20 1. HEURISTICS

converges to 0 in probability for any δ 0. But in this case many transitions occur

in practice, and the trajectory looks chaotical instead of periodic. So we have to

choose smaller periods even if we cannot assure that the particle stays close to the

global minimum since it needs some time to cross the barrier. Let us study the

transition times. For this we assume that the starting point is −1 corresponding

to the bottom of the deepest well. If the depth of the well is always larger than

μ = ε ln T (ε), then the particle does not have enough time during one period to

climb the barrier and should therefore stay in the starting well. On the contrary if

the depth of the starting well becomes smaller than μ, the transition can and will

happen. More precisely, for μ ∈ (inft≥0 v−(t), supt≥0 v−(t)) we define

aμ

−(s)

= inf{t ≥ s: v−(t) ≤ μ}.

The first transition time from −1 to 1 denoted τ+ has the following asymptotic

behavior in the small noise limit: τ+/T (ε) → aμ

−(0).

The second transition which

lets the particle return to the starting well will appear near the deterministic time

aμ

+(aμ −(s))T

(ε). The definitions of the coeﬃcients aμ

−

and aμ

+

are similar, the depth

of the left well just being replaced by that of the right well. In order to observe

periodic behavior of the trajectory, the particle has to stay a little time in the right

well before going back. This will happen under the assumption v+(aμ(0)) μ,

that is, the right well is the deepest one at the transition time. In fact we can then

define the resonance interval IR, the set of all values μ such that the trajectories

look periodic in the small noise limit:

IR = max

i=±

inf

t≥0

vi(t), inf

t≥0

max

i=±

vi(t) .

On this interval trajectories approach some deterministic periodic limit. We now

outline the construction of a quality measure that is based on these observations, to

be optimized in order to obtain stochastic resonance as the best possible response

to periodic forcing. The measure we consider is based on the probability that a

random transition of the diffusion happens during a small time window around the

limiting deterministic transition time. Recall the transition times τ±1

ε

of

Xε

to ±1.

For h 0,ε 0,T ≥ 0 let

Mh(ε,

T ) = min

i=±

Pi

τ∓1

ε

T (ε)

∈ [aμ

i

− h, aμ

i

+ h] .

In the small noise limit, this quality measure tends to 1 and optimal tuning can be

obtained due to its asymptotic behavior described by the formula

lim

ε→0

ε ln(1 −

Mh(ε,

T )) = max{μ

i=±

− vi(aμ

i

− h)}

for μ ∈ IR, uniformly on each compact subset. This property results from classical

large deviation techniques applied to an approximation of the diffusion which is

supposed to be locally time homogeneous, and will be derived in Chapter 4. Now

we minimize the term on the left hand side in the preceding equality. In fact, if

the window length 2h is small then μ − vi(aμ

i

− h) ≈ 2hvi(aμ)

i

since vi(aμ)

i

= μ by

definition. The value vi(aμ)

i

is of course negative. Thus the position in which its

absolute value is maximal should be identified. At this position the depth of the

starting well drops most rapidly below the level μ.

It is clear that for h small the eventually existing global minimizer μR(h) is a

good candidate for the resonance point. To get rid of the dependence on h, we shall

consider the limit of μR(h) as h → 0 denote by μR. This limit, if it exists, is called