8 1. HEURISTICS

The variable period of the input will be denoted by some positive number T . We

therefore consider the stochastic differential equation

(1.3)

d

dt

Xt

ε

= f

t

T

, Xt

ε

+

√

ε

˙

W t,

with a one-dimensional Wiener process W (white noise

˙

W ). We may circumscribe

a more mathematical concept of stochastic resonance like this: given T (ω =

1

T

),

find the parameter ε = ε(T ) such that Xε is optimally tuned with the periodic

input f(

t

T

, ·). We pose the problem in the following (almost equivalent) way: given

ε 0, find the good scale T = T (ε) such that optimal tuning of Xε with the

periodic input is given, at least in the limit ε → 0.

1.2.1. Random motion of a strongly damped Brownian particle. The

analogy with the motion of a physical particle in a periodically changing double

well potential landscape alluded to in (1.3) (see also Mazo [72] and Schweitzer [97])

motivates us to pause for a moment and give it a little more thought. As in the

previous section, let us concentrate on a one-dimensional setting, remarking that

our treatment easily generalizes to a finite-dimensional setting. Due to Newton’s

law, the motion of a particle is governed by the impact of all forces acting on it.

Let us denote F the sum of these forces, m the mass, x the space coordinate and

v the velocity of the particle. Then

m ˙ v = F.

Let us first assume the potential to be turned off. In their pioneering work at the

turn of the twentieth century, Marian Smoluchowski and Paul Langevin introduced

stochastic concepts to describe the Brownian particle motion by claiming that at

time t

F (t) = −γv(t) + 2kTγ

˙

W t.

The first term results from friction γ and is velocity dependent. An additional sto-

chastic force represents random interactions between Brownian particles and their

simple molecular random environment. The white noise

˙

W (the formal derivative

of a Wiener process) plays the crucial role. The diffusion coeﬃcient (standard

deviation of the random impact) is composed of Boltzmann’s constant k, friction

and environmental temperature T . It satisfies the condition of the fluctuation-

dissipation theorem expressing the balance of energy loss due to friction and energy

gain resulting from noise. The equation of motion becomes

⎧

⎨

⎩

˙(t) x = v(t),

˙(t) v = −

γ

m

v(t) +

√

2kTγ

m

˙

W

t

.

In equilibrium, the stationary Ornstein–Uhlenbeck process provides its solution:

v(t) = v(0)

e−

γ

m

t

+

√

2kTγ

m

t

0

e−

γ

m

(t−s)

dWs.

The ratio β :=

γ

m

determines the dynamic behavior. Let us focus on the over-

damped situation with large friction and very small mass. Then for t

1

β

= τ

(relaxation time), the first term in the expression for velocity can be neglected,

while the stochastic integral represents a Gaussian process. By integrating, we