where is a smooth function on and the usual Laplace operator on . Show that with respect to the measure , the operator is essentially self-adjoint on .
- Compute for the previous operator.
- Show that if, as a bilinear form, for some , then the semigroup is stochastically complete.
- Let now be smooth vector fields on such that is a basis of for every . We denote by the Laplace-Beltrami operator associated with the corresponding Riemannian metric and the diffusion operator where is a smooth function. Show that is symmetric with respect to a measure that shall be computed.
- Show that if the vector fields ‘s are globally Lipschitz, then is essentially self-adjoint.
- Compute where is the carre du champ of .
- Deduce a criterion for the stochastic completeness of the semigroup .
- Let be an elliptic diffusion operator with smooth coefficients on . Show that can be written as , where is a smooth vector field and is the Laplace Beltrami operator of some Riemannian metric.
- Show that if is symmetric with respect to a measure equivalent to the Lebesgue measure, then it can be written as (see question 4) for some and some .