STOCHASTIC CALCULUS 08

The Fokker–Planck Equation

The Fokker–Planck equation describes how the probability density of a stochastic process changes through time.

The main idea

A stochastic differential equation describes individual random trajectories.

For example,

\[ dX_t = a(X_t,t)\,dt + b(X_t,t)\,dW_t. \]

But instead of following one trajectory, we may ask:

\[ \boxed{ \text{How is probability distributed across possible values of }X_t? } \]

The Fokker–Planck equation describes the evolution of that probability density.

1. Probability density

Let

\[ p(x,t) \]

be the probability density of \(X_t\).

Then approximately,

\[ p(x,t)\,dx \]

represents the probability that \(X_t\) lies in a small interval near \(x\).

More precisely,

\[ P(a\le X_t\le b) = \int_a^b p(x,t)\,dx. \]

2. Total probability

The density must satisfy

\[ \boxed{ \int_{-\infty}^{\infty} p(x,t)\,dx = 1 } \]

for every time \(t\), provided the process is defined over the whole real line.

3. Density changes through time

Because \(X_t\) is stochastic, its distribution may change with time.

So the density depends on both state and time:

\[ p=p(x,t). \]

The quantity

\[ \frac{\partial p}{\partial t} \]

describes how the probability density changes locally through time.

4. The one-dimensional Fokker–Planck equation

For the SDE

\[ dX_t = a(X_t,t)\,dt + b(X_t,t)\,dW_t, \]

the corresponding Fokker–Planck equation is

\[ \boxed{ \frac{\partial p}{\partial t} = - \frac{\partial}{\partial x} \left[ a(x,t)p(x,t) \right] + \frac12 \frac{\partial^2}{\partial x^2} \left[ b(x,t)^2p(x,t) \right] } \]

5. Two parts of the equation

The first term,

\[ - \frac{\partial}{\partial x} (ap), \]

comes from the drift.

The second term,

\[ \frac12 \frac{\partial^2}{\partial x^2} (b^2p), \]

comes from diffusion.

\[ \boxed{ \text{drift} \rightarrow \text{probability transport} } \] \[ \boxed{ \text{diffusion} \rightarrow \text{probability spreading} } \]

6. Drift moves probability

Suppose

\[ dX_t=\mu\,dt. \]

There is no stochastic term.

The process moves systematically at speed \(\mu\).

Its probability distribution therefore shifts through state space without being spread by Brownian motion.

7. Diffusion spreads probability

Now consider pure Brownian motion:

\[ dX_t=\sigma\,dW_t. \]

There is no deterministic drift.

The Fokker–Planck equation becomes

\[ \boxed{ \frac{\partial p}{\partial t} = \frac{\sigma^2}{2} \frac{\partial^2p}{\partial x^2} } \]

This is the diffusion equation.

8. Brownian motion example

For standard Brownian motion,

\[ dW_t=dW_t, \]

\[ a=0, \qquad b=1. \]

The density satisfies

\[ \boxed{ \frac{\partial p}{\partial t} = \frac12 \frac{\partial^2p}{\partial x^2} } \]

with initial condition concentrated at

\[ x=0. \]

9. Brownian probability density

For \(t>0\), the Brownian density is

\[ \boxed{ p(x,t) = \frac{1}{\sqrt{2\pi t}} \exp \left( -\frac{x^2}{2t} \right) } \]

This is the density of

\[ N(0,t). \]

10. What happens as time increases?

At early times, the density is concentrated near the initial value.

As time increases, it becomes wider.

This reflects

\[ \operatorname{Var}(W_t)=t. \]

Probability is spreading through state space.

11. Density-spreading diagram

12. Constant drift and diffusion

Consider

\[ dX_t = \mu\,dt + \sigma\,dW_t, \qquad X_0=x_0. \]

Its density satisfies

\[ \frac{\partial p}{\partial t} = -\mu\frac{\partial p}{\partial x} + \frac{\sigma^2}{2} \frac{\partial^2p}{\partial x^2}. \]

The solution is normally distributed:

\[ \boxed{ X_t \sim N( x_0+\mu t,\, \sigma^2t ) } \]

13. Interpretation

The centre of the density moves according to

\[ x_0+\mu t. \]

The spread increases according to

\[ \sigma^2t. \]

\[ \boxed{ \mu \rightarrow \text{movement of the centre} } \] \[ \boxed{ \sigma \rightarrow \text{rate of spreading} } \]

14. Probability flow

The Fokker–Planck equation can be written as a conservation law:

\[ \frac{\partial p}{\partial t} = -\frac{\partial J}{\partial x}, \]

where the probability flux is

\[ \boxed{ J = ap - \frac12 \frac{\partial}{\partial x} \left( b^2p \right) } \]

Probability is not created arbitrarily. It flows through state space.

15. Conservation interpretation

Consider an interval

\[ [a,b]. \]

The probability inside is

\[ \int_a^b p(x,t)\,dx. \]

Its rate of change depends on probability flowing across the boundaries.

This is analogous to conservation laws in fluid mechanics.

16. From trajectories to density

The SDE viewpoint follows individual possible trajectories:

\[ X_t^{(1)}, \quad X_t^{(2)}, \quad \ldots \]

The Fokker–Planck viewpoint describes the density of all possible states directly:

\[ p(x,t). \]

\[ \boxed{ \text{SDE} \rightarrow \text{sample-path description} } \] \[ \boxed{ \text{Fokker--Planck equation} \rightarrow \text{probability-density description} } \]

17. These describe the same stochastic model

The SDE and its Fokker–Planck equation are two mathematical descriptions of the same diffusion process.

One describes random motion path by path.

The other describes how probability is distributed across states.

18. Obtaining probabilities

Once \(p(x,t)\) is known, probabilities can be calculated directly.

For example,

\[ \boxed{ P(X_t>L) = \int_L^\infty p(x,t)\,dx } \]

This gives the probability that the process exceeds a threshold at time \(t\).

19. Obtaining expectation

The expected value is

\[ \boxed{ E[X_t] = \int_{-\infty}^{\infty} x\,p(x,t)\,dx } \]

20. Obtaining the second moment

Similarly,

\[ \boxed{ E[X_t^2] = \int_{-\infty}^{\infty} x^2p(x,t)\,dx } \]

and therefore

\[ \operatorname{Var}(X_t) = E[X_t^2] - (E[X_t])^2. \]

21. Connection with moment equations

Moment equations can be derived from the Fokker–Planck equation by multiplying by powers of \(x\) and integrating.

For example, multiplying by \(x\) and integrating can produce an equation for

\[ E[X_t]. \]

Multiplying by \(x^2\) can produce an equation for

\[ E[X_t^2]. \]

\[ \boxed{ \text{Fokker--Planck equation} \rightarrow \text{moment equations} } \]

22. Multidimensional processes

Suppose

\[ \mathbf{X}_t \]

is an \(n\)-dimensional diffusion:

\[ d\mathbf{X}_t = \mathbf{a}(\mathbf{X}_t,t)\,dt + B(\mathbf{X}_t,t)\,d\mathbf{W}_t. \]

Define the diffusion covariance matrix

\[ D=BB^T. \]

Then the density satisfies the multidimensional Fokker–Planck equation

\[ \boxed{ \frac{\partial p}{\partial t} = - \sum_i \frac{\partial}{\partial x_i} (a_i p) + \frac12 \sum_{i,j} \frac{\partial^2}{\partial x_i\partial x_j} (D_{ij}p) } \]

23. Why covariance appears

When several stochastic variables are driven by Brownian motion, their random changes may be related.

The matrix

\[ D=BB^T \]

describes the local covariance structure of those changes.

This determines how probability spreads in different directions.

24. Epidemic interpretation

Suppose a stochastic epidemic model has random susceptible and infectious states.

Instead of only studying individual trajectories

\[ (S_t,I_t), \]

we may consider a joint density

\[ p(s,i,t). \]

This density describes how probability is distributed over possible epidemic states.

25. Threshold probabilities

If \(I_t\) represents infectious prevalence and \(I_c\) is a critical threshold, then a density-based model can be used to study

\[ P(I_t>I_c). \]

This is directly relevant when decisions depend on the probability of exceeding a capacity or safety threshold.

26. Fokker–Planck versus Monte Carlo

Monte Carlo simulation approximates the distribution by generating many sample paths.

The Fokker–Planck equation attempts to evolve the probability density directly.

\[ \boxed{ \text{Monte Carlo} : \text{simulate many trajectories} } \] \[ \boxed{ \text{Fokker--Planck} : \text{solve directly for }p } \]

27. Advantages and difficulties

The Fokker–Planck equation can provide the full probability density without requiring a large number of independent simulations.

However, solving it numerically becomes much harder as the number of state variables increases.

This is one reason moment methods and Monte Carlo simulation remain important.

28. Stationary densities

Sometimes the density approaches a time-independent form

\[ p_s(x). \]

Then

\[ \frac{\partial p}{\partial t}=0. \]

The stationary density satisfies

\[ \boxed{ 0 = - \frac{d}{dx}(ap_s) + \frac12 \frac{d^2}{dx^2}(b^2p_s) } \]

when such a stationary distribution exists.

29. A crucial distinction

The Fokker–Planck equation is deterministic.

The quantity being solved for,

\[ p(x,t), \]

is a deterministic probability density.

It describes the distribution of the random process \(X_t\).

\[ \boxed{ X_t = \text{random} } \] \[ \boxed{ p(x,t) = \text{deterministic description of its probability distribution} } \]

30. The structure to remember

\[ \boxed{ dX_t = a\,dt+b\,dW_t } \] \[ \downarrow \] \[ \boxed{ \frac{\partial p}{\partial t} = -\frac{\partial}{\partial x}(ap) + \frac12 \frac{\partial^2}{\partial x^2}(b^2p) } \]

The drift transports probability, while the diffusion term spreads it.

31. The central idea

An SDE tells us how one possible random state evolves.

The Fokker–Planck equation tells us how the entire probability distribution evolves.

\[ \boxed{ \text{random trajectories} \longleftrightarrow \text{probability density} } \]

This provides a direct bridge between stochastic differential equations and probability distributions.