If we want to get some useful information out of this, we have to average over all possible realizations of F(t), with the initial velocity as a condition. A useful quantity for example is
⟨v(t)⋅v(t)⟩v0=v−ξ2t0+2∫t0dτe−ξ(2t−τ)v0⋅⟨F(τ)⟩v0 +∫t0dτ′∫t0dτe−ξ(2t−τ−τ′)⟨F(τ)⋅F(τ′)⟩v0.