Let $d,k\in\mathbb N$, $p:[0,1)^d\to[0,\infty)$ be a probability density function with respect to the $d$-dimensional Lebesuge measure $\lambda^{\otimes d}$, $\sigma^2>0$, $$A(x,y):=\sum_{i=1}^kp(x_i)e^{-\frac{\left\|x-y\right\|^2}{\sigma^2}}\;\;\;\text{for }x\in[0,1)^{dk}\text{ and }y\in[0,1)^d$$ and $$E(x):=\int\left|A(x,\;\cdot\;)-p\right|^2\:{\rm d}\lambda^{\otimes dk}.$$ Let $T>0$, $$\varrho:=e^{-\frac ET}$$ and $$c:=\int\varrho\:{\rm d}\lambda^{\otimes dk}.$$
Assume $X$ is a $[0,1)^{dk}$-valued random variable with $X\sim\frac1c\varrho\lambda^{\otimes dk}$.
Can we give a closed form expression for the distribution of the points $X_i\in[0,1)^d$?
Obviously, when $E(x)=0$, then $A(x,\;\cdot\;)$ is an approximation of $p$ ...