Take the process ${x}_t$ following geometric Brownian motion (GBM) $$x_t=\mu x_t \,dt+\sigma x_t \,dW_t$$ with $x_0>0$ known. It has first moment equal to $$\text{E}[x_t]=x_0 e^{\mu t}$$ and second moment going to zero as $x_0 \to 0$.
Taking a more general process \begin{equation}dx_t=μ(t,x_t) \,dt+\sigma(t,x_t) \,dW_t, (1)\end{equation} is it possible to prove that the second moment is going to zero as $x_0 \to 0$? And, in general, for a Markovian process under a martingale measure, what are the conditions for having second moment converging to zero when $x_0 \to 0$?