Let $W_t$ is standard Brownian motion under probability measure $P$. Consider 1-D stochastic differential equation $$ dY_t = dt + \sigma(Y_t) dW_t, \ Y_0 = y\ge 0.$$ We assume $\sigma(0) = 0$, and $\sigma(x)$ is locally Holder-1/2, i.e. for any bounded subinterval $I\subset [0,\infty)$, we assume there exists constant $K_I$ s.t. $$|\sigma(x_1)- \sigma(x_2)| \le K_I |x_1 - x_2|^{1/2}, \ \forall x_1, x_2 \in I.$$ Note that, the above SDE has strong non-negative solution by comparison with $d X_t = \sigma(X_t) dW_t$.

[Q1] Define $\tau = \inf[t>0: Y_t = 0]$. Is $\tau>0$ almost surely?

[Q2] Can one show that $Y_t > 0$ almost surely for arbitrary given $t>0$?

In fact, it's enough to show the above results with $y= 0$.

My guess is that, [Q2] is too strong to be true, but [Q1] is correct. It will be helpful to get a proof of [Q1] at least.