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It is known (see, for example, [1]- [3]) that compound Poisson processes with drift find wide application in numerous applied branches of probability theory (in queueing and reliability theory, inventory control, in ruin problems, and so forth). The main goal of these papers is the study of boundary functionals, i.e., functionals of trajectories of a process related to its arrival at certain boundaries. Examples of such functionals are the first passage time of a given level, the maximum value of the process, the overshoot of some threshold, and so on. The existing general analytical and combinatorial methods allow one to find effectively the distributions of such boundary functionals.The study of boundary functionals leads to the necessity of solving Wiener-Hopf type equations. It turns out that different functionals possess common properties which follow from properties of the original process.In this paper we present a new method for investigating boundary functionals first we construct and study the potential of the original equation (Part I); then (Part II) with the help of the potential the boundary functionals are described. This method allows one to classify boundary functionals and hence unify various boundary problems. For the expectation of events related to the attainment of a positive level by a compound Poisson process with positive jumps and negative drift we present a general expression of the form R[G6 + (p], in which Rx is the resolvent of the original process on the positive real line, 6 is the delta function and the constant C and the function (p are determined by the particular boundary functional in question. In particular, we clarify the special role of the distribution of the absolute maximum of the process. It turns out that with accuracy up to a constant multiple the distribution of the absolute maximum coincides with the potential.As examples we treat the following boundary functionals the absolute maximum of the process, the first passage time to a positive level, the probability of ruin (see, for example [3]), and the distribution of the maximum of the virtual waiting time process (see [2]).

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