  On a locally compact group $E$ with countable base, we consider a random walk
$X$ that has a unique (up to a positive factor) $r$-invariant measure for some
$r>0$. Under some weak conditions on the measure, there is a unique continuous
exponential on $E$ naturally associated with $X$. It follows that there exists
an $R$-recurrent random walk in the sense of Tweedie on $E$ if and only if $E$
is a recurrent group and there exists a Harris random walk on~$E$.
