21,414 research outputs found
A Model for Optimal Human Navigation with Stochastic Effects
We present a method for optimal path planning of human walking paths in
mountainous terrain, using a control theoretic formulation and a
Hamilton-Jacobi-Bellman equation. Previous models for human navigation were
entirely deterministic, assuming perfect knowledge of the ambient elevation
data and human walking velocity as a function of local slope of the terrain.
Our model includes a stochastic component which can account for uncertainty in
the problem, and thus includes a Hamilton-Jacobi-Bellman equation with
viscosity. We discuss the model in the presence and absence of stochastic
effects, and suggest numerical methods for simulating the model. We discuss two
different notions of an optimal path when there is uncertainty in the problem.
Finally, we compare the optimal paths suggested by the model at different
levels of uncertainty, and observe that as the size of the uncertainty tends to
zero (and thus the viscosity in the equation tends to zero), the optimal path
tends toward the deterministic optimal path
A time-fractional mean field game
We consider a Mean Field Games model where the dynamics of the agents is
subdiffusive. According to the optimal control interpretation of the problem,
we get a system involving fractional time-derivatives for the
Hamilton-Jacobi-Bellman and the Fokker-Planck equations. We discuss separately
the well-posedness for each of the two equations and then we prove existence
and uniqueness of the solution to the Mean Field Games syste
A survey of random processes with reinforcement
The models surveyed include generalized P\'{o}lya urns, reinforced random
walks, interacting urn models, and continuous reinforced processes. Emphasis is
on methods and results, with sketches provided of some proofs. Applications are
discussed in statistics, biology, economics and a number of other areas.Comment: Published at http://dx.doi.org/10.1214/07-PS094 in the Probability
Surveys (http://www.i-journals.org/ps/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Diffusion and localization of relative strategy scores in the Minority Game
We study the equilibrium distribution of relative strategy scores of agents
in the asymmetric phase () of the basic Minority
Game using sign-payoff, with agents holding two strategies over
histories. We formulate a statistical model that makes use of the gauge freedom
with respect to the ordering of an agent's strategies to quantify the
correlation between the attendance and the distribution of strategies. The
relative score of the two strategies of an agent is described
in terms of a one dimensional random walk with asymmetric jump probabilities,
leading either to a static and asymmetric exponential distribution centered at
for fickle agents or to diffusion with a positive or negative drift for
frozen agents. In terms of scaled coordinates and the
distributions are uniquely given by and in quantitative agreement with
direct simulations of the game. As the model avoids the reformulation in terms
of a constrained minimization problem it can be used for arbitrary payoff
functions with little calculational effort and provides a transparent and
simple formulation of the dynamics of the basic Minority Game in the asymmetric
phase
Coupled effects of local movement and global interaction on contagion
By incorporating segregated spatial domain and individual-based linkage into
the SIS (susceptible-infected-susceptible) model, we investigate the coupled
effects of random walk and intragroup interaction on contagion. Compared with
the situation where only local movement or individual-based linkage exists, the
coexistence of them leads to a wider spread of infectious disease. The roles of
narrowing segregated spatial domain and reducing mobility in epidemic control
are checked, these two measures are found to be conducive to curbing the spread
of infectious disease. Considering heterogeneous time scales between local
movement and global interaction, a log-log relation between the change in the
number of infected individuals and the timescale is found. A theoretical
analysis indicates that the evolutionary dynamics in the present model is
related to the encounter probability and the encounter time. A functional
relation between the epidemic threshold and the ratio of shortcuts, and a
functional relation between the encounter time and the timescale are
found
- …