On a statistical problem connected with a random walk

Let $Z^d$ be a $d$-dimensional lattice and $P(0,z)$, $z\in Z^d$, be given transition function of a random walk. If there is no particle on the lattice (hypothesis $H_0$) we observe independent gaussian noises at each moment $t=0,1,\dots$ for every $z\in Z^d$. If there is a particle on the lattice (hypothesis $H_1$) then it moves according to a random walk with the transition function $P(0,z)$ and we observe the additional constant signal $\mu$ at each moment for the point where the particle is situated. For what $P(0,z)$ and $\mu$ is it possible to test the hypotheses $H_0$ and $H_1$ without error for infinite time of observation? We show that for $d=1$ and for $d=2$ it is possible to distinguish $H_0$ and $H_1$ for any $\mu\ne 0$, but for $d\ge 3$ there exists a «critical» value $\mu_0$ of $|\mu|$. Some lower and upper bounds for $\mu_0$ are obtained.