Due to a printing error the above mentioned article [Annals of Applied Probability 10 (2000) 75--103, doi:10.1214/aoap/1019737665] had numerous equations appearing incorrectly in the print version of this paper. The entire article follows as it should have appeared. IMS apologizes to the author and the readers for this error. A recent paper by Harrison and Van Mieghem explained in general mathematical terms how one forms an ``equivalent workload formulation'' of a Brownian network model. Denoting by $Z(t)$ the state vector of the original Brownian network, one has a lower dimensional state descriptor $W(t)=MZ(t)$ in the equivalent workload formulation, where $M$ can be chosen as any basis matrix for a particular linear space. This paper considers Brownian models for a very general class of open processing networks, and in that context develops a more extensive interpretation of the equivalent workload formulation, thus extending earlier work by Laws on alternate routing problems. A linear program called the static planning problem is introduced to articulate the notion of ``heavy traffic'' for a general open network, and the dual of that linear program is used to define a canonical choice of the basis matrix $M$. To be specific, rows of the canonical $M$ are alternative basic optimal solutions of the dual linear program. If the network data satisfy a natural monotonicity condition, the canonical matrix $M$ is shown to be nonnegative, and another natural condition is identified which ensures that $M$ admits a factorization related to the notion of resource pooling.
We consider a dynamic control problem associated with a generalized Brownian network, the objective being to minimize expected discounted cost over an infinite planning horizon. In this Brownian control problem (BCP), both the system manager's control and the associated cumulative cost process may be locally of unbounded variation. Due to this aspect of the cost process, both the precise statement of the problem and its analysis involve delicate technical issues. We show that the BCP is equivalent, in a certain sense, to a reduced Brownian control problem (RBCP) of lower dimension. The RBCP is…
We construct a measure valued Markov process which we call infinite canonical super-Brownian motion, and which corresponds to the canonical measure of super-Brownian motion conditioned on non-extinction. Infinite canonical super-Brownian motion is a natural candidate for the scaling limit of various random branching objects on $\Z^d$ when these objects are (a) critical; (b) mean-field and (c) infinite. We prove that ICSBM is the scaling limit of the spread-out oriented percolation incipient infinite cluster above 4 dimensions and of incipient infinite branching random walk in any dimension. We…
We consider a stationary fluid queue with fractional Brownian motion input. Conditional on the workload at time zero being greater than a large value $b$, we provide the limiting distribution for the amount of time that the workload process spends above level $b$ over the busy cycle straddling the origin, as $b\to\infty$. Our results can be interpreted as showing that long delays occur in large clumps of size of order $b^{2-1/H}$. The conditional limit result involves a finer scaling of the queueing process than fluid analysis, thereby departing from previous related literature.
We study a single-server Markovian queueing model with $N$ customer classes in which priority is given to the shortest queue. Under a critical load condition, we establish the diffusion limit of the workload and queue length processes in the form of a Walsh Brownian motion (WBM) living in the union of the $N$ nonnegative coordinate axes in $\mathbb{R}^N$ and a linear transformation thereof. This reveals the following asymptotic behavior. Each time that queues begin to build starting from an empty system, one of them becomes dominant in the sense that it contains nearly all the workload in the …
We obtain probability measures on the canonical space penalizing the Wiener measure by a function of its maximum (resp. minimum, local time). We study the law of the canonical process under these new probability measures.
The original density is 1 for $t\in (0,1)$, $b$ is an integer base ($b\geq 2$%), and $p\in (0,1)$ is a parameter. The first construction stage divides the unit interval into $b$ subintervals and multiplies the density in each subinterval by either 1 or -1 with the respective frequencies of $\frac{1% }{2}+\frac{p}{2}$ and ${1/2}-\frac{p}{2}$. It is shown that the resulting density can be renormalized so that, as $n\to \infty $ ($n$ being the number of iterations) the signed measure converges in some sense to a non-degenerate limit. If $H=1+\log_{b}$ $p>{1}/{2}$, hence $p>b^{{-1}/{% 2}}$, renorm…
The canonical Brownian motion that P. Malliavin in 1999 and then S. Fang in 2002 constructed lives in the group of Holderian homeomorphisms of the circle. In this paper, we present another way to construct a Brownian motion that lives exactly in the group of orientation preserving smooth diffeomorphisms of the circle.
An asymptotic framework for optimal control of multiclass stochastic processing networks, using formal diffusion approximations under suitable temporal and spatial scaling, by Brownian control problems (BCP) and their equivalent workload formulations (EWF), has been developed by Harrison (1988). This framework has been implemented in many works for constructing asymptotically optimal control policies for a broad range of stochastic network models. To date all asymptotic optimality results for such networks correspond to settings where the solution of the EWF is a reflected Brownian motion in t…
Two-sided heat kernel bounds for $\sqrt{8/3}$-Liouville Brownian motion
Technische Universität Braunschweig（布伦瑞克工业大学）；Kobe University（神户大学）；Research Institute for Mathematical Sciences, Kyoto University（京都大学数理科学研究所在）；Massachusetts Institute of Technology（麻省理工学院）；University of Cambridge（剑桥大学）
A generalized bridge is the law of a stochastic process that is conditioned on N linear functionals of its path. We consider two types of representations of such bridges: orthogonal and canonical. The orthogonal representation is constructed from the entire path of the underlying process. Thus, future knowledge of the path is needed. The orthogonal representation is provided for any continuous Gaussian process. In the canonical representation the filtrations and the linear spaces generated by the bridge process and the underlying process coincide. Thus, no future information of the underlying …
This paper presents a new approach to the analysis of mixed processes \[X_t=B_t+G_t,\qquad t\in[0,T],\] where $B_t$ is a Brownian motion and $G_t$ is an independent centered Gaussian process. We obtain a new canonical innovation representation of $X$, using linear filtering theory. When the kernel \[K(s,t)=\frac{\partial^2}{\partial s\,\partial t}\mathbb{E}G_tG_s,\qquad s\ne t\] has a weak singularity on the diagonal, our results generalize the classical innovation formulas beyond the square integrable setting. For kernels with stronger singularity, our approach is applicable to processes with…
Given $2n$ unit equilateral triangles, there are finitely many ways to glue each edge to a partner. We obtain a random sphere-homeomorphic surface by sampling uniformly from the gluings that produce a topological sphere. As $n$ tends to infinity, these random surfaces (appropriately scaled) converge in law. The limit is a "canonical" sphere-homeomorphic random surface, much the way Brownian motion is a canonical random path. Depending on how the surface space and convergence topology are specified, the limit is the Brownian sphere, the peanosphere, the pure Liouville quantum gravity sphere, or…
