= Solution
For the first proposed generator,
$$
q_{x,y}=(1+|x|)^{-2}e^{-|x-y|^2},
$$
the state-dependent factor cancels when the jump probabilities are normalized. Hence
$$
P(x,y)=\frac{e^{-|x-y|^2}}{\sum_{z\ne0}e^{-z^2}}
$$
depends 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.
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