# A two-dimensional diffusion process¶

## Theory¶

A two-dimensional diffusion process is a stochastic process that comprises two \(W(t)\) and allows for a mixing of these noise terms across its two dimensions.

with \(N\) the drift vector and \(g\) the diffusion matrix, which can be state dependent. We define, as the previous example, a process identical to the Ornstein—Uhlenbeck process, with

and we take \(N_1=2.0\) and \(N_2=1.0\). For this particular case a more involved diffusion matrix \(g\) will be used. Let the matrix \(g\) be state-dependent, i.e., dependent of the actual values of \(y_1\) and \(y_2\) via

and we will take \(g_{1,1} = g_{2,2}=0.5\) and \(g_{1,2} = g_{2,1} = 0\).

## Integrating a 2-dimensional process¶

Taking the above parameters and writing again an Euler–Maruyama integration method

```
# integration time and time sampling
t_final = 2000
delta_t = 0.001
# Define the drift vector N
N = np.array([2.0, 1.0])
# Define the diffusion matrix g
g = np.array([[0.5, 0.0], [0.0, 0.5]])
# The time array of the trajectory
time = np.arange(0, t_final, delta_t)
# Initialise the array y
y = np.zeros([time.size, 2])
# Generate two Wiener processes with a scale of np.sqrt(delta_t)
dW = np.random.normal(loc = 0, scale = np.sqrt(delta_t), size = [time.size, 2])
# Integrate the process (takes about 20 secs)
for i in range(1, time.size):
y[i,0] = y[i-1,0] - N[0] * y[i-1,0] * delta_t + g[0,0]/(1 + np.exp(y[i-1,0]**2)) * dW[i,0] + g[0,1] * dW[i,1]
y[i,1] = y[i-1,1] - N[1] * y[i-1,1] * delta_t + g[1,0] * dW[i,0] + g[1,1]/(1 + np.exp(y[i-1,1]**2)) * dW[i,1]
```

The stochastic trajectory in 2 dimensions for 10 time units (10000 data points)

## Back to `kramersmoyal`

and the Kramers—Moyal coefficients¶

First notice that all the results now will be two-dimensional surfaces, so we will need to plot them as such

```
# Choose the size of your target space in two dimensions
bins = np.array([300, 300])
# Introduce the desired orders to calculate, but in 2 dimensions
powers = np.array([[0,0], [1,0], [0,1], [1,1], [2,0], [0,2], [2,2]])
# insert into kmc: 0 1 2 3 4 5 6
# Notice that the first entry in [,] is for the first dimension, the
# second for the second dimension...
# Choose a desired bandwidth bw
bw = 0.1
# Calculate the Kramers−Moyal coefficients
kmc, edges = km(y, bw = bw, bins = bins, powers = powers)
# The K−M coefficients are stacked along the first dim of the
# kmc array, so kmc[1,...] is the first K−M coefficient, kmc[2,...]
# is the second. These will be 2-dimensional matrices.
```

Now one can visualise the Kramers–Moyal coefficients (surfaces) in green and the
respective theoretical surfaces in black. (Don’t forget to normalise:
`kmc / delta_t`

).