81 research outputs found
Stationary distributions for diffusions with inert drift
Consider reflecting Brownian motion in a bounded domain in that acquires drift in proportion to the amount of local time spent on the boundary of the domain. We show that the stationary distribution for the joint law of the position of the reflecting Brownian motion and the value of the drift vector has a product form. Moreover, the first component is uniformly distributed on the domain, and the second component has a Gaussian distribution. We also consider more general reflecting diffusions with inert drift as well as processes where the drift is given in terms of the gradient of a potential
Invariant densities for dynamical systems with random switching
We consider a non-autonomous ordinary differential equation on a smooth
manifold, with right-hand side that randomly switches between the elements of a
finite family of smooth vector fields. For the resulting random dynamical
system, we show that H\"ormander type hypoellipticity conditions are sufficient
for uniqueness and absolute continuity of an invariant measure.Comment: 16 pages; we replaced our original article to point out and close a
gap in the discussion of the Lorenz system in Section 7 (see Remark 2); this
gap is only present in the journal version of this article --- it wasn't
present in the previous arxiv versio
A Sublinear Variance Bound for Solutions of a Random Hamilton Jacobi Equation
We estimate the variance of the value function for a random optimal control
problem. The value function is the solution of a Hamilton-Jacobi
equation with random Hamiltonian
in dimension . It is known that homogenization occurs as , but little is known about the statistical fluctuations of .
Our main result shows that the variance of the solution is bounded
by . The proof relies on a modified Poincar\'e
inequality of Talagrand
Positive temperature versions of two theorems on first-passage percolation
The estimates on the fluctuations of first-passsage percolation due to
Talagrand (a tail bound) and Benjamini--Kalai--Schramm (a sublinear variance
bound) are transcribed into the positive-temperature setting of random
Schroedinger operators.Comment: 15 pp; to appear in GAFA Seminar Note
A dynamical trichotomy for structured populations experiencing positive density-dependence in stochastic environments
Positive density-dependence occurs when individuals experience increased
survivorship, growth, or reproduction with increased population densities.
Mechanisms leading to these positive relationships include mate limitation,
saturating predation risk, and cooperative breeding and foraging. Individuals
within these populations may differ in age, size, or geographic location and
thereby structure these populations. Here, I study structured population models
accounting for positive density-dependence and environmental stochasticity i.e.
random fluctuations in the demographic rates of the population. Under an
accessibility assumption (roughly, stochastic fluctuations can lead to
populations getting small and large), these models are shown to exhibit a
dynamical trichotomy: (i) for all initial conditions, the population goes
asymptotically extinct with probability one, (ii) for all positive initial
conditions, the population persists and asymptotically exhibits unbounded
growth, and (iii) for all positive initial conditions, there is a positive
probability of asymptotic extinction and a complementary positive probability
of unbounded growth. The main results are illustrated with applications to
spatially structured populations with an Allee effect and age-structured
populations experiencing mate limitation
Modelling Non-linear Crowd Dynamics in Bio-PEPA
Emergent phenomena occur due to the pattern of non-linear and distributed local interactions between the elements of a system over time. Surprisingly, agent based crowd models, in which the movement of each individual follows a limited set of simple rules, often re-produce quite closely the emergent behaviour of crowds that can be observed in reality. An example of such phenomena is the spontaneous self-organisation of drinking parties in the squares of cities in Spain, also known as "El Botellon" [20]. We revisit this case study providing an elegant stochastic process algebraic model in Bio-PEPA amenable to several forms of analyses, among which simulation and fluid flow analysis. We show that a fluid flow approximation, i.e. a deterministic reading of the average behaviour of the system, can provide an alternative and efficient way to study the same emergent behaviour as that explored in [20] where simulation was used instead. Besides empirical evidence, also an analytical justification is provided for the good correspondence found between simulation results and the fluid flow approximation
Applying Mean-Field Approximation to Continuous Time Markov Chains
The mean-field analysis technique is used to perform analysis of a system with a large number of components to determine the emergent deterministic behaviour and how this behaviour modifies when its parameters are perturbed. The computer science performance modelling and analysis community has found the mean-field method useful for modelling large-scale computer and communication networks. Applying mean-field analysis from the computer science perspective requires the following major steps: (1) describing how the agent populations evolve by means of a system of differential equations, (2) finding the emergent deterministic behaviour of the system by solving such differential equations, and (3) analysing properties of this behaviour. Depending on the system under analysis, performing these steps may become challenging. Often, modifications of the general idea are needed. In this tutorial we consider illustrating examples to discuss how the mean-field method is used in different application areas. Starting from the application of the classical technique, moving to cases where additional steps have to be used, such as systems with local communication. Finally, we illustrate the application of existing model checking analysis techniques
Stochastic population growth in spatially heterogeneous environments
Classical ecological theory predicts that environmental stochasticity
increases extinction risk by reducing the average per-capita growth rate of
populations. To understand the interactive effects of environmental
stochasticity, spatial heterogeneity, and dispersal on population growth, we
study the following model for population abundances in patches: the
conditional law of given is such that when is small the
conditional mean of is approximately , where and are the abundance and per
capita growth rate in the -th patch respectivly, and is the
dispersal rate from the -th to the -th patch, and the conditional
covariance of and is approximately . We show for such a spatially extended population that if
is the total population abundance, then ,
the vector of patch proportions, converges in law to a random vector
as , and the stochastic growth rate equals the space-time average per-capita growth rate
\sum_i\mu_i\E[Y_\infty^i] experienced by the population minus half of the
space-time average temporal variation \E[\sum_{i,j}\sigma_{ij}Y_\infty^i
Y_\infty^j] experienced by the population. We derive analytic results for the
law of , find which choice of the dispersal mechanism produces an
optimal stochastic growth rate for a freely dispersing population, and
investigate the effect on the stochastic growth rate of constraints on
dispersal rates. Our results provide fundamental insights into "ideal free"
movement in the face of uncertainty, the persistence of coupled sink
populations, the evolution of dispersal rates, and the single large or several
small (SLOSS) debate in conservation biology.Comment: 47 pages, 4 figure
Influence of opinion dynamics on the evolution of games
Under certain circumstances such as lack of information or bounded
rationality, human players can take decisions on which strategy to choose in a
game on the basis of simple opinions. These opinions can be modified after each
round by observing own or others payoff results but can be also modified after
interchanging impressions with other players. In this way, the update of the
strategies can become a question that goes beyond simple evolutionary rules
based on fitness and become a social issue. In this work, we explore this
scenario by coupling a game with an opinion dynamics model. The opinion is
represented by a continuous variable that corresponds to the certainty of the
agents respect to which strategy is best. The opinions transform into actions
by making the selection of an strategy a stochastic event with a probability
regulated by the opinion. A certain regard for the previous round payoff is
included but the main update rules of the opinion are given by a model inspired
in social interchanges. We find that the dynamics fixed points of the coupled
model is different from those of the evolutionary game or the opinion models
alone. Furthermore, new features emerge such as the resilience of the fraction
of cooperators to the topology of the social interaction network or to the
presence of a small fraction of extremist players.Comment: 7 pages, 5 figure
Stochastic Approximation to Understand Simple Simulation Models
This paper illustrates how a deterministic approximation of a stochastic process
can be usefully applied to analyse the dynamics of many simple simulation models. To
demonstrate the type of results that can be obtained using this approximation, we present two
illustrative examples which are meant to serve as methodological references for researchers
exploring this area. Finally, we prove some convergence results for simulations of a family
of evolutionary games, namely, intra-population imitation models in n-player games with
arbitrary payoffs.Ministerio de Educación (JC2009- 00263), Ministerio de Ciencia e Innovación (CONSOLIDER-INGENIO 2010: CSD2010-00034, DPI2010-16920
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