For each $x \in \mathbf{Z}$ let $(\eta_t(x))_{t\geq0}$ denote independent copies of a process $(\eta_t(0))_{t\geq0}$ defined as follows. The process $\eta_t(0)$ takes values in $\{-1,1\}$, where $-1$ and $1$ denote the values of a scenery with two possible states, and

• $\eta_0(0) = -1$ with probability $1/2$ and $1$ otherwise;
• at the arrival times of a Poisson process $N_t(0)$ with rate $\lambda > 0$, $\eta(0)$ switches state, ie \begin{equation} \eta_t(0) = (-1)^{N_t(0)}\eta_0(0). \end{equation}

Let $X = (X_t)_{t\geq0}$ denote an independent continuous-time nearest neighbour random walk on $\mathbf{Z}$, ie at rate $1$, $X$ jumps to one of its nearest neighbours chosen with equal probability.

What is the mean first time that $X$ encounters a scenery different from the one in which it started?

That is, denoting \begin{equation} T := \inf\{ t > 0 : \eta_t(X_t) = - \eta_0(X_0) \}, \end{equation} what is the value of $\mathbf{E}T$?

There are potentially loads of applications but the ones I have in mind are biological. Imagine the diffusing particle represents the motion of an organism through a one-dimensional habitat and $\eta_t(x)$ denotes the presence of a virus at $x$ at time $t$. How long until the organism encounters the virus?