Past exam of the field theories and collective phenomena in condensed matter course of the University of Oxford / 2025 / 1 / d

The integrated stochastic differential equation readsThe Gaussian white noise has zero expected value, soIts Dirac delta covariance givesThus $\boxed{⟨[y(t)-y(0){]}^2⟩=\Gamma t}$, which is also the variance because the displacement has zero expected value. More generally its covariance is $⟨Y(s)Y(t)⟩=\Gamma \min (s,t)$. Equivalently $Y(t)=\sqrt{\Gamma }{W}_t$ for standard Brownian motion, and the diffusion coefficient is $D=\Gamma /2$.