30.
Abstract: We prove an analogue of Ehrhard’s inequality for the two dimensional isotropic Cauchy measure. In contrast to the Gaussian case, the inequality is not valid for non-convex sets. We provide the proof for rectangles which are symmetric with respect to one coordinate axis.
29.
Abstract: We prove Archimedes’ principle for a macroscopic ball in ideal gas consisting of point particles with non-zero mass. The main result is an asymptotic theorem, as the number of point particles goes to infinity and their total mass remains constant. We also show that, asymptotically, the gas has an exponential density as a function of height. We find the asymptotic inverse temperature of the gas. We derive an accurate estimate of the volume of the phase space using the local central limit theorem.
28.
Abstract: We consider matrix-valued processes described as solutions to stochastic differential equations of very general form. We study the family of the empirical measure-valued processes constructed from the corresponding eigenvalues. We show that the family indexed by the size of the matrix is tight under very mild assumptions on the coefficients of the initial SDE. We characterize the limiting distributions of its subsequences as solutions to an integral equation. We use this result to study some universality classes of random matrix flows. These generalize the classical results related to Dyson Brownian motion and squared Bessel particle systems. We study some new phenomena as the existence of the generalized Marchenko–Pastur distributions supported on the real line. We also introduce universality classes related to generalized geometric matrix Brownian motions and Jacobi processes. Finally we study, under some conditions, the convergence of the empirical measure-valued process of eigenvalues associated to matrix flows to the law of a free diffusion.
27.
Abstract: Let p_t(x), f_t(x) and q_t^∗(x) be the densities at time t of a real Lévy process, its running supremum and the entrance law of the reflected excursions at the infimum. We provide relationships between the asymptotic behaviour of p_t(x), f_t(x) and q_t^∗(x), when t is small and x is large. Then for large x, these asymptotic behaviours are compared to this of the density of the Lévy measure. We show in particular that, under mild conditions, if p_t(x) is comparable to tν(x), as t→0 and x→∞, then so is f_t(x).
26.
Abstract: For an open subset Ω of R^d, symmetric with respect to a hyperplane and with positive part Ω_+, we consider the Neumann/Dirichlet Laplacians −Δ_{N/D,Ω} and −Δ_{N/D,Ω_+}. Given a Borel function Φ on [0,∞) we apply the spectral functional calculus and consider the pairs of operators Φ(−Δ_{N,Ω}) and Φ(−Δ_{N,Ω_+}), or Φ(−Δ_{D,Ω}) and Φ(−Δ_{D,Ω_+}). We prove relations between the integral kernels for the operators in these pairs, which in the particular cases of Ω_+=R^{d−1}×(0,∞) and Φ_t(u)=exp(−tu), u≥0, t>0, were known as reflection principles for the Neumann/Dirichlet heat kernels. These relations are then generalized to the context of symmetry with respect to a finite number of mutually orthogonal hyperplanes.
25.
Abstract: We provide integral formulae for the Laplace transform of the entrance law of the reflected excursions for symmetric Lévy processes in terms of their characteristic exponent. For subordinate Brownian motions and stable processes we express the density of the entrance law in terms of the generalized eigenfunctions for the semigroup of the process killed when exiting the positive half-line. We use the formulae to study in-depth properties of the density of the entrance law such as asymptotic behavior of its derivatives in time variable.
24.
Abstract: We provide sharp two-sided estimates on the Dirichlet heat kernel k1(t, x, y) for the Laplacian in a ball. The result accurately describes the exponential behaviour of the kernel for small times and significantly improves the qualitatively sharp results known so far. As a consequence we obtain the full description of the kernel k1(t, x, y) in terms of its global two-sided sharp estimates. Such precise estimates were possible to obtain due to the enrichment of analytical methods with probabilistic tools.
23.
Abstract: We study the existence and uniqueness of solutions of SDEs describing squared Bessel particle systems in full generality. We define nonnegative and non-colliding squared Bessel particle systems and we study their properties. Particle systems dissatisfying non-colliding and unicity properties are pointed out. The structure of squared Bessel particle systems is described.
22.
Abstract: We provide short and simple proofs of the continuous time ballot theorem for processes with cyclically interchangeable increments and Kendall’s identity for spectrally positive Lévy processes. We obtain the later result as a direct consequence of the former. The ballot theorem is extended to processes having possible negative jumps. Then we prove through straightforward arguments based on the law of bridges and Kendall’s identity, Theorem 2.4 in [20] which gives an expression for the law of the supremum of spectrally positive Lévy processes. An analogous formula is obtained for the supremum of spectrally negative Lévy processes.
21.
Abstract: A characterization of the existence of non-central Wishart distributions (with shape and non-centrality parameter) as well as the existence of solutions to Wishart stochastic differential equations (with initial data and drift parameter) in terms of their exact parameter domains is given. These two families are the natural extensions of the non-central chi-square distributions and the squared Bessel processes to the positive semidefinite matrices.
20.
Abstract: We prove a two-term Weyl-type asymptotic law, with error term O(1/n), for the eigenvalues of the operator ψ(-Δ) in an interval, with zero exterior condition, for complete Bernstein functions ψ such that \psi ψ'(\psi) converges to infinity as \psi→∞. This extends previous results obtained by the authors for the fractional Laplace operator (ψ(\psi)=\psi^α/2) and for the Klein–Gordon square root operator (ψ(\psi)=(1+\psi)^1/2-1). The formula for the eigenvalues in (-a,a) is of the form λ_n = \psi(\mu_n^2)+O(1/n), where \mu_n is the solution of \mu_n = n π/2a - 1/a \nu(\mu_n), and \nu(\mu) \in [0,π/2) is given as an integral involving ψ.
19.
Abstract: We provide sharp two-sided estimates of the Fourier–Bessel heat kernel and we give sharp two-sided estimates of the transition probability density for the Bessel process in (0,1) killed at 1 and killed or reflected at 0.
18.
Abstract: In the paper we consider the Bessel differential operator L^{(\mu)} = \dfrac{d^2}{dx^2}+\frac{2\mu+1}{2}\dfrac{d}{dx} in half-line [a,∞), a>0, and its Dirichlet heat kernel. For \mu=0, by combining analytical and probabilistic methods, we provide sharp two-sided estimates of the heat kernel for the whole range of the space parameters x,y>a and every t>0, which complements the recent results given in [1], where the case \mu\neq 0 was considered.
17.
Abstract: Let us consider a real valued Lévy process X, whose transition probabilities are absolutely continuous and have bounded densities. Then the law of the past supremum of X before any deterministic time t is absolutely continuous on (0,∞). We show that its density f_t(x) is continuous on (0,∞) if and only if the potential density h' of the upward ladder height process is continuous on (0,∞). Then we prove that f_t behaves at 0 as h′. We also describe the asymptotic behaviour of f_t, when t tends to infinity. Then some related results are obtained for the density of the meander and this of the entrance law of the Lévy process conditioned to stay positive.
16.
Abstract: The main objective of the work is to provide sharp two-sided estimates of the λ-Green function, λ≥0, of the hyperbolic Brownian motion of a half-space. We rely on the recent results obtained by K. Bogus and J. Małecki (2015), regarding precise estimates of the Bessel heat kernel for half-lines. We also substantially use the results of H. Matsumoto and M. Yor (2005) on distributions of exponential functionals of Brownian motion.
15.
Abstract: In this paper we study the Bessel process R_t^(\mu) with index μ ≠ 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 for the whole range of space parameters x,y > a and every t > 0.
14.
Abstract: We study systems of stochastic differential equations describing positions x_1,..., x_p of p ordered particles, with inter-particles repulsions of the form H_ij(x_i,x_j)/(x_i−x_j). We show the existence of strong and pathwise unique non-colliding solutions of the system with a colliding initial point x_1(0)≤...≤x_p(0) in the whole generality, under natural assumptions on the coefficients of the equations.
13.
Abstract: The eigenvalues and eigenfunctions of the one-dimensional quasi-relativistic Hamiltonian (-ℏ^2c^2d^2/dx^2 + m^2c^4)^(1/2) + V_well(x) (the Klein–Gordon square-root operator with electrostatic potential) with the infinite square well potential V_well(x) are studied. Eigenvalues represent energies of a "massive particle in the box" quasi-relativistic model. Approximations to eigenvalues λ_n are given, uniformly in n, ℏ, m, c and a, with error less than C_1ℏca^(-1)exp(-C_2ℏ-1mca)n^(-1). Here 2a is the width of the potential well. As a consequence, the spectrum is simple and the nth eigenvalue is equal to (nπ/2 - π/8)ℏc/a + O(1/n) as n → ∞. Non-relativistic, zero mass and semi-classical asymptotic expansions are included as special cases. In the final part, some L^2 and L^∞ properties of eigenfunctions are studied.
12.
Abstract: In this paper we study the supremum functional M_t=sup(X_s: 0≤s≤t), where X_t, t≥0, is a one-dimensional Lévy process. Under very mild assumptions we provide a simple, uniform estimate of the cumulative distribution function of M_t. In the symmetric case we find an integral representation of the Laplace transform of the distribution of M_t if the Lévy–Khintchin exponent of the process increases on (0,∞).
11.
Abstract: Let T_1^(\mu) be the first hitting time of the point 1 by the Bessel process with index μ ∈ ℝ starting from x > 1. Using an integral formula for the density q_x^(\mu)(t) of T_1^(\mu), obtained in Byczkowski and Ryznar (Stud Math 173(1):19–38, 2006), we prove sharp estimates of the density of T_1^(\mu), which exhibit the dependence both on time and space variables. Our result provides optimal uniform estimates for the density of the hitting time of the unit ball by the Brownian motion in ℝ^n, which improve existing bounds. Another application is to provide sharp estimates for the Poisson kernel for half-spaces for hyperbolic Brownian motion in real hyperbolic spaces.
10.
Abstract: Let X_t be a subordinate Brownian motion, and suppose that the Lévy measure of the underlying subordinator has a completely monotone density. Under very mild conditions, we find integral formulae for the tail distribution P(\tau_x>t) of first passage times \tau_x through a barrier at x>0, and its derivatives in t. As a corollary, we examine the asymptotic behaviour of P(\tau_x>t) and its t-derivatives, either as t→∞ or x→0.
9.
Abstract: We prove a multidimensional version of the Yamada-Watanabe theorem, i.e., a theorem giving conditions on coefficients of a stochastic differential equation for existence and pathwise uniqueness of strong solutions. It implies an existence and uniqueness theorem for the eigenvalue and eigenvector processes of matrix-valued stochastic processes, called a “spectral” matrix Yamada-Watanabe theorem. The multidimensional Yamada-Watanabe theorem is also applied to particle systems of squared Bessel processes, corresponding to matrix analogues of squared Bessel processes, Wishart and Jacobi matrix processes. The β-versions of these particle systems are also considered.
8.
Abstract: The purpose of the paper is to provide a general method for computing the hitting distributions of some regular subsets D for Ornstein–Uhlenbeck type operators of the form \frac{1}{2}Δ+F⋅∇, with F bounded and orthogonal to the boundary of D. As an important application we obtain integral representations of the Poisson kernel for a half-space and balls for hyperbolic Brownian motion and for the classical Ornstein–Uhlenbeck process. The method developed in this paper is based on stochastic calculus and on the skew product representation of multidimensional Brownian motion and yields more complete results than those based on the Feynman–Kac technique.
7.
Abstract: The spectral properties of the pseudo-differential operator (-d^2/dx^2)^(1/2) + x^2 are analyzed by a combination of functional integration methods and direct analysis. We obtain a representation of its eigenvalues and eigenfunctions, prove precise asymptotic formulae, and establish various analytic properties. We also derive trace asymptotics and heat kernel estimates.
6.
Abstract: Let X^(μ)={X^(μ)_t;t≥0}, μ >0, be then-dimensional hyperbolic Brownian motion with drift, that is a diffusion on the real hyperbolic spaceHnhaving the Laplace–Beltrami operator with drift as its generator. We prove the reflection principle for X^(μ), which enables us to study the process X^(μ) killed when exiting the hyperbolichalf-space, that is the set D={x∈H^n:x_1>0}. We provide formulae, uniform estimates and describe asymptotic behavior of the Green function and the Poisson kernel of D for the process X^(μ). Finally, we derive formula for the λ-Poisson kernel of the set D.
5.
Abstract: We study the spectral properties of the transition semigroup of the killed one-dimensional Cauchy process on the half-line (0, ∞) and the interval (−1, 1). This process is related to the square root of one-dimensional Laplacian A=−(-d^2/dx^2)^(1/2) with a Dirichlet exterior condition (on a complement of a domain), and to a mixed Steklov problem in the half-plane. For the half-line, an explicit formula for generalized eigenfunctions ψλ of 𝒜 is derived, and then used to construct a spectral representation of 𝒜. Explicit formulas for the transition density of the killed Cauchy process on the half-line (or the heat kernel of 𝒜 in (0, ∞)), and for the distribution of the first exit time from the half-line follow. The formula for ψλ is also used to construct approximations to eigenfunctions of 𝒜 in the interval. For the eigenvalues λn of 𝒜 in the interval the asymptotic formula λn = n π/2 − π/8 + O(1/n) is derived, and all eigenvalues λn are proved to be simple. Finally, efficient numerical methods of estimation of eigenvalues λn are applied to obtain lower and upper numerical bounds for the first few eigenvalues up to the ninth decimal point.
4.
Abstract: We apply the Feynman–Kac formula to compute the λ-Poisson kernels and λ-Green functions for half-spaces or balls in hyperbolic spaces. We present known results in a unified way and also provide new formulas for the λ-Poisson kernels and λ-Green functions of half-spaces in H^n and for balls in real and complex hyperbolic spaces.
3.
Abstract: The purpose of the paper is to find explicit formulas describing the joint distributions of the first hitting time and place for half-spaces of codimension one for a diffusion in ℝ^{n+1}, composed of one-dimensional Bessel process and independent n-dimensional Brownian motion. The most important argument is carried out for the two-dimensional situation. We show that this amounts to computation of distributions of various integral functionals with respect to a two-dimensional process with independent Bessel components. As a result, we provide a formula for the Poisson kernel of a half-space or of a strip for the operator (I − Δ)^α/2, 0 < α < 2. In the case of a half-space, this result was recently found, by different methods, in Byczkowski et al. (Trans Am Math Soc 361:4871–4900, 2009). As an application of our method we also compute various formulas for first hitting places for the isotropic stable Lévy process.
2.
Abstract: The purpose of the paper is to find explicit formulas for basic objects pertaining to the potential theory of the operator (I − Δ)^α/2, which is based on Bessel potentials J_α = (I − Δ)^(-α/2), 0 < α < 2. We compute the harmonic measure of the half-space and obtain a concise form for the corresponding Green function of the operator (I − Δ)^α/2. As an application we provide sharp estimates for the Green function of the half-space for the relativistic process.
1.
Abstract: Let (X_t)_(t⩾0) be the n-dimensional hyperbolic Brownian motion, that is the diffusion on the real hyperbolic space D^n having the Laplace–Beltrami operator as its generator. The aim of the paper is to derive the formulas for the Gegenbauer transform of the Poisson kernel and the Green function of the ball for the process (X_t)_(t⩾0). Under additional hypotheses we prove integral representations for the Poisson kernel. This yields explicit formulas in D^4 and D^6 spaces for the Poisson kernel and the Green function as well.