Abelian网络中的本质增强：连续性与一致严格单调性

Continuity and strict positivity of the multi-layer extension of the stochastic heat equation

We prove the continuity and strict positivity of the multi-layer extension to the stochastic heat equation introduced in [OW11] which form a hierarchy of partition functions for the continuum directed random polymer. This shows that the corresponding free energy (logarithm of the partition function) is well defined. This is also a step towards proving the conjecture stated at the end of the above paper that an array of such partition functions has the Markov property.

Strict monotonicity of percolation thresholds under covering maps

We answer a question of Benjamini and Schramm by proving that under reasonable conditions, quotienting a graph strictly increases the value of its percolation critical parameter $p_c$. More precisely, let $\mathcal{G}=(V,E)$ be a quasi-transitive graph with $p_c(\mathcal{G})<1$, and let $G$ be a nontrivial group that acts freely on $V$ by graph automorphisms. Assume that $\mathcal{H}:=\mathcal{G}/G$ is quasi-transitive. Then one has $p_c(\mathcal{G})<p_c(\mathcal{H})$. We provide results beyond this setting: we treat the case of general covering maps and provide a similar result for the unique…

反应性赌博可能产生反效果： ruin概率随着p增加而增加，初始财富呈Hölder连续

Survival probability in reactive gambling: monotonicity, singularity, and Hölder continuity

该研究探讨了反应性赌博策略的反效果，证明当p<1/2且初始财富x>2时，赌徒有正概率存活，且破产概率随p增加而增加，但对x呈现奇异的Hölder连续性。

Strict monotonicity of principal eigenvalues of elliptic operators in $\mathbb{R}^d$ and risk-sensitive control

This paper studies the eigenvalue problem on $\mathbb{R}^d$ for a class of second order, elliptic operators of the form $\mathscr{L} = a^{ij}\partial_{x_i}\partial_{x_j} + b^{i}\partial_{x_i} + f$, associated with non-degenerate diffusions. We show that strict monotonicity of the principal eigenvalue of the operator with respect to the potential function $f$ fully characterizes the ergodic properties of the associated ground state diffusion, and the unicity of the ground state, and we present a comprehensive study of the eigenvalue problem from this point of view. This allows us to extend or s…

Strict continuity of the transition semigroup for the solution of a well-posed martingale problem

In this note we connect the notion of solutions of a martingale problem to the notion of a strongly continuous and locally equi-continuous semigroup on the space of bounded continuous functions equipped with the strict topology. This extends the classical connection of semigroups to Markov processes that was used successfully in the context of compact spaces to the context of Polish spaces. In addition, we consider the context of locally compact spaces and show how the transition semigroup on the space of functions vanishing at infinity can be extended to the space of bounded continuous functi…

Monotonicity of the speed for biased random walk on Galton-Watson tree

Ben Arous, Fribergh and Sidoravicius \cite{GAV2014} proved that speed of biased random walk $RW_λ$ on a Galton-Watson tree without leaves is strictly decreasing for $λ\leq \frac{m_1}{1160},$ where $m_1$ is minimal degree of the Galton-Watson tree. And A\"ıdékon \cite{EA2013} improved this result to $λ\leq \frac{1}{2}.$ In this paper, we prove that for the $RW_λ$ on a Galton-Watson tree without leaves, its speed is strictly decreasing for $λ\in \left[0,\frac{m_1}{1+\sqrt{1-\frac{1}{m_1}}}\right]$ when $m_1\geq 2;$ and we owe the proof to A\"ıdékon \cite{EA2013}.

The fractal dimension of Liouville quantum gravity: universality, monotonicity, and bounds

We prove that for each $γ\in (0,2)$, there is an exponent $d_γ> 2$, the "fractal dimension of $γ$-Liouville quantum gravity (LQG)", which describes the ball volume growth exponent for certain random planar maps in the $γ$-LQG universality class, the exponent for the Liouville heat kernel, and exponents for various continuum approximations of $γ$-LQG distances such as Liouville graph distance and Liouville first passage percolation. We also show that $d_γ$ is a continuous, strictly increasing function of $γ$ and prove upper and lower bounds for $d_γ$ which in some cases greatly improve on previ…

二维定向渗流中临界点的单调性

Monotonicity of the critical point in two-dimensional oriented percolation with enhancement

本文研究了二维定向渗流中临界点的单调性，通过引入对角线边的概率参数ε，证明了临界参数随ε的增加而严格递减。

Convex ordering of Pólya random variables and monotonicity of the error estimate of Bernstein-Stancu operators

In the present paper we show that in Pólya's urn model, for an arbitrarily fixed initial distribution of the urn, the corresponding random variables satisfy a convex ordering with respect to the replacement parameter. As an application, we show that in the class of convex functions, the absolute value of the error of Bernstein-Stancu operators is a non-decreasing (strictly increasing under an additional hypothesis) function of the corresponding parameter. The proof relies on two results of independent interest: an interlacing lemma of three sets and the monotonicity of the (partial) first mome…

Monotonicity of facet numbers of random convex hulls

Let $X_1,\ldots,X_n$ be independent random points that are distributed according to a probability measure on $\mathbb{R}^d$ and let $P_n$ be the random convex hull generated by $X_1,\ldots,X_n$ ($n\geq d+1$). Natural classes of probability distributions are characterized for which, by means of Blaschke-Petkantschin formulae from integral geometry, one can show that the mean facet number of $P_n$ is strictly monotonically increasing in $n$.

One-sided continuity properties for the Schonmann projection

We consider the plus-phase of the two-dimensional Ising model below the critical temperature. In $1989$ Schonmann proved that the projection of this measure onto a one-dimensional line is not a Gibbs measure. After many years of continued research which have revealed further properties of this measure, the question whether or not it is a Gibbs measure in an almost sure sense remains open. In this paper we study the same measure by interpreting it as a temporal process. One of our main results is that the Schonmann projection is almost surely a regular $g$-measure. That is, it does possess the …

具有随机单调性和一般增长生成元的随机时间 Horizon BSDEs 的加权解及相关 PDEs

Weighted solutions of random time horizon BSDEs with stochastic monotonicity and general growth generators and related PDEs

研究具有随机终端时间、随机单调性及一般增长生成元的多维 BSDEs，在无有限矩限制下通过创新分析技术建立了加权解的存在唯一性，统一强化了现有结果，并给出连续依赖性、稳定性定理及 PDEs 的非线性 Feynman-Kac 公式。

Continuity Results and Estimates for the Lyapunov Exponent of Brownian Motion in Random Potential

We collect some applications of the variational formula established by Schröder (1988) and Rueß(2013) for the quenched Lyapunov exponent of Brownian motion in stationary and ergodic nonnegative potential. We show for example that the Lyapunov exponent for nondeterministic potential is strictly lower than the Lyapunov exponent for the averaged potential. The behaviour of the Lyapunov exponent under independent perturbations of the underlying potential is examined. And with the help of counterexamples we are able to give a detailed picture of the continuity properties of the Lyapunov exponent.