein-Uhlenbeck process on a submanifold of finite codimension in X, and the projection on \({\mathbb{R}}^ k\) of the infinitesimal generator of this process.

Reviewer: H.Airault

MSC:

60J60	Diffusion processes
60J35	Transition functions, generators and resolvents
35K99	Parabolic equations and parabolic systems

Keywords:

Wiener space; Ornstein-Uhlenbeck process; Wiener measure; Riemannian metric; parabolic equation; Heisenberg diffusion; infinitesimal generator

Citations:

Zbl 0122.401; Zbl 0155.231; Zbl 0656.60046

References:

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