Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 2 / 28K / c / Solution

For the first proposed generator,the state-dependent factor cancels when the jump probabilities are normalized. Hencedepends only on $y-x$ and is symmetric. The jump chain is therefore a one-dimensional symmetric random walk with finite variance, and is recurrent. By part (b), the corresponding continuous-time process is recurrent. This proves that at least one of the two generators has the required property.