Consider the following Markov chain on $\mathbb{Z}$: $$ \Pr[X_{t+1}=x+1|X_t=x]=1-\Pr[X_{t+1}=x-1|X_t=x]=\frac{1}{2}+e^{-|x|}\cdot \mathbf{1}_{\{x\neq 0\}} $$$$ P(x,x+1)=1-P(x,x-1)=\frac{1}{2}+e^{-|x|}\cdot \mathbf{1}_{\{x\neq 0\}} $$
Do there exist constants $c,C>0$ such that $$ c\cdot P^t(z,z) \leq P^t(0,0)\leq C\cdot P^t(z,z) $$
for all $z\in\mathbb{Z}$ and $t\in\mathbb{N}$?
$$ %c\cdot \Pr[X_{2t}=z|X_0=z]\leq \Pr[X_{2t}=0|X_0=0]\leq C\cdot %\Pr[X_{2t}=z|X_0=z] %$$ $$ %\Pr[X_{t+1}=x+1|X_t=x]=1-\Pr[X_{t+1}=x-1|X_t=x]=\frac{1}{2}+e^{-|x|}\cdot %\mathbf{1}_{\{x\neq 0\}} %$$