Jump processes as generalized gradient flows

We have created a functional framework for a class of non-metric gradient systems. The state space is a space of nonnegative measures, and the class of systems includes the Forward Kolmogorov equations for the laws of Markov jump processes on Polish spaces. This framework comprises a definition of a notion of solutions, a method to prove existence, and an archetype uniqueness result. We do this by using only the structure that is provided directly by the dissipation functional, which need not be homogeneous, and we do not appeal to any metric structure.

Coarse groups, and the isomorphism problem for oligomorphic groups

Let [Formula: see text] denote the topological group of permutations of the natural numbers. A closed subgroup [Formula: see text] of [Formula: see text] is called oligomorphic if for each [Formula: see text], its natural action on [Formula: see text]-tuples of natural numbers has only finitely many orbits. We study the complexity of the topological isomorphism relation on the oligomorphic subgroups of [Formula: see text] in the setting of Borel reducibility between equivalence relations on Polish spaces. Given a closed subgroup [Formula: see text] of [Formula: see text], the coarse group [Formula: see text] is the structure with domain the cosets of open subgroups of [Formula: see text], and a ternary relation [Formula: see text]. This structure derived from [Formula: see text] was introduced in [A. Kechris, A. Nies and K. Tent, The complexity of topological group isomorphism, J. Symbolic Logic 83(3) (2018) 1190–1203, Sec. 3.3]. If [Formula: see text] has only countably many open subgroups, then [Formula: see text] is a countable structure. Coarse groups form our main tool in studying such closed subgroups of [Formula: see text]. We axiomatize them abstractly as structures with a ternary relation. For the oligomorphic groups, and also the profinite groups, we set up a Stone-type duality between the groups and the corresponding coarse groups. In particular, we can recover an isomorphic copy of [Formula: see text] from its coarse group in a Borel fashion. We use this duality to show that the isomorphism relation for oligomorphic subgroups of [Formula: see text] is Borel reducible to a Borel equivalence relation with all classes countable. We show that the same upper bound applies to the larger class of closed subgroups of [Formula: see text] that are topologically isomorphic to oligomorphic groups.

Topological Manifolds and Polish Spaces

We give,as a preliminary result, some topological characterizations of locally compact second-countable Hausdorff spaces. Then we show that a topological manifold, with boundary or not,is precisely a Polish space with a coordinate open cover; this connects geometry with descriptive set theory.

Duality Theorem for a Minimax Vector Optimization Problem

A duality theorem is stated and proved for a minimax vector optimization problem where the vectors are elements of the set of products of compact Polish spaces. A special case of this theorem is derived to show that two metrics on the space of probability distributions on countable products of Polish spaces are identical. The appendix includes a proof that, under the appropriate conditions, the function studied in the optimisation problem is indeed a metric. The optimisation problem is comparable to multi-commodity optimal transport where there is dependence between commodities. This paper builds on the work of R.S. MacKay who introduced the metrics in the context of complexity science in [4] and [5]. The metrics have the advantage of measuring distance uniformly over the whole network while other metrics on probability distributions fail to do so (e.g total variation, Kullback–Leibler divergence, see [5]). This opens up the potential of mathematical optimisation in the setting of complexity science.

Proximity inductive dimension and Brouwer dimension agree on compact Hausdorff spaces

We show that the proximity inductive dimension defined by Isbell agrees with the Brouwer dimension originally described by Brouwer (for Polish spaces without isolated points) on the class of compact Hausdorff spaces. This shows that Fedorchuk?s example of a compact Hausdorff space whose Brouwer dimension exceeds its Lebesgue covering dimension is an example of a space whose proximity inductive dimension exceeds its proximity dimension as defined by Smirnov. This answers Isbell?s question of whether or not proximity inductive dimension and proximity dimension coincide.