Siva Athreya

Branching-coalescing particle systems

Abstract.We study the ergodic behavior of systems of particles performing independent random walks, binary splitting, coalescence and deaths. Such particle systems are dual to systems of linearly interacting Wright-Fisher diffusions, used to model a population with resampling, selection and mutations. We use this duality to prove that the upper invariant measure of the particle system is the only homogeneous nontrivial invariant law and the limit started from any homogeneous nontrivial initial law.

On the coverage of space by random sets

Let ξ1, ξ2,… be a Poisson point process of density λ on (0,∞) d , d ≥ 1, and let ρ, ρ1, ρ2,… be i.i.d. positive random variables independent of the point process. Let C := ⋃ i≥1 {ξ i + [0,ρ i ] d }. If, for some t > 0, (0,∞) d ⊆ C, then we say that (0,∞) d is eventually covered by C. We show that the eventual coverage of (0,∞) d depends on the behaviour of xP(ρ > x) as x → ∞ as well as on whether d = 1 or d ≥ 2. These results may be compared to those known for complete coverage of ℝ d by such Poisson Boolean models. In addition, we consider the set ⋃{i≥1:X i =1} [i,i+ρ i ], where X 1, X 2,… is a {0,1}-valued Markov chain and ρ1, ρ2,… are i.i.d. positive-integer-valued random variables independent of the Markov chain. We study the eventual coverage properties of this random set.

Invariance principle for variable speed random walks on trees

We consider stochastic processes on complete, locally compact tree-like metric spaces (T,r)(T,r) on their “natural scale” with boundedly finite speed measure νν. Given a triple (T,r,ν)(T,r,ν) such a speed-νν motion on (T,r)(T,r) can be characterized as the unique strong Markov process which if restricted to compact subtrees satisfies for all x,y∈Tx,y∈T and all positive, bounded measurable ff, Ex[∫τy0dsf(Xs)]=2∫Tν(dz)r(y,c(x,y,z))f(z)<∞, Ex[∫0τydsf(Xs)]=2∫Tν(dz)r(y,c(x,y,z))f(z)<∞, where c(x,y,z)c(x,y,z) denotes the branch point generated by x,y,zx,y,z. If (T,r)(T,r) is a discrete tree, XX is a continuous time nearest neighbor random walk which jumps from vv to v′∼vv′∼v at rate 12⋅(ν({v})⋅r(v,v′))−112⋅(ν({v})⋅r(v,v′))−1. If (T,r)(T,r) is path-connected, XX has continuous paths and equals the νν-Brownian motion which was recently constructed in [Trans. Amer. Math. Soc. 365 (2013) 3115–3150]. In this paper, we show that speed-νnνn motions on (Tn,rn)(Tn,rn) converge weakly in path space to the speed-νν motion on (T,r)(T,r) provided that the underlying triples of metric measure spaces converge in the Gromov–Hausdorff-vague topology introduced in [Stochastic Process. Appl. 126 (2016) 2527–2553].

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

In Athreya et al. (2015) 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.

On a Singular Semilinear Elliptic Boundary Value Problem and the Boundary Harnack Principle

AbstractOn a bounded C2-domain

Hölder norm estimates for elliptic operators on finite and infinite-dimensional spaces

D \subset {\mathbb R}^d

Respondent-Driven Sampling and Sparse Graph Convergence

we consider the singular boundary-value problem 1/2Δu=f(u) in D, u∂D=φ, where d≥3, f:(0,∞)→(0,∞) is a locally Hölder continuous function such that f(u)→∞ as u→0 at the rate u−α, for some α∈(0,1), and φ is a non-negative 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