Contributed 10

## Reflecting Diffusion Processes, Stochastic Networks and Their Applications (Organizer: Amber Puha)

Conference
10:30 PM — 11:00 PM KST
Local
Jul 20 Tue, 9:30 AM — 10:00 AM EDT

### Measure valued processes characterized by a field of reflecting Brownian motions arising from certain queuing problems

Amarjit Budhiraja (University of North Carolina)

5
We study a class of queuing models in which the state of the system at any instant is given by a finite nonnegative Borel measure on the nonnegative real line which puts a unit atom at the remaining processing time of each job in system. The settings where the processing time distributions of jobs have bounded support or light tails have been investigated in previous works. In the current work we study the case where these distributions have finite second moments and regularly varying tails. By considering a parameter given in terms of the tails of processing time distributions, we consider a novel time, volume, and spatial scaling for the measure valued process and show that the scaled measure valued process converges in distribution (in the space of paths of measures). In a sharp contrast to results for bounded support and light tailed service time distributions, this time there is no state space collapse and the limiting random measures are not concentrated on a single atom. Nevertheless, the description of the limit is simple and given explicitly in terms of a certain random field of reflected Brownian motions.

This is joint work with Sayan Banerjee and Amber Puha.

### Asymptotic behavior of a critical fluid model for bandwidth sharing with general file size distributions

Yingjia Fu (University of California San Diego)

5
This work concerns the asymptotic behavior of solutions to a critical fluid model for a data communication network, where file sizes are generally distributed and the network operates under a fair bandwidth sharing policy, chosen from the family of (weighted) $\alpha$-fair policies introduced by Mo and Walrand. Solutions of the fluid model are measure-valued functions of time. Under law of large numbers scaling, Gromoll and Williams proved that these solutions approximate dynamic solutions of a flow level model for congestion control in data communication networks, introduced by Massoulié and Roberts. In a recent work, we proved stability of the strictly subcritical version of this fluid model under mild assumptions. In this talk, we study the asymptotic behavior (as time goes to infinity) of solutions of the critical fluid model, in which the nominal load on each network resource is less than or equal to its capacity and at least one resource is fully loaded. For this we introduce a new Lyapunov function, inspired by the work of Kelly and Williams, Mulvany et al. and Paganini et al. Using this, under moderate conditions on the file size distributions, we prove that critical fluid model solutions converge uniformly to the set of invariant states as time goes to infinity, when started in suitable relatively compact sets. We expect that this result will play a key role in developing a diffusion approximation for the critically loaded flow level model of Massoulié and Roberts. Furthermore, the techniques developed here may be useful for studying other stochastic network models with resource sharing.

### Error bounds for the one-dimensional constrained Langevin approximation for density dependent Markov chains

Felipe Campos (University of California, San Diego)

5
The stochastic dynamics of chemical reaction networks are often modeled using continuous-time Markov chains. However, except in very special cases, these processes cannot be analysed exactly and their simulation can be computationally intensive. An approach to this problem is to consider a diffusion approximation. The Constrained Langevin Approximation (CLA) is a reflected diffusion approximation for stochastic chemical reaction networks proposed by Leite & Williams. In this work, we extend this approximation to (nearly) density dependent Markov chains, when the diffusion state space is one-dimensional. Then, we provide a bound for the error of the CLA in a strong approximation. Finally, we discuss some applications for chemical reaction networks and epidemic models, illustrating these with examples.

Joint work with Ruth Williams.

### Obliquely reflecting diffusions in nonsmooth domains: some new uniqueness results

Cristina Costantini (University of Chieti-Pescara)

5
Exhaustive existence and uniqueness results are available for Brownian motion reflecting in a polyhedron with constant direction of reflection on each face (Varadhan and Williams, 1984; Dai and Williams, 1995) or in a smooth cone with radially constant direction of reflection (Kwon and Williams, 1991). Only partial results are available for reflecting diffusions in nonsmooth domains with curved boundaries and varying directions of reflection, although these situations come up in applications (see, e.g., Kang, Kelly, Lee and Williams, 2009 or Kang and Williams, 2012). This talk will present some recent, published and unpublished, existence and uniqueness results. We consider semimartingale reflecting diffusions, characterized as solutions of Stochastic Differential Equations with Reflection (SDERs). We obtain existence and uniqueness of the solution in a piecewise smooth, 2-dimensional domain, with a varying direction of reflection on each "side", under easily verifiable, geometrically meaningful conditions. In the case of a polygon with a constant direction of reflection on each side, our conditions coincide with Dai and Williams'. Moreover we allow for cusps (Costantini and Kurtz, 2018) and for situations where two ''sides'' meet smoothly but the direction of reflection is discontinuos. We also obtain existence and uniqueness in a d-dimensional domain with one singular point (such as a smooth cone or ''horn''), with a varying direction of reflection, under similar assumptions. The keystone of our arguments is a new reverse ergodic theorem for nonhomogeneous, possibly killed, Markov chains (Costantini and Kurtz, 2021), which is used in combination with a result on existence of strong Markov solutions to SDERs (Costantini and Kurtz, 2019).

### Q&A for Contributed Session 10

0
This talk does not have an abstract.

###### Session Chair

Ruth J. Williams (University of California at San Diego)

Contributed 16

## Probability Theory and Statistical Mechanics

Conference
10:30 PM — 11:00 PM KST
Local
Jul 20 Tue, 9:30 AM — 10:00 AM EDT

### Coexistence of localized Gibbs measures and delocalized gradient Gibbs measures on trees

Florian Henning (Ruhr-University Bochum)

2
In statistical mechanics, Gibbs measures for a spin system (a stochastic process) indexed by a countable graph under the influence of an interaction potential describe equilibrium distributions. They are defined in terms of being compatible with the Gibbsian specification associated with the potential, a system of prescribed conditional distributions built from the potential. In case of an unbounded local spin space, existence of Gibbs measures does not directly follow from compactness arguments. In this talk we focus on the situation where the underlying graph is a regular tree, spins take values in the integers (or an integer lattice) and the interaction potential is spatially homogeneous and of gradient type, i.e., depends only on the difference of spins values at neighboring sites.
We provide general conditions in terms of the relevant p-norms of the associated transfer operator Q (the exponential of the interaction potential) which ensure the existence of a countable family of spatially homogeneous Gibbs measures, describing localization at different heights. Next we prove existence of spatially homogeneous gradient Gibbs measures, describing increments of spin values along the edges of the tree. We construct these gradient Gibbs measures in terms of an edge-wise independent resampling process for $Z_q$-valued Gibbs measures for a suitable transformed fuzzy transfer operator $Q^q$. Then we prove that they are delocalized. Finally, we show that the two conditions on Q can be fulfilled at the same time, which then implies coexistence of both types of measures.

The talk is based on joint work with Christof Kuelske, which is accepted for publication in the Annals of Applied Probability.

Reference: arXiv:2002.09363

### Inhomogeneous gradient Gibbs measures on regular trees with homogeneous interactions

Christof Kuelske (Ruhr-University Bochum)

2
It is known that some statistical mechanics models with homogeneous interactions on regular lattices may admit inhomogeneous infinite-volume states. A famous example for this phenomenon are the Dobrushin-states for the Ising model which lack translation-invariance in three or more lattice dimensions. We investigate whether states which lack translation-invariance also exist on regular trees for Z-valued spin models with nearest-neighbor gradient interactions. Our analysis includes the SOS-model and the discrete Gaussian, which are important models of mathematical statistical mechanics, where they are mostly studied on the lattice. We show that, under rather general assumptions on the interaction, such inhomogeneous gradient states do exist. Our proof uses probabilistic methods in close combination with dynamical systems methods. In a first part we extend the probabilistic approach of our earlier work ("Coexistence of localized Gibbs measures and delocalized gradient Gibbs measures on trees", to be published in the Annals of Applied Probability). This allows to draw a relation between the gradient Gibbs states we are aiming at, and the Gibbs states of certain internal q-state spin-models with discrete rotation symmetry, which holds also for inhomogeneous states. In a second part we investigate these q-spin models on the regular tree via their associated discrete dynamical systems. The proofs of existence and lack of translation invariance of infinite-volume gradient states are then specifically based on properties of the local pseudo-unstable manifold of the corresponding discrete dynamical systems of these internal models, around the free state, at large q.

Reference: arXiv:2102.11899, Existence of gradient Gibbs measures on regular trees which are not translation invariant

### Statistical mechanical model of adsorption at the surface interface contacting with an ideal gas

Changho Kim (University of California, Merced)

3
We develop a statistical mechanical model for an ideal gas interfaced with a lattice surface where adsorption and desorption of gas particles occur. While this type of model has been investigated, we revisit it for the development of a thermodynamically consistent particle-continuum hybrid model for stochastic simulations of gas-solid interfacial systems as described below. Following the Langmuir adsorption model, we assume that the mean adsorption rate is proportional to the mean impinge rate of gas particles onto the surface and the mean desorption rate is given as a function of surface temperature. As a result, thermodynamic equilibrium is expected to be established for a given pressure of the ideal gas and temperature of the system. By investigating the detailed balance conditions, we derive the equilibrium fluctuational properties of the ideal gas state and surface coverage. We consider several velocity models, including the spectacular reflection and thermal wall models, from which the velocity of each desorbed or colliding particle is drawn. Based on this statistical mechanical model, we find how the ideal gas state and surface coverage after a finite short time delta t can be updated using the adsorption and desorption counts during delta t. For the momentum and energy update, we confirm that the same thermodynamic equilibrium is established whether adsorption and desorption are considered. The resulting time update model gives essential information on how to construct a particle-continuum hybrid model, where positions of all adsorbed particles are tracked whereas only the aggregated information such as the total mass, momentum, and energy densities are tracked in the gas. We present preliminary simulation results of the particle-continuum hybrid simulation method and demonstrate the importance of using a thermodynamically consistent statistical mechanical model.

### Q&A for Contributed Session 16

0
This talk does not have an abstract.

###### Session Chair

HyunJae Yoo (Hankyong National University)

Contributed 19

## Detection and Segmentation

Conference
10:30 PM — 11:00 PM KST
Local
Jul 20 Tue, 9:30 AM — 10:00 AM EDT

### Detection of outliers in compositional data on disabled people in the São Paulo State

Paulo Oliveira (University of São Paulo)

2
Outliers are observations that, for some reason, di_er from the other observations belonging to the data set. In univariate and bivariate data sets, outliers can be detected analyzing the scatter plot. Observations distant from the cloud formed by the data set are considered unusual. In multivariate data sets, the detection of outliers using graphics is more difficult because we have to analysis a couple of variables each time, which results is a long and less reliable process because we can find an observation that is unusual for one variable and not unusual for the others, masking the results. Compositional data are vectors, called compositions, whose components are all positive, it satisfies the sum equal one and has a Simplex space. The sum constraint induces the correlation between the components and this requires that the statistical methods for the analysis of datasets consider this fact. The theory for compositional data was developed mainly by Aitchison in the 1980s, and since then, several techniques and methods have been developed for compositional data modelling. Disabled Person is any person who presents loss or abnormality of a psychological or anatomical structure or function that generates incapacity for the performance of activities, that is, they have different characteristics from most people who are part of society and these characteristics make it difficult to their social inclusion. Disabilities can be permanent or temporary and limit the ability to perform one or more activities such as seeing, listening, walking and intellectual. It is characterized as a complex multidimensional experience and imposes several measurement challenges. Worldwide, disabled people have worse health prospects, lower levels of education, lower economic participation and higher poverty rates compared to people without disabilities. This is partly due to the fact that disabled people face barriers to access services that many of us have long considered guaranteed, such as health, education, employment. In statistical terms, were considered data from 3681111 respondents from the complete questionnaire of the IBGE (Statistics and Geography Brazilian Institute) census aggregated in 645 municipalities in the State of S_o Paulo, Brazil, considering as variables the 16 levels of disabled people with and the following methods were used to detect multivariate outliers’ detection: 95% confidence ellipse based on the first two main components; The Forward Search; Based on MCD (Minimum Covariance Determinant). and finally; Based on MED (Mas Eigen Difference) for comparative study between outlier detection performance by different methods.

### Consistent change-point detection for general distributions

Florencia Leonardi (University of São Paulo)

2
We propose a method based on regularized maximum likelihood for change point detection of general multivariate distributions under independent sampling. We show that the estimator is consistent and almost surely recovers the set of change-points under usual and easy to verify conditions. These conditions apply to a large variety of models, such as categorical or normal random variables and finite-state Markov chains. We also show that we can efficiently compute the estimator through a dynamic programming algorithm under a decomposable penalty term.

This is joint work with Lucas Prates de Oliveira.

### Change point detection under linear model: use of MOSUM approach

Joonpyo Kim (Seoul National University)

6
This talk presents a new detection method for structural change points based on a MOSUM approach under a piecewise linear model. Most existing methods focus on mean changes, assuming that the underlying model is piecewise constant. However, this stringent assumption cannot be applicable to many real-world processes, such as manufacturing. The proposed method significantly extends the scope of the change points structure by employing the linear regression model so that it is capable of identifying slope changes or smoothness of the processes beyond their mean changes. The proposed method is computationally efficient and easily used for real-time detection due to the inherent feature of the moving window approach. Furthermore, some theoretical properties of the proposed change point estimator are investigated. Results from the real data analysis and simulation examples show the promising empirical performance of the proposed method.

### Interval-censored least-squares regressions

Taehwa Choi (Korea University)

2
This article suggests the linear regression model under interval-censored data, where exact event times are unobserved but fall in observed censoring intervals. It is commonly arisen in longitudinal studies such as breast cosmesis data, where periodic monitoring is progressed to check the patient clinical status. Many of previous researches has been mostly focused on probability-based methods such as Cox and transformation models in terms of both theoretical and practical approaches. In contrast, there has not been received much attention on accelerated failure time model, despite direct interpretation on event time is possible. In this article, we generalize the Buckley-James method to explain the accelerated lifetime effects under the interval-censored data. Coupled with regression estimating procedure, a novel EM-algorithm for nonparametric likelihood estimation is devised for nuisance function parameter. Asymptotic behaviors are established, along with slower rate of convergence for nuisance function parameter due to absence of exact data. Simulation studies demonstrate the finite sample performance, and the method is applied to the real data to illustrate the practical usage.

### Q&A for Contributed Session 19

0
This talk does not have an abstract.

###### Session Chair

Myung Hee Lee (Weil Cornell Medicine)

Contributed 23

## Bayesian Nonparametric Inference

Conference
10:30 PM — 11:00 PM KST
Local
Jul 20 Tue, 9:30 AM — 10:00 AM EDT

### Bernstein - von Mises type theorem for a scale hyperparameter in Bayesian nonparametric inference

Natalia Bochkina (University of Edinburgh)

2
We consider the problem of estimating a smooth function adaptively from a Bayesian perspective in a nonparametric regression model, observed either directly or indirectly. We consider the model in the sequence space, with a smoothing Gaussian prior on the unknown coefficients, and a hyperprior on the prior scale to achieve an adaptive estimator. We show, that under some conditions on the true function, such as self-similarity, the MAP estimator of the scale hyperparameter converges to its oracle value, and the posterior distribution of the scale can be approximated by a Gaussian distribution as the number of observations grows. As far as we are aware, it is the first result of Gaussian approximation of the posterior distribution of a hyperparameter. We will show that this can be interpreted as estimation of the scale parameter from data under model misspecification, and that in the considered setting the posterior variance and the Fisher information of the scale parameter are of different order. We will illustrate these results on an inverse problem with Volterra operator.

### Convergence of unadjusted Hamiltonian Monte Carlo for mean-field models

Katharina Schuh (University Bonn)

5
In this talk we study the unadjusted Hamiltonian Monte Carlo algorithm applied to high-dimensional probability distributions of mean-field type. We evolve dimension-free convergence and discretization error bounds in Wasserstein distance. These bounds require the discretization step to be sufficiently small, but do not require strong convexity of either the unary or pairwise potential terms present in the mean-field model. To handle high dimensionality, we use a particlewise coupling that is contractive in a complementary particlewise metric.
This talk is based on joint work with Nawaf Bou-Rabee.

### Nonparametric Bayesian volatility estimation for gamma-driven stochastic differential equations

Peter Spreij (University of Amsterdam)

2
We study a nonparametric Bayesian approach to estimation of the volatility function of a stochastic differential equation (SDE) driven by a gamma process. The volatility function is assumed to be positive and Hölder continuous. We show that the SDE admits a weak solution, unique in law. The volatility function is modelled a priori as piecewise constant on a partition of the real line, and we specify a gamma prior on its coefficients. This leads to a straightforward procedure for posterior inference via the Gibbs sampler. We give the contraction rate of the posterior distribution in terms of the Hölder exponent and the sample size.

Joint work with Denis Belomestny, Shota Gugushvili, Moritz Schauer.

### Hamiltonian Monte Carlo in high dimensions

Andreas Eberle (University of Bonn)

5
Hamiltonian Monte Carlo (HMC) is a Markov Chain Monte Carlo method that is widely used in applications. It is based on a combination of Hamiltonian dynamics and momentum randomizations. The Hamiltonian dynamics is discretized, and the discretization bias can either be taken into account (unadjusted HMC) or corrected by a Metropolis-Hastings accept-reject step (Metropolis adjusted HMC). Despite its empirical success, until a few years ago there have been almost no convergence bounds for the algorithm. This has changed in the last years where approaches to quantify convergence to equilibrium based on coupling, conductance and hypocoercivity have been developed. In this talk, I will present the coupling approach, and show how it can be used to obtain an understanding of the dimension dependence for unadjusted HMC in several high dimensional model classes. I will also mention some open questions.

### Q&A for Contributed Session 23

0
This talk does not have an abstract.

###### Session Chair

Kyoungjae Lee (Inha University)