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Strong mixing measures for $C_0$-semigroups

Marina Murillo-Arcila, Alfredo Peris

Abstract

Our purpose is to obtain a very effective and general method to prove that certain $C_0$-semigroups admit invariant strongly mixing measures. More precisely, we show that the Frequent Hypercyclicity Criterion for $C_0$-semigroups ensures the existence of invariant mixing measures with full support. We will several examples, that range from birth-and-death models to the Black-Scholes equation, which illustrate these results.

Strong mixing measures for $C_0$-semigroups

Abstract

Our purpose is to obtain a very effective and general method to prove that certain -semigroups admit invariant strongly mixing measures. More precisely, we show that the Frequent Hypercyclicity Criterion for -semigroups ensures the existence of invariant mixing measures with full support. We will several examples, that range from birth-and-death models to the Black-Scholes equation, which illustrate these results.

Paper Structure

This paper contains 3 sections, 4 theorems, 46 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Invariant measures and the frequent hypercyclicity criterion
  3. Applications

Key Result

Theorem 1

Let $(T_t)_t$ be a $C_0$-semigroup on a separable Banach space $X$. If there exist, $X_0\subset X$ dense in $X$, and maps $S_t:X_0\rightarrow X_0$, $t>0$, such that then $(T_t)_{t\geq 0}$ is frequently hypercyclic.

Figures (1)

  • Figure 1: Graph of a typical function $f\in A$

Theorems & Definitions (11)

  • Theorem 1: mangino_peris2011frequently
  • Theorem 2: murillo_peris_jmaa_13
  • Theorem 3
  • proof
  • Remark 4
  • Corollary 5
  • Example 6
  • Example 7
  • Example 8
  • Example 11
  • ...and 1 more