Assume $\varepsilon \in [0,1/2]$. Consider the discrete-time random walk $X_0 = 0$, $X_{t+1} - X_t \sim f(X_t) \delta_0 + (1-f(X_t))\operatorname{Rademacher}$, where $\operatorname{Rademacher}$ is the Rademacher distribution, and $$ f(x) = \begin{cases} 1-\varepsilon & x > 0 \\ \frac{1}{2} & x=0 \\ \varepsilon & x<0. \end{cases} $$ I am interested in the probability $O_T = \mathbb{P}[X_T > 0]$, and would like to show that $O_T > 1/2$ for sufficiently large $T$.

I tried to prove this via a continuum approximation to SDE and generalized arcsine laws and it seemed very hard for me.

I am wondering whether there is more general theory, as it seems to me that higher diffusion in some part of the space and appropriate boundary behavior should lead to the random walk spending more time in the "less diffusive" part.

I'm grateful for any pointers or proof ideas.

Assume $\varepsilon \in [0,1/2]$. Consider the discrete-time random walk $X_0 = 0$, $X_{t+1} - X_t \sim f(X_t) \delta_0 + (1-f(X_t))\operatorname{Rademacher}$, where $\delta_0$ is the Dirac delta on zero and $\operatorname{Rademacher}$ is the Rademacher distribution, and $$ f(x) = \begin{cases} 1-\varepsilon & x > 0 \\ \frac{1}{2} & x=0 \\ \varepsilon & x<0. \end{cases} $$ I am interested in the probability $O_T = \mathbb{P}[X_T > 0]$, and would like to show that $O_T > 1/2$ for sufficiently large $T$.

I tried to prove this via a continuum approximation to SDE and generalized arcsine laws and it seemed very hard for me.

I am wondering whether there is more general theory, as it seems to me that higher diffusion in some part of the space and appropriate boundary behavior should lead to the random walk spending more time in the "less diffusive" part.

I'm grateful for any pointers or proof ideas.