The squared gradient operator: A counterexample. (L’opérateur carré du champ: Un contre-exemple.) (French) Zbl 0760.60071

Mokobodzki, G.

Séminaire de probabilités XXIII, Lect. Notes Math. 1372, 324-325 (1989).

Let \(L\) be the infinitesimal generator of a Markov process. If the extended domain of \(L\) is an algebra (with ordinary multiplication), one can define the “squared gradient operator” by \(\Gamma(f,g) \equiv L(fg) - fL(g)-gL(f)\), and in this case we say that the resolvent of the process (or its transition semigroup) “admits a squared gradient operator.” It is known that the extended domain is an algebra iff it contains a complete algebra which is stable under the resolvent, and P. A. Meyer had conjectured (but failed to prove) that the stability requirement could be removed. The author produces a counterexample to Meyer’s conjecture: a certain time change of a killed Brownian motion which fails to admit a squared gradient operator although the extended domain of its generator contains a complete (but not stable) algebra.
For the entire collection see [Zbl 0738.28002].

Reviewer: J.Mitro (Cincinnati)

MSC:

60J35

Transition functions, generators and resolvents

Keywords:

infinitesimal generator of a Markov process; time change of a killed Brownian motion; gradient operator