Branching Processes in Random Environment (BPREs) $(Z_n:n\ge 0)$ are the generalization of Galton–Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. In the supercritical case, the process survives with positive probability and then almost surely grows geometrically. This paper focuses on rare events when the process takes positive but small values for large times. We describe the asymptotic behavior of $\mathbb{P}(1\le Z_n\le k|Z_0=i)$, $k,i\in \mathbb{N}$ as $n\to \infty$. More precisely, we characterize...

In this paper we consider processes Xₜ with values in $L^p$, p ≥ 1 on subsets T of a unit cube in ℝⁿ satisfying a natural condition of boundedness of increments, i.e. a process has bounded increments if for some non-decreasing f: ℝ₊ → ℝ₊ ||Xₜ-Xₛ||ₚ ≤ f(||t-s||), s,t ∈ T. We give a sufficient criterion for a.s. continuity of all processes with bounded increments on subsets of a given set T. This criterion turns out to be necessary for a wide class of functions f. We use a geometrical Paszkiewicz-type...

We study the behavior of a continuous flow near a boundary. We prove that if φ is a flow on $E={ℝ}^{n+1}$ for which $\partial E={ℝ}^n\times 0$ is an invariant set and S ⊂ ∂E is an isolated invariant set, with non-zero homological Conley index, then there exists an x in EE such that either α(x) or ω(x) is in S. We also prove an index theorem for a flow on ${ℝ}^n\times [0,\infty )$.

The paper deals with the existence of a Kneser solution of the $n$-th order nonlinear differential inclusion $$\begin{array}{cc}\hfill x^{\left(n\right)}\left(t\right)\in -A_1(t,x\left(t\right),...,x^{(n-1)}\left(t\right))x^{(n-1)}\left(t\right)-...-A_n(t,x\left(t\right),...,& x^{(n-1)}\left(t\right))x\left(t\right)\\ & \text{for}\text{a.a.}t\in [a,\infty ),\end{array}$$ where $a\in (0,\infty )$, and $A_i:[a,\infty )\times {ℝ}^n\to ℝ$, $i=1,...,n,$ are upper-Carathéodory mappings. The derived result is finally illustrated by the third order Kneser problem.

We prove smoothing properties of nonlocal transition semigroups associated to a class of stochastic differential equations (SDE) in ${ℝ}^d$ driven by additive pure-jump Lévy noise. In particular, we assume that the Lévy process driving the SDE is the sum of a subordinated Wiener process $Y$ (i.e. $Y=W\circ T$, where $T$ is an increasing pure-jump Lévy process starting at zero and independent of the Wiener process $W$) and of an arbitrary Lévy process independent of $Y$, that the drift coefficient is continuous...

Let $p$ be an odd prime, and let $a$ be an integer not divisible by $p$. When $m$ is a positive integer with $p\equiv 1(mod2m)$ and $2$ is an $m$th power residue modulo $p$, we determine the value of the product ${\prod }_{k\in R_m\left(p\right)}(1+tan(\pi ak/p))$, where $$R_m\left(p\right)=\{0<k<p:k\in \mathbb{Z}\text{is}\text{an}m\text{th}\text{power}\text{residue}\text{modulo}p\}.$$ In particular, if $p=x^2+64y^2$ with $x,y\in \mathbb{Z}$, then $$\prod _{k\in R_4\left(p\right)}\left(1+tan\pi \frac{ak}{p}\right)={(-1)}^y{(-2)}^{(p-1)/8}.$$

The $\sigma$-finite measure ${𝒫}_{sup}$ which unifies supremum penalisations for a stable Lévy process is introduced. Silverstein’s coinvariant and coharmonic functions for Lévy processes and Chaumont’s $h$-transform processes with respect to these functions are utilized for the construction of ${𝒫}_{sup}$.

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...

We construct some nondecreasing quantities associated to the first eigenvalue of $-{\Delta }_{\phi }+cR$ $(c\ge \frac{1}{2}(n-2)/(n-1))$ along the Yamabe flow, where ${\Delta }_{\phi }$ is the Witten-Laplacian operator with a $C^2$ function $\phi$. We also prove a monotonic result on the first eigenvalue of $-{\Delta }_{\phi }+\frac{1}{4}(n/(n-1))R$ along the Yamabe flow. Moreover, we establish some nondecreasing quantities for the first eigenvalue of $-{\Delta }_{\phi }+c{R}^a$ with $a\in (0,1)$ along the Yamabe flow.

In the present work, we consider spectrally positive Lévy processes $(X_t,t\ge 0)$ not drifting to $+\infty$ and we are interested in conditioning these processes to reach arbitrarily large heights (in the sense of the height process associated with $X$) before hitting $0$. This way we obtain a new conditioning of Lévy processes to stay positive. The (honest) law ${\mathbb{P}}_x^{☆}$ of this conditioned process (starting at $xgt;0$) is defined as a Doob $h$-transform via a martingale. For Lévy processes with infinite variation paths,...

We consider the problem $$\left\{\begin{array}{cc}-\Delta u=u^{\frac{N+2}{N-2}+\lambda }\hfill & \text{in}\Omega \setminus \epsilon \omega ,\hfill \\ u\&gt;0\hfill & \text{in}\Omega \setminus \epsilon \omega ,\hfill \\ u=0\hfill & \text{on}\partial \left(\Omega \setminus \epsilon \omega \right),\hfill \end{array}\right.$$ where $\Omega$ and $\omega$ are smooth bounded domains in ${ℝ}^N$, $N\ge 3$, $\epsilon \&gt;0$ and $\lambda \in ℝ.$ We prove that if the size of the hole $\epsilon$ goes to zero and if, simultaneously, the parameter $\lambda$ goes to zero at the appropriate rate, then the problem has a solution which blows up at the origin.

We consider the spatial $𝛬$-Fleming–Viot process model ( (2010) 162–216) for frequencies of genetic types in a population living in ${ℝ}^d$, in the special case in which there are just two types of individuals, labelled $0$ and $1$. At time zero, everyone in a given half-space has type 1, whereas everyone in the complementary half-space has type $0$. We are concerned with patterns of frequencies of the two types at large space and time scales. We consider two cases, one in which the...

Let $\mathbb{N}$ be the set of positive integers and let $s\in \mathbb{N}$. We denote by $d^s$ the arithmetic function given by $d^s\left(n\right)={\left(d\left(n\right)\right)}^s$, where $d\left(n\right)$ is the number of positive divisors of $n$. Moreover, for every $\ell ,m\in \mathbb{N}$ we denote by ${\delta }^{s,\ell ,m}\left(n\right)$ the sequence $$\underset{m\text{-times}}{\underbrace{d^s(d^s(...d^s(d^s\left(n\right)+\ell )+\ell ...)+\ell )}}=\left\{\begin{array}{cc}{d}^s\left(n\right)\hfill & \text{for}m=1,\hfill \\ d^s(d^s\left(n\right)+\ell )\hfill & \text{for}m=2,\hfill \\ d^s(d^s(d^s\left(n\right)+\ell )+\ell )\hfill & \text{for}m=3,\hfill \\ ⋮\hfill \end{array}\right.$$ We present classical and nonclassical notes on the sequence ${\left({\delta }^{s,\ell ,m}\left(n\right)\right)}_{m\ge 1}$, where $\ell$, $n$, $s$ are understood as parameters.

We study a one-dimensional stochastic differential equation driven by a stable Lévy process of order $\alpha$ with drift and diffusion coefficients $b$, $\sigma$. When $\alpha \in (1,2)$, we investigate pathwise uniqueness for this equation. When $\alpha \in (0,1)$, we study another stochastic differential equation, which is equivalent in law, but for which pathwise uniqueness holds under much weaker conditions. We obtain various results, depending on whether $\alpha \in (0,1)$ or $\alpha \in (1,2)$ and on whether the driving stable process is symmetric or not. Our...