Leanne Dong | University of South Australia (original) (raw)

Papers by Leanne Dong

Building upon the well-posedness results in snse1, in this note we prove the existence of invaria... more Building upon the well-posedness results in snse1, in this note we prove the existence of invariant measures for the stochastic Navier-Stokes equations with stable Lévy noise. The crux of our proof relies on the assumption of finite dimensional Lévy noise.

Hawkes processes are a popular means of modeling the event times of self-exciting phenomena, such... more Hawkes processes are a popular means of modeling the event times of self-exciting phenomena, such as earthquake strikes or tweets on a topical subject. Classically, these models are fit to historical event time data via likelihood maximization. However, in many scenarios, the exact times of historical events are not recorded for either privacy (e.g., patient admittance to hospitals) or technical limitations (e.g., most transport data records the volume of vehicles passing loop detectors but not the individual times). The interval-censored setting denotes when only the aggregate counts of events at specific time intervals are observed. Fitting the parameters of interval-censored Hawkes processes requires designing new training objectives that do not rely on the exact event times. In this paper, we propose a model to estimate the parameters of a Hawkes process in interval-censored settings. Our model builds upon the existing Hawkes Intensity Process (HIP) of in several important direc...

arXiv: Analysis of PDEs, 2018

In this paper we first prove the existence and uniqueness of weak and strong solution (in PDE sen... more In this paper we first prove the existence and uniqueness of weak and strong solution (in PDE sense) to the stochastic Navier-Stokes equations on the rotating 2-dimensional unit sphere perturbed by stable Levy noise. Then we show the existence of invariant measures under the assumption of finite dimensional noise.

arXiv: Analysis of PDEs, 2018

In this paper we prove the existence and uniqueness of a strong solution (in PDE sense) to the st... more In this paper we prove the existence and uniqueness of a strong solution (in PDE sense) to the stochastic Navier-Stokes equations on the rotating 2-dimensional unit sphere perturbed by stable L\'evy noise. This strong solution turns out to exist globally in time.

Journal of Mathematical Analysis and Applications, 2020

The Navier-Stokes equation with rough data arises in many problems of fluid dynamics but mathemat... more The Navier-Stokes equation with rough data arises in many problems of fluid dynamics but mathematical analysis of such problems is notoriously difficult. In this paper we consider a two-dimensional fluid moving on the surface of a rotating sphere under the influence of an impulsive force that is very irregular in time. More precisely, we assume that the impulsive force is associated to a Brownian Motion subordinated by a stable subordinator. Then we prove the existence and uniqueness of a strong solution (in PDE sense) to the stochastic Navier-Stokes equations on the rotating 2-dimensional unit sphere perturbed by a stable Levy noise. This strong solution turns out to exist globally in time.

This paper considers a numeric algorithm to solve the equation \begin{align*} y(t)=f(t)+\int^t_0 ... more This paper considers a numeric algorithm to solve the equation \begin{align*} y(t)=f(t)+\int^t_0 g(t-\tau)y(\tau)\,d\tau \end{align*} with a kernel ggg and input fff for yyy. In some applications we have a smooth integrable kernel but the input fff could be a generalised function, which could involve the Dirac distribution. We call the case when f=deltaf=\deltaf=delta, the Dirac distribution centred at 0, the fundamental solution EEE, and show that E=delta+hE=\delta+hE=delta+h where hhh is integrable and solve \begin{align*} h(t)=g(t)+\int^t_0 g(t-\tau)h(\tau)\,d\tau \end{align*} The solution of the general case is then \begin{align*} y(t)=f(t)+(h*f)(t) \end{align*} which involves the convolution of hhh and fff. We can approximate ggg to desired accuracy with piecewise constant kernel for which the solution hhh is known explicitly. We supply an algorithm for the solution of the integral equation with specified accuracy.

ArXiv, 2021

This work builds a novel point process and tools for fitting Hawkes processes with interval-censo... more This work builds a novel point process and tools for fitting Hawkes processes with interval-censored data. Such data only records the aggregated counts of events during specific time intervals – such as the number of patients admitted to the hospital, or the volume of vehicles passing traffic loop detectors – and not the exact occurrence time of the events. First, we define the Mean Behavior Poisson (MBP) process, a novel Poisson process with a direct parameter correspondence to the popular self-exciting Hawkes process. We fit MBPs in the interval-censored setting using an interval-censored Poisson log-likelihood (IC-LL). We use the parameter equivalence to uncover the parameters of the associated Hawkes process. Second, we introduce two novel exogenous functions to distinguish the exogenous from the endogenous events. We propose the multi-impulse exogenous function – for when the exogenous events are observed as event time – and the latent homogeneous Poisson process exogenous func...

Building upon the well-posedness results in \cite{snse1}, in this note we prove the existence of ... more Building upon the well-posedness results in \cite{snse1}, in this note we prove the existence of invariant measures for the stochastic Navier-Stokes equations with stable L\'evy noise. The crux of our proof relies on the assumption of finite dimensional L\'evy noise.

arXiv: Probability, 2018

In this paper we prove that the stochastic Navier-Stokes equations with additive stable Levy nois... more In this paper we prove that the stochastic Navier-Stokes equations with additive stable Levy noise generates a random dynamical systems. Then we prove the existence of random attractor for the Navier-Stokes equations on 2D spheres under stable Levy noise. We also deduce the existence of Feller Markov invariant measure.

Bulletin of the Australian Mathematical Society

Building upon the well-posedness results in snse1, in this note we prove the existence of invaria... more Building upon the well-posedness results in snse1, in this note we prove the existence of invariant measures for the stochastic Navier-Stokes equations with stable Lévy noise. The crux of our proof relies on the assumption of finite dimensional Lévy noise.

Hawkes processes are a popular means of modeling the event times of self-exciting phenomena, such... more Hawkes processes are a popular means of modeling the event times of self-exciting phenomena, such as earthquake strikes or tweets on a topical subject. Classically, these models are fit to historical event time data via likelihood maximization. However, in many scenarios, the exact times of historical events are not recorded for either privacy (e.g., patient admittance to hospitals) or technical limitations (e.g., most transport data records the volume of vehicles passing loop detectors but not the individual times). The interval-censored setting denotes when only the aggregate counts of events at specific time intervals are observed. Fitting the parameters of interval-censored Hawkes processes requires designing new training objectives that do not rely on the exact event times. In this paper, we propose a model to estimate the parameters of a Hawkes process in interval-censored settings. Our model builds upon the existing Hawkes Intensity Process (HIP) of in several important direc...

arXiv: Analysis of PDEs, 2018

In this paper we first prove the existence and uniqueness of weak and strong solution (in PDE sen... more In this paper we first prove the existence and uniqueness of weak and strong solution (in PDE sense) to the stochastic Navier-Stokes equations on the rotating 2-dimensional unit sphere perturbed by stable Levy noise. Then we show the existence of invariant measures under the assumption of finite dimensional noise.

arXiv: Analysis of PDEs, 2018

In this paper we prove the existence and uniqueness of a strong solution (in PDE sense) to the st... more In this paper we prove the existence and uniqueness of a strong solution (in PDE sense) to the stochastic Navier-Stokes equations on the rotating 2-dimensional unit sphere perturbed by stable L\'evy noise. This strong solution turns out to exist globally in time.

Journal of Mathematical Analysis and Applications, 2020

The Navier-Stokes equation with rough data arises in many problems of fluid dynamics but mathemat... more The Navier-Stokes equation with rough data arises in many problems of fluid dynamics but mathematical analysis of such problems is notoriously difficult. In this paper we consider a two-dimensional fluid moving on the surface of a rotating sphere under the influence of an impulsive force that is very irregular in time. More precisely, we assume that the impulsive force is associated to a Brownian Motion subordinated by a stable subordinator. Then we prove the existence and uniqueness of a strong solution (in PDE sense) to the stochastic Navier-Stokes equations on the rotating 2-dimensional unit sphere perturbed by a stable Levy noise. This strong solution turns out to exist globally in time.

This paper considers a numeric algorithm to solve the equation \begin{align*} y(t)=f(t)+\int^t_0 ... more This paper considers a numeric algorithm to solve the equation \begin{align*} y(t)=f(t)+\int^t_0 g(t-\tau)y(\tau)\,d\tau \end{align*} with a kernel ggg and input fff for yyy. In some applications we have a smooth integrable kernel but the input fff could be a generalised function, which could involve the Dirac distribution. We call the case when f=deltaf=\deltaf=delta, the Dirac distribution centred at 0, the fundamental solution EEE, and show that E=delta+hE=\delta+hE=delta+h where hhh is integrable and solve \begin{align*} h(t)=g(t)+\int^t_0 g(t-\tau)h(\tau)\,d\tau \end{align*} The solution of the general case is then \begin{align*} y(t)=f(t)+(h*f)(t) \end{align*} which involves the convolution of hhh and fff. We can approximate ggg to desired accuracy with piecewise constant kernel for which the solution hhh is known explicitly. We supply an algorithm for the solution of the integral equation with specified accuracy.

ArXiv, 2021

This work builds a novel point process and tools for fitting Hawkes processes with interval-censo... more This work builds a novel point process and tools for fitting Hawkes processes with interval-censored data. Such data only records the aggregated counts of events during specific time intervals – such as the number of patients admitted to the hospital, or the volume of vehicles passing traffic loop detectors – and not the exact occurrence time of the events. First, we define the Mean Behavior Poisson (MBP) process, a novel Poisson process with a direct parameter correspondence to the popular self-exciting Hawkes process. We fit MBPs in the interval-censored setting using an interval-censored Poisson log-likelihood (IC-LL). We use the parameter equivalence to uncover the parameters of the associated Hawkes process. Second, we introduce two novel exogenous functions to distinguish the exogenous from the endogenous events. We propose the multi-impulse exogenous function – for when the exogenous events are observed as event time – and the latent homogeneous Poisson process exogenous func...

Building upon the well-posedness results in \cite{snse1}, in this note we prove the existence of ... more Building upon the well-posedness results in \cite{snse1}, in this note we prove the existence of invariant measures for the stochastic Navier-Stokes equations with stable L\'evy noise. The crux of our proof relies on the assumption of finite dimensional L\'evy noise.

arXiv: Probability, 2018

In this paper we prove that the stochastic Navier-Stokes equations with additive stable Levy nois... more In this paper we prove that the stochastic Navier-Stokes equations with additive stable Levy noise generates a random dynamical systems. Then we prove the existence of random attractor for the Navier-Stokes equations on 2D spheres under stable Levy noise. We also deduce the existence of Feller Markov invariant measure.

Bulletin of the Australian Mathematical Society