Search results for " Differential equations"
showing 10 items of 146 documents
Infinite rate mutually catalytic branching in infinitely many colonies: The longtime behavior
2012
Consider the infinite rate mutually catalytic branching process (IMUB) constructed in [Infinite rate mutually catalytic branching in infinitely many colonies. Construction, characterization and convergence (2008) Preprint] and [Ann. Probab. 38 (2010) 479-497]. For finite initial conditions, we show that only one type survives in the long run if the interaction kernel is recurrent. On the other hand, under a slightly stronger condition than transience, we show that both types can coexist.
Quantitative ergodicity for some switched dynamical systems
2012
International audience; We provide quantitative bounds for the long time behavior of a class of Piecewise Deterministic Markov Processes with state space Rd × E where E is a finite set. The continuous component evolves according to a smooth vector field that switches at the jump times of the discrete coordinate. The jump rates may depend on the whole position of the process. Under regularity assumptions on the jump rates and stability conditions for the vector fields we provide explicit exponential upper bounds for the convergence to equilibrium in terms of Wasserstein distances. As an example, we obtain convergence results for a stochastic version of the Morris-Lecar model of neurobiology.
Simulation of BSDEs with jumps by Wiener Chaos Expansion
2016
International audience; We present an algorithm to solve BSDEs with jumps based on Wiener Chaos Expansion and Picard's iterations. This paper extends the results given in Briand-Labart (2014) to the case of BSDEs with jumps. We get a forward scheme where the conditional expectations are easily computed thanks to chaos decomposition formulas. Concerning the error, we derive explicit bounds with respect to the number of chaos, the discretization time step and the number of Monte Carlo simulations. We also present numerical experiments. We obtain very encouraging results in terms of speed and accuracy.
Donsker-Type Theorem for BSDEs: Rate of Convergence
2019
In this paper, we study in the Markovian case the rate of convergence in Wasserstein distance when the solution to a BSDE is approximated by a solution to a BSDE driven by a scaled random walk as introduced in Briand, Delyon and Mémin (Electron. Commun. Probab. 6 (2001) Art. ID 1). This is related to the approximation of solutions to semilinear second order parabolic PDEs by solutions to their associated finite difference schemes and the speed of convergence. peerReviewed
Random walk approximation of BSDEs with H{\"o}lder continuous terminal condition
2018
In this paper, we consider the random walk approximation of the solution of a Markovian BSDE whose terminal condition is a locally Hölder continuous function of the Brownian motion. We state the rate of the L2-convergence of the approximated solution to the true one. The proof relies in part on growth and smoothness properties of the solution u of the associated PDE. Here we improve existing results by showing some properties of the second derivative of u in space. peerReviewed
Spatio-temporal behaviour of the deep chlorophyll maximum in Mediterranean Sea: Development of a stochastic model for picophytoplankton dynamics
2013
In this paper, by using a stochastic reaction-diffusion-taxis model, we analyze the picophytoplankton dynamics in the basin of the Mediterranean Sea, characterized by poorly mixed waters. The model includes intraspecific competition of picophytoplankton for light and nutrients. The multiplicative noise sources present in the model account for random fluctuations of environmental variables. Phytoplankton distributions obtained from the model show a good agreement with experimental data sampled in two different sites of the Sicily Channel. The results could be extended to analyze data collected in different sites of the Mediterranean Sea and to devise predictive models for phytoplankton dynam…
Modeling of Sensory Characteristics Based on the Growth of Food Spoilage Bacteria
2016
During last years theoretical works shed new light and proposed new hypothesis on the mechanisms which regulate the time behaviour of biological populations in different natural systems. Despite of this, the role of environmental variables in ecological systems is still an open question. Filling this gap of knowledge is a crucial task for a deeper comprehension of the dynamics of biological populations in real ecosystems. In this work we study how the dynamics of food spoilage bacteria influences the sensory characteristics of fresh fish specimens. This topic is crucial for a better understanding of the role played by the bacterial growth on the organoleptic properties, and for the quality …
Mean-field games and two-point boundary value problems
2014
A large population of agents seeking to regulate their state to values characterized by a low density is considered. The problem is posed as a mean-field game, for which solutions depend on two partial differential equations, namely the Hamilton-Jacobi-Bellman equation and the Fokker-Plank-Kolmogorov equation. The case in which the distribution of agents is a sum of polynomials and the value function is quadratic is considered. It is shown that a set of ordinary differential equations, with two-point boundary value conditions, can be solved in place of the more complicated partial differential equations associated with the problem. The theory is illustrated by a numerical example.
Oscillation of second-order neutral differential equations
2015
Author's version of an article in the journal: Funkcialaj Ekvacioj. Also available from the publisher at: http://www.math.kobe-u.ac.jp/~fe/
Stochastic Differential Equations
2020
Stochastic differential equations describe the time evolution of certain continuous n-dimensional Markov processes. In contrast with classical differential equations, in addition to the derivative of the function, there is a term that describes the random fluctuations that are coded as an Ito integral with respect to a Brownian motion. Depending on how seriously we take the concrete Brownian motion as the driving force of the noise, we speak of strong and weak solutions. In the first section, we develop the theory of strong solutions under Lipschitz conditions for the coefficients. In the second section, we develop the so-called (local) martingale problem as a method of establishing weak so…