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The diffusion semigroup connected with the trajectories. (Le semi-groupe d’une diffusion en liaison avec les trajectoires.) (French) Zbl 0747.60070
Séminaire de probabilités XXIII, Lect. Notes Math. 1372, 326-342 (1989).
[For the entire collection see Zbl 0722.00030.]
Let \(V\) be a \(C^ \infty\) manifold and \(L\) be a \(C^ \infty\) second order elliptic operator with \(L1=0\). One can associate with \(L\) a sub- Markovian semigroup \(P_ t\) acting on the bounded measurable functions. It is known that the three following properties are equivalent:
1. The semigroup \(P_ t\) acts on \(C_ 0(V)\) and \(t\hookrightarrow P_ t\) is continuous.
2. Let \({\mathcal D}\) be the domain of \(L\) acting on \(C_ 0(V)\) in the distribution sense. Then \((L,{\mathcal D}(L))\) is the infinitesimal generator of a contraction semigroup on \(C_ 0(V)\).
3. There exists a barrier function for \(L-\lambda I\) for any (or for one) positive \(\lambda\).
The classical proofs use condition 3 to prove the equivalence between 1 and 2. One of the main purposes of this paper is to give a direct proof of this last equivalence, without using barrier functions. The regularity properties of the trajectories of the flow of solutions of the stochastic differential equation associated with \(L\) with respect to the starting point \(x\in V\) is mainly investigated. Namely let \(\hat V\) be the completion of \(V\) with a dead-point and \(\xi^ x\) be the lifetime of the process \(X^ x\) starting from \(x\in \hat V\). Then one of the essential properties is that 1 is equivalent to the stochastic continuity of \((t,x)\hookrightarrow X^ x_ t\) and that the stochastic continuity of the mapping \(x\hookrightarrow\xi^ x\) implies that \(P_ t\) maps \(CB(V)\) in \(C_ 0(V)\). One can also find numerous theorems relating the regularity of the action of \(P_ t\) on various functional spaces with the properties of the trajectories of the corresponding diffusion.

60J35 Transition functions, generators and resolvents
60J45 Probabilistic potential theory
60J60 Diffusion processes
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