Take a centered random walk $S_n$ and consider the sequence of its partial sums $A_n:={\sum }_{i=1}^n{S}_i$. Suppose $S_1$ is in the domain of normal attraction of an $\alpha$-stable law with $1lt;\alpha \le 2$. Assuming that $S_1$ is either right-exponential (i.e. $\mathbb{P}(S_{1}gt;x|S_{1}gt;0)={\mathrm{e}}^{-ax}$ for some $agt;0$ and all $xgt;0$) or right-continuous (skip free), we prove that $$\mathbb{P}\{A_{1}gt;0,\cdots ,A_{N}gt;0\}\sim C_{\alpha }{N}^{1/\left(2\alpha \right)-1/2}$$ as $N\to \infty$, where $C_{\alpha }gt;0$ depends on the distribution of the walk. We also consider a conditional version of this problem and study positivity of integrated discrete bridges.

We consider the random vector $u(t,\underline{x})=(u(t,x_1),\cdots ,u(t,x_d))$, where $t\>0,x_1,\cdots ,x_d$ are distinct points of ${ℝ}^2$ and $u$ denotes the stochastic process solution to a stochastic wave equation driven by a noise white in time and correlated in space. In a recent paper by Millet and Sanz–Solé [10], sufficient conditions are given ensuring existence and smoothness of density for $u(t,\underline{x})$. We study here the positivity of such density. Using techniques developped in [1] (see also [9]) based on Analysis on an abstract Wiener space, we characterize the set of...

We consider the random vector $u(t,\underline{x})=(u(t,x_1),\cdots ,u(t,x_d))$, where t > 0, x1,...,xd are distinct points of ${ℝ}^2$ and u denotes the stochastic process solution to a stochastic wave equation driven by a noise white in time and correlated in space. In a recent paper by Millet and Sanz–Solé [10], sufficient conditions are given ensuring existence and smoothness of density for $u(t,\underline{x})$. We study here the positivity of such density. Using techniques developped in [1] (see also [9]) based on Analysis on an abstract Wiener space, we characterize...

We give a generalization in the non-compact case to various positivity theorems obtained by Malliavin Calculus in the compact case.

In [14] we formalized probability and probability distribution on a finite sample space. In this article first we propose a formalization of the class of finite sample spaces whose element’s probability distributions are equivalent with each other. Next, we formalize the probability measure of the class of sample spaces we have formalized above. Finally, we formalize the sampling and posterior probability.

We consider the parabolic Anderson model, the Cauchy problem for the heat equation with random potential in ℤd. We use i.i.d. potentials ξ:ℤd→ℝ in the third universality class, namely the class of almost bounded potentials, in the classification of van der Hofstad, König and Mörters [Commun. Math. Phys.267 (2006) 307–353]. This class consists of potentials whose logarithmic moment generating function is regularly varying with parameter γ=1, but do not belong to the class of so-called double-exponentially...

We introduce potential spaces on fractal metric spaces, investigate their embedding theorems, and derive various Besov spaces. Our starting point is that there exists a local, stochastically complete heat kernel satisfying a two-sided estimate on the fractal considered.

The purpose of the paper is to extend results of the potential theory of the classical Schrödinger operator to the α-stable case. To obtain this we analyze a weak version of the Schrödinger operator based on the fractional Laplacian and we prove the Conditional Gauge Theorem.

Let X = X(t); t ≥ 0 be the hyperbolic Brownian motion on the real hyperbolic space ℍⁿ = x ∈ ℝⁿ:xₙ > 0. We study the Green function and the Poisson kernel of tube domains of the form D × (0,∞)⊂ ℍⁿ, where D is any Lipschitz domain in ${ℝ}^{n-1}$. We show how to obtain formulas for these functions using analogous objects for the standard Brownian motion in ${ℝ}^{2n}$. We give formulas and uniform estimates for the set $D_a=x\in ℍⁿ:x₁\in (0,a)$. The constants in the estimates depend only on the dimension of the space.

The purpose of this paper is to find optimal estimates for the Green function and the Poisson kernel for a half-line and intervals of the geometric stable process with parameter α ∈ (0,2]. This process has an infinitesimal generator of the form $-log(1+{(-\Delta )}^{\alpha /2})$. As an application we prove the global scale invariant Harnack inequality as well as the boundary Harnack principle.

Expected suprema of a function f observed along the paths of a nice Markov process define an excessive function, and in fact a potential if f vanishes at the boundary. Conversely, we show under mild regularity conditions that any potential admits a representation in terms of expected suprema. Moreover, we identify the maximal and the minimal representing function in terms of probabilistic potential theory. Our results are motivated by the work of El Karoui and Meziou (2006) on the max-plus decomposition...

We develop potential-theoretical methods in the construction of measure-valued branching processes.We complete results of P. J. Fitzsimmons and E. B. Dynkin on the construction, regularity and other properties of the superprocess associated with a given right process and a branching mechanism.