Let $(X_t)_{t\geq0}$ be a simple symmetric random walk on $\mathbb{Z}^d$ in continuous time.
This means that when the walker $X$ lands on a site $x\in\mathbb{Z}^d$, an $Exp(1)$-clock starts counting time; when this clock rings, the walker jumps to a randomly chosen neighbour $y\in\mathbb{Z}^d$.
Let's suppose that the walk starts at the origin, $X_0=0$. Let $x\in\mathbb{Z}^d$ and $t\geq0$ be a fixed site and instant in time, respectively. Let us consider the quantity
$$ T_x=\int_0^t1_{\{X_s=x\}}ds. $$
I encountered the above quantity in a paper I am reading. It is supposed to be intended as the (random) amount of time spent by the walker $X$ on the site $x$, up until time $t$. This is quite clear (the integrand equals $1$ whenever $X_s=x$, and $0$ otherwise).
Nevertheless, I have some perplexities regarding the formal definition of $T_x$, e.g.:
- is the integral defining $T_x$ supposed to be intended as a Riemann integral?
- the integrand is a random process (i.e. a collection of random variables, indexed by the continuous parameter $s$), and $T_x$ is thus a random variable; is everything well defined? If yes, how? What is the formal theory behind such objects?
I would be grateful to anybody pointing to references for formal definitions of the above mentioned concepts. I am familiar with collections of countably many random variables, not with collections of uncountably many of them, let alone with integrals of such quantities.
Moreover, then paper goes on stating that
$$ \mathbb{P}(T_x>0)=\frac{\mathbb{E}(T_x)}{\mathbb{E}(T_x|T_x>0)}, $$
and regarding it as a triviality. Is it the case? Could anybody help me working out the above equality?
Thank you.