Brownian motion
Brownian motion is the basic random process used to model continuously evolving stochastic fluctuations in many stochastic differential equations.
Start with the simplest question
Suppose we know the current value \(W(t)\). What happens during the next short time interval \(\Delta t\)?
The change is
\[\boxed{\Delta W=W(t+\Delta t)-W(t)}.\]For Brownian motion,
\[\boxed{\Delta W\sim N(0,\Delta t)}.\]This one statement contains the central rule.
| Part | Meaning |
|---|---|
| mean \(0\) | the increment has no preferred upward or downward direction |
| variance \(\Delta t\) | the spread of the random increment grows with the time interval |
| standard deviation \(\sqrt{\Delta t}\) | the typical size of a Brownian increment is of order \(\sqrt{\Delta t}\) |
Why do we write \(\Delta W=\sqrt{\Delta t}\,Z\)?
If
\[Z\sim N(0,1),\]then multiplying by \(\sqrt{\Delta t}\) gives
\[\boxed{\Delta W=\sqrt{\Delta t}\,Z}.\]Its mean remains zero:
\[E[\Delta W]=0,\]and its variance becomes
\[\operatorname{Var}(\Delta W)=\Delta t.\]A numerical example
Suppose
\[\Delta t=0.04.\]Then
\[\sqrt{\Delta t}=0.2.\]If a standard normal draw gives
\[Z=1.3,\]then
\[\Delta W=0.2(1.3)=0.26.\]If instead \(Z=-0.8\), then
\[\Delta W=0.2(-0.8)=-0.16.\]So the process may move either upward or downward.
What does a Brownian path look like?
Every path starts at
\[W(0)=0.\]After that, each path follows its own random sequence of increments.
Brownian motion is continuous, not discrete
A simulation may use points such as
\[0,\Delta t,2\Delta t,3\Delta t,\ldots\]to approximate the process. That does not make Brownian motion a discrete-time process.
The true mathematical process is indexed by continuous time:
\[\boxed{W(t),\qquad t\ge0}.\]The discrete grid is only a computational approximation used to draw or simulate the continuous-time process.
The formal properties
Standard Brownian motion satisfies four key properties.
| Property | Meaning |
|---|---|
| \(W(0)=0\) | the process starts at zero |
| independent increments | changes over non-overlapping time intervals are independent |
| normal increments | \(W(t)-W(s)\sim N(0,t-s)\) |
| continuous paths | the path has no jumps |
Independent increments
Consider two non-overlapping intervals:
\[[t_1,t_2]\qquad\text{and}\qquad[t_3,t_4],\qquad t_2\le t_3.\]The increments
\[W(t_2)-W(t_1)\]and
\[W(t_4)-W(t_3)\]are independent.
Knowing what Brownian motion did in one interval does not tell us the random increment in a separate future interval.
The distribution depends only on interval length
For \(0\le s The distribution depends on the interval length \(t-s\), not on the absolute starting time \(s\). Because \(W(0)=0\), taking \(s=0\) gives Therefore Its standard deviation is So the possible Brownian values spread out as time increases. At small \(t\), the distribution is narrow. At larger \(t\), it is wider because the variance equals \(t\). The expected value is zero: But this does not mean that one Brownian path stays near zero. It means that if we average many independent paths at the same time \(t\), the positive and negative values balance on average. For a small interval \(\Delta t\), For example: As the time interval shrinks, Brownian increments shrink—but only like the square root of time. An ordinary differentiable function changes approximately like Brownian motion changes on the larger scale Therefore whose typical magnitude grows without bound as \(\Delta t\to0\). These two statements are not contradictory: This unusual combination is exactly why ordinary calculus must be modified when Brownian motion appears in differential equations. In ordinary calculus, a small time increment is \(dt\). Brownian motion has a random increment \(dW_t\). Their characteristic sizes are different: This means Brownian fluctuations are much larger than \(dt\) on very small time scales. Since a Brownian increment has typical size \(\sqrt{dt}\), squaring gives In Itô calculus this becomes the formal rule This does not mean that ordinary algebra has suddenly made a random number exactly equal to time. It is shorthand for a precise limiting property called quadratic variation. Divide \([0,T]\) into many short intervals. Ordinary smooth increments satisfy squared changes that vanish when summed over a very fine partition. Brownian increments behave differently: This non-zero quadratic variation is the mathematical reason that second-order terms survive in Itô's formula. Brownian motion can be viewed informally as the continuous-time limit of many tiny independent random shocks. This viewpoint explains why Brownian motion is useful for representing unresolved environmental or demographic fluctuations when a continuous approximation is appropriate. A stochastic differential equation often has the form The Brownian term does not mean that the biological variable itself is Brownian motion. It means Brownian increments are being used to drive random fluctuations in that variable. For over a small interval \(\Delta t\), Using gives This is the basic structure behind the Euler–Maruyama method developed later. A CTMC may be more natural when individual infections or recoveries matter explicitly. Brownian-driven SDEs can be useful when populations are large enough that a continuous random approximation is appropriate. Brownian motion is a mathematical model of continuous stochastic fluctuation. It does not claim that every biological source of randomness is literally Gaussian, independent or memoryless. Whether Brownian noise is appropriate depends on the biological mechanism and the modelling scale.Brownian motion at a fixed time
The distribution at different times
Why Brownian motion does not simply average back to zero
The size of a Brownian increment
\(\Delta t\) standard deviation of \(\Delta W\) 1 1 0.01 0.1 0.0001 0.01 Why Brownian paths are so rough
Continuous but nowhere differentiable
Continuous Not differentiable there are no jumps in the path the path is too irregular to have a finite ordinary slope nearby times have nearby values zooming in does not make the path look smooth Brownian motion and ordinary time behave differently
Why does \((dW_t)^2\) behave like \(dt\)?
Quadratic variation intuition
Brownian motion as accumulated random shocks
Connection to stochastic differential equations
Term Role \(f(X_t,t)dt\) deterministic drift \(g(X_t,t)dW_t\) random fluctuation \(W_t\) Brownian motion driving the noise A simple SDE step
Brownian motion versus a CTMC path
CTMC Brownian motion state often changes by discrete jumps path is continuous event times are individually identifiable fluctuation is represented continuously useful for event-level population models useful for diffusion and SDE approximations What Brownian motion does not mean