Stochastic processes and antiderivational equations on non-Archimedean manifolds

Stochastic processes on manifolds over non-Archimedean fields and with transition measures having values in the field C of complex numbers are studied. Stochastic antideriva-tional equations (with the non-Archimedean time parameter) on manifolds are investigated. 1. Introduction. Stochastic processes and stochastic differential equations on real Banach spaces and manifolds on them were intensively studied (see, e. and the references therein). The stochastic processes considered there were with values in either real Banach spaces or manifolds on them. The results of these investigations were used in many mathematical and theoretical physical problems. In particular, stochastic processes on some Lie groups were studied. On the other hand, the development of non-Archimedean functional analysis, non-Archimedean quantum physical theories and quantum mechanics poses problems of developing measure theory, and stochastic processes on non-Archimedean Banach In those articles, real-valued and complex-valued stochastic processes were considered, also stochastic processes with values in non-Archimedean fields and linear spaces over them, but with compact or locally compact supports of transition measures, were considered. There, pseudodifferential stochas-tic equations based on pseudodifferential operators in the sense of Vladimirov [40] were also considered. These pseudodifferential operators are quite different from an-tiderivational operators of Schikhof [37]. The latter serve as the non-Archimedean ana-log of the indefinite integration, while the former serve as non-Archimedean analogs of the classical pseudodifferential operators. There can be different variants of non-Archimedean stochastic processes, depending on whether the time parameter is either non-Archimedean or real, a space is of functions either complex-valued or with values in a non-Archimedean field. Then transition measures may be either complex-valued or with values in a non-Archimedean field. It totally gives eight variants. The case of the non-Archimedean time and a space of functions with values in a non-Archimedean space was not practically The present paper is devoted to the latter variant and its meaning is primarily in its applicability for investigations of unitary representations of totally disconnected nonlocally compact groups. It also permits the construction of volume elements associated with transition measures