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A classic method for computing the mass function of dark matter halos isprovided by excursion set theory, where density perturbations evolvestochastically with the smoothing scale, and the problem of computing theprobability of halo formation is mapped into the so-called first-passage timeproblem in the presence of a barrier. While the full dynamical complexity ofhalo formation can only be revealed through N-body simulations, excursion settheory provides a simple analytic framework for understanding various aspectsof this complex process. In this series of paper we propose improvements ofboth technical and conceptual aspects of excursion set theory, and we exploreup to which point the method can reproduce quantitatively the data from N-bodysimulations. In paper I of the series we show how to derive excursion settheory from a path integral formulation. This allows us both to deriverigorously the absorbing barrier boundary condition, that in the usualformulation is just postulated, and to deal analytically with the non-markoviannature of the random walk. Such a non-markovian dynamics inevitably enters wheneither the density is smoothed with filters such as the top-hat filter incoordinate space (which is the only filter associated to a well defined halomass) or when one considers non-Gaussian fluctuations. In these cases, beside``markovian'' terms, we find ``memory'' terms that reflect the non-markovianityof the evolution with the smoothing scale. We develop a general formalism forevaluating perturbatively these non-markovian corrections, and in this paper weperform explicitly the computation of the halo mass function for gaussianfluctuations, to first order in the non-markovian corrections due to the use ofa tophat filter in coordinate space.

The Halo Mass Function from Excursion Set Theory. I. Gaussian fluctuations with non-markovian dependence on the smoothing scale