Motion of the centroid of $k$ Brownian particles?

Question

Imagine we have $k$ Brownian particles diffusing in a three-dimensional solution, where each particle has the same diffusion coefficient $D$ (measured in $\mu^2/sec$). Now imagine that we have a hypothetical particle $C$ that sits at the centroid of the $k$ Brownian particles. What diffusion coefficient / properties does $C$ have?

Answer 1

The displacement $B_t^j-B_s^j$ of the $j$th particle is a normal random variables with mean zero and variance $D(t-s)$. These random variables, $j=1,\dots,k$, are independent. Therefore, their average is also normal, with mean zero and variance $\frac{D}{k}(t-s)$. (Variance adds up under summation, and then gets divided by $k^2$).

Conclusion: the centroid is also experiencing Brownian motion, but slower, with coefficient $D/k$.