Bounds for the expected value of one-step processes

Mean-field models are often used to approximate Markov processes with large state-spaces. One-step processes, also known as birth-death processes, are an important class of such processes and are processes with state space $\{0,1,\ldots,N\}$ and where each transition is of size one. We derive explicit bounds on the expected value of such a process, bracketing it between the mean-field model and another simple ODE. Our bounds require that the Markov transition rates are density dependent polynomials that satisfy a sign condition. We illustrate the tightness of our bounds on the SIS epidemic process and the voter model.Comment: 14 pages, 4 figures, revise

Sampling-based Approximations with Quantitative Performance for the Probabilistic Reach-Avoid Problem over General Markov Processes

This article deals with stochastic processes endowed with the Markov (memoryless) property and evolving over general (uncountable) state spaces. The models further depend on a non-deterministic quantity in the form of a control input, which can be selected to affect the probabilistic dynamics. We address the computation of maximal reach-avoid specifications, together with the synthesis of the corresponding optimal controllers. The reach-avoid specification deals with assessing the likelihood that any finite-horizon trajectory of the model enters a given goal set, while avoiding a given set of undesired states. This article newly provides an approximate computational scheme for the reach-avoid specification based on the Fitted Value Iteration algorithm, which hinges on random sample extractions, and gives a-priori computable formal probabilistic bounds on the error made by the approximation algorithm: as such, the output of the numerical scheme is quantitatively assessed and thus meaningful for safety-critical applications. Furthermore, we provide tighter probabilistic error bounds that are sample-based. The overall computational scheme is put in relationship with alternative approximation algorithms in the literature, and finally its performance is practically assessed over a benchmark case study

Lepton Flavour Violating top decays at the LHC

We consider lepton flavour violating decays of the top quark, mediated by four-fermion operators. We compile constraints on a complete set of SU(3)*U(1)-invariant operators, arising from their loop contributions to rare decays and from HERA's single top search. The bounds on e-mu flavour change are more restrictive than l-tau; nonetheless the top could decay to a jet $+ e \bar{\mu}$ with a branching ratio of order $10^{-3}$. We estimate that the currently available LHC data (20 inverse-fb at 8 TeV) could be sensitive to $BR(t \to e \bar{\mu}$+ jet) $\sim 6\times 10^{-5}$, and extrapolate that 100 inverse-fb at 13 TeV could reach a sensitivity of $\sim 1 \times 10^{-5}$.Comment: 10 pages + Appendice