Consider a 1 dimensional random walk with variable transition probabilities. Suppose the probability of stepping right is given by the sigmoid function $p(x)=\frac{1}{1+e^{-Cx}}$ for $C\geq 0$. When $x<0$, $p(x)>1/2$, so it is likely that $x$ will increase, whereas when $x>0$, it is more likely that $x$ will decrease. In that sense, $x=0$ is an attractor, because no matter where we start, eventually we will end up and stay around $0$. Consider this walk as a Markov process and let $\pi$ be the invariant distribution that the process converges to. Pick a random variable $X$ from this distribution. By symmetry, $\mathbb{E}(X)=0$. Moreover, I believe $\text{Var}[X]$ is finite, but I am unsure how to find it in terms of $C$. Finally, it would be nice if I had a complete description of $\pi$ as well.
If we let $q_n$ be the probability that $X=n$, then the sequence $q$ satisfies:
$$ q_n = p(n-1)q_{n-1} + \left(1 - p(n+1)\right)q_{n+1} $$
Unfortunately, because $p(x)$ is the sigmoid function, I'm finding it difficult to solve this recurrence. Approximations/upper bounds on the variance are acceptable too!