= Solution
For $\theta>0$, $E_s=\exp(\theta B_s-\theta^2s/2)$ is an <Exponential martingale for Brownian motion>, as follows directly from <independent> <Gaussian> increments. Stop it at $\tau_\delta\wedge t$, where $\tau_\delta$ is the first time $B$ reaches $\delta$. On $\{\tau_\delta\le t\}$,
$$
E_{\tau_\delta}\ge\exp(\theta\delta-\theta^2t/2).
$$
The stopped <expectation> is one and the other contribution is nonnegative, so
$$
\mathbb P(\sup_{s\le t}B_s\ge\delta)\le e^{-\theta\delta+\theta^2t/2}.
$$
For $t>0$, minimize the exponent at $\theta=\delta/t$. Apply the same argument to $-B$ and use the union bound to obtain
$$
\boxed{\mathbb P(\sup_{s\le t}|B_s|>\delta)\le2e^{-\delta^2/(2t)}.}
$$
At $t=0$ the <probability> is zero; the right side can be interpreted by its limit. This proves the <Gaussian maximal bound for Brownian motion> without losing an extra factor from a two-sided reflection argument.
Let $L$ be a global Lipschitz constant of $b$. Realize the diffusion with the stated generator by $dX_s^\varepsilon=b(X_s^\varepsilon)ds+\varepsilon dW_s$, where $W$ is a standard $d$-dimensional <Brownian motion>. Global Lipschitz <continuity> gives existence and uniqueness; its generator is exactly the printed operator. Couple it with the ordinary differential equation having the same initial value $x(0)=x_0$. Subtraction gives, with $D_t=\sup_{s\le t}|X_s^\varepsilon-x_s|$,
$$
D_t\le\varepsilon\sup_{s\le t}|W_s|+L\int_0^tD_sds.
$$
The <Gronwall inequality> implies $D_t\le\varepsilon e^{Lt}\sup_{s\le t}|W_s|$. If a <Euclidean norm> exceeds $u$, some coordinate has absolute value exceeding $u/\sqrt d$. The scalar bound and a union over the coordinates therefore give
$$
\mathbb P(D_t>\delta)\le2d\exp\!\left(-\frac{\delta^2e^{-2Lt}}{2dt\varepsilon^2}\right),\qquad t>0.
$$
Taking the <logarithm> and multiplying by $\varepsilon^2$ yields the explicit answer
$$
\boxed{\limsup_{\varepsilon\downarrow0}\varepsilon^2\log\mathbb P(D_t>\delta)\le-\frac{\delta^2e^{-2Lt}}{2dt}<0.}
$$
For $t=0$ the <probability> is zero and the limit is $-\infty$, with $\log0=-\infty$. This is <exponential small-noise concentration for a Lipschitz diffusion>.
\b[Matching the deterministic initial value is necessary.] The printed last line specifies the differential equation but does not separately write this initial condition. Without it, take $b=0$ and a deterministic solution displaced from $x_0$ by $2\delta$ in one fixed direction. The deviation event already holds at time zero, giving logarithmic rate zero instead of a strictly negative rate.
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