Exponential convergence of Markovian semigroups and their spectra on \(L^p\)-spaces
Let \((M,\mathcal B,m)\) be a probability space and \(T_t\), \(t\geq 0\), be a strongly continuous Markovian semigroup on \(L^2=L^2(m)\) such that the \(L^2\)-dual is again Markovian. Under some assumptions, it is shown that this gives rise to semigroups on all spaces \(L^p(m)\), \(1\leq p\leq\infty\), and the authors study the exponential convergence behaviour of these semigroups in dependence on \(p\). Write \(m(f) := \int f\,dm\) and define \[ \gamma_{p\to q} = -\limsup_{t\to\infty} \frac 1t\log\|T_t - m(\cdot)\|_{L^p(m)\to L^q(m)}. \] The main results of the paper are as follows: {\parindent=6mm \begin{itemize} \item[(1)]\(p\mapsto \gamma_{p\to p}\) is continuous on \((1,\infty)\) and if it is strictly positive for some \(p\), it is strictly positive for all \(p\in (1,\infty)\). For self-adjoint semigroups, the maximum is attained at \(p=2\). This is an abstract (Riesz-Thorin interpolation) argument. \item [(2)] If \(T_t\) is \((p,q)\)-hyperbounded (i.e.\ \(\|T_Kf\|_{L^q}\leq C\|f\|_{L^p}\) for some \(K>0\)) and one pair \(1<p<q<\infty\), then it is hyperbounded for all \(1<p<q<\infty\). A similar statement holds for hypercontractivity (i.e.\ hyperboundedness with \(C=1\)). \item [(3)] \(T_t\) is hyperbounded iff \(\gamma_{p\to q}\geq 0\) for some \(1<p<q<\infty\) iff \(\gamma_{p\to q}=\gamma_{2\to 2}\) for all \(p,q\in (1,\infty)\). A similar statement holds for hypercontractivity. \item [(4)] If \(T_t\) is \((p,2)\)-hyperbounded for some \(p>2\), then the spectrum of the generator is \(p\)-independent. More precisely, this is even true for the continuous and point spectra, respectively. A concrete example of non-independence of the spectrum is given. \item [(5)] For non-symmetric semigroups on \(L^2\) it is shown that the validity of a log-Sobolev inequality and a further energy estimate is sufficient for hyperboundedness. \item [(6)] Inclusions of point and continuous spectra of the generator in various \(L^p\)-settings are discussed. \end{itemize}}
scientific article; zbMATH DE number 6308337
scientific article