Three-dimensional Lagrangian Tracer Modelling in Wadden Sea ...

CHAPTER 2. THEORY 29
where dt denotes an infinitesimal step ∆t and � ξ(t)dt is defined as an increment
of the Wiener process � W (t). This random process was named after
the American mathematician Norbert Wiener who studied the phenomenon
of Brownian motion and gave its mathematical design. The Wiener process
W (t) describes the path of a particle due to Brownian motion with time
t and consists of an accumulation of independently distributed stochastic
increments dW (t). If W (t) and W (t + dt) are the values of the function at
times t and t + dt, respectively, then dW (t) stands for the increment of the
process in the infinitesimal interval dt
dW (t) = W (t + dt) − W (t) = ξ(t) dt. (2.4.37)
Furthermore, W (t) has the following properties (see Gardiner [1983]):
1. Start: W (t = 0) ≡ 0 (unless a different starting point is specified),
2. Trajectories: paths (trajectories) are continuous functions of t ∈ [0, ∞),
3. Mean: 〈W (t)〉 ≡ 0,
4. Correlation function: 〈W (t) W (s)〉 = min(a, b),
5. Gaussian distribution: for any t1, · · · , tn the random vector (W (t1), · · · , W (tn))
is Gaussian,
6. For any s,t: a) 〈W (t) 2 〉 ≡ t,
b) 〈W (t) − W (s)〉 ≡ 0,
c) 〈(W (t) − W (s)) 2 〉 = 〈W (t) 2 〉+〈W (s) 2 〉−2 〈W (t) W (s)〉
⎧
⎫
⎪⎨ t − s t > s ⎪⎬
= t + s − 2 min(t, s) = 0 t = s
⎪⎩
⎪⎭
s − t t < s
= � �
�t − s�, 7. Variance: 〈(W (t) − 〈W (t)〉) 2 〉 = 〈W (t) 2 〉 = t,
8. Increment: from property 5 and 6 b), c) it follows that
dW (t) = W (s) − W (t + dt) ∈ N (0, √ dt) (2.4.38)
where N (0, √ dt) is a set of Gaussian random numbers with zero mean
and a standard deviation of √ dt,
9. Increment: all increments W (t2) − W (t1), · · · , W (tn) − W (tn−1) are
statistically independent of each other for t1 ≤ t2 ≤ · · · ≤ tn−1 ≤ tn.