Classical Brownian Motion

import numpy as np
import matplotlib.pyplot as plt
from matplotlib import animation, rc
from IPython.display import HTML

np.random.seed(140)

Lets begin by defining Brownian Motion.

Definition: Brownian motion, denoted $B_t$ for $t>0$, is a stochastic process such that

$B_0$ = 0
$B_t$ is continous almost surely
The increments of the process are normally distributed. $B_t - B_s \sim N(0, |t-s|)$
If $0 \leq t_1 < t_2 < t_3 $ then $B_{t_2} - B_{t_1}$ is independent of $B_{t_3} - B_{t_2}$

Suppose we want to simulate Brownian Motion from $0$ to $t_{\text{max}} = 2$ with 500 frames.

t_max = 2
frames = 500

This means we will update our time every $\frac{t_{max}}{\text{frames}}$ seconds for $0 \leq t \leq t_{\text{max}}$

t = np.linspace(0, t_max, frames)

Next we create an array of normally distributed numbers which will represent the increments at each time step. The variance of these numbers is $\frac{t_{\text{max}}}{\text{frames}}$.

interval_displacement = np.random.normal(scale = np.sqrt(t_max/frames), size=len(t))

The motion of our process is simply the cumulative sum of this array. After all, the position of a sequence is the sequence of partial sums of displacement.

BM = np.cumsum(interval_displacement)
BM[:4]

array([-0.02954629, -0.01082589,  0.00090491, -0.05146864])

We can plot the process evolve through time

fig, ax = plt.subplots()
ax.set_xlim((0, t_max))
vertical_height = max(abs(max((BM))), abs(min(BM)))
ax.set_ylim(min(BM),max(BM))
plt.title("Path of Brownian Motion")
plt.xlabel("t")
plt.ylabel("$B_t$")
plt.close();

line, = ax.plot([], [], lw=2)

def BM_path(time):
    line.set_data(t[:time], BM[:time])
    return (line,)

path_animation = animation.FuncAnimation(fig, BM_path, frames=frames, interval=10, blit=True)
HTML(path_animation.to_html5_video())