1.2. HEURISTICS OF OUR MATHEMATICAL APPROACH 9

obtain in the over-damped limit (β → ∞) that v and thus x is Gaussian with

almost constant mean

m(t) = x(0) +

1 −

e−βt

β

v(0) ≈ x(0)

and covariance close to the covariance of white noise, see Nelson [82]:

K(s, t) =

2kT

γ

min(s, t) +

kT

γβ

− 2 +

2e−βt

+

2e−βs

−

e−β|t−s|

−

e−β(t+s)

≈

2kT

γ

min(s, t), s, t ≥ 0.

Hence the time-dependent change of the velocity of the Brownian particle can be

neglected, the velocity rapidly converges to thermal equilibrium (˙ v ≈ 0), while the

spatial coordinate remains far from it. In the so-called adiabatic transformation,

the evolution of the particle’s position is thus given by the transformed Langevin

equation

˙(t) x =

√

2kT

γ

˙

W t.

Let us next suppose that we face a Brownian particle in an external field of force,

associated with a potential U(t, x), t ≥ 0, x ∈ R. This then leads to the Langevin

equation

⎧

⎨

⎩m

˙(t) x = v(t),

˙(t) v = −γ v(t) −

∂U

∂x

(t, x(t)) + 2kTγ

˙

W t.

In the over-damped limit, after relaxation time, the adiabatic elimination of the

fast variables (see Gardiner [46]) then leads to an equation similar to the one

encountered in the previous section, namely

˙(t) x = −

1

γ

∂U

∂x

(t, x(t)) +

√

2kT

γ

˙

W

t

.

1.2.2. Time independent potential. We now continue discussing the heuris-

tics of stochastic resonance for systems described by equations of the type encoun-

tered in the previous two sections. To motivate the link to the theory of large

deviations, we first study the case in which U(t, ·) is given by some time inde-

pendent potential function U for all t. Following Freidlin and Wentzell [40], the

description of the asymptotics contained in the large deviations principle requires

the crucial notion of action functional. It is defined for T 0 and absolutely

continuous functions ϕ: [0,T ] → R with derivative ˙ ϕ by

S0T (ϕ) =

1

2

T

0

˙ ϕ

s

− −

∂

∂x

U (ϕs)

2

ds.

By means of the action functional we can define the quasipotential function

V (x, y) = inf{S0T (ϕ): ϕ0 = x, ϕT = y, T 0},

for x, y ∈ R. Intuitively, V (x, y) describes the minimal work to be done in the

potential landscape given by U to pass from x to y. Keeping this in mind, the

relationship between U and V is easy to understand (for a more formal argument

see Chapter 3). If x and y are in the same potential well, we have

(1.4) V (x, y) = 2(U(y) −

U(x))+,