1,721,030 research outputs found
On convergence of population processes in random environments to the stochastic heat equation with colored noise
No full textOn convergence of population processes in random environments to the stochastic heat equation with colored noise
No full text
New results on pathwise uniqueness for the heat equation with colored noise
No full textWe consider strong uniqueness and thus also existence of strong solutions for the stochastic heat equation with a multiplicative colored noise term. Here, the noise is white in time and colored in q dimensional space (q ! 1) with a singular correlation kernel. The noise coefficient is Hölder continuous in the solution. We discuss improvements of the sufficient conditions obtained in Mytnik, Perkins and Sturm (2006) that relate the Hölder coefficient with the singularity of the correlation kernel of the noise. For this we use new ideas of Mytnik and Perkins (2011) who treat the case of strong uniqueness for the stochastic heat equation with multiplicative white noise in one dimension. Our main result on pathwise uniqueness confirms a conjecture that was put forward in their paper
Mechanisms to synchronize neuronal activity
No full text Temporal aspects of neuronal activity have received increasing attention in recent years. Oscillatory dynamics and the synchronization of neuronal activity are hypothesized to be of functional relevance to information processing in the brain. Here we review theoretical studies of single neurons at different levels of abstraction, with an emphasis on the implications for properties of networks composed of such units. We then discuss the influence of different types of couplings and choices of parameters to the existence of a stable state of synchronous or oscillatory activity. Finally we relate these theoretical studies to the available experimental data, and suggest future lines of research
On spatial coalescents with multiple mergers in two dimensions
No full textWe consider the genealogy of a sample of individuals taken from a spatially structured population when the variance of the offspring distribution is relatively large. The space is structured into discrete sites of a graph G. If the population size at each site is large, spatial coalescents with multiple mergers, so called spatial Lambda-coalescents, for which ancestral lines migrate in space and coalesce according to some Lambda-coalescent mechanism, are shown to be appropriate approximations to the genealogy of a sample of individuals. We then consider as the graph G the two dimensional torus with side length 2L+1 and show that as L tends to infinity, and time is rescaled appropriately, the partition structure of spatial Lambda-coalescents of individuals sampled far enough apart converges to the partition structure of a non-spatial Kingman coalescent. From a biological point of view this means that in certain circumstances both the spatial structure as well as larger variances of the underlying offspring distribution are harder to detect from the sample. However, supplemental simulations show that for moderately large L the different structure is still evident
The spatial Λ -coalescent
No full textThis paper extends the notion of the Λ-coalescent of Pitman (1999) to the spatial setting. The partition elements of the spatial Λ-coalescent migrate in a (finite) geographical space and may only coalesce if located at the same site of the space. We characterize the Λ-coalescents that come down from infinity, in an analogous way to Schweinsberg (2000). Surprisingly, all spatial coalescents that come down from infinity, also come down from infinity in a uniform way. This enables us to study space-time asymptotics of spatial Λ-coalescents on large tori in d≥3 dimensions. Some of our results generalize and strengthen the corresponding results in Greven et al. (2005) concerning the spatial Kingman coalescent
On spatial coalescents with multiple mergers in two dimensions
No full textWe consider the genealogy of a sample of individuals taken from a spatially structured population when the variance of the offspring distribution is relatively large. The space is structured into discrete sites of a graph G. If the population size at each site is large, spatial coalescents with multiple mergers, so called spatial Lambda-coalescents, for which ancestral lines migrate in space and coalesce according to some Lambda-coalescent mechanism, are shown to be appropriate approximations to the genealogy of a sample of individuals. We then consider as the graph G the two dimensional torus with side length 2L+1 and show that as L tends to infinity, and time is rescaled appropriately, the partition structure of spatial Lambda-coalescents of individuals sampled far enough apart converges to the partition structure of a non-spatial Kingman coalescent. From a biological point of view this means that in certain circumstances both the spatial structure as well as larger variances of the underlying offspring distribution are harder to detect from the sample. However, supplemental simulations show that for moderately large L the different structure is still evident
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