Continuous-time Gaussian white noise ($⟨A_i(t)A_j(s)⟩=q{\delta }_{ij}\delta (t-s)$)

Continuous-time Gaussian white noise is a random distribution rather than an ordinary finite-variance function at every time. Integrating it against deterministic kernels gives Gaussian random variables with covariance $q\int f(s)g(s)ds$. For independent isotropic components, a three-dimensional dot covariance has amplitude $3q$. An ordinary rapidly decaying force covariance approximates this model only above its correlation time.