Brownian motion

Idea

Think of Brownian motion $W_{t}$ as the limit of a random walk when:

The step size becomes infinitesimally small, and
The time between steps shrinks to zero, but
The variance scales properly with time.

Definition

Brownian motion $W (t)$ , also called Wiener process, is a fundamental stochastic process with the following properties:

$W (0) = 0$
Independent increments
$W (t + Δ t) - W (t) \sim N (0, Δ t)$
Continuous paths, but nowhere differentiable

Visualization: we fix a particular time increment $Δ t$ and create a draw for $W (t + Δ t) - W (t)$ . Then we "integrate" (cumulative sums) the results to obtain a sample of $W$ .

Heuristic Interpretation of $\frac{d W}{d t}$

Although $\frac{d W}{d t}$ does not exist in the classical sense (because Brownian paths are too irregular), we can informally treat it as a kind of random increment:

\frac{d W}{d t} \sim Gaussian noise \sim N (0, \infty)

So over a small interval $d t$ , we write:

d W (t) = \frac{d W}{d t} \cdot d t \sim N (0, d t)

This forms the basis of numerical simulation of Brownian motion and stochastic differential equations (SDEs).

Next level:

Functions of a Brownian motion

When we have a closed-form expression like:

X (t) = f (t, W (t))

where $W (t)$ is Brownian motion, the result is still a random function, because $W (t)$ itself is a random process.

We can visualize $X$ : we can generate several samples of $W (t)$ for a particular time increment $d t$ , and then we compute $f (t, X (t))$ .

Pure Brownian Motion (No Potential)

Brownian motion $W (t)$ has independent Gaussian increments:

d W (t) \sim N (0, d t)

This means each small step is purely random.
There is no bias — the particle drifts nowhere in particular.

This is equivalent to the free particle case in physics: no forces, no potential, just randomness.

Brownian Motion in a Potential $U (q)$

Once you add a potential $U (q)$ , something changes:

The particle feels a force $F (q) = - \nabla U (q)$
That force biases the particle’s random motion toward lower potential (more likely paths go "downhill")

This leads to a stochastic ordinary differential equation of the form:

d q (t) = - \nabla U (q (t)) d t + \sqrt{2 κ} d W (t)

So each step is still Gaussian, but centered around a drift:

d q (t) \sim N (- \nabla U (q (t)) d t, 2 κ d t)

Connecting to the Path Integral View

This connect with Feynman's path integral formulation. And it is related to Itô integral. I have to understand this better yet.