Anomalies in spaces of even and odd dimensions in the scheme of stochastic quantization

On the basis of stochastic quantization method with the generalised scheme of stochastic regularization, axial anomalies in even- and odd-dimensional spaces are studied. Although stochastic regularization preserves formally axial and gauge symmetries, the standard even-dimensional axial anomalies of Dirac fermions are correctly reproduced in the limit of taking-off the regularization, while the anomalies of chiral fermions are reproduced in the covariant form. General conditions of existence and compensation of the anomalies of massless fermions violating the parity in odd-dimensional spaces are analysed. The stochastic scheme in odd-dimensional spaces in working when the anomalies violating the parity are absent. $P$-anomalous part of the effective action of infinitely heavy fermions in odd-dimensional spaces is explicitely calculated.