The velocity v ( t ) of the Brownian particle can be integrated to yield its trajectory r ( t ) . If it is initially located at the origin with probability 1, then the result is r ( t ) = v ( 0 ) τ τ ( 1 − − e − − t / τ τ ) + τ τ ∫ ∫ 0 t a ( t ′ ) [ 1 − − e − − ( t − − t ′ ) / τ τ ] d t ′ .