The regularized generating functional in Euclidean stochastic quantization of Dirac theory in a background field is derived in both Markovian and non-Markovian stochastic regularization schemes to show the existence of the equilibrium limit under the same condition as in the unregularized theory. Contrary to the latter case, however, the equilibrium limit depends on an arbitrary kernel in the fermion Langevin equation, which confirms our previous result on the kernel dependence of anomalies in an external gauge field. In addition to the known kernels yielding the consistent and covariant anomalies, a new kernel is proposed, which leads to a consistent form in $D=2\mathrm{}$ dimensions and a new form in $D=4\mathrm{}$. The relation to Fujikawa's path-integral method is also discussed, where the kernel introduces, in general, non-Hermitian operators which are treated by the plane-wave fermion measure and the regularized transformations of Bern et al., now depending on the kernel.

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