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Siva Athreya

Publications and source records attributed to Siva Athreya.

36 records · Page 2Linked to original sources
arXiv2019

Central limit theorem for statistics of subcritical configuration models

We consider sub-critical configuration models and show that the central limit theorem for any additive statistic holds when the statistics satisfies a fourth moment assumption, a variance lower bound and the degree sequence of graph satisfies a growth condition. If the degree sequence is bounded, for well known statistics like component counts, log-partition function, and maximum cut-size which are Lipschitz under addition of an edge or switchings then the assumptions reduce to linear growth condition for the variance of the statistic. Our proof is based on an application of the central limit theorem for martingale-difference arrays due to Mcleish (1974) to a suitable exploration process.

math.PR↗
arXiv2018

Strong existence and uniqueness for stable stochastic differential equations with distributional drift

We consider the stochastic differential equation $$ dX_t = b(X_t) dt + dL_t,$$ where the drift $b$ is a generalized function and $L$ is a symmetric one dimensional $α$-stable Lévy processes, $α\in (1, 2)$. We define the notion of solution to this equation and establish strong existence and uniqueness whenever $b$ belongs to the Besov--Hölder space $\mathcal{C}^β$ for $β>1/2-α/2$.

math.PR↗
arXiv2017

Smoothness of Flow and Path-by-Path Uniqueness in Stochastic Differential Equations

We consider the stochastic differential equation $$ X_t = x_0 + \int_0^t f(X_s)ds + \int_0^tσ(X_s)dB^{H}_s,$$ with $x_0 \in \mathbb{R}^d$, $d \geq 1$, $f: \mathbb{R}^d \rightarrow \mathbb{R}^d$ is bounded continuous, $σ: \mathbb{R}^d \rightarrow \mathbb{R}^{d\times d}$ is a uniformly elliptic, bounded, twice continuously differentiable conservative vector field and $B^H$ is fractional Brownian motion with $H \in (\frac{1}{3}, \frac{1}{2}]$. When $d=1$, $H= \frac{1}{2}$, and $f$ is Hölder continuous, in the spirit of Davie [D07], we establish the existence of a null set $\mathcal{N}$ depending only on $f, σ$ such that for all $x_0\in \mathbb{R}$ and $ω\in Ω\setminus \mathcal{N}$, the above equation admits a path-by-path unique solution. Our proof is based on establishing the uniform continuous differentiability of the flow associated with the equation. We also establish the path-by-path uniqueness for $d \geq 1$ and $H \in (\frac{1}{3}, \frac{1}{2}]$, but the null set may depend on $x_0$, thus extending a result of Catellier-Gubinelli [CG12].

math.PR↗
arXiv2017

Harnack inequality for non-local Schrödinger operators

Let $x \in \mathbb{R}^d$, $d \geq 3,$ and $f: \mathbb{R}^d \rightarrow \mathbb{R}$ be a twice differentiable function with all second partial derivatives being continuous. For $1\leq i,j \leq d$, let $a_{ij} : \mathbb{R}^d \rightarrow \mathbb{R}$ be a differentiable function with all partial derivatives being continuous and bounded. We shall consider the Schrödinger operator associated to \begin{eqnarray*} \mathcal{L}f(x) &=& \frac12 \sum_{i=1}^d \sum_{j=1}^d \frac{\partial}{\partial x_i} \left(a_{ij}(\cdot) \frac{\partial f}{\partial x_j}\right)(x) + \int_{\mathbb{R}^d\setminus{\{0\}}} [f(y) - f(x) ]J(x,y)dy. \end{eqnarray*} where $J: \mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}$ is a symmetric measurable function. Let $q: \mathbb{R}^d \rightarrow \mathbb{R}.$ We specify assumptions on $a,q,$ and $J$ so that non-negative bounded solutions to $${\mathcal L}f + qf = 0$$ satisfy a Harnack inequality. As tools we also prove a Carleson estimate, a Uniform Boundary Harnack Principle and a 3G inequality for solutions to ${\mathcal L}f = 0.$

math.PR↗
arXiv2017

Respondent driven sampling and sparse graph convergence

We consider a particular respondent-driven sampling procedure governed by a graphon. By a specific clumping procedure of the sampled vertices we construct a sequence of sparse graphs. If the sequence of the vertex-sets is stationary then the sequence of sparse graphs converge to the governing graphon in the cut-metric. The tools used are concentration inequality for Markov chains and the Stein-Chen method.

math.PR↗
arXiv2017

Invariance principle for variable speed random walks on trees

We consider stochastic processes on complete, locally compact tree-like metric spaces $(T,r)$ on their "natural scale" with boundedly finite speed measure $ν$. Given a triple $(T,r,ν)$ such a speed-$ν$ motion on $(T,r)$ can be characterized as the unique strong Markov process which if restricted to compact subtrees satisfies for all $x,y\in T$ and all positive, bounded measurable $f$, \[ \mathbb{E}^x [ \int^{τ_y}_0\mathrm{d}s\, f(X_s) ] = 2\int_Tν(\mathrm{d}z)\, r(y,c(x,y,z))f(z) < \infty, \] where $c(x,y,z)$ denotes the branch point generated by $x,y,z$. If $(T,r)$ is a discrete tree, $X$ is a continuous time nearest neighbor random walk which jumps from $v$ to $v'\sim v$ at rate $\tfrac{1}{2}\cdot (ν(\{v\})\cdot r(v,v'))^{-1}$. If $(T,r)$ is path-connected, $X$ has continuous paths and equals the $ν$-Brownian motion which was recently constructed in [AthreyaEckhoffWinter2013]. In this paper we show that speed-$ν_n$ motions on $(T_n,r_n)$ converge weakly in path space to the speed-$ν$ motion on $(T,r)$ provided that the underlying triples of metric measure spaces converge in the Gromov-Hausdorff-vague topology introduced recently in [AthreyaLohrWinter2016].

math.PR↗
arXiv2016

Dense graph limits under respondent-driven sampling

We consider certain respondent-driven sampling procedures on dense graphs. We show that if the sequence of the vertex-sets is ergodic then the limiting graph can be expressed in terms of the original dense graph via a transformation related to the invariant measure of the ergodic sequence. For specific sampling procedures, we describe the transformation explicitly.

math.PR↗
arXiv2016

The gap between Gromov-vague and Gromov-Hausdorff-vague topology

In Athreya, Löhr, Winter (2016), an invariance principle is stated for a class of strong Markov processes on tree-like metric measure spaces. It is shown that if the underlying spaces converge Gromov vaguely, then the processes converge in the sense of finite dimensional distributions. Further, if the underlying spaces converge Gromov-Hausdorff vaguely, then the processes converge weakly in path space. In this paper we systematically introduce and study the Gromov-vague and the Gromov-Hausdorff-vague topology on the space of equivalence classes of metric boundedly finite measure spaces. The latter topology is closely related to the Gromov-Hausdorff-Prohorov metric which is defined on different equivalence classes of metric measure spaces. We explain the necessity of these two topologies via several examples, and close the gap between them. That is, we show that convergence in Gromov-vague topology implies convergence in Gromov-Hausdorff-vague topology if and only if the so-called lower mass-bound property is satisfied. Furthermore, we prove and disprove Polishness of several spaces of metric measure spaces in the topologies mentioned above (summarized in Figure~1). As an application, we consider the Galton-Watson tree with critical offspring distribution of finite variance conditioned to not get extinct, and construct the so-called Kallenberg-Kesten tree as the weak limit in Gromov-Hausdorff-vague topology when the edge length are scaled down to go to zero.

math.PR↗
arXiv2016

Subdiffusivity of a random walk among a Poisson system of moving traps on ${\mathbb Z}$

We consider a random walk among a Poisson system of moving traps on ${\mathbb Z}$. In earlier work [DGRS12], the quenched and annealed survival probabilities of this random walk have been investigated. Here we study the path of the random walk conditioned on survival up to time $t$ in the annealed case and show that it is subdiffusive. As a by-product, we obtain an upper bound on the number of so-called thin points of a one-dimensional random walk, as well as a bound on the total volume of the holes in the random walk's range.

math.PR↗
arXiv2014

Random Walks in I.I.D. Random Environment on Cayley Trees

We consider the random walk in an \emph{i.i.d.} random environment on the infinite $d$-regular tree for $d \geq 3$. We consider the tree as a Cayley graph of free product of finitely many copies of $\Zbold$ and $\Zbold_2$ and define the i.i.d. environment as invariant under the action of this group. Under a mild non-degeneracy assumption we show that the walk is always transient.

math.PR↗
arXiv2012

Systems of branching, annihilating, and coalescing particles

This paper studies systems of particles following independent random walks and subject to annihilation, binary branching, coalescence, and deaths. In the case without annihilation, such systems have been studied in our 2005 paper "Branching-coalescing particle systems". The case with annihilation is considerably more difficult, mainly as a consequence of the non-monotonicity of such systems and a more complicated duality. Nevertheless, we show that adding annihilation does not significantly change the long-time behavior of the process and in fact, systems with annihilation can be obtained by thinning systems without annihilation.

math.PR↗
arXiv2011

Brownian motion on R trees

The real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree. We use Dirichlet form methods to construct Brownian motion on any given locally compact $R$-tree {$(T,r)$} equipped with a Radon measure $ν$ {on $(T,{\mathcal B}(T))$}. We specify a criterion under which the Brownian motion is recurrent or transient. For compact recurrent $R$-trees we provide bounds on the mixing time. In this revised version, assumption (A3) for an $R$-tree has been removed.

math.PR↗
arXiv2010

Correlated Drainage Model

In this article we present an example of a random oriented tree model on d-dimensional lattice, that is a forest in d=3 with positive probability. This is in contrast with the other random tree models in the literature which are a forest only when d strictly greater than 3.

math.PR↗
arXiv2007

Critical Age Dependent Branching Markov Processes and Their Scaling Limits

This paper studies: (i) the long time behaviour of the empirical distribution of age and normalised position of an age dependent critical branching Markov process conditioned on non-extinction; and (ii) the super-process limit of a sequence of age dependent critical branching Brownian motions.

math.PR↗
arXiv1999

On a singular semilinear elliptic boundary value problem and the boundary Harnack principle

We consider the singular boundary-value problem Δu = f(u) in D; u|_dD= phi, where 1. D is a bounded C^2-domain of R^d, d >= 3 2. f: (0,1) -> (0,1) is a locally Hölder continuous function such that f(u) -> 1 as u -> 0 at the rate u^{-α}, for some αin (0,1), 3. and phi is a positive continuous function satisfying certain growth assumptions. We show existence of solutions bounded below by a positive harmonic function, which are smooth in D and continuous in D-bar. Such solutions are shown to satisfy a boundary Harnack principle. Probabilistic techniques are used in proving the main results.

math.PR↗
arXiv1999

Uniqueness for a class of one-dimensional stochastic PDEs using moment duality

For independent nearest-neighbour bond percolation on Z^d with d >> 6, we prove that the incipient infinite cluster's two-point function and three-point function converge to those of integrated super-Brownian excursion (ISE) in the scaling limit. The proof is based on an extension of the new expansion for percolation derived in a previous paper, and involves treating the magnetic field as a complex variable. A special case of our result for the two-point function implies that the probability that the cluster of the origin consists of n sites, at the critical point, is given by a multiple of n^{-3/2}, plus an error term of order n{-3/2-ε} with ε> 0. This is a strong statement that the critical exponent δis given by δ= 2.

math.PR↗