Minimal Martin Boundary Research Articles

Green function estimate for censored stable processes

Sharp two-sided estimates for Green functions of censored α-stable process Y in a bounded C 1,1 open set D are obtained, where α  (1, 2). It is shown that the Martin boundary and minimal Martin boundary of Y can all be identified with the Euclidean boundary of D. Sharp two-sided estimates for the Martin kernel of Y are also derived.

Semilinear perturbations of harmonic spaces and Martin-Orlicz capacities: an approach to the trace of moderate $U$-functions

Let $(X,\mathcal H)$ be a harmonic space in the sense of H. Bauer [7] which has a Green function $G_X$. It is known [27] that to every reference measure $r$ there corresponds a suitable integral representation of functions in ${{\mathcal H}^+(X)}\cap L^1(X,r)$. Let $Y$ be the minimal Martin boundary, $P$ the Martin kernel, and denote by ${\mathcal M(Y)}$ the set of all signed Borel measures on $Y$ with bounded variation. In this paper we consider the perturbed (semilinear) structure $(X,\mathcal U)$ obtained from $(X,\mathcal H)$ by means of $({\gamma},\Psi)$ where ${\gamma}$ is a local Kato measure on $X$ and $\Psi$ belongs to a class of real-valued functions on $X\times{\mathbb R}$ containing $(x,t)\mapsto t|t|^{{\alpha}-1}$ for any ${\alpha}>1$. We show that for every function $u\in{\mathcal U_r(X)}:=\{u\in{\mathcal U(X)}:|u|\leq h\;\mbox{for some}\;h\in {{\mathcal H}^+(X)}\cap L^1(X,r)\}$ there exists a unique signed measure $\nu\in{\mathcal M(Y)}$ such that $$ u+\int_X G_X(\cdot,{\zeta})\Psi({\zeta},u({\zeta}))\,d{\gamma}({\zeta})=\int_Y P(\cdot,y)\,d\nu(y). $$ Conversely, we prove that this integral equation admits a solution $u\in{\mathcal U_r(X)}$ whenever $\nu$ does not charge compact sets $K\subset Y$ of zero Martin-Orlicz capacity, that is, the integral $$ \int_X\int_X G_X(x,{\zeta}) \Psi({\zeta},\int_Y P({\zeta},y)\,d\mu(y))\,d{\gamma}({\zeta})\,d r(x) $$ is equal to $0$ or $\infty$ for every $\mu\in{\mathcal M^+(Y)}$ such that $\mu(Y{\backslash} K)=0$. In Section 6, we use our approach to investigate the trace of moderate solutions to some semilinear equations.

Sierpiński Gasket as a Martin Boundary I: Martin Kernels; Dedicated to Professor Masatoshi Fukushima on the occasion of his 60th birthday

We show that a Sierpinski gasket in N dimension is homeomorphic to the minimal Martin boundary of some canonical Markov chain. This provides a new class of examples for the boundary theory of Markov chains and the basis for a harmonic analysis on p.c.f. fractal structures.

Martin boundary of unlimited covering surfaces

LetW be an open Riemann surface and\(\bar W\) ap-sheeted (1<p<∞) unlimited covering surface ofW. Denote by Δ1 (resp.,\(\bar \Delta _1 \)) the minimal Martin boundary ofW (resp.,\(\bar W\)). For ζ ∈ Δ, let\(\nu _{\bar W} \)ζ be the (cardinal) number of the set of pionts\(\bar \varsigma \in \bar \Delta _1 \) which lie over ζ and\(\mathcal{M}_\varsigma \) the class of open connected subsetsM ofW such thatM∪{ζ} is a minimal fine neighborhood of ζ. Our main result is the following:\(\nu _{\bar W} \left( \varsigma \right) = \max _{M \in \mathcal{M}_\varsigma n} n_{\bar W} (M)\), where\(n_{\bar W} (M)\) is the number of components of π-1 M and π is the projection of\(\bar W\) ontoW. Moreover, some applications of the above results are discussed whenW is the unit disc.

Martin Boundaries of Quasi-Sectorial Domains

Cranston and Salisbury have obtained an integral test for the existence of a maximal number of minimal Martin boundary points at 0 in certain domains (cf. [8]). This paper will extend the result as follows: let D ⊂ R d be an open Greenian set (with respect to the Laplacian) consisting of n disjoint open connected cones with Lipschitz boundary and a subset of the boundary of these cones. Let μ be some local Kato measure supported by the boundary of the cones and consider the Schrodinger operator 1/2Δ-µ. We will assume a boundary Harnack principle and give a sufficient integral criterion for the existence of exactly n minimal Martin boundary points at 0. In certain cases there is a necessary criterion, too. When the sufficient integral criterion holds we will give a necessary and a sufficient condition for the existence of a certain process related to the Schrodinger operator that connects two different admissible boundary points. In the paper of Cranston and Salisbury the case μ = 0, d = 2 is treated, but many of the arguments work as well in the general situation.

On the thinness of non-tangential sets

Let Ω⊄Rd and ξ∈∂Ω. LetE be any non-tangential subset of Ω. we prove that ifE is internal thin at ξ, then it is minimal thin at any minimal Martin boundary point of Ω.

Martin boundaries of sectorial domains

LetD be a domain in R2 whose complement is contained in a pair of rays leaving the origin. That is, D contains two sectors whose base angles sum to 2π. We use balayage to give an integral test that determines if the origin splits into exactly two minimal Martin boundary points, one approached through each sector. This test is related to other integral tests due to Benedicks and Chevallier, the former in the special case of a Denjoy domain. We then generalise our test, replacing the pair of rays by an arbitrary number.

Martin boundaries of cartesian products of markov chains

Let P and Q be the stochastic transition operators of two time-homogeneous, irreducible Markov chains with countable, discrete state spaces X and Y, respectively. On the Cartesian product Z = X x Y, define a transition operator of the form Ra = a·P + (1 — a) · Q, 0 &lt; a &lt; 1, where P is considered to act on the first variable and Q on the second. The principal purpose of this paper is to describe the minimal Martin boundary of Ra (consisting of the minimal positive eigenfunctions of Ra with respect to some eigenvalue t, also called t-harmonic functions) in terms of the minimal Martin boundaries of P and Q.

Boundaries of random walks on graphs and groups with infinitely many ends

Consider an irreducible random walk {Z n} on a locally finite graphG with infinitely many ends, and assume that its transition probabilities are invariant under a closed group Γ of automorphisms ofG which acts transitively on the vertex set. We study the limiting behaviour of {Z n} on the spaceΩ of ends ofG. With the exception of a degenerate case,Ω always constitutes a boundary of Γ in the sense of Furstenberg, and {Z n} converges a.s. to a random end. In this case, the Dirichlet problem for harmonic functions is solvable with respect toΩ. The degenerate case may arise when Γ is amenable; it then fixes a unique end, and it may happen that {Z n} converges to this end. If {Z n} is symmetric and has finite range, this may be excluded. A decomposition theorem forΩ, which may also be of some purely graph-theoretical interest, is derived and applied to show thatΩ can be identified with the Poisson boundary, if the random walk has finite range. Under this assumption, the ends with finite diameter constitute a dense subset in the minimal Martin boundary. These results are then applied to random walks on discrete groups with infinitely many ends.

On Brownian Paths Connecting Boundary Points

There exists a Greenian domain $D \subset \mathbb{R}^2$ such that for every set $U$ of attainable minimal Martin boundary points which has null harmonic measure, there exist attainable minimal Martin boundary points $u, \nu \not\in U$ which cannot be connected by an $h$-process in $D$ starting from $u$ and converging to $\nu$.

Conditional Brownian Motion in Rapidly Exhaustible Domains

Let $D$ be a domain in $\mathbb{R}^d$ and let $\Delta_1$ be the set of minimal points of the Martin boundary of $D$. For $x \in D$ and $z \in \Delta_1$, let $(X_t)$ under the law $P^{x; z}$ be Brownian motion in $D$, starting at $x$ and conditioned to converge to $z$. Let $\tau$ be the lifetime of $(X_t)$, so $X_{\tau-} = z P^{x; z}$ a.s. Let $q \in L^p(D)$ for some $p > d/2$. Under the assumption that $D$ is what we call rapidly exhaustible, which is essentially a very weak boundary smoothness condition, we show that if the quantity $E^{x;z}\big\{\exp\big\lbrack\int^\tau_0 q(X_s) ds\big\rbrack\big\}$ is finite for one $x \in D$ and one $z \in \Delta_1$, then this quantity is bounded on $D \times \Delta_1$. This result may be viewed as saying, in a fairly strong sense, that the amount of time $(X_t)$ spends in each part of $D$ does not depend very much on the minimal Martin boundary point $z$ to which $(X_t)$ is conditioned to converge.