Suppose the dynamics of X follows: $$dX_t = \mu(X_t)dt + \sigma(X_t)dZ_t$$ Where $Z_t$ is the standard Brownian motion. $\sigma(X)>0$ everywhere, and $\mu(1)>0$.
Starting from $X_0 = 1$, what would be the probability of $X_t$ hitting 1 again before time t? How should I calculate that? If I write it as a PDE for $P(X,t-\tau)$, where $P(X,t-\tau)$ denotes at time $\tau<t$, $X_t = X$, I will have the boundary condition that $P(1,t-\tau) = 1$ for any $\tau$.
Does it mean the probability always equal to $1$ for any $t>0$?