= Exponential small-noise concentration for a Lipschitz diffusion
{title2=$\limsup_{\varepsilon\downarrow0}\varepsilon^2\log\mathbb P(\sup_{s\le t}|X_s^\varepsilon-x_s|>\delta)<0$}
Couple $dX_t^\varepsilon=b(X_t^\varepsilon)dt+\varepsilon dW_t$ with $\dot x_t=b(x_t)$ and the same initial value. If $b$ has Lipschitz constant $L$, the <Gronwall inequality> gives $\sup_{s\le t}|X_s^\varepsilon-x_s|\le\varepsilon e^{Lt}\sup_{s\le t}|W_s|$. Applying the <Gaussian maximal bound for Brownian motion> coordinatewise in dimension $d$ bounds the deviation <probability> by $2d\exp[-\delta^2e^{-2Lt}/(2dt\varepsilon^2)]$ for $t>0$. Matching initial values is essential.
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