## 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.

### MSC:

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

Zbl 0722.00030
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