Let any site of $\mathbb{Z}$ host a number of particles $\eta_0(x)$ which is distributed according to some probability distribution independently and identically for any site $x \in \mathbb{Z}$. At time $0$ every particle starts an independent continuos-time random walk. Let $\eta_t(x)$ be the number of particles at $x$ at time $t$.
Does the process converges to a measure that is independent on the distribution of particles at time $0$?
I don't expect to receive a complete answer here, but where could I find an answer to this question?
Let us start with one particle moving randomly at each time step with probability $\frac12$ to the left or to the right. The probability of having the particule at position $x$ at time $t$ follows the finite difference equation $$p(x,t+1)=\frac12 p(x+1,t)+\frac12 p(x-1,t).\tag{1}$$ Let us solve this equation with initial condition $p(x,0)=1$ if $x=0$ and $p(x,0)=0$ otherwise. The solution is the binomial distribution $$p(2x,2t)=\frac1{2^{2t}}\binom {2t}{t+x}\qquad\qquad p(2x+1,2t+1)=\frac{1}{2^{2t+1}}\binom{2t+1}{t+x+1}.$$ It is known that for large $t$, this distribution converges to the Gaussian distribution such that $$p(x,t)\simeq \sqrt{\frac{2}{\pi t}}\exp\left(-\frac{2x^2}{t}\right)$$
The general solution for the average number of particle at time $t$ and position $x$ is given by $$\overline{\eta}_t(x)=\sum_{y\in\mathbb Z}\eta_0(y)p(x-y,t).$$ The values $\eta_0(y)$ are fixed. For a large $t$, the Gaussian distribution is large and therefore $p$ is relatively large on $[-\sqrt{t}, \sqrt{t}]$. This means that $\overline \eta_t(x)$ is the average of the values of $\eta_0(y)$ for $y\in[x-\sqrt{t},x+\sqrt{t}]$, which is, if it exists $\bar\eta$, the mean of the starting distribution. If the second order statistics of $\eta$ is defined, the fluctuations are of order $\frac{\sigma_\eta^2}{t}$ by the central limit theorem.
If the mean of $\eta$ does not exist, the convergence is not guaranteed globally.