493 research outputs found
Accelerated nonlocal nonsymmetric dispersion for monostable equations on the real line
We consider the accelerated propagation of solutions to equations with a nonlocal linear dispersion on the real line and monostable nonlinearities (both local or nonlocal, however, not degenerated at ), in the case when either of the dispersion kernel or the initial condition has regularly heavy tails at both , perhaps different. We show that, in such case, the propagation in the right direction is fully determined by the right tails of either the kernel or the initial condition. We describe both cases of integrable and monotone initial conditions which may give different orders of the acceleration. Our approach is based, in particular, on the extension of the theory of sub-exponential distributions, which we introduced early in [D. Finkelshtein, P. Tkachov. 'Kesten's bound for sub-exponential densities on the real line and its multi-dimensional analogues', Advances in Applied Probability, 2018, 50(2), 373-395]
Propagation of chaos for rank-based interacting diffusions and long time behaviour of a scalar quasilinear parabolic equation
We study a quasilinear parabolic Cauchy problem with a cumulative
distribution function on the real line as an initial condition. We call
'probabilistic solution' a weak solution which remains a cumulative
distribution function at all times. We prove the uniqueness of such a solution
and we deduce the existence from a propagation of chaos result on a system of
scalar diffusion processes, the interactions of which only depend on their
ranking. We then investigate the long time behaviour of the solution. Using a
probabilistic argument and under weak assumptions, we show that the flow of the
Wasserstein distance between two solutions is contractive. Under more stringent
conditions ensuring the regularity of the probabilistic solutions, we finally
derive an explicit formula for the time derivative of the flow and we deduce
the convergence of solutions to equilibrium.Comment: Stochastic partial differential equations: analysis and computations
(2013) http://dx.doi.org/10.1007/s40072-013-0014-
Random attractors for degenerate stochastic partial differential equations
We prove the existence of random attractors for a large class of degenerate
stochastic partial differential equations (SPDE) perturbed by joint additive
Wiener noise and real, linear multiplicative Brownian noise, assuming only the
standard assumptions of the variational approach to SPDE with compact
embeddings in the associated Gelfand triple. This allows spatially much rougher
noise than in known results. The approach is based on a construction of
strictly stationary solutions to related strongly monotone SPDE. Applications
include stochastic generalized porous media equations, stochastic generalized
degenerate p-Laplace equations and stochastic reaction diffusion equations. For
perturbed, degenerate p-Laplace equations we prove that the deterministic,
infinite dimensional attractor collapses to a single random point if enough
noise is added.Comment: 34 pages; The final publication is available at
http://link.springer.com/article/10.1007%2Fs10884-013-9294-
Brownian bridges to submanifolds
We introduce and study Brownian bridges to submanifolds. Our method involves
proving a general formula for the integral over a submanifold of the minimal
heat kernel on a complete Riemannian manifold. We use the formula to derive
lower bounds, an asymptotic relation and derivative estimates. We also see a
connection to hypersurface local time. This work is motivated by the desire to
extend the analysis of path and loop spaces to measures on paths which
terminate on a submanifold
Analysis of symmetries in models of multi-strain infections
In mathematical studies of the dynamics of multi-strain diseases caused by antigenically diverse pathogens, there is a substantial interest in analytical insights. Using the example of a generic model of multi-strain diseases with cross-immunity between strains, we show that a significant understanding of the stability of steady states and possible dynamical behaviours can be achieved when the symmetry of interactions between strains is taken into account. Techniques of equivariant bifurcation theory allow one to identify the type of possible symmetry-breaking Hopf bifurcation, as well as to classify different periodic solutions in terms of their spatial and temporal symmetries. The approach is also illustrated on other models of multi-strain diseases, where the same methodology provides a systematic understanding of bifurcation scenarios and periodic behaviours. The results of the analysis are quite generic, and have wider implications for understanding the dynamics of a large class of models of multi-strain diseases
The effects of symmetry on the dynamics of antigenic variation
In the studies of dynamics of pathogens and their interactions with a host
immune system, an important role is played by the structure of antigenic
variants associated with a pathogen. Using the example of a model of antigenic
variation in malaria, we show how many of the observed dynamical regimes can be
explained in terms of the symmetry of interactions between different antigenic
variants. The results of this analysis are quite generic, and have wider
implications for understanding the dynamics of immune escape of other
parasites, as well as for the dynamics of multi-strain diseases.Comment: 21 pages, 4 figures; J. Math. Biol. (2012), Online Firs
On the wellposedness of some McKean models with moderated or singular diffusion coefficient
We investigate the well-posedness problem related to two models of nonlinear
McKean Stochastic Differential Equations with some local interaction in the
diffusion term. First, we revisit the case of the McKean-Vlasov dynamics with
moderate interaction, previously studied by Meleard and Jourdain in [16], under
slightly weaker assumptions, by showing the existence and uniqueness of a weak
solution using a Sobolev regularity framework instead of a Holder one. Second,
we study the construction of a Lagrangian Stochastic model endowed with a
conditional McKean diffusion term in the velocity dynamics and a nondegenerate
diffusion term in the position dynamics
Review: Allelochemicals as multi-kingdom plant defence compounds: towards an integrated approach
© 2020 The Authors. Pest Management Science published by John Wiley & Sons Ltd on behalf of Society of Chemical Industry. The capability of synthetic pesticides to manage weeds, insect pests and pathogens in crops has diminished due to evolved resistance. Sustainable management is thus becoming more challenging. Novel solutions are needed and, given the ubiquity of biologically active secondary metabolites in nature, such compounds require further exploration as leads for novel crop protection chemistry. Despite improving understanding of allelochemicals, particularly in terms of their potential for use in weed control, their interactions with multiple biotic kingdoms have to date largely been examined in individual compounds and not as a recurrent phenomenon. Here, multi-kingdom effects in allelochemicals are introduced by defining effects on various organisms, before exploring current understanding of the inducibility and possible ecological roles of these compounds with regard to the evolutionary arms race and dose–response relationships. Allelochemicals with functional benefits in multiple aspects of plant defence are described. Gathering these isolated areas of science under the unified umbrella of multi-kingdom allelopathy encourages the development of naturally-derived chemistries conferring defence to multiple discrete biotic stresses simultaneously, maximizing benefits in weed, insect and pathogen control, while potentially circumventing resistance. © 2020 The Authors. Pest Management Science published by John Wiley & Sons Ltd on behalf of Society of Chemical Industry
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