Suppose $P(x,dy)$ and $Q(x,dy)$ are two Markov transition kernels on a topological space $E$ equipped with Borel $\sigma$-algebra $\mathcal B(E)$. Suppose for every $x \in E$, $P(x,\cdot)$ and $Q(x, \cdot)$ are equivalent, i.e. for any $A \in \mathcal B(E)$ and $x \in E$, $P(x,A) > 0 \Leftrightarrow Q(x,A) > 0$.


Suppose there exists a unique measure which is invariant for $Q$ (up to multiplicative constant). I explicitly do not wish to exclude the case that this invariant measure has infinite mass.



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> Is it true that, up to a multiplicative constant, there exists at most one invariant measure for $P$?
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Uniqueness of invariant measure for equivalent transition probabilities
Since you don't want to exclude the case of infinite stationary measures, it is very easy to produce a counterexample. The simple random walk on $\mathbb Z$ has a unique stationary measure (which coincides with the counting one), whereas for $p\neq q$ the random walk with the transition probabilities $p(n,n+1)=p$ and $p(n,n-1)=q$ has two minimal stationary measures. 


