Branching random walk with trapping zones

We consider a branching random walk with values in a certain set $\mathbb{S}$, where the branching mechanism is different according to whether particles (individuals) are in a certain so called trapping set $A?\mathbb{S}$ or not. We are then interested, under different scenarios, in properties of either the transient random measure describing distribution of individuals on $\mathbb{S}$ over time or its asymptotic behaviour.     

Weak Dirichlet processes with jumps

This paper develops systematically the stochastic calculus via regularization in the case of jump processes. In particular one continues the analysis of real-valued càdlàg weak Dirichlet processes with respect to a given filtration. Such a process is the sum of a local martingale and an adapted process $A$ such that $[N,A]=0$, for any continuous local martingale $N$. Given a function $u:[0,T]\times \mathbb{R}\to \mathbb{R}$, which is of class $C^{0,1}$ (or sometimes less), we provide a chain rule type expansion for $u(t,X_t)$ which stands in applications for a chain Itô type rule.     

Ideal amenability of module Lau Banach algebra

Given Banach algebras $A$ and $B$, where $A$ be a Banach $B$-bimodule. In this paper we study the ideal amenability, approximate ideal and cyclic ideal amenability of module Lau Banach algebra $A{\times }_{\alpha }B$.     

Covariance of stochastic integrals with respect to fractional Brownian motion

We find an explicit expression for the cross-covariance between stochastic integral processes with respect to a $d$-dimensional fractional Brownian motion (fBm) $B_t$ with Hurst parameter $H>\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$, where the integrands are vector fields applied to $B_t$. It provides, for example, a direct alternative proof of Y. Hu and D. Nualart’s result that the stochastic integral component in the fractional Bessel process decomposition is not itself a fractional Brownian motion.     

Vertex-minors and the Erdős–Hajnal conjecture

We prove that for every graph $H$, there exists $\epsilon >0$ such that every $n$-vertex graph with no vertex-minors isomorphic to $H$ has a pair of disjoint sets $A$, $B$ of vertices such that $\left|A\right|,\left|B\right|\ge \epsilon n$ and $A$ is complete or anticomplete to $B$. We deduce this from recent work of Chudnovsky, Scott, Seymour, and Spirkl (2018). This proves the analog of the Erd?s–Hajnal conjecture for vertex-minors.     

Nonlinear Fourier transforms for the sine-Gordon equation in the quarter plane

Using the Unified Transform, also known as the Fokas method, the solution of the sine-Gordon equation in the quarter plane can be expressed in terms of the solution of a matrix Riemann–Hilbert problem whose definition involves four spectral functions $a,b,A,B$. The functions $a(k)$ and $b(k)$ are defined via a nonlinear Fourier transform of the initial data, whereas $A(k)$ and $B(k)$ are defined via a nonlinear Fourier transform of the boundary values. In this paper, we provide an extensive study of these nonlinear Fourier transforms and the associated eigenfunctions under weak regularity and decay assumptions on the initial and boundary values. The results can be used to determine the long-time asymptotics of the sine-Gordon quarter-plane solution via nonlinear steepest descent techniques.     

Mutually orthogonal matrices from division algebras

Matrices A and B in $M_n(\mathbb{C})$ are said to be mutually orthogonal if $A{B}^{?}+B{A}^{?}=0$, where ^? denotes the conjugate transpose. We study cardinalities of certain $\mathbb{R}$-linearly independent families of matrices arising from matrix embeddings of a division algebra of index m with center a number field Z, satisfying the property that matrices from different families are mutually orthogonal. The question is of importance in the context of coding for certain wireless channels, where the cardinalities of such sets is connected to the maximum code rate consistent with low decoding complexity. It follows from our results that the maximum code rate for the codes we consider is severely limited.     

Pointwise estimates for first passage times of perpetuity sequences

We consider first passage times ${\tau }_u=inf\{n:Y_n>u\}$ for the perpetuity sequence

$Y_n=B_1+A_1{B}_2+?+(A_1\dots A_{n?1})B_n,$

where $(A_n,B_n)$ are i.i.d. random variables with values in ${\mathbb{R}}^{+}\times \mathbb{R}$. Recently, a number of limit theorems related to ${\tau }_u$ were proved including the law of large numbers, the central limit theorem and large deviations theorems (see Buraczewski et al., in press). We obtain a precise asymptotics of the sequence $\mathbb{P}[{\tau }_u=logu∕\rho ]$, $\rho >0$, $u\to \infty$ which considerably improves the previous results of Buraczewski et al. (in press). There, probabilities $\mathbb{P}[{\tau }_u\in I_u]$ were identified, for some large intervals $I_u$ around $k_u$, with lengths growing at least as $loglogu$. Remarkable analogies and differences to random walks (Buraczewski and Ma?lanka, in press; Lalley, 1984) are discussed.     

Hamilton differential Harnack inequality and W-entropy for Witten Laplacian on Riemannian manifolds

In this paper, we prove the Hamilton differential Harnack inequality for positive solutions to the heat equation of the Witten Laplacian on complete Riemannian manifolds with the $CD(?K,m)$-condition, where $m\in [n,\infty )$ and $K\ge 0$ are two constants. Moreover, we introduce the W-entropy and prove the W-entropy formula for the fundamental solution of the Witten Laplacian on complete Riemannian manifolds with the $CD(?K,m)$-condition and on compact manifolds equipped with $(?K,m)$-super Ricci flows.     

Large deviations for generalized Polya urns with arbitrary urn function

We consider a generalized two-color Polya urn (black and white balls) first introduced by Hill et al. (1980), where the urn composition evolves as follows: let $\pi :\left[0,1\right]\to \left[0,1\right]$, and denote by $x_n$ the fraction of black balls after step $n$, then at step $n+1$ a black ball is added with probability $\pi \left(x_n\right)$ and a white ball is added with probability $1?\pi \left(x_n\right)$. Originally introduced to mimic attachment under imperfect information, this model has found applications in many fields, ranging from Market Share modeling to polymer physics and biology.In this work we discuss large deviations for a wide class of continuous urn functions $\pi$. In particular, we prove that this process satisfies a Sample-Path Large Deviations principle, also providing a variational representation for the rate function. Then, we derive a variational representation for the limit

$?\left(s\right)=\underset{n\to \infty }{lim}\frac{1}{n}log\mathbb{P}\left(n{x}_n=?sn?\right),s\in \left[0,1\right],$

where $n{x}_n$ is the number of black balls at time $n$, and use it to give some insight on the shape of $?\left(s\right)$. Under suitable assumptions on $\pi$ we are able to identify the optimal trajectory. We also find a non-linear Cauchy problem for the Cumulant Generating Function and provide an explicit analysis for some selected examples. In particular we discuss the linear case, which embeds the Bagchi–Pal Model [6], giving the exact implicit expression for $?$ in terms of the Cumulant Generating Function.     

On purely discontinuous additive functionals of subordinate Brownian motions

Let $A_t={\sum }_{s\le t}F(X_{s?},X_s)$ be a purely discontinuous additive functional of a subordinate Brownian motion $X=(X_t,{\mathbb{P}}_x)$. We give a sufficient condition on the non-negative function $F$ that guarantees that finiteness of $A_{\infty }$ implies finiteness of its expectation. This result is then applied to study the relative entropy of ${\mathbb{P}}_x$ and the probability measure induced by a purely discontinuous Girsanov transform of the process $X$. We prove these results under the weak global scaling condition on the Laplace exponent of the underlying subordinator.     

On the martingale problem and Feller and strong Feller properties for weakly coupled Lévy type operators

This paper considers the martingale problem for a class of weakly coupled Lévy type operators. It is shown that under some mild conditions, the martingale problem is well-posed and uniquely determines a strong Markov process $(X,\Lambda )$. The process $(X,\Lambda )$, called a regime-switching jump diffusion with Lévy type jumps, is further shown to possess Feller and strong Feller properties under non-Lipschitz conditions via the coupling method.     

Characterization of derivations on B(X) by local actions

Let A be a unital algebra and M be a unital A-bimodule. A linear map δ : A →M is said to be Jordan derivable at a nontrivial idempotent P ∈ A if δ(A) ? B + A ? δ(B) =δ(A ? B) for any A, B ∈ A with A ? B = P, here A ? B = AB + BA is the usual Jordan product. In this article, we show that if A = Alg N is a Hilbert space nest algebra and M = B(H), or A = M = B(X), then, a linear map δ : A → M is Jordan derivable at a nontrivial projection P ∈ N or an arbitrary but fixed nontrivial idempotent P ∈ B(X) if and only if it is a derivation. New equivalent characterization of derivations on these operator algebras was obtained.     

On the girth and diameter of generalized Johnson graphs

Let $v>k>i$ be non-negative integers. The generalized Johnson graph, $J(v,k,i)$, is the graph whose vertices are the $k$-subsets of a $v$-set, where vertices $A$ and $B$ are adjacent whenever $|A\cap B|=i$. In this article, we derive general formulas for the girth and diameter of $J(v,k,i)$. Additionally, we provide a formula for the distance between any two vertices $A$ and $B$ in terms of the cardinality of their intersection.     

Total positivity of Narayana matrices

We prove the total positivity of the Narayana triangles of type $A$ and type $B$, and thus affirmatively confirm a conjecture of Chen, Liang and Wang and a conjecture of Pan and Zeng. We also prove the strict total positivity of the Narayana squares of type $A$ and type $B$.     

Existence of renormalized solutions to elliptic equation in Musielak–Orlicz space

We prove existence of renormalized solutions to general nonlinear elliptic equation in Musielak–Orlicz space avoiding growth restrictions. Namely, we consider

$?\mathrm{div}A(x,\mathrm{?}u)=f\in L^1(\mathrm{\Omega }),$

on a Lipschitz bounded domain in ${\mathbb{R}}^N$. The growth of the monotone vector field A is controlled by a generalized nonhomogeneous and anisotropic N-function M. The approach does not require any particular type of growth condition of M or its conjugate $M^{?}$ (neither ${\mathrm{\Delta }}_2$, nor ${\mathrm{?}}_2$). The condition we impose is log-Hölder continuity of M, which results in good approximation properties of the space. The proof of the main results uses truncation ideas, the Young measures methods and monotonicity arguments.     

Smooth density and its short time estimate for jump process determined by SDE

We study a nondegenerate jump process on Euclidean space determined by SDE. We show the existence of the smooth density $p(s,x;t,y)$ of its transition probability and its short time asymptotics as $t?s\to 0$. Assumptions required for these facts are relaxed considerably from past works by Picard and Ishikawa–Kunita. We show these facts using Malliavin calculus on Poisson space. Our calculus is simpler and more efficient than previous works.