RELATIVE ENTROPY AND LARGE DEVIATIONS UNDER SUBLINEAR EXPECTATIONS

We give a definition of relative entropy with respect to a sublinear expectation and establish large deviation principle for the empirical measures for independent random variables under the sublinear expectation.

A SIGN CHANGING SOLUTION FOR ELLIPTIC EQUATION WITH SUBLINEAR TERM AT ORIGIN

It is proved that the semilinear elliptic problem with zero boundary value -Δ u=λu-|u| q-1 u has a changing sign solution, as q∈(0,1) and λ>λ 2 , where λ 2 is the second eigenvalue of the operator -Δ in the space H 1 0(Ω).

A New Demonstration of a Theorem for Sublinear Approximations

The demonstration of a theorem for sublinear approximations, due to Demyanov and Rubinov, is modified as a new one in this paper.

OSCILLATION CRITERION FOR A CLASS OFSECOND ORDER SUBLINEAR DIFFERENTIALEQUATIONS

In this paper some oscillation criteria are established for equation [q(t)y'] +a(t)f(y) = 0, where a(t) is not assumed to be non-negative and f(y) is nodecreasing in y, and yf(y) > 0 for y≠0, f(y) also satisfics a sublinear condition, q(t) is a positive function on [0, ). These results extend earlier oscillation theorems of Philos and Wong.

A general central limit theorem under sublinear expectations

Under some weaker conditions,we give a central limit theorem under sublinear expectations,which extends Peng's central limit theorem.

POSITIVE SOLUTIONS OF SINGULAR SUBLINEAR SECOND ORDER BOUNDARY VALUE PROBLEMS

This paper mainly studies the existence of positive solutions of singular sub-linear boundary value problems concerning the generalized Emden-Fowler equations. Anecessary and sufficient condition for the existence of positive solutions to this problemhas been obtained by using the method of lower and upper solutions with the fixed poilltt heorems.

The minimal sublinear expectations and their related properties

In this paper, we prove that for a sublinear expectation ?[·] defined on L 2(Ω, $ \mathcal{F} $ ), the following statements are equivalent: ? is a minimal member of the set of all sublinear expectations defined on L 2(Ω, $ \mathcal{F} $ )? is linearthe two-dimensional Jensen’s inequality for ? holds.Furthermore, we prove a sandwich theorem for subadditive expectation and superadditive expectation.

MATHER SETS FOR SUBLINEAR DUFFING EQUATIONS

The existence of Mather sets(generalized quasiperiodic solutions and uNlinked periodicsolutions)for sublinear Duffing equations is shown. Here the approach is based on the use ofaction-angle variables and the application of a generalized version of Aubry-Mather theoremon semi-cylinder with finite twist assumption.

Moment bounds for IID sequences under sublinear expectations

With the notion of independent identically distributed(IID) random variables under sublinear expectations introduced by Peng,we investigate moment bounds for IID sequences under sublinear expectations. We obtain a moment inequality for a sequence of IID random variables under sublinear expectations. As an application of this inequality,we get the following result:For any continuous functionsatisfying the growth condition |(x) | C(1 + |x|p) for some C > 0,p 1 depending on ,the central limit theorem under sublinear expectations obtained by Peng still holds.

Multi-dimensional Central Limit Theorems and Laws of Large Numbers under Sublinear Expectations

In this paper, we present some multi-dimensional central limit theorems and laws of large numbers under sublinear expectations, which extend some previous results.

On the Sublinear Convergence Rate of Multi-block ADMM

The alternating direction method of multipliers(ADMM)is widely used in solving structured convex optimization problems.Despite its success in practice,the convergence of the standard ADMM for minimizing the sum of N(N≥3)convex functions,whose variables are linked by linear constraints,has remained unclear for a very long time.Recently,Chen et al.(Math Program,doi:10.1007/s10107-014-0826-5,2014)provided a counter-example showing that the ADMM for N≥3 may fail to converge without further conditions.Since the ADMM for N≥3 has been very successful when applied to many problems arising from real practice,it is worth further investigating under what kind of sufficient conditions it can be guaranteed to converge.In this paper,we present such sufficient conditions that can guarantee the sublinear convergence rate for the ADMM for N≥3.Specifically,we show that if one of the functions is convex(not necessarily strongly convex)and the other N-1 functions are strongly convex,and the penalty parameter lies in a certain region,the ADMM converges with rate O(1/t)in a certain ergodic sense and o(1/t)in a certain non-ergodic sense,where t denotes the number of iterations.As a by-product,we also provide a simple proof for the O(1/t)convergence rate of two-blockADMMin terms of both objective error and constraint violation,without assuming any condition on the penalty parameter and strong convexity on the functions.

The Existence Theorems for a Class of Sublinear Elliptic Equations in R^N

An existence theorem for the solution to the equation -△u+b(x)u=f(x,u),in R^N is given by means of variational method where b(x)→∞,as丨x丨→∞ and f(x,s)has linear growth in s at infinity and sublinear growth in s at zero.For a special case,some multiplicity result is proved.

A REMARK ON PERIODIC SOLUTIONS OF SINGULAR HAMILTONIAN SYSTEMS WITH SUBLINEAR TERMS

In this paper,we prove the existence of multiple periodic solutionsfor a class of singular Hamiltonian systems with sublinear terms via variationalmethods.

ON INDEFINITE SUBLINEAR ELLIPTIC EQUATIONS

1.IntroductionLetnCRN,N23,beasmoothboundeddomainandconsiderthesemilinearellipticequationswhereatbELoc(Q),AERIisaparameter.Itisasimplifiedmodelforsomeproblemsinpopulationdynamicsandinreaction-diffusionprocessess(seethereferencesof[1]).Bytheresultsof[...

Convergences of Random Variables Under Sublinear Expectations

In this note, the authors survey the existing convergence results for random variables under sublinear expectations, and prove some new results. Concretely, under the assumption that the sublinear expectation has the monotone continuity property, the authors prove that convergence in capacity is stronger than convergence in distribution,and give some equivalent characterizations of convergence in distribution. In addition,they give a dominated convergence theorem under sublinear expectations, which may have its own interest.

Sublinear operators on block-type spaces

This study establishes the boundedness of sublinear operators on block spaces built on Banach function spaces. These results are used to study the boundedness of the Marcinkiewicz integrals, singular integral operators and fractional integral operators with homogeneous kernels on block-type spaces.

Strong Laws of Large Numbers for Sublinear Expectation under Controlled 1st Moment Condition

This paper deals with strong laws of large numbers for sublinear expectation under controlled 1st moment condition. For a sequence of independent random variables,the author obtains a strong law of large numbers under conditions that there is a control random variable whose 1st moment for sublinear expectation is finite. By discussing the relation between sublinear expectation and Choquet expectation, for a sequence of i.i.d random variables, the author illustrates that only the finiteness of uniform 1st moment for sublinear expectation cannot ensure the validity of the strong law of large numbers which in turn reveals that our result does make sense.

Some Inequalities and Limit Theorems Under Sublinear Expectations

In this note, we study inequality and limit theory under sublinear expectations. We mainly prove Doob＇s inequality for submartingale and Kolmogrov＇s inequality. By Kolmogrov＇s inequality, we obtain a special version of Kolmogrov＇s law of large numbers. Finally, we present a strong law of large numbers for independent and identically distributed random variables under one-order type moment condition.

G-Lévy processes under sublinear expectations

We introduce G-Lévy processes which develop the theory of processes with independent and stationary increments under the framework of sublinear expectations.We then obtain the Lévy-Khintchine formula and the existence for G-Lévy processes.We also introduce G-Poisson processes.

Convergence rate of Peng’s law of large numbers under sublinear expectations

This short note provides a new and simple proof of the convergence rate for the Peng’s law of large numbers under sublinear expectations,which improves the results presented by Song[15]and Fang et al.[3].