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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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