MathDB

Brownian motion (for example, pollen grains in water randomly pushed by collisions from water molecules) simplified to one dimension and beginning at the origin has several interesting properties. If

B(t)

denotes the position of the particle at time

t

, the average of

B(t)

x=0

, but the averate of

B(t)^{2}

t

, and these properties of course still hold if we move the space and time origins (

x=0

and

t=0

) to a later position and time of the particle (past and future are independent). What is the average of the product

B(t)B(s)

Problem Statement