Ergodic theorems for transient one-dimensional diffusions
For a one-dimensional diffusion \(X = \{X(t), t \geq 0\}\) satisfying \[ dX(t) = \mu \bigl( X(t) \bigr)dt+ \sigma \bigl( X(t) \bigr) dB(t), \;t \geq 0,\quad X(0) = x_0, \] \(B\) a Brownian motion on the line, the following problem is considered: Under which conditions on \(\mu\), \(\sigma\) and a Borel set \(B\) a relation of the form \[ \int^t_0 1_B \bigl( X(u) \bigr) du \sim \text{const.} t \;(t \to \infty)\;P_x \text{-a.s.} \] holds? The conditions on \(\mu\) and \(\sigma\) are not explicitly formulated in the context of Feller's or Dynkin's treatment of diffusions but here for short the terminology of them is used. Basically it is assumed that, with respect to the scale \(S\) and the speed measure \(m\) of \(X\), \(- \infty\) is an entrance or natural boundary and \(+ \infty\) is a natural boundary with \(S(+ \infty) < + \infty\) (implying transience of \(X\), infinite lifetime of \(X\) and \(X(t) \to + \infty\), \(P_x\)-a.s.). Then some further conditions are given relating the scale, the speed measure and the set \(B\) near \(+ \infty\) in such an appropriate manner that ergodic relations of the wanted kind hold.
scientific article; zbMATH DE number 803384
scientific article