Given a integral $\int\limits_{\Omega}{f(\mathbf{x})}$ with a well-posed integrand $f(\mathbf{x})$, where $\mathbf{x}$ is a point from some multidimensional space $\Omega$, one can integrate it successfully with a Markov Chain Monte Carlo method, in particular using Metropolis-Hastings method. Detailed balance and ergodicity are achieved with any integrand. The value of the desired integral always exists and is finite, even in presence of delta distributions in the integrand. This is given.

However in my case the integrand $f(\mathbf{x})$ is defined such, that (one-dimensional) delta distributions $\delta_{t_0}(t)$ occur in it in different dimensions at different (unknown) locations $t_0$. Moreover the locations of such delta distributions cannot be found by any sampling strategies or numerical optimization (proven). This makes the integrand unsamplable in terms of random walk or probabilistic sampling.

Since I have an access to the evaluation of these delta distributions during sampling and mutation process, I attempt to regularize them using some smoothing kernel (convolution with a normalized kernel function $k_r(t)=\frac{1}{r}k\left(\frac{t}{r}\right)$ with some bandwidth $r$). This leads to a tempered integrand $f_r(\mathbf{x})$. During the integration at every step $n$ I gradually shrink the kernel bandwidth $r_n$ to zero in order to achieve $f_{r_n} \to f$ as $n \to \infty$ in spirit of serial tempering. Thus I expect the integration method to be consistent, meaning that the error introduced by tempering will vanish faster than the integral converges.

Thus I have two rather similar questions:

1. Would the integral converge to proper value in presence of delta distributions in the integrand with this method? What confuses me is that the chain either tends to stuck in the vicinity of the delta distribution or would otherwise just miss it, as the mode of it becomes narrower when $n$ is large.
2. What should be the cool-down schedule of the integrand $f_{r_n}$ (i.e. the shrinkage rate of the sequence $r_n$) in order to guarantee the consistent convergence of the MCMC? In other words, what are the conditions for the cool-down schedule that allows the integral to converge before the integrand $f_{r_n}$ becomes unsamplable?