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Register a new userWhat is a stochastic Hamiltonian process on finite graph? An optimal transport answer
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We present a definition of stochastic Hamiltonian process on finite graph via its corresponding density dynamics in Wasserstein manifold. We demonstrate the existence of stochastic Hamiltonian process in many classical discrete problems, such as the optimal transport problem, Schrodinger equation and Schrodinger bridge problem (SBP). The stationary and periodic properties of Hamiltonian processes are also investigated in the framework of SBP.
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In this paper we give an answer to Furstenbergs problem on topological disjointness. Namely, we show that a transitive system $(X,T)$ is disjoint from all minimal systems if and only if $(X,T)$ is weakly mixing and there is some countable dense subset $D$ of $X$ such that for any minimal system $(Y,S)$, any point $yin Y$ and any open neighbourhood $V$ of $y$, and for any nonempty open subset $Usubset X$, there is $xin Dcap U$ satisfying that ${nin{ mathbb Z}_+: T^nxin U, S^nyin V}$ is syndetic. Some characterization for the general case is also described. As applications we show that if a transitive system $(X,T)$ is disjoint from all minimal systems, then so are $(X^n,T^{(n)})$ and $(X, T^n)$ for any $nin { mathbb N}$. It turns out that a transitive system $(X,T)$ is disjoint from all minimal systems if and only if the hyperspace system $(K(X),T_K)$ is disjoint from all minimal systems.
We present a new stochastic framework for studying ship capsize. It is a synthesis of two strands of transition state theory. The first is an extension of deterministic transition state theory to dissipative non-autonomous systems, together with a probability distribution over the forcing functions. The second is stochastic reachability and large deviation theory for transition paths in Markovian systems. In future work we aim to bring these together to make a tool for predicting capsize rate in different stochastic sea states, suggesting control strategies and improving designs.
In contrast to existing works on stochastic averaging on finite intervals, we establish an averaging principle on the whole real axis, i.e. the so-called second Bogolyubov theorem, for semilinear stochastic ordinary differential equations in Hilbert space with Poisson stable (in particular, periodic, quasi-periodic, almost periodic, almost automorphic etc) coefficients. Under some appropriate conditions we prove that there exists a unique recurrent solution to the original equation, which possesses the same recurrence property as the coefficients, in a small neighborhood of the stationary solution to the averaged equation, and this recurrent solution converges to the stationary solution of averaged equation uniformly on the whole real axis when the time scale approaches zero.
LTE networks are commonplace nowadays; however, comparatively little is known about where (and why) they are deployed, and the demand they serve. We shed some light on these issues through large-scale, crowd-sourced measurement. Our data, collected by users of the WeFi app, spans multiple operators and multiple cities, allowing us to observe a wide variety of deployment patterns. Surprisingly, we find that LTE is frequently used to improve the {em coverage} of network rather than the capacity thereof, and that no evidence shows that video traffic be a primary driver for its deployment. Our insights suggest that such factors as pre-existing networks and commercial policies have a deeper impact on deployment decisions than purely technical considerations.
Let $DeltasubsetneqV$ be a proper subset of the vertices $V$ of the defining graph of an aperiodic shift of finite type $(Sigma_{A}^{+},S)$. Let $Delta_{n}$ be the union of cylinders in $Sigma_{A}^{+}$ corresponding to the points $x$ for which the first $n$-symbols of $x$ belong to $Delta$ and let $mu$ be an equilibrium state of a Holder potential $phi$ on $Sigma_{A}^{+}$. We know that $mu(Delta_{n})$ converges to zero as $n$ diverges. We study the asymptotic behaviour of $mu(Delta_{n})$ and compare it with the pressure of the restriction of $phi$ to $Sigma_{Delta}$. The present paper extends some results in cite{CCC} to the case when $Sigma_{Delta}$ is irreducible and periodic. We show an explicit example where the asymptotic behaviour differs from the aperiodic case.
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