7 Non-asymptotic-time Dynamics
119
a Brownian motion), a Brownian bridge (B t ) t∈[0,τ ] on [0, τ ] with parameter σ is
approximated by taking an i.i.d. sample S = {T 1 , . . . , T N } of large size N from the
uniform distribution on [0, τ ], and setting
B t = σ
√
N τ
P S ([0, t]) −
t
τ
where P S ([0, t]) is the proportion of the sample S that lies within [0, t], i.e.
P S ([0, t]) =
#(S∩[0,t])
N
. The well-definedness of the limit as N → ∞ is given by
Donsker’s theorem [4, Theorem 1.1.1]. Note that B 0 = B τ = 0.
Let us now outline the connection between a Brownian bridge and a Brownian
motion. A (zero-drift) Brownian motion with diffusion parameter σ is an infinite-time
stochastic process (M t ) t∈R with continuous time-dependence, such that increments
in M t over consecutive time-intervals are independent of each other, and M t has mean
0 and standard deviation σ
√ |t|. Now given a finite time τ > 0, one can construct a
portion (M t ) t∈[0,τ ] of a Brownian motion (M t ) t∈R with diffusion parameter σ as the
sum of two independent components,
M t = t X + B t ,
(7.5)
where (B t ) t∈[0,τ ] is a Brownian bridge on [0, τ ] with parameter σ , and X is a normal
random variable independent of (B t ) t∈[0,τ ] with mean 0 and standard deviation
σ
√
τ
.
There is no way to define B t for t > τ so as to extend the construction (7.5) of
Brownian motion beyond time τ : for any t > τ, the standard deviation
σ t
√ τ
of the
component t X is already larger than the standard deviation σ
√
t of M t .
7.3.2 Finite-Time Non-autonomous Adler Equation
We consider the model (7.3) with g : [0, τ ] → R as shown in Fig. 7.1, where τ =
2π × 10
5 s. Since the construction of g (as detailed shortly) is based on a sample
realisation of a Brownian bridge, and Brownian bridges admit no natural extension
to infinite time, the model we consider exemplifies point (1) of the three main points
in Sect. 7.1.2. In this section, we will compare numerical simulation of this model
with the picture presented in Sect. 7.2.4 and observe clear agreement. This will serve
to exemplify point (2) of the abovementioned three points in Sect. 7.1.2.
The function g(t) was constructed as follows. First, a sample realisation (M t )
of a zero-drift Brownian motion with diffusion parameter
1
√
τ
was constructed on
[0, τ ], by cumulative addition of independent and identically distributed Gaussian
increments, with a time-step of 0.01 s. (The purpose of taking
1
√ τ
for the diffusion is
simply to help “bound” this long-duration process; it means that M τ has a standard
deviation of 1.) A corresponding sample realisation (B t ) t∈[0,τ ] of a Brownian bridge
of parameter
1
√ τ
was then constructed as B t = M t −
t
τ
M τ . Finally, (g(t)) t∈[0,τ ] was
119
a Brownian motion), a Brownian bridge (B t ) t∈[0,τ ] on [0, τ ] with parameter σ is
approximated by taking an i.i.d. sample S = {T 1 , . . . , T N } of large size N from the
uniform distribution on [0, τ ], and setting
B t = σ
√
N τ
P S ([0, t]) −
t
τ
where P S ([0, t]) is the proportion of the sample S that lies within [0, t], i.e.
P S ([0, t]) =
#(S∩[0,t])
N
. The well-definedness of the limit as N → ∞ is given by
Donsker’s theorem [4, Theorem 1.1.1]. Note that B 0 = B τ = 0.
Let us now outline the connection between a Brownian bridge and a Brownian
motion. A (zero-drift) Brownian motion with diffusion parameter σ is an infinite-time
stochastic process (M t ) t∈R with continuous time-dependence, such that increments
in M t over consecutive time-intervals are independent of each other, and M t has mean
0 and standard deviation σ
√ |t|. Now given a finite time τ > 0, one can construct a
portion (M t ) t∈[0,τ ] of a Brownian motion (M t ) t∈R with diffusion parameter σ as the
sum of two independent components,
M t = t X + B t ,
(7.5)
where (B t ) t∈[0,τ ] is a Brownian bridge on [0, τ ] with parameter σ , and X is a normal
random variable independent of (B t ) t∈[0,τ ] with mean 0 and standard deviation
σ
√
τ
.
There is no way to define B t for t > τ so as to extend the construction (7.5) of
Brownian motion beyond time τ : for any t > τ, the standard deviation
σ t
√ τ
of the
component t X is already larger than the standard deviation σ
√
t of M t .
7.3.2 Finite-Time Non-autonomous Adler Equation
We consider the model (7.3) with g : [0, τ ] → R as shown in Fig. 7.1, where τ =
2π × 10
5 s. Since the construction of g (as detailed shortly) is based on a sample
realisation of a Brownian bridge, and Brownian bridges admit no natural extension
to infinite time, the model we consider exemplifies point (1) of the three main points
in Sect. 7.1.2. In this section, we will compare numerical simulation of this model
with the picture presented in Sect. 7.2.4 and observe clear agreement. This will serve
to exemplify point (2) of the abovementioned three points in Sect. 7.1.2.
The function g(t) was constructed as follows. First, a sample realisation (M t )
of a zero-drift Brownian motion with diffusion parameter
1
√
τ
was constructed on
[0, τ ], by cumulative addition of independent and identically distributed Gaussian
increments, with a time-step of 0.01 s. (The purpose of taking
1
√ τ
for the diffusion is
simply to help “bound” this long-duration process; it means that M τ has a standard
deviation of 1.) A corresponding sample realisation (B t ) t∈[0,τ ] of a Brownian bridge
of parameter
1
√ τ
was then constructed as B t = M t −
t
τ
M τ . Finally, (g(t)) t∈[0,τ ] was