Let's say we have a stochastic differential equation: $$ \mathrm{d}S_t = \mu \mathrm{d}t + \sigma|S^\beta| \mathrm{d}W_t $$ where $W_t$ is a standard brownian motion.

When $\beta=0$, the whole thing continues to be a brownian motion (albeit with drift and with non-unit variance). In particular, there is a non-zero probability that $S_t$ will take on negative values. Conversely, when $\beta=1$ this describes a particular instance of geometric brownian motion and as a result, there process almost certainly avoids taking on negative values.

My question: for $0 < \beta < 1$ (note: strict inequalties), what statements can be made about the probability of the process taking on negative values? In particular, are negative values almost never achieved for whenever $\beta < 1$, or is there some critical $0 < \beta_c < 1$ such that processes with $\beta < \beta_c$ are able to take on negative values with non-zero probability while those with $\beta > \beta_c$ are not?

Please note edits after original post changing the specific form of the setup

Let's say we have a stochastic differential equation: $$ \mathrm{d}S_t = |S^\beta| {(\mu \mathrm{d}t + \sigma\mathrm{d}W_t)} $$ where $W_t$ is a standard brownian motion.

When $\beta=0$, the whole thing continues to be a brownian motion (albeit with drift and with non-unit variance). In particular, there is a non-zero probability that $S_t$ will take on negative values. Conversely, when $\beta=1$ this describes a particular instance of geometric brownian motion and as a result, there process almost certainly avoids taking on negative values.

My question: for $0 < \beta < 1$ (note: strict inequalties), what statements can be made about the probability of the process taking on negative values? In particular, are negative values almost never achieved for whenever $\beta < 1$, or is there some critical $0 < \beta_c < 1$ such that processes with $\beta < \beta_c$ are able to take on negative values with non-zero probability while those with $\beta > \beta_c$ are not? Does the answer depend on the value of $\mu$? I'm most interested in the case where $\mu=0$ but left the drift term there to see if someone could offer insight about the more general setup.