Random real trees

We survey recent developments about random real trees, whose prototype is the Continuum Random Tree (CRT) introduced by Aldous in 1991. We briefly explain the formalism of real trees, which yields a neat presentation of the theory and in particular of the relations between discrete Galton-Watson trees and continuous random trees. We then discuss the particular class of self-similar random real trees called stable trees, which generalize the CRT. We review several important results concerning stable trees, including their branching property, which is analogous to the well-known property of Galton-Watson trees, and the calculation of their fractal dimension. We then consider spatial trees, which combine the genealogical structure of a real tree with spatial displacements, and we explain their connections with superprocesses. In the last section, we deal with a particular conditioning problem for spatial trees, which is closely related to asymptotics for random planar quadrangulations.Comment: 25 page

Stochastic Continuous Time Neurite Branching Models with Tree and Segment Dependent Rates

In this paper we introduce a continuous time stochastic neurite branching model closely related to the discrete time stochastic BES-model. The discrete time BES-model is underlying current attempts to simulate cortical development, but is difficult to analyze. The new continuous time formulation facilitates analytical treatment thus allowing us to examine the structure of the model more closely. We derive explicit expressions for the time dependent probabilities p(\gamma, t) for finding a tree \gamma at time t, valid for arbitrary continuous time branching models with tree and segment dependent branching rates. We show, for the specific case of the continuous time BES-model, that as expected from our model formulation, the sums needed to evaluate expectation values of functions of the terminal segment number \mu(f(n),t) do not depend on the distribution of the total branching probability over the terminal segments. In addition, we derive a system of differential equations for the probabilities p(n,t) of finding n terminal segments at time t. For the continuous BES-model, this system of differential equations gives direct numerical access to functions only depending on the number of terminal segments, and we use this to evaluate the development of the mean and standard deviation of the number of terminal segments at a time t. For comparison we discuss two cases where mean and variance of the number of terminal segments are exactly solvable. Then we discuss the numerical evaluation of the S-dependence of the solutions for the continuous time BES-model. The numerical results show clearly that higher S values, i.e. values such that more proximal terminal segments have higher branching rates than more distal terminal segments, lead to more symmetrical trees as measured by three tree symmetry indicators.Comment: 41 pages, 2 figures, revised structure and text improvement

Recent results of quantum ergodicity on graphs and further investigation

We outline some recent proofs of quantum ergodicity on large graphs and give new applications in the context of irregular graphs. We also discuss some remaining questions.Comment: To appear in "Annales de la facult\'e des Sciences de Toulouse

Sequential Design for Optimal Stopping Problems

We propose a new approach to solve optimal stopping problems via simulation. Working within the backward dynamic programming/Snell envelope framework, we augment the methodology of Longstaff-Schwartz that focuses on approximating the stopping strategy. Namely, we introduce adaptive generation of the stochastic grids anchoring the simulated sample paths of the underlying state process. This allows for active learning of the classifiers partitioning the state space into the continuation and stopping regions. To this end, we examine sequential design schemes that adaptively place new design points close to the stopping boundaries. We then discuss dynamic regression algorithms that can implement such recursive estimation and local refinement of the classifiers. The new algorithm is illustrated with a variety of numerical experiments, showing that an order of magnitude savings in terms of design size can be achieved. We also compare with existing benchmarks in the context of pricing multi-dimensional Bermudan options.Comment: 24 page

Real-time Planning as Decision-making Under Uncertainty

In real-time planning, an agent must select the next action to take within a fixed time bound. Many popular real-time heuristic search methods approach this by expanding nodes using time-limited A* and selecting the action leading toward the frontier node with the lowest f value. In this thesis, we reconsider real-time planning as a problem of decision-making under uncertainty. We treat heuristic values as uncertain evidence and we explore several backup methods for aggregating this evidence. We then propose a novel lookahead strategy that expands nodes to minimize risk, the expected regret in case a non-optimal action is chosen. We evaluate these methods in a simple synthetic benchmark and the sliding tile puzzle and find that they outperform previous methods. This work illustrates how uncertainty can arise even when solving deterministic planning problems, due to the inherent ignorance of time-limited search algorithms about those portions of the state space that they have not computed, and how an agent can benefit from explicitly meta-reasoning about this uncertainty