Let $(E,\mathcal E,\lambda)$ be a measure space, $Q$ be a Markov kernel on $(E,\mathcal E)$ with $$Q(x,B)=\int_B\lambda({\rm d}y)q(x,y)\;\;\;\text{for all }(x,B)\in E\times\mathcal E$$ for some symmetric $q:E\times E\to[0,\infty)$, $p$ be a probability density on $(E,\mathcal E,\lambda)$, $\mu:=p\lambda$ and $\kappa$ denote the transition kernel of the Markov chain generated by the Metropolis-Hastings algorithm with proposal kernel $Q$ and target distribution $\mu$.
Let $x\in E$. Can we find a closed form expression for the total variation norm $\left\|\kappa(x,\;\cdot\;)-\mu\right\|$? (My goal is to show geometric ergodicity in a particular instance of the described setting.)
By definition, $$\kappa(x,B)=\tilde\kappa(x,B)+\underbrace{(1-\tilde\kappa(x,E))}_{=:\:r(x)}1_B(x)\;\;\;\text{for all }(x,B)\in E\times\mathcal E$$ with $$\tilde\kappa(x,B):=\int_BQ(x,{\rm d}y)\alpha(x,y)\;\;\;\text{for }(x,B)\in E\times\mathcal E$$ and $$\alpha(x,y):=\left.\begin{cases}\displaystyle\min\left(1,\frac{p(y)}{p(x)}\right)&\text{, if }p(x)>0\\0&\text{, otherwise}\end{cases}\right\}\;\;\;\text{for }x,y\in E.$$
If $k(x,y):=q(x,y)\alpha(x,y)$ for $x,y\in E$, we may write $$\kappa(x,B)-\mu(B)=\int(p(y)-k(x,y))1_B(x)+(k(x,y)-p(y))1_B(y)\:\lambda({\rm d}y)\;\;\;\text{for all }B\in\mathcal E\tag1.$$
If necessary, assume that $(E,\mathcal E)$ is a $\mathbb R$-vector space, $(E,\mathcal E,\lambda)$ is translation-invariant and $q(x,y)=\tilde q(x-y)$ for some $\mathcal E$-measurable $\tilde q:E\to[0,\infty)$. Moreover, you may assume that $\left\{p>0\right\}\subseteq\left\{q(x,\;\cdot\;)>0\right\}$ for all $x\in E$.