Related papers: The density of the $(\alpha,\beta)$-superprocess a…

An upper quasi-density on $\bf H$ (the integers or the non-negative integers) is a real-valued subadditive function $\mu^\ast$ defined on the whole power set of $\mathbf H$ such that $\mu^\ast(X) \le \mu^\ast({\bf H}) = 1$ and $\mu^\ast(k…

Let $\alpha\in(0,2)$ and $d\in\mathbb{N}$. Consider the following stochastic differential equation (SDE) driven by $\alpha$-stable process in $\mathbb{R}^d$: $$ dX_t=b(X_t)dt+\sigma(X_{t-})d L^{\alpha}_t, \quad X_0=x\in\mathbb{R}^d, $$…

Let $\mathrm{d}(A)$ be the asymptotic density (if it exists) of a sequence of integers $A$. For any real numbers $0\leq\alpha\leq\beta\leq 1$, we solve the question of the existence of a sequence $A$ of positive integers such that…

We derive explicit asymptotic expansions of the density of the supremum of a strictly stable process when the index $\alpha$ is not rational. In the case when parameters $\alpha$ and $\rho=\p(X_1>0)$ satisfy $\rho+k=l/\alpha$ for some…

If $X$ is a spectrally positive stable process of index $\alpha\in(1,2)$ whose L\'{e}vy measure has density $cx^{-\alpha-1}$ on $(0,\infty),$ and $S_1=\sup_{0<t\leq1}X_t,$ it is known that $P(S_1>x)\backsim c\alpha^{-1}x^{-\alpha}$ as…

We study the density of functions which are holomorphic in a neighbourhood of the closure $\overline{\Omega}$ of a bounded non-smooth pseudoconvex domain $\Omega$, in the Bergman space $ H^2(\Omega ,\varphi)$ with a plurisubharmonic weight…

We show that under very mild conditions on a measure $\mu$ on the interval $[0,\infty)$, the span of $\{x^k\}_{k=n}^{\infty}$ is dense in $L^2(\mu)$ for any $n=0,1,\ldots$. We present two different proofs of this result, one based on the…

Let $\xi=(\xi_t)$ be a locally finite $(2,\beta)$-superprocess in $\RR^d$ with $\beta<1$ and $d>2/\beta$. Then for any fixed $t>0$, the random measure $\xi_t$ can be a.s. approximated by suitably normalized restrictions of Lebesgue measure…

We consider the integro-differential equation ${\rm I}^{\alpha}_{0+}f= x^m f$ on the half-line. We show that there exists a density solution, which is then unique and can be expressed in terms of the Beta distribution, if and only if $m>…

We give lower bounds for the density $p_T(x,y)$ of the law of $X_t$, the solution of $dX_t=\sigma (X_t) dB_t+b(X_t) dt,X_0=x,$ under the following local ellipticity hypothesis: there exists a deterministic differentiable curve $x_t, 0\leq…

In this paper we study the Bessel process R_t^{(\mu)} with index \mu\neq 0 starting from x>0 and killed when it reaches a positive level a, where x>a>0. We provide sharp estimates of the transition probability density p_a^{(\mu)}(t,x,y) for…

We further develop the relationship between $\beta$-numbers and discrete curvatures to provide a new proof that under weak density assumptions, finiteness of the pointwise discrete curvature $\operatorname{curv}^{\alpha}_{\mu;2}(x,r)$ at…

Let $X=(X_t)_{t\ge0}$ be a stable L\'{e}vy process of index $\alpha \in(1,2)$ with no negative jumps and let $S_t=\sup_{0\le s\le t}X_s$ denote its running supremum for $t>0$. We show that the density function $f_t$ of $S_t$ can be…

We study two one-parameter families of point processes connected to random matrices: the Sine_beta and Sch_tau processes. The first one is the bulk point process limit for the Gaussian beta-ensemble. For beta=1, 2 and 4 it gives the limit…

For $\{X(t), t \in G_\delta\}$ a centered Gaussian process with stationary increments and a.s. sample paths on a discrete grid $G_\delta=\{0,\delta,2\delta, ...\}$, where $\delta>0$, we investigate the stationary reflected process…

We consider the Halfin-Whitt diffusion process $X_d(t)$, which is used, for example, as an approximation to the $m$-server $M/M/m$ queue. We use recently obtained integral representations for the transient density $p(x,t)$ of this diffusion…

According to a theorem of S. Schumacher, for a diffusion X in an environment determined by a stable process that belongs to an appropriate class and has index a, it holds that X_t/(log t)^a converges in distribution, as t goes to infinity,…

We investigate, using the spherical Jeans equation, self-gravitating dynamical equilibria satisfying a relation rho/sigma_r^3 propto r^-alpha, which holds for simulated dark-matter haloes over their whole resolved radial range. Considering…

We assume that we observe $N$ independent copies of a diffusion process on a time-interval $[0,2T]$. For a given time $t$, we estimate the transition density $p_t(x,y)$, namely the conditional density of $X_{t + s}$ given $X_s = x$, under…

In this paper, we show the weak and strong well-posedness of density dependent stochastic differential equations driven by $\alpha$-stable processes with $\alpha \in(1,2)$. The existence part is based on Euler's approximation as…