Moderate deviations

In van den Berg, Bolthausen and den Hollander [1] the results in (4-5) were extended to deviations on the scale of the mean. The main result reads:

Theorem 1: Let $d \geqslant 3$ and $a>0$. For every $b>0$

$\lim_{t \to \infty} \frac{1}{t^{(d-2)/d}} \log P(\vert W^a(t)\vert \leqslant bt) = - I_d^{\kappa_a}(b),$ (6)

where

$I_d^{\kappa_a}(b) = \inf_{\phi \in \Phi_d^{\kappa_a}(b)} \Big[ \frac{1}{2} \int_{\mathbb{R}^d} \vert\nabla\phi\vert^2(x) dx \Big]$ (7)

with

$\Phi_d^{\kappa_a}(b) = \Big\{\phi \in H^1(\mathbb{R}^d) \colon \int_{\mathbb{... ...,~ \int_{\mathbb{R}^d}\Big(1-e^{-\kappa_a\phi^2(x)}\Big)dx \leqslant b \Big\}.$ (8)

The idea behind Theorem 1 is that the optimal strategy for the Brownian motion to realise the moderate deviation event in (6) is to behave like a Brownian motion in a drift field $xt^{1/d} \mapsto (\nabla \phi/\phi)(x)$ for some smooth $\phi \colon \mathbb{R}^d \mapsto [0,\infty)$. The cost of adopting this drift during a time $t$ is the exponential of $t^{(d-2)/d}$ times the integral in (7). The effect of the drift is to push the Brownian motion towards the origin. Conditioned on adopting the drift, the Brownian motion spends time $\phi^2(x)$ per unit volume in the neighbourhood of $xt^{1/d}$, and it turns out that the Wiener sausage covers a fraction $1-\exp[-\kappa_a\phi^2(x)]$ of the space in that neighborhood. The best choice of the drift field is therefore given by a minimiser of the variational problem in (7), or by a minimising sequence.

We thus see that the optimal strategy for the Wiener sausage is to cover only part of the space and to leave random holes whose sizes are of order 1 and whose density varies on scale $t^{1/d}$. This strategy is more complicated than for (4).

Note that, apparently, a deviation on the scale of the mean `does not squeeze all the empty space out of the Wiener sausage'. Also note that the limit in (6) does depend on $a$.

It is clear from (6) that the case $d=2$ is critical. The following parallel of Theorem 1 is also proved in [1]:

Theorem 2:  Let $d=2$ and $a>0$. For every $b>0$

$\lim_{t \to \infty} \frac{1}{\log t} \log P(\vert W^a(t)\vert \leqslant bt/\log t) = - I_2^{2\pi}(b),$ (9)

where $I_2^{2\pi}(b)$ is given by the same formulas as in (7-8), except that $(d,\kappa_a)$ is replaced by $(2,2\pi)$.

Theorem 2 shows that for $d=2$ the moderate deviations have a polynomially small rather than an exponentially small probability. The optimal strategy is of the same type, but now the Wiener sausage lives on scale $\sqrt{t/\log t}$, which is only slightly below the diffusive scale. Contrary to the case $d \geqslant 3$, the rate function does not depend on $a$. This means that the random holes have a typical size and a typical mutual distance that tend to infinity as $t \to \infty$, washing out the dependence on the radius of the Wiener sausage.