The Perron-Frobenius Theorem for Markov Semigroups

Preprint English OPEN
Hijab, Omar;
(2014)
  • Subject: 47D07 (Primary) 58J65, 52A40 (Secondary) | Mathematics - Functional Analysis
    arxiv: Mathematics::Representation Theory | Mathematics::Classical Analysis and ODEs | Mathematics::Analysis of PDEs | Mathematics::Probability | Mathematics::Operator Algebras

Let $P^V_t$, $t\ge0$, be the Schrodinger semigroup associated to a potential $V$ and Markov semigroup $P_t$, $t\ge0$, on $C(X)$. Existence is established of a left eigenvector and right eigenvector corresponding to the spectral radius $e^{\lambda_0t}$ of $P^V_t$, simult... View more
  • References (22)
    22 references, page 1 of 3

    dπ ≤ Z dµ ≥ 0 f ∈ C(X).

    [1] S. Aida (1998) “Uniform Positivity Improving Property, Sobolev Inequalities, and Spectral Gaps” J. Functional Analysis 158, 152-185.

    [2] D. Bakry (2004) “Functional inequalities for Markov semigroups,” Probability measures on groups, Probability measures on groups, Mumbai, India.

    [3] J.-D. Deuschel and D. W. Stroock (1984) “Large Deviations,” Pure and Applied Mathematics Series 137, Academic Press.

    [4] M. D. Donsker and S. R. S. Varadhan (1975) “On a variational formula for the principal eigenvalue for operators with maximum principle,” Proceedings of the National Academy of Sciences USA 72, 780-783 (http://www.pnas.org/content/72/3/780.full.pdf).

    [5] M. D. Donsker and S. R. S. Varadhan (1975) “Asymptotic evaluation of certain Markov process expectations for large time, I,”' Communications on Pure and Applied Mathematics XXVIII, 1-47.

    [6] R. S. Ellis (1985) “Entropy, Large Deviations, and Statistical Mechanics,” Grundlehren der mathematischen Wissenschaften 271, Springer-Verlag.

    [7] S. Friedland (1981) “Convex spectral functions,” Linear and Multilinear Algebra 9, 299-316.

    [8] S. Friedland and S. Karlin (1975) “Some inequalities for the spectral radius of non-negative matrices and applications,” Duke Mathematical Journal 42, 459-490.

    [9] G. Frobenius (1908) “U¨ ber Matrizen aus positiven Elementen,” S.-B. Preuss. Akad. wiss. Berlin, 471-476.

  • Metrics
Share - Bookmark