Browsing by Author "Nolen, James"

In this paper, we introduce several unbalanced optimal transport problems between two Radon measures with different total masses. Initially, we explore a generalization of the Benamou-Brenier problem, incorporating a growth constraint to accommodate a non-decreasing total mass during transportation. This leads to the formulation of a modified Hellinger-Kantorovich (mHK) problem. Our investigation reveals quasi-metric properties of this novel problem and characterizes it within a cone setting through a newly defined quasi-cone metric, resulting in an equivalent formulation of the mHK problem. This formulation simplifies the demonstration of the existence of optimal solutions and facilitates explicit calculations for transport problems between two Dirac measures.

A significant advancement in our work is the construction of a dual problem for the mHK problem, a topic previously unexplored. We confirm the duality and identify optimality conditions for transport plans, successfully deriving a one-to-one (Monge) map under certain regularity conditions for the initial measure. Furthermore, we propose a dynamic formulation for the mHK problem within a cone setting, focusing on minimization over dynamic plans involving absolutely continuous curves between cone points. This approach not only projects a dynamic plan onto an absolutely continuous curve between initial and target measures but also establishes a close relationship with solutions to continuity equations.

Motivated by dynamic models of biological growth, our study extends to practical applications, providing an equivalent convex formulation of the mHK problem and developing numerical schemes based on the Douglas-Rachford algorithm and the Alternating Direction Method of Multipliers algorithm. We apply these schemes to synthetic data, demonstrating the utility of our theoretical findings.

This dissertation examines the asymptotic behavior of three branching processes. The first is a branching process with selection; the selection is dictated by a fitness function which is the sum of a linear part and a periodic part. It is shown that the system has an asymptotic speed and that there is a stationary distribution in an appropriate moving frame. This is done through an examination of tightness of the process and application of an ergodic theorem. The second process studied is a branching process with selection driven by a symmetric function with a single local maximum at the origin and which monotonically decreases away from the origin. For this process, a large particle limit of the system is proven and related to the solution to a free boundary partial differential equation. Finally, a branching process is studied in which the branch rate of particles is a function of the empirical measure. Weak convergence to the solution of a non-local partial differential equation is proven. Tightness is proven first, and then the limit object is characterized by its behavior when applied to test functions.

This thesis examines the Fleming-Viot process, a particle system which provides an approximation method for killed Markov processes conditioned on survival and their quasi-stationary distributions (QSDs). In the first part, we establish that the Fleming-Viot process also provides for an approximation method for killed McKean-Vlasov processes conditioned on survival and their QSDs. We prove that the law conditioned on survival of a given McKean-Vlasov process killed on the boundary of its domain may be obtained from the hydrodynamic limit of the corresponding Fleming-Viot particle system. We then show that if the target killed McKean-Vlasov process converges to a QSD as t→ ∞, such a QSD may be obtained from the stationary distributions of the corresponding N-particle Fleming-Viot system as N→∞. The main techniques we employ are two coupling constructions, martingale methods, and an analysis of the dynamical historical processes, which together enable a compactness-uniqueness argument.

In the second part, we fix our given killed Markov process as a normally reflected diffusion in a compact domain, killed according to a position-dependent Poisson clock. We obtain the Fleming-Viot super process as a scaling limit of the Fleming-Viot multi-colour process. This has three applications: it enables us to prove a conjecture due to Bieniek and Burdzy on the asymptotic distribution of the spine of the Fleming-Viot process, it provides for a particle representation for the principal right eigenfunction of the infinitesimal generator, and it provides an approximation method for the Q-process - the killed Markov process conditioned never to be killed. The main technique employed is to observe that tilting the empirical measure by the principal right eigenfunction of the infinitesimal generator allows for fast-variable elimination.

One of the key mechanisms that mediate renal autoregulation is the tubuloglomerular feedback (TGF) system, which is a negative feedback loop in the kidney that balances glomerular filtration with tubular reabsorptive capacity. In this dissertation, we develop several mathematical models of the TGF system to study TGF-mediated model dynamics.

First, we develop a mathematical model of compliant thick ascending limb (TAL) of a short loop of Henle in the rat kidney, called TAL model, to investigate the effects of spatial inhomogeneous properties in TAL on TGF-mediated dynamics. We derive a characteristic equation that corresponds to a linearized TAL model, and conduct a bifurcation analysis by finding roots of that equation. Results of the bifurcation analysis are also validated via numerical simulations of the full model equations.

We then extend the TAL model to explicitly represent an entire short-looped nephron including the descending segments and having compliant tubular walls, developing a short-looped nephron model. A bifurcation analysis for the TGF loop-model equations is similarly performed by computing parameter boundaries, as functions of TGF gain and delay, that separate differing model behaviors. We also use the loop model to better understand the effects of transient as well as sustained flow perturbations on the TGF system and on distal NaCl delivery.

To understand the impacts of internephron coupling on TGF dynamics, we further develop a mathematical model of a coupled-TGF system that includes any finite number of nephrons coupled through their TGF systems, coupled-nephron model. Each model nephron represents a short loop of Henle having compliant tubular walls, based on the short-looped nephron model, and is assumed to interact with nearby nephrons through electrotonic signaling along the pre-glomerular vasculature. The characteristic equation is obtained via linearization of the loop-model equations as in TAL model. To better understand the impacts of parameter variability on TGF-mediated dynamics, we consider special cases where the relation between TGF delays and gains among two coupled nephrons is specifically chosen. By solving the characteristic equation, we determine parameter regions that correspond to qualitatively differing model behaviors.

TGF delays play an essential role in determining qualitatively and quantitatively different TGF-mediated dynamic behaviors. In particular, when noise arising from external sources of system is introduced, the dynamics may become significantly rich and complex, revealing a variety of model behaviors owing to the interaction with delays. In our next study, we consider the effect of the interactions between time delays and noise, by developing a stochastic model. We begin with a simple time-delayed transport equation to represent the dynamics of chloride concentration in the rigid-TAL fluid. Guided by a proof for the existence and uniqueness of the steady-state solution to the deterministic Dirichlet problem, obtained via bifurcation analysis and the contraction mapping theorem, an analogous proof for stochastic system with random boundary conditions is presented. Finally we conduct multiscale analysis to study the effect of the noise, specifically when the system is in subcritical region, but close enough to the critical delay. To analyze the solution behaviors in long time scales, reduced equations for the amplitude of solutions are derived using multiscale method.

Problems in stochastic homogenization theory typically deal with approximating differential operators with rapidly oscillatory random coefficients by operators with homogenized deterministic coefficients. Even though the convergence of these operators in multiple scales is well-studied in the existing literature in the form of a Law of Large Numbers, very little is known about their rate of convergence or their large deviations.

In the first part of this thesis, we we establish analytic results for the Gaussian correction in homogenization of an elliptic differential equation with random diffusion in randomly layered media. We also derive a Central Limit Theorem for a diffusion in a weakly random media.

In the second part of this thesis devise a technique for obtaining large deviation results for homogenization problems in random media. We consider the special cases of an elliptic equation with random potential, the random diffusion problem in randomly layered media and a reaction-diffusion equation with highly oscillatory reaction term.