Ergodic limits for inhomogeneous evolution equations

Abstract

Let $u$ satisfy an inhomogeneous wave equation such as \begin{align*} u''(t)+A^{2}u(t)=h(t),\qquad u(0)=f,\quad u'(0)=g. \end{align*} We show that in many cases, the limit as $t\rightarrow \infty$ of $\frac{1}{t}\int_{0}^{t}u(s)ds$ exists, and can be calculated explicitly.

Keywords

• ergodic theory
• inhomogeneous wave equations
• uniformly bounded groups
• asymptotics of linear evolution equations