Representation of Markov chains by random maps: existence and regularity conditions

We systematically investigate the problem of representing Markov chains by families of random maps, and which regularity of these maps can be achieved depending on the properties of the probability measures. Our key idea is to use techniques from optimal transport to select optimal such maps. Optimal transport theory also tells us how convexity properties of the supports of the measures translate into regularity properties of the maps via Legendre transforms. Thus, from this scheme, we cannot only deduce the representation by measurable random maps, but we can also obtain conditions for the representation by continuous random maps. Finally, we present conditions for the representation of Markov chain by random diffeomorphisms.Comment: 22 pages, several changes from the previous version including extended discussion of many detail

Dynamics and bifurcations of random circle diffeomorphisms

We discuss iterates of random circle diffeomorphisms with identically distributed noise, where the noise is bounded and absolutely continuous. Using arguments of B. Deroin, V.A. Kleptsyn and A. Navas, we provide precise conditions under which random attracting fixed points or random attracting periodic orbits exist. Bifurcations leading to an explosion of the support of a stationary measure from a union of intervals to the circle are treated. We show that this typically involves a transition from a unique random attracting periodic orbit to a unique random attracting fixed point

Reconstructing firm-level interactions in the Dutch input-output network from production constraints

Recent crises have shown that the knowledge of the structure of input-output networks, at the firm level, is crucial when studying economic resilience from the microscopic point of view of firms that try to rewire their connections under supply and demand constraints. Unfortunately, empirical inter-firm network data are protected by confidentiality, hence rarely accessible. The available methods for network reconstruction from partial information treat all pairs of nodes as potentially interacting, thereby overestimating the rewiring capabilities of the system and the implied resilience. Here, we use two big data sets of transactions in the Netherlands to represent a large portion of the Dutch inter-firm network and document its properties. We, then, introduce a generalized maximum-entropy reconstruction method that preserves the production function of each firm in the data, i.e. the input and output flows of each node for each product type. We confirm that the new method becomes increasingly more reliable in reconstructing the empirical network as a finer product resolution is considered and can, therefore, be used as a realistic generative model of inter-firm networks with fine production constraints. Moreover, the likelihood of the model directly enumerates the number of alternative network configurations that leave each firm in its current production state, thereby estimating the reduction in the rewiring capability of the system implied by the observed input-output constraints.Theoretical Physic