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Diffusions hypercontractives. (French) Zbl 0561.60080
Sémin. de probabilités XIX, Univ. Strasbourg 1983/84, Proc., Lect. Notes Math. 1123, 177-206 (1985).
[For the entire collection see Zbl 0549.00007.]
Let $$(P_ t)$$ be a Markovian semigroup of diffusion with infinitesimal generator L and stationary probability $$\mu$$. The hypercontractivity of $$(P_ t)$$ means that there exists a constant $$\lambda >0$$ such that for all $$p\geq 1$$, $$q\geq 1$$, $$t>0$$, satisfying $$q-1\leq (p-1)e^{\lambda t}$$, $$\| P_ tf\|_{L^ q}\leq \| f\|_{L^ p}$$, $$f\in L^ p(\mu)$$. After proving some equivalent formulations of hypercontractivity, including Sobolev’s logarithmic inequalities, in terms of $$\Gamma$$ and $$\Gamma_ 2$$, called square field respectively iterated square field operators by the authors: $\Gamma (f,g)=[L(fg)- fL(g)-gL(f)],$
$\Gamma_ 2(f,g)=[L\Gamma (f,g)-\Gamma (Lf,g)-\Gamma (f,Lg)],$ sufficient conditions for hypercontractivity are established. Some examples are discussed to illustrate the availability of these conditions. As useful tools, the operators $$\Gamma$$ and $$\Gamma_ 2$$ are investigated and calculated for some cases.
Reviewer: Sh.W.He

##### MSC:
 60J60 Diffusion processes
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