= Solution
Put $\kappa=2(1-H)>0$. For the <Gaussian process> $Z$, <self-similarity of a stochastic process> gives the identity in distribution
$$
\frac{Z(N\,\cdot)}{N}\ \overset d=\ N^{H-1}Z
=\frac{Z}{\sqrt{N^\kappa}}.
$$
Apply the given small-noise <large deviation principle> with $L=N^\kappa$. The answer is
$$
\boxed{\text{speed }N^{2(1-H)},\qquad\text{good rate function }I.}
$$
No <Legendre-Fenchel transform> needs to be recomputed: the scaled paths have exactly the small-noise law at this scale.
If the stated small-noise <large deviation principle> is indexed only by integer $L$, use $L_N=\lfloor N^\kappa\rfloor$. This gives speed $N^\kappa$ as well, since $L_N/N^\kappa\to1$. The multiplying factor $\sqrt{L_N/N^\kappa}\to1$ does not change the <large deviation principle>. To justify this last step, the error in the path norm is $o(1)\|Z/\sqrt{L_N}\|$. For each fixed $M$, <good rate functions> have bounded sublevel sets, so $\inf_{\|f\|\geq M}I(f)\to\infty$ as $M\to\infty$. The closed-set upper bound then makes the <probability> of an error greater than any fixed $\eta>0$ superexponentially small. Thus <exponential equivalence> applies. This also justifies positive real, rather than only integer, rescalings used in part (c).
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