We consider diffusions corresponding to the generatorWe show uniqueness for the corresponding martingale problem under certain non-degeneracy conditions on b i , γ i and present a counter-example when these conditions are not satisfied. As a special case, we establish uniqueness in law for some classes of super-Markov chains with state dependent branching rates and spatial motions.

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.

Abstract. In [ALW15] 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 Gromovvague 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 GromovHausdorff-vague topology when the edge length are scaled down to go to zero.

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We consider stochastic processes on complete, locally compact tree-like metric spaces $(T,r)$ on their "natural scale" with boundedly finite speed measure $\nu$. Given a triple $(T,r,\nu)$ such a speed-$\nu$ 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 $f$, \[ \mathbb{E}^x [ \int^{\tau_y}_0\mathrm{d}s\, f(X_s) ] = 2\int_T\nu(\mathrm{d}z)\, r(y,c(x,y,z))f(z) < \infty, \] 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 $\tfrac{1}{2}\cdot (\nu(\{v\})\cdot r(v,v'))^{-1}$. If $(T,r)$ is path-connected, $X$ has continuous paths and equals the $\nu$-Brownian motion which was recently constructed in [AthreyaEckhoffWinter2013]. In this paper we show that speed-$\nu_n$ motions on $(T_n,r_n)$ converge weakly in path space to the speed-$\nu$ motion on $(T,r)$ provided that the underlying triples of metric measure spaces converge in the Gromov-Hausdorff-vague topology introduced recently in [AthreyaLohrWinter2016].Comment: 45 page

Abstract. We introduce a new method for proving the estimatewhere u solves the equation ∆u − λu = f . The method can be applied to the Laplacian on R ∞ . It also allows us to obtain similar estimates when we replace the Laplacian by an infinite-dimensional Ornstein-Uhlenbeck operator or other elliptic operators. These operators arise naturally in martingale problems arising from measure-valued branching diffusions and from stochastic partial differential equations.

BackgroundGlobally, the routinely used case-based reporting and IgG serosurveys underestimate the actual prevalence of COVID-19. Simultaneous estimation of IgG antibodies and active SARS-CoV-2 markers can provide a more accurate estimation.MethodsA cross-sectional survey of 16416 people covering all risk groups was done between 3-16 September 2020 using the state of Karnataka’s infrastructure of 290 hospitals across all 30 districts. All participants were subjected to simultaneous detection of SARS-CoV-2 IgG using a commercial ELISA kit, SARS-CoV-2 antigen using a rapid antigen detection test (RAT), and reverse transcription-polymerase chain reaction (RT-PCR) for RNA detection. Maximum-likelihood estimation was used for joint estimation of the adjusted IgG, active, and total prevalence, while multinomial regression identified predictors.FindingsThe overall adjusted prevalence of COVID-19 in Karnataka was 27 ·3% (95% CI: 25 ·7-28 ·9), including IgG 16 ·4% (95% CI: 15 ·1 - 17 ·7) and active infection 12 ·7% (95% CI: 11 ·5-13 ·9). The case-to-infection ratio was 1:40, and the infection fatality rate was 0 ·05%. Influenza-like symptoms or contact with a COVID-19 positive patient are good predictors of active infection. The RAT kits had higher sensitivity (68%) in symptomatic participants compared to 47% in asymptomatic.InterpretationThis is the first comprehensive survey providing accurate estimates of the COVID-19 burden anywhere in the world. Further, our findings provide a reasonable approximation of population immunity threshold levels. Using the RAT kits and following the syndromic approach can be useful in screening and monitoring COVID-19. Leveraging existing surveillance platforms, coupled with appropriate methods and sampling framework, renders our model replicable in other settings.

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, 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.