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Zbl 0561.60080
Bakry, D.; Emery, M.
Diffusions hypercontractives.
(French)
[A] SÃ©min. de probabilitÃ©s XIX, Univ. Strasbourg 1983/84, Proc., Lect. Notes Math. 1123, 177-206 (1985).

[For the entire collection see Zbl 0549.00007.] \par Let $(P\sb t)$ be a Markovian semigroup of diffusion with infinitesimal generator L and stationary probability $\mu$. The hypercontractivity of $(P\sb t)$ means that there exists a constant $\lambda >0$ such that for all $p\ge 1$, $q\ge 1$, $t>0$, satisfying $q-1\le (p-1)e\sp{\lambda t}$, $\Vert P\sb tf\Vert\sb{L\sp q}\le \Vert f\Vert\sb{L\sp p}$, $f\in L\sp p(\mu)$. After proving some equivalent formulations of hypercontractivity, including Sobolev's logarithmic inequalities, in terms of $\Gamma$ and $\Gamma\sb 2$, called square field respectively iterated square field operators by the authors: $$\Gamma (f,g)=[L(fg)- fL(g)-gL(f)],$$ $$\Gamma\sb 2(f,g)=[L\Gamma (f,g)-\Gamma (Lf,g)-\Gamma (f,Lg)],$$ sufficient conditions for hypercontractivity are established. Some examples are discussed to illustrate the availability of these conditions. As useful tools, the operators $\Gamma$ and $\Gamma\sb 2$ are investigated and calculated for some cases.
[Sh.W.He]
MSC 2000:
*60J60 Diffusion processes

Keywords: hypercontractivity; Sobolev's logarithmic inequalities

Citations: Zbl 0549.00007

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