Simple Search

Query:

Enter a query and click »Search«...

Format:

Display: entries per page entries

Zbl 0561.60080
Bakry, D.; Emery, M.
Diffusions hypercontractives. (French)
[A] Sémin. de probabilités 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

Cited in: Zbl 1247.05139 Zbl 1246.53056 Zbl 1183.58028 Zbl 1155.53026 Zbl 1094.53030 Zbl 1073.35127 Zbl 1040.46042 Zbl 0953.60067 Zbl 0794.60064 Zbl 0759.60079 Zbl 0707.60068

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Article

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Simple Search

Zentralblatt MATH Berlin [Germany]