Publication Date

2016-07-14

Availability

Open access

Embargo Period

2016-07-14

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PHD)

Department

Mathematics (Arts and Sciences)

Date of Defense

2016-06-23

First Committee Member

Ilie Grigorescu,

Second Committee Member

Victor Pestien

Third Committee Member

Ming-liang Cai

Fourth Committee Member

Mehdi Shadmehr

Abstract

We study a hydrodynamic limit for a system of N diffusions moving in an open domain D ⊆ Rd undergoing branching when one particle reaches a certain subset of the boundary. The particle at the boundary and another random neighbor are eliminated and replaced with two new particles created instantaneously at a random point with distribution γ(dx) in D. This thesis proves the d = 1 case with D = (0,1), γ(dx) = δc(dx), c ∈ (0,1) while the general case is done in an upcoming paper. The mechanism represents a hybrid between the Fleming- Viot branching and a mean-field version of the Bak-Sneppen fitness model where the absorbing boundary represents the not viable, or minimal configuration. The limiting profile is the normalization of the solution of a heat equation with mass creation, which is studied using its representation via an auxiliary measure valued supercritical process. Self-criticality is manifested by the presence of the quasi- stationary distributions emerging as profiles under equilibrium.

Keywords

Jump diffusion process; Fleming-Viot; Bak-Sneppen; Hydrodynamic limit; Catalytic branching; Quasi-stationary distribution

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