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Interpret the multiplicative Gaussian white noise as the limit of a smooth short-correlation-time velocity field, hence in the Stratonovich integral convention. Put ${\partial }_i=\partial /\partial B_i$. For a realization, differentiating the field-space Dirac delta gives its field-space continuity equationThe velocity-gradient covariance has zero trace variance: contracting $T_{mn}^{ij}$ with $m=i,n=j$ gives $3-(9+3)/4=0$. Thus the gradient is trace free, consistently with the incompressible ideal induction equation.

A noise impulse in component ${\sigma }_{jn}$ changes ${\tilde{B}}_j$ by ${\tilde{B}}_n$ times that impulse. Its instantaneous retarded action on the Dirac delta is thereforeOnly earlier times affect the present field. In the white-noise limit the integral over this causal interval has ${\int }_0^t\delta (t-t^{\prime })d{t}^{\prime }=1/2$, corresponding to the symmetric short-time regularization. Applying the Furutsu–Novikov formula and then averaging givesSubstitute into the averaged field-space continuity equation:The differentiated $B_n$ term vanishes because ${\sum }_j{T}_{mj}^{ij}=0$, and the remaining contraction isConsequently the closed vector Fokker-Planck equation for Gaussian white-noise magnetic stretching isThis is a positive diffusion operator: the displayed matrix has eigenvalues $B^2$ on the two transverse directions and $B^2/2$ along $\mathbf{B}$. Replacing the endpoint half-weight by one would double the diffusion and fail to reproduce the later rate.

For an isotropic initial ensemble in Gaussian white-noise magnetic stretching, $F(B)=4\pi B^{2}P(B)$ is the magnitude probability density function. The radial vector equation $P_t=D(B^2{P}_{BB}+4B{P}_B)$ gives the displayed conservative Fokker-Planck equation. Its integral is conserved under zero endpoint probability flux. For an anisotropic initial vector distribution, use the angularly averaged magnitude density instead; isotropic noise alone does not make arbitrary initial data isotropic.