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We derive two-sided bounds for the Newton and Poisson kernels of the $W$-invariant Dunkl Laplacian in geometric complex case when the multiplicity $k(\alpha)=1$, i.e. for flat complex symmetric spaces. For the invariant Dunkl-Poisson kernel $P^W(x,y)$, the estimates are $$ P^W(x,y)\asymp \frac{P^{{\bf R}^d}(x,y)}{\prod_{\alpha > 0 \ }|x-\sigma_\alpha y|^{2k(\alpha)}},$$ where the $\alpha$'s are the positive roots of a root system acting in ${\bf R}^d$, the $\sigma_\alpha$'s are the corresponding symmetries and $P^{{\bf R}^d}$ is the classical Poisson kernel in ${{\bf R}^d}$. Analogous bounds are proven for the Newton kernel when $d\ge 3$. The same estimates are derived in the rank one direct product case $\mathbb Z_2^N$ and conjectured for general $W$-invariant Dunkl processes. As an application, we get a two-sided bound for the Poisson and Newton kernels of the classical Dyson Brownian motion and of the Brownian motions in any Weyl chamber., Comment: 31 pages

Within the framework of balayage spaces (the analytical equivalent of nice Hunt processes), we prove equicontinuity of bounded families of harmonic functions and apply it to obtain criteria for compactness of potential kernels.

We introduce a set of tools which simplify and streamline the proofs of limit theorems concerning near-critical particles in branching random walks under optimal assumptions. We exemplify our method by giving another proof of the Seneta-Heyde norming for the critical additive martingale, initially due to A\"id\'ekon and Shi. The method involves in particular the replacement of certain second moment estimates by truncated first moment bounds, and the replacement of ballot-type theorems for random walks by estimates coming from an explicit expression for the potential kernel of random walks killed below the origin. Of independent interest might be a short, self-contained proof of this expression, as well as a criterion for convergence in probability of non-negative random variables in terms of conditional Laplace transforms., Comment: 23 pages, v3: presentation changes, accepted version for EJP

We study potential operators (Riesz and Bessel potentials) associated with classical Jacobi and Fourier-Bessel expansions. We prove sharp estimates for the corresponding potential kernels. Then we characterize those $1 \le p,q \le \infty$, for which the potential operators are of strong type $(p,q)$, of weak type $(p,q)$ and of restricted weak type $(p,q)$. These results may be thought of as analogues of the celebrated Hardy-Littlewood-Sobolev fractional integration theorem in the Jacobi and Fourier-Bessel settings. As an ingredient of our line of reasoning, we also obtain sharp estimates of the Poisson kernel related to Fourier-Bessel expansions., 28 pages, 4 figures; v2 (some comments on Bessel potentials added)

This paper presents the solution of an integral equation of a mixed type in three-dimensions in the space L2 (W) \times C[0,T], where T < \infty, and W is the domain of integration with respect to position. The kernel of position integral term is considered in the potential function form, while the kernel of time is considered as a continuous kernel. A linear system of Fredholm integral equations of the first and second kinds are obtained and solved. Krein's method is used to solve the Fredholm integral equation of the first kind, while the second kind is solved numerically.

We show that a certain axisymmetric hypersingular integral equation arising in problems of cracks in the elasticity theory may be explicitly solved in the case where the crack occupies a plane circle. We give three different forms of the resolving formula. Two of them involve regular kernels, while the third one involves a singular kernel, but requires less regularity assumptions on the the right-hand side of the equation., MSC 2010: 45DB05, 45E05, 78A45

The potential kernel of a positive left additive functional (of a strong Markov process $X$) maps positive functions to strongly supermedian functions and satisfies a variant of the classical domination principle of potential theory. Such a kernel $V$ is called a regular strongly supermedian kernel in recent work of L. Beznea and N. Boboc. We establish the converse: Every regular strongly supermedian kernel $V$ is the potential kernel of a random measure homogeneous on $[0,\infty[$. Under additional finiteness conditions such random measures give rise to left additive functionals. We investigate such random measures, their potential kernels, and their associated characteristic measures. Given a left additive functional $A$ (not necessarily continuous), we give an explicit construction of a simple Markov process $Z$ whose resolvent has initial kernel equal to the potential kernel $U_{\!A}$. The theory we develop is the probabilistic counterpart of the work of Beznea and Boboc. Our main tool is the Kuznetsov process associated with $X$ and a given excessive measure $m$.

In the frame of Borel right Markov processes, we investigate, following an analytical point of view, the Revuz correspondence between classes of potential kernels and their associated measures, improving upon the results of Revuz, Azema, Getoor and Sharpe, Fitzsimmons, Fitzsimmons and Getoor and Dellacherie, Maisonneuve and Meyer. In the probabilistic approach of the problem, the kernels that occur are the potential operators of different types of homogeneous random measures. We completely characterize the hypothesis (B) of Hunt in terms of Revuz measures.

It is proved that the potential kernel of a recurrent, aperiodic random walk on the integer lattice $\mathbb{Z}^2$ admits an asymptotic expansion of the form $$(2 \pi \sqrt{|Q|})^{-1} \ln Q(x_2, -x_1) + \const + |x|^{-1} U_1 (\omega^x) + |x|^{-2} U_2 (\omega^x) + \dots ,$$ where $|Q|$ and $Q(\theta)$ are, respectively, the determinant and the quadratic form of the covariance matrix of the increment X of the random walk, $\omega^x = x/|x|$ and the $U_k (\omega)$ are smooth functions of $\omega, |\omega| = 1$, provided k that all the moments of X are finite. Explicit forms of $U_1$ and $U_2$ are given in terms of the moments of X.

Let $X = \{X_t = (X_t^1,\cdots, X_t^d), t \geqq 0\}$ be an isotropic stochastic process with stationary independent increments having its values in $d$-dimensional Euclidean space, $d \geqq 2$. Let $R_t = |X_t|$ be the radial process. It is proved (except for a rather trivial exception) that the Markov process $\{R_t\}$ hits points if and only if the real process $\{X_t^1\}$ hits points; a simple analytic criterion for the latter possibility has been known now for some time. If $x > 0$, the sets $\{t: R_t = x\}$ and $\{t: X_t^1 = 0\}$ are then shown to have the same size in the sense that there is an exact Hausdorff measure function that works for both. Finally, if $X^1$ hits points, it is shown that then $X$ will hit any reasonable smooth surface.

This is an attempt to show that a potential theory is associated with certain recurrent Markov processes in a natural way. For transient Markov processes this fact has been studied intensely. Thus Hunt [9] bases a general potential theory on transient continuous parameter processes, Doob [7] and Hunt [10] use the theory to construct boundaries for discrete parameter processes, Ito and McKean [11] solve the problem of characterizing the recurrent sets for simple random walk in three and higher dimension within the framework of the associated potential theory.