The contact process is a well-studied Markov process. I'm just concerned with the one-dimensional nearest-neighbor version here.

The state space is $\eta\in\{0,1\}^\mathbb Z$, and for state $\eta$ at each site $x\in\mathbb Z$ the dynamics are: $0\rightarrow 1$ at rate $\lambda\sum_{y:|x-y|=1}\eta(y)$ and $1\rightarrow0$ at rate $1$.

There is a critical rate $\lambda_c$ such that for $\lambda>\lambda_c$ we have a positve probability of survival for any initial distribution that does not place unit mass of the state of all zeros. We have convergence to a stationary $\nu_\lambda$ such that $$\rho_\lambda=\nu_\lambda\left(\left\{\eta : \eta(x)=1\right\}\right)>0$$ for $\lambda>\lambda_c$ and is independent of $x.$

What I am interested in is this process restricted to the half-line, $\mathbb Z^+_0=\{0,1,2,\ldots\}$. In particular, I want to know what the stationary distribution is. It seems like we would have $$u_\lambda(x)=\nu_\lambda\left(\left\{\eta : \eta(x)=1\right\}\right)>0$$ an increasing function of $x$ with $u_\lambda(0)>0$ and $u_\lambda(x)\rightarrow\rho_\lambda$ as $x\rightarrow\infty$.

I can't seem to find any papers that discuss this. Does anyone have any relevant references or maybe can provide some relevant calculations that help to understand $u_\lambda(x)?$

**Edit:** I'm specifically interested in the questions:

1) Does $u_\lambda(x)\rightarrow\rho_\lambda$ as $x\rightarrow\infty?$

2) Is $u_\lambda(x)$ strictly increasing in $x?$

3) What is the relationship between $u_\lambda(0)$ and $\displaystyle\lim_{x\rightarrow\infty}u_\lambda(x)?$

I'm fairly certain (1) and (2) are true, but I'd especially like quantitative results for (3). The more specific about $u_\lambda(x)$ the better, however I'm not asking for anything close to an exact formula as that is likely not possible.

You can assume I'm familiar with all the basics (monotonicity, coupling, graphical representation, duality, etc., whatever is in Liggett's two books at least).