Lesson 1.3: The "Weird" Scaling Property (√Δt)

Let's ask a simple question: What is the "typical size" of one tiny, random step?

1. Our Step: We'll look at a tiny, "infinitesimal" step in time. We'll call its duration $\Delta t$ (e.g., $\Delta t = 0.01$ seconds).
2. Our Random Change: The random change in our path $W_t$ during this time is $\Delta W$ (e.g., $\Delta W = W_{0.01} - W_0$).
3. Apply Rule #4: Let's plug our tiny step into the "recipe" from Rule #4. With $s = 0$ and $t = \Delta t$, the rule tells us the distribution for our tiny step $\Delta W$ is:
   $$\Delta W \sim N(0, \Delta t)$$
4. Connect to Module 0: We learned that the notation $N(\mu, \sigma^2)$ means: Mean ($\mu$) = 0 and Variance ($\sigma^2$) = $\Delta t$.
5. The "Wow" Moment: We learned that the "intuitive" or "typical" size of a random outcome is its Standard Deviation ($\sigma$). We also learned the iron-clad relationship:
   $$\text{Standard Deviation} = \sqrt{\text{Variance}}$$

Let's plug in our Variance from Step 4:

In the "Normal World" of predictable calculus, a small change $\Delta f$ is proportional to $\Delta t$. But in our new "Random World," the "typical size" of the change $\Delta W$ is proportional to $\sqrt{\Delta t}$.

This scaling rule is the mathematical proof for why normal calculus fails. In normal calculus, a derivative (or "velocity") is found by calculating $\frac{\text{Change in Position}}{\text{Change in Time}}$, or $\frac{\Delta f}{\Delta t}$.

In the Random World, the "Typical Velocity" is $\approx \frac{\text{Typical } \Delta W}{\Delta t}$. Let's plug in our new scaling rule:

As our time step $\Delta t$ gets infinitely small ($\Delta t \to 0$), the denominator $\sqrt{\Delta t}$ goes to 0, which means the whole fraction $\frac{1}{\sqrt{\Delta t}}$ goes to infinity. The instantaneous velocity is infinite.