Semigroup inequalities, stochastic domination, Hardy's inequality, and strong ergodicity
For the classical L p -spaces of signed measures on N, we devise a framework in which bounds for a sub-Markovian semigroup of interest can be obtained, up to a constant factor, from bounds for another tractable semigroup that dominates stochastically the first one. The main tools are the Hardy inequality, the definition of related auxiliary L p spaces suited to take advantage of the domination, and the proof that the norms are equivalent to the classical ones if the reference measure is quasi-geometrically decreasing. We illustrate the results using birth-death and single-birth processes.
Functional central limit theorems for a large network in which customers join the shortest of several queues
Abstract.We consider N single server infinite buffer queues with service rate β. Customers arrive at rate Nα, choose L queues uniformly, and join the shortest. We study the processes  for large N, where RNt(k) is the fraction of queues of length at least k at time t. Laws of large numbers (LLNs) are known, see Vvedenskaya et al. [15], Mitzenmacher [12] and Graham [5]. We consider certain Hilbert spaces with the weak topology. First, we prove a functional central limit theorem (CLT) under the a priori assumption that the initial data RN0 satisfy the corresponding CLT. We use a compactness-uniqueness method, and the limit is characterized as an Ornstein-Uhlenbeck (OU) process. Then, we study the RN in equilibrium under the stability condition α<β, and prove a functional CLT with limit the OU process in equilibrium. We use ergodicity and justify the inversion of limits  lim N→∞ lim t→∞= lim t→∞ lim N→∞ by a compactness-uniqueness method. We deduce a posteriori the CLT for RN0 under the invariant laws, an interesting result in its own right. The main tool for proving tightness of the implicitly defined invariant laws in the CLT scaling and ergodicity of the limit OU process is a global exponential stability result for the nonlinear dynamical system obtained in the functional LLN limit.
Bentham's Equality-Sensitive Utilitarianism
Rosen argues that Bentham's utilitarian doctrine was sensitive to distributive concerns and would not countenance sacrifice of fundamental individual interests for aggregate gains in happiness in society. This essay seeks to extend and deepen Rosen's argument. It is argued that Bentham's equality-sensitive principle of utility is an expression of an individualist conception of human happiness which contrasts sharply with the orthodox utilitarian abstract conception. Evidence for this interpretation of the basic motivation of Bentham's doctrine is drawn from his view of the relationship between happiness and expectations, from various expressions of his ‘each to count for one’ formula, and from his reformulations of the principle of utility itself late in his career.