Publication Date

2019-04-01

Availability

Open access

Embargo Period

2019-04-01

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PHD)

Department

Mathematics (Arts and Sciences)

Date of Defense

2018-12-12

First Committee Member

Ilie Grigorescu

Second Committee Member

Victor Pestien

Third Committee Member

Mingliang Cai

Fourth Committee Member

Burton Rosenberg

Abstract

In this paper we study the fluctuation limit of a particle system in non-equilibrium. Each individual among n particles with current position x(t) moves on the positive axis according to a Poisson clock. With probability 1 − p, depending on the average position of the particle configuration, it moves to x(t) + 1 and with probability p to γ x(t), γ ∈ (0, 1). This is the Additive Increase Multiplicative Decrease (AIMD) internet traffic protocol, where x(t) is the data transmission rate of a given user. Under proper scaling, when n → ∞, the system has a deterministic fluid limit described as the solution of an ordinary differential equation. We are looking at the functional second approximation ξ(t), i.e. departures from this limit, on a Central Limit Theorem scale. The random field ξ(t) is identified by its action on special test functions φ. For polynomial test functions, the central limit theorem fluctuation field is tight and we identify its limit explicitly. Labeling the random field Z(k, t) for each monomial φ(x) = x^k of degree k ≥ 1 we obtain a hierarchical system of diffusions, in the following sense: the vector (Z(1, t), . . . , Z(m, t)) is a linear diffusion with time dependent explicit coefficients and the system for m0 > m is consistent with the system for m in that the matrix is subdiagonal. When the initial data is Gaussian, the infinite-dimensional process, indexed by φ(x) polynomials, is a Gaussian process. The abstract random field limit is formulated as a generalized Ornstein-Uhlenbeck process and we discuss some open problems.

Keywords

Fluctuation, diffusion

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