On the derivation of the Nakajima-Zwanzig probability density function via white noise analysis

This paper presents an application of white noise analysis in obtaining the probability density function associated with Nakajima-Zwanzig equation. We revisit the derivation of the Nakajima-Zwanzig equation and solve the probability density function. Moreover, with the parametrization, x(t)=xO+∫t0t∫sosκ(s′,s)ω(s′)ds′ds, we show that in the absence of memory effects, κ(t, s) ≈ δ(t − s), the obtained probability density for the Nakajima-Zwanzig equation reduces to that of the Gaussian distribution with σ2 = (t−t0).