 Gaussian white-noise magnetic stretching

For a trace-free isotropic Gaussian white noise velocity gradient with covariance $⟨{\sigma }_{im}(t){\sigma }_{jn}(t^{\prime })⟩={\kappa }_2\delta (t-t^{\prime })[{\delta }_{ij}{\delta }_{mn}-({\delta }_{im}{\delta }_{jn}+{\delta }_{in}{\delta }_{jm})/4]$, interpret the multiplicative equation as the Stratonovich integral limit of smooth velocity fluctuations. The causal Furutsu–Novikov formula has endpoint weight one half. It gives $P_t=({\kappa }_2/2){\partial }_i[(B^2{\delta }_{ij}-B_i{B}_j/2){\partial }_{j}P]$ for the vector probability density function.