I'm trying to understand a proof that involves the dual of stochastic processes.
The definition: Let $X^x_t$ and $Y^y_t$ be two stochastic processes starting from $x$ and $y$ respectively. They are not necessarily on the same state space. $X^x_t$ and $Y^y_t$ are dual w.r.t. a function $f$ if $$ \mathbb{E}[f(X^x_t, y)] = \mathbb{E}[f(x,Y^y_t)]$$ where the first expectation is w.r.t. the law of $X^x_t$ and the second is w.r.t. the law of $Y^y_t$. Usually $f$ is taken to be $f(x,y) = 1_{x\geq y}$ so the dual definition becomes $$ P(X^x_t, \geq y) = P(x \geq Y^y_t).$$
Chapter 3 in Liggett's textbook states the following properties of dual processes "illustrate in a simple way how information about one of the processes can be used to deduce properties of the other process."
- $P^0(X_t =0) = 1$ for all $t \geq 0$.
- $P^y(Y_t = 0)=0$ for all $t\geq 0$ and $Y>0$. (Here $P^y$ is the law of the process $Y_t$ given that it starts at $y$.)
- $L = \lim_{x \rightarrow \infty} \lim_{t\rightarrow \infty} P^x(X_t=0)$ exists and is either 0 or 1.
- If $L=1$, then $Y_t \rightarrow \infty$ in probability as $t\rightarrow \infty$ for any initial point $y\geq 0$.
- If $L=0$, then the distribution of $Y_t$ given $Y_0=0$ has a limit as $t \rightarrow \infty$. If, in addition, $\lim_{t\rightarrow \infty} P^X[0<X_t\leq y] = 0$ for all $y\geq 0$, then the distribution of $Y_t$ given $Y_0=y$ has a limit as $t\rightarrow \infty$ which is independent of $y$.
I get the proofs of the first two, which are based on the definition of duality. But the last three I don't understand. The book says that the last three parts "say that $X_t$ escapes to $\infty$ with positive probability for initial states other than zero iff $Y_t$ is positive recurrent.
I just don't get the connection between $X_t$ and $Y_t$. Is there a real-world example of dual processes? Or some intuitive explanation?