## 1 Introduction  
In the _stable marriage problem_, every vertex of an undirected bipartite graph, \(G\), represents either a woman or a man. Their sympathy for each person of the opposite gender is expressed by their _preference lists_: the more beloved person has the higher rank. A marriage scheme is a matching, \(M\) on \(G\). We say that such a scheme is _stable_, if there is no pair of participants willing to leave their partners in order to marry each other. More formally: an edge, \(uv\), _blocks_\(M\), if \(u\) and \(v\) both are unpaired or prefer each other to their partners in \(M\). A matching \(M\) is _stable_, if there is no blocking edge in \(G\). [PERSON] and [PERSON] [1] were the first to state that a stable matching always exists. Their well-known deferred-acceptance algorithm finds a stable marriage in strongly polynomial time. One of the most advanced extensions of the stable marriage problem is the _stable allocation problem_. It was introduced by [PERSON] and [PERSON] [2] in 2002. Here, we talk about jobs and machines instead of men and women. Edges have capacity and participants have quota on their matched edges. This quota stands for the time a job needs to get done and the time a machine is able to work in total. Edges are used for assigning jobs to machines, such that none of the machines spends more time on a job than the edge capacity allows them. The goal is to find a feasible set of contracts, such that no machine-job pair exists where both could improve their states by breaking the scheme. Edge \(e\) is _blocking_, if it is unsaturated, and neither end vertices of \(e\) could fill up its quota with at least as good edges as \(e\). An allocation, \(x\), is _stable_, if none of the edges of \(G\) are blocking. [PERSON] and [PERSON] [2] give two algorithms to solve the problem: while their augmenting path algorithm runs in strongly polynomial time, the refined Gale-Shapley algorithm is more efficient in simple cases, e.g., on instances where all jobs get one of their best choices, it only needs sublinear time. [PERSON] _et al_. [3,4] succeeded in speeding up the first method, relying on sophisticated data structures, such as dynamic trees. They also extended the second one to the case of irrational data. Stable allocations have been further generalized to _stable flows_ by [PERSON] [5] and [PERSON] [6]. [PERSON] also gave a constructive proof for the existence of a stable flow in every network by solving a stable allocation problem in a modified network. In an instance of stable flow, special vertices are designated as terminals, and each non-terminal vertex has a preference list of its incoming and outgoing edges that specifies from which edges it prefers to receive flow and which edges it prefers to send flow along. Stable flows are well suited for modeling real-world market situations, as they capture the different trading preferences of vendors and customers. We extend this model to the setting of _flows over time_. Flows over time were introduced by [PERSON] and [PERSON] [7,8]. In addition to the stable flow instance, we have transit times on the edges and a time horizon specifying the end of the process. We prove the existence of stable flows over time and, moreover, show that, for the case of an infinite (or sufficiently large) time horizon, there is a stable flow over time, which converges to a static stable flow. By introducing time to the stable flow setting, one can achieve a considerably more realistic and more interesting description of real market situations. With flows over time and transit times on the edges, we can model transportation problems or illustrate distances amongst the vendors. Structure of the Paper:In Section 2, we introduce the stable flow problem and present an elegant version of the Gale-Shapley algorithm that operates directly on the given network. We also give a simple proof of its correctness. In Section 3, we present the stable flow over time problem and show the existence of stable flows, even in the case of networks with an infinite time horizon.