Publication Date



Open access

Embargo Period


Degree Type


Degree Name

Doctor of Philosophy (PHD)


Mathematics (Arts and Sciences)

Date of Defense


First Committee Member

Ilie Grigorescu

Second Committee Member

Victor Pestien

Third Committee Member

Ming-liang Cai

Fourth Committee Member

Mehdi Shadmehr


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.


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