3
$\begingroup$

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?

Improve this question
$\endgroup$
4
  • $\begingroup$ Can you tell where this scenery comes from? I think the question would be more interesting. $\endgroup$
    – Dieter Kadelka
    Commented Jul 21, 2020 at 21:43
  • $\begingroup$ 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? $\endgroup$
    – as1
    Commented Jul 21, 2020 at 22:08
  • $\begingroup$ Interesting. I think you should edit your question and add this. $\endgroup$
    – Dieter Kadelka
    Commented Jul 21, 2020 at 23:27
  • $\begingroup$ Some trivial comments: Clearly something between $\lambda^{-1}$ (the first change of state along the path of $X_t$ occurs at rate $\lambda$) and $(\lambda+1)^{-1}$ (nothing will change until either $X_t$ jumps or the state at $0$ changes). For $\lambda = 1$, something close to $0.681$ (based on a sample of one million trials). $\endgroup$
    – Mateusz Kwaśnicki
    Commented Jul 25, 2020 at 20:41

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .