1,053 research outputs found
Some simple but challenging Markov processes
In this note, we present few examples of Piecewise Deterministic Markov
Processes and their long time behavior. They share two important features: they
are related to concrete models (in biology, networks, chemistry,. . .) and they
are mathematically rich. Their math-ematical study relies on coupling method,
spectral decomposition, PDE technics, functional inequalities. We also relate
these simple examples to recent and open problems
Asymptotic preserving schemes for highly oscillatory kinetic equation
This work is devoted to the numerical simulation of a Vlasov-Poisson model
describing a charged particle beam under the action of a rapidly oscillating
external electric field. We construct an Asymptotic Preserving numerical scheme
for this kinetic equation in the highly oscillatory limit. This scheme enables
to simulate the problem without using any time step refinement technique.
Moreover, since our numerical method is not based on the derivation of the
simulation of asymptotic models, it works in the regime where the solution does
not oscillate rapidly, and in the highly oscillatory regime as well. Our method
is based on a "double-scale" reformulation of the initial equation, with the
introduction of an additional periodic variable
Mutual Information and Minimum Mean-square Error in Gaussian Channels
This paper deals with arbitrarily distributed finite-power input signals
observed through an additive Gaussian noise channel. It shows a new formula
that connects the input-output mutual information and the minimum mean-square
error (MMSE) achievable by optimal estimation of the input given the output.
That is, the derivative of the mutual information (nats) with respect to the
signal-to-noise ratio (SNR) is equal to half the MMSE, regardless of the input
statistics. This relationship holds for both scalar and vector signals, as well
as for discrete-time and continuous-time noncausal MMSE estimation. This
fundamental information-theoretic result has an unexpected consequence in
continuous-time nonlinear estimation: For any input signal with finite power,
the causal filtering MMSE achieved at SNR is equal to the average value of the
noncausal smoothing MMSE achieved with a channel whose signal-to-noise ratio is
chosen uniformly distributed between 0 and SNR
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