Ergodic limits for inhomogeneous evolution equations
Behzad Djafari Rouhani, Gisèle Ruiz Goldstein, Jerome A. Goldstein · Electronic journal of qualitative theory of differential equations · 2020
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.