  We investigate the effect on the Big Bang Nucleosynthesis (BBN) from the
presence of a stochastic primordial magnetic field (PMF) whose strength is
spatially inhomogeneous. We assume a uniform total energy density and a
gaussian distribution of field strength. In this case, domains of different
temperatures exist in the BBN epoch due to variations in the local PMF. We show
that in such case, the effective distribution function of particle velocities
averaged over domains of different temperatures deviates from the
Maxwell-Boltzmann distribution. This deviation is related to the scale
invariant strength of the PMF energy density $\rho_{\rm Bc}$ and the
fluctuation parameter $\sigma_{\rm B}$. We perform BBN network calculations
taking into account the PMF strength distribution, and deduce the element
abundances as functions of the baryon-to-photon ratio $\eta$, $\rho_{\rm Bc}$,
and $\sigma_{\rm B}$. We find that the fluctuations of the PMF reduces the
$^7$Be production and enhances D production. We analyze the averaged
thermonuclear reaction rates compared with those of a single temperature, and
find that the averaged charged-particle reaction rates are very different.
Finally, we constrain the parameters $\rho_{\rm Bc}$ and $\sigma_{\rm B}$ from
observed abundances of $^4$He and D, and find that the $^7$Li abundance is
significantly reduced. We also find that if the $\eta$ value during BBN was
larger than the present-day value due to a dissipation of the PMF or a
radiative decay of exotic particles after BBN or if the stellar depletion of
$^7$Li occurred, abundances of all light elements can be consistent with
observational constraints.
