Exponential small-noise concentration for a Lipschitz diffusion (${\mathrm{limsup}}_{\epsilon \downarrow 0}{\epsilon }^2\log \mathbb{P}({\sup }_{s\le t}|X_s^{\epsilon }-x_s|>\delta )<0$)

Couple $d{X}_t^{\epsilon }=b(X_t^{\epsilon })dt+\epsilon d{W}_t$ with ${\dot{x}}_t=b(x_t)$ and the same initial value. If $b$ has Lipschitz constant $L$, the Gronwall inequality gives ${\sup }_{s\le t}|X_s^{\epsilon }-x_s|\le \epsilon e^{Lt}{\sup }_{s\le t}|W_s|$. Applying the Gaussian maximal bound for Brownian motion coordinatewise in dimension $d$ bounds the deviation probability by $2d\exp [-{\delta }^2{e}^{-2Lt}/(2dt{\epsilon }^2)]$ for $t>0$. Matching initial values is essential.