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

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

### Citations:

Zbl 0122.401; Zbl 0155.231; Zbl 0656.60046
Full Text:

### References:

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