Sufficient condition for invariant sets of stochastic differential equations

Question

Suppose I have an ($n$-dimensional) SDE of the form $dx_t = \mu(x_t) dt + \sigma(x_t)\,dW_t$, and suppose I have some set $S$ defined as the set $x: g(x) = 0$ for some function $g$. Using Itô's lemma I find that \begin{align} dg_t = \bigg(\nabla g(x_t) \cdot \mu(x_t) + \frac{1}{2} \text{Tr}[\sigma^\dagger(x_t) Hg(x_t) \sigma(x_t)] \bigg)dt + \nabla g(x_t) \cdot \sigma(x_t)dW_t \end{align} Suppose that $dg = 0$ for $x_t \in S$, i.e. that both the mean and the stochastic term in the above evolution equation vanish for $x \in S$ (this is a fairly non-generic situation for an SDE but the equations I have in mind have this property for some functions $g$). My question is: is this condition sufficient to show that the stochastic dynamics leave $S$ invariant, or are more conditions on the dynamics/the function $g$ necessary? The analogous condition in the fully deterministic case is sufficient to show invariance, but I am not sure about whether it is holds when stochasticity is introduced.