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Register a new userThe self-similar evolution of stationary point processes via persistent homology
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Persistent homology provides a robust methodology to infer topological structures from point cloud data. Here we explore the persistent homology of point clouds embedded into a probabilistic setting, exploiting the theory of point processes. We provide variants of notions of ergodicity and investigate measures on the space of persistence diagrams. In particular we introduce the notion of self-similar scaling of persistence diagram expectation measures and prove a packing relation for the occurring dynamical scaling exponents. As a byproduct we generalize the strong law of large numbers for persistent Betti numbers proven in [Hiraoka et al., Ann. Appl. Probab. 28(5), 2018] for averages over cubes to arbitrary convex averaging sequences.
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This paper addresses the question of predicting when a positive self-similar Markov process X attains its pathwise global supremum or infimum before hitting zero for the first time (if it does at all). This problem has been studied in Glover et al. (2013) under the assumption that X is a positive transient diffusion. We extend their result to the class of positive self-similar Markov processes by establishing a link to Baurdoux and van Schaik (2013), where the same question is studied for a Levy process drifting to minus infinity. The connection to Baurdoux and van Schaik (2013) relies on the so-called Lamperti transformation which links the class of positive self-similar Markov processes with that of Levy processes. Our approach will reveal that the results in Glover et al. (2013) for Bessel processes can also be seen as a consequence of self-similarity.
We prove that a positive self-similar Markov process $(X,mathbb{P})$ that hits 0 in a finite time admits a self-similar recurrent extension that leaves 0 continuously if and only if the underlying L{e}vy process satisfies Cram{e}rs condition.
For a positive self-similar Markov process, X, we construct a local time for the random set, $Theta$, of times where the process reaches its past supremum. Using this local time we describe an exit system for the excursions of X out of its past supremum. Next, we define and study the ladder process (R,H) associated to a positive self-similar Markov process X, namely a bivariate Markov process with a scaling property whose coordinates are the right inverse of the local time of the random set $Theta$ and the process X sampled on the local time scale. The process (R,H) is described in terms of a ladder process linked to the L{e}vy process associated to X via Lampertis transformation. In the case where X never hits 0, and the upward ladder height process is not arithmetic and has finite mean, we prove the finite-dimensional convergence of (R,H) as the starting point of X tends to 0. Finally, we use these results to provide an alternative proof to the weak convergence of X as the starting point tends to 0. Our approach allows us to address two issues that remained open in Caballero and Chaumont [Ann. Probab. 34 (2006) 1012-1034], namely, how to remove a redundant hypothesis and how to provide a formula for the entrance law of X in the case where the underlying L{e}vy process oscillates.
We develop the theory of strong stationary duality for diffusion processes on compact intervals. We analytically derive the generator and boundary behavior of the dual process and recover a central tenet of the classical Markov chain theory in the diffusion setting by linking the separation distance in the primal diffusion to the absorption time in the dual diffusion. We also exhibit our strong stationary dual as the natural limiting process of the strong stationary dual sequence of a well chosen sequence of approximating birth-and-death Markov chains, allowing for simultaneous numerical simulations of our primal and dual diffusion processes. Lastly, we show how our new definition of diffusion duality allows the spectral theory of cutoff phenomena to extend naturally from birth-and-death Markov chains to the present diffusion context.
In many scientific and technological contexts we have only a poor understanding of the structure and details of appropriate mathematical models. We often, therefore, need to compare different models. With available data we can use formal statistical model selection to compare and contrast the ability of different mathematical models to describe such data. There is, however, a lack of rigorous methods to compare different models emph{a priori}. Here we develop and illustrate two such approaches that allow us to compare model structures in a systematic way {by representing models in terms of simplicial complexes}. Using well-developed concepts from simplicial algebraic topology, we define a distance between models based on their simplicial representations. Employing persistent homology with a flat filtration provides for alternative representations of the models as persistence intervals, which represent the structure of the models, from which we can also obtain the distances between models. We then expand on this measure of model distance to study the concept of model equivalence in order to determine the conceptual similarity of models. We apply our methodology for model comparison to demonstrate an equivalence between a positional-information model and a Turing-pattern model from developmental biology, constituting a novel observation for two classes of models that were previously regarded as unrelated.
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