Martin boundaries of diffusions and random walks on hyperbolic spaces
We discuss certain kinds of diffusions on hyperbolic spaces, associated random walks on discrete groups of isometries of the latter, and their Martin boundaries.
math.DG↗
arXiv subjects
Debanjan Nandi
Publications and source records attributed to Debanjan Nandi.
Exact approximation order and well-distributed sets
We prove that for any proper metric space $X$ and a function $ψ:(0,\infty)\to(0,\infty)$ from a suitable class of approximation functions, the Hausdorff dimensions of the set $W_ψ(Q)$ of all points $ψ$-well-approximable by a well-distributed subset $Q\subset X$, and the set $E_ψ(Q)$ of points that are exactly $ψ$-approximable by $Q$, coincide. This answers in a general setting, a question of Beresnevich-Dickinson-Velani in the case of approximation of reals by rationals, and answered by Bugeaud in that case using the continued-fraction expansion of reals. Our main result applies in particular to approximation by orbits of fixed points of a wide class of discrete groups of isometries acting on the boundary of hyperbolic metric spaces.
math.NT↗
Harmonic measures on Bowditch boundaries of groups hyperbolic relative to virtually nilpotent subgroups
For a group hyperbolic relative to virtually nilpotent subgroups, on a cusped graph associated to the group, we construct a random walk whose Martin boundary is the Bowditch boundary of the group. Moreover, the harmonic measure is a conformal density corresponding to a hyperbolic Green metric and is exact dimensional on the Bowditch boundary. The latter equipped with a visual distance induced by the Green metric is an Ahlfors-regular metric measure space. The dimension is given in terms of the drift, a Green drift and the asymptotic entropy. The Patterson-Sullivan density for the action on the cusped graph in this case is doubling, its dimension is obtained by looking at cusp excursions of geodesics.
math.GR↗
Diophantine approximations, large intersections and geodesics in negative curvature
In this paper we prove quantitative results about geodesic approximations to submanifolds in negatively curved spaces. Among the main tools is a new and general Jarník-Besicovitch type theorem in Diophantine approximation. The framework we develop is flexible enough to treat manifolds of variable negative curvature, a variety of geometric targets, and logarithm laws as well as spiraling phenomena in both measure and dimension aspect. Several of the results are new also for manifolds of constant negative sectional curvature. We further establish a large intersection property of Falconer in this context.
math.MG↗
A density result for homogeneous Sobolev spaces
We show that in a bounded Gromov hyperbolic domain $Ω$ smooth functions with bounded derivatives $C^\infty(Ω)\cap W^{k,\infty}(Ω)$ are dense in the homogeneous Sobolev spaces $L^{k,p}(Ω)$.
math.FA↗
Accessible Parts of Boundary for Simply Connected Domains
For a bounded simply connected domain $Ω\subset\mathbb{R}^2$, any point $z\inΩ$ and any $0<α<1$, we give a lower bound for the $α$-dimensional Hausdorff content of the set of points in the boundary of $Ω$ which can be joined to $z$ by a John curve with a suitable John constant depending only on $α$, in terms of the distance of $z$ to $\partialΩ$. In fact this set in the boundary contains the intersection $\partialΩ_z\cap\partialΩ$ of the boundary of a John sub-domain $Ω_z$ of $Ω$, centered at $z$, with the boundary of $Ω$. This may be understood as a quantitative version of a result of Makarov. This estimate is then applied to obtain the pointwise version of a weighted Hardy inequality.
math.CA↗
A density result for homogeneous Sobolev spaces on planar domains
We show that in a bounded simply connected planar domain $Ω$ the smooth Sobolev functions $W^{k,\infty}(Ω)\cap C^\infty(Ω)$ are dense in the homogeneous Sobolev spaces $L^{k,p}(Ω)$.
math.FA↗
Characterizations of generalized John domains in $\mathbb{R}^n$ via metric duality
In this paper, we extend the characterization of John disks obtained by N\"akki and V\"ais\"al\"a [Exp. Math. 1991] to generalized John domains in higher dimensions under mild assumptions. The main ingredient in this characterization is to use the higher dimensional analogues of the local linear connectivity (LLC) and homological bounded turning properties introduced by V\"ais\"al\"a in his study of metric duality theory [Math. Scan. 1997]. Somewhat surprisingly, we constructed a uniform domain in $\R^3$, which is topologically simple, such that the complementary domain fails to be homotopically $1$-bounded turning. In particular, this shows that a similar characterization of generalized John domains in terms of higher dimensional homotopic bounded turning does not hold in dimension three.
math.GN↗