We derive asymptotics for the probability that the origin is an extremal point of a random walk in ${ℝ}^n$. We show that in order for the probability to be roughly $1/2$, the number of steps of the random walk should be between ${\mathrm{e}}^{n/(Clogn)}$ and ${\mathrm{e}}^{Cnlogn}$ for some constant $Cgt;0$. As a result, we attain a bound for the $\frac{\pi }{2}$-covering time of a spherical Brownian motion.

We study supercritical branching Brownian motion on the real line starting at the origin and with constant drift $c$. At the point $xgt;0$, we add an absorbing barrier, i.e. individuals touching the barrier are instantly killed without producing offspring. It is known that there is a critical drift $c_0$, such that this process becomes extinct almost surely if and only if $c\ge c_0$. In this case, if $Z_x$ denotes the number of individuals absorbed at the barrier, we give an asymptotic for $P(Z_x=n)$ as $n$ goes to infinity....

Our purpose is to investigate properties for processes with stationary and independent increments under $G$-expectation. As applications, we prove the martingale characterization of $G$-Brownian motion and present a pathwise decomposition theorem for generalized $G$-Brownian motion.

We consider generalized Wigner ensembles and general $\beta$-ensembles with analytic potentials for any $\beta \ge 1$. The recent universality results in particular assert that the local averages of consecutive eigenvalue gaps in the bulk of the spectrum are universal in the sense that they coincide with those of the corresponding Gaussian $\beta$-ensembles. In this article, we show that local averaging is not necessary for this result, i.e. we prove that the single gap distributions in the bulk are universal....

Motivated by applications in queueing fluid models and ruin theory, we analyze the asymptotics of $\mathbb{P}(su{p}_{t\in [0,T]}({\sum }_{i=1}^n{\lambda }_i{B}_{H_i}\left(t\right)-ct)>u)$, where $B_{H_i}\left(t\right):t\ge 0$, i = 1,...,n, are independent fractional Brownian motions with Hurst parameters $H_i\in (0,1]$ and λ₁,...,λₙ > 0. The asymptotics takes one of three different qualitative forms, depending on the value of $mi{n}_{i=1,...,n}{H}_i$.

We consider the hexagonal circle packing with radius $1/2$ and perturb it by letting the circles move as independent Brownian motions for time $t$. It is shown that, for large enough $t$, if ${𝛱}_t$ is the point process given by the center of the circles at time $t$, then, as $t\to \infty$, the critical radius for circles centered at ${𝛱}_t$ to contain an infinite component converges to that of continuum percolation (which was shown – based on a Monte Carlo estimate – by Balister, Bollobás and Walters to be strictly...

Let $$T\left(q\right)=\sum _{k=1}^{\infty }d\left(k\right)q^k,\left|q\right|<1,$$ where $d\left(k\right)$ denotes the number of positive divisors of the natural number $k$. We present monotonicity properties of functions defined in terms of $T$. More specifically, we prove that $$H\left(q\right)=T\left(q\right)-\frac{log(1-q)}{log\left(q\right)}$$ is strictly increasing on $(0,1)$, while $$F\left(q\right)=\frac{1-q}{q}H\left(q\right)$$ is strictly decreasing on $(0,1)$. These results are then applied to obtain various inequalities, one of which states that the double inequality $$\alpha \frac{q}{1-q}+\frac{log(1-q)}{log\left(q\right)}<T\left(q\right)<\beta \frac{q}{1-q}+\frac{log(1-q)}{log\left(q\right)},0<q<1,$$ holds with the best possible constant factors $\alpha =\gamma$ and $\beta =1$. Here, $\gamma$ denotes Euler’s constant. This refines a result of Salem, who...

In this paper we give a construction of operators satisfying q-CCR relations for q > 1: $A\left(f\right)A*\left(g\right)-A*\left(g\right)A\left(f\right)=q^N⟨f,g⟩I$ and also q-CAR relations for q < -1: $B\left(f\right)B*\left(g\right)+B*\left(g\right)B\left(f\right)={\left|q\right|}^N⟨f,g⟩I$, where N is the number operator on a suitable Fock space ${ℱ}_q\left(\right)$ acting as Nx₁ ⊗ ⋯ ⊗ xₙ = nx₁ ⊗ ⋯ ⊗xₙ. Some applications to combinatorial problems are also given.

Assume $f_1,...,f_N$ a finite set of functions in $H^{\infty }\left(D\right)$, the space of bounded analytic functions on the open unit disc. We give a sufficient condition on a function $f$ in $H^{\infty }\left(D\right)$ to belong to the norm-closure of the ideal $I(f_1,...,f_N)$ generated by $f_1,...,f_N$, namely the property $$\left|f\right(z\left)\right|\le \alpha \left(\right|f_1\left(z\right)|+...+|f_N\left(z\right)\left|\right)\text{for}z\in D$$ for some function $\alpha$ : ${\mathbf{R}}_{+}\to {\mathbf{R}}_{+}$ satisfying ${lim}_{t\to 0}\alpha \left(t\right)/t=0.$ The main feature in the proof is an improvement in the contour-construction appearing in L. Carleson’s solution of the corona-problem. It is also shown that the property $$\left|f\left(z\right)\right|\le C\underset{1\le j\le N}{max}\left|f_j\left(z\right)\right|\text{for}z\in D$$ ...

We show that if $\alpha >1$, then the logarithmically weighted Bergman space $A_{{log}^{\alpha }}^2$ is mapped by the Libera operator $ℒ$ into the space $A_{{log}^{\alpha -1}}^2$, while if $\alpha >2$ and $0<\epsilon \le \alpha -2$, then the Hilbert matrix operator $H$ maps $A_{{log}^{\alpha }}^2$ into $A_{{log}^{\alpha -2-\epsilon }}^2$.We show that the Libera operator $ℒ$ maps the logarithmically weighted Bloch space ${ℬ}_{{log}^{\alpha }}$, $\alpha \in ℝ$, into itself, while $H$ maps ${ℬ}_{{log}^{\alpha }}$ into ${ℬ}_{{log}^{\alpha +1}}$.In Pavlović’s paper (2016) it is shown that $ℒ$ maps the logarithmically weighted Hardy-Bloch space ${ℬ}_{{log}^{\alpha }}^1$, $\alpha >0$, into ${ℬ}_{{log}^{\alpha -1}}^1$. We show that this result is sharp. We also show that $H$ maps ${ℬ}_{{log}^{\alpha }}^1$, $\alpha \ge 0$,...

We show that the only flow solving the stochastic differential equation (SDE) on $ℝ$ $$\mathrm{d}{X}_t=1_{\{X_{t}gt;0\}}{W}^{+}\left(\mathrm{d}t\right)+1_{\{X_{t}lt;0\}}\mathrm{d}{W}^{-}\left(\mathrm{d}t\right),$$ where $W^{+}$ and $W^{-}$ are two independent white noises, is a coalescing flow we will denote by ${\varphi }^{\pm }$. The flow ${\varphi }^{\pm }$ is a Wiener solution of the SDE. Moreover, $K^{+}=𝖤\left[{\delta }_{{\varphi }^{\pm }}\right|W^{+}]$ is the unique solution (it is also a Wiener solution) of the SDE $$K_{s,t}^{+}f\left(x\right)=f\left(x\right)+{\int }_s^t{K}_{s,u}\left(1_{ℝ}^{+}{f}^{\text{'}}\right)\left(x\right)W^{+}\left(\mathrm{d}u\right)+\frac{1}{2}{\int }_s^t{K}_{s,u}f``\left(x\right)\mathrm{d}u$$ for $slt;t$, $x\in ℝ$ and $f$ a twice continuously differentiable function. A third flow ${\varphi }^{+}$ can be constructed out of the $n$-point motions of $K^{+}$. This flow is coalescing and its $n$-point motion...

Let We prove that $F$ is completely monotonic on $(0,\mathrm{\infty })$. This complements a result of Miller and Moskowitz (2006), who proved that $F$ is positive and strictly decreasing on $(0,\mathrm{\infty })$. The sequence $\{S(k)\}$ $(k=1,2,\mathrm{\dots })$ plays a role in information theory.

Let $f:I\to I$ be a $C^2$ multimodal interval map satisfying polynomial growth of the derivatives along critical orbits. We prove the existence and uniqueness of equilibrium states for the potential ${\phi }_t:x\mapsto -tlog\left|Df\left(x\right)\right|$ for $t$ close to $1$, and also that the pressure function $t\mapsto P\left({\phi }_t\right)$ is analytic on an appropriate interval near $t=1$.