arXiv subjects

Xinxin Chen

Publications and source records attributed to Xinxin Chen.

At least 37 records · Page 2Linked to original sources

arXiv2023

Domain of attraction of the fixed points of Branching Brownian motion

We give a complete characterisation of the domain of attraction of fixed points of branching Brownian motion (BBM) with critical drift. Prior to this classification, we introduce a suitable metric space of locally finite point measures on which we prove 1) that the BBM with critical drift is a well-defined Markov process and 2) that it satisfies the Feller property. Several applications of this characterisation are given.

math.PR↗

arXiv2023

Nanoparticles Passive Targeting Allows Optical Imaging of Bone Diseases

Bone health related skeletal disorders are commonly diagnosed by X-ray imaging, but the radiation limits its use. Light excitation and optical imaging through the near-infrared-II window (NIR-II, 1000-1700 nm) can penetrate deep tissues without radiation risk, but the targeting of contrast agent is non-specific. Here, we report that lanthanide-doped nanocrystals can be passively transported by endothelial cells and macrophages from the blood vessels into bone marrow microenvironment. We found that this passive targeting scheme can be effective for longer than two months. We therefore developed an intravital 3D and high-resolution planar imaging instrumentation for bone disease diagnosis. We demonstrated the regular monitoring of 1 mm bone defects for over 10 days, with resolution similar to X-ray imaging result, but more flexible use in prognosis. Moreover, the passive targeting can be used to reveal the early onset inflammation at the joints as the synovitis in the early stage of rheumatoid arthritis. Furthermore, the proposed method is comparable to μCT in recognizing symptoms of osteoarthritis, including the mild hyperostosis in femur which is ~100 μm thicker than normal, and the growth of millimeter-scale osteophyte in the knee joint, which further proves the power and universality of our approach in diagnosis of bone diseases

physics.bio-ph↗

arXiv2021

Total number of births on the negative half-line of the binary branching Brownian motion in the boundary case

The binary branching Brownian motion in the boundary case is a particle system on the real line behaving as follows. It starts with a unique particle positioned at the origin at time $0$. The particle moves according to a Brownian motion with drift $μ= 2$ and diffusion coefficient $σ^2 = 2$, until an independent exponential time of parameter $1$. At that time, the particle dies giving birth to two children who then start independent copies of the same process from their birth place. It is well-known that in this system, the cloud of particles eventually drifts to $\infty$. The aim of this note is to provide a precise estimate for the total number of particles that were born on the negative half-line, investigating in particular the tail decay of this random variable.

math.PR↗

arXiv2020

HR-Depth: High Resolution Self-Supervised Monocular Depth Estimation

Self-supervised learning shows great potential in monoculardepth estimation, using image sequences as the only source ofsupervision. Although people try to use the high-resolutionimage for depth estimation, the accuracy of prediction hasnot been significantly improved. In this work, we find thecore reason comes from the inaccurate depth estimation inlarge gradient regions, making the bilinear interpolation er-ror gradually disappear as the resolution increases. To obtainmore accurate depth estimation in large gradient regions, itis necessary to obtain high-resolution features with spatialand semantic information. Therefore, we present an improvedDepthNet, HR-Depth, with two effective strategies: (1) re-design the skip-connection in DepthNet to get better high-resolution features and (2) propose feature fusion Squeeze-and-Excitation(fSE) module to fuse feature more efficiently.Using Resnet-18 as the encoder, HR-Depth surpasses all pre-vious state-of-the-art(SoTA) methods with the least param-eters at both high and low resolution. Moreover, previousstate-of-the-art methods are based on fairly complex and deepnetworks with a mass of parameters which limits their realapplications. Thus we also construct a lightweight networkwhich uses MobileNetV3 as encoder. Experiments show thatthe lightweight network can perform on par with many largemodels like Monodepth2 at high-resolution with only20%parameters. All codes and models will be available at https://github.com/shawLyu/HR-Depth.

cs.CV↗

arXiv2020

A new proof of Liggett's theorem for non-interacting Brownian motions

In this note, we give a new proof of Liggett's theorem on the invariant measures of independent particle systems from [Lig78] in the particular case of independent drifted Brownian motions. This particular case has received a lot of attention recently due to its applications for the analysis of the local extrema of discrete Gaussian free field. The novelty of our proof is that it identifies directly the expected Poisson Point Process with exponential intensity without relying on the Choquet-Deny convolution equation $μ* P=μ$ ([ChoquetDeny60,Deny60]).

math.PR↗

arXiv2020

The fixed points of Branching Brownian Motion

In this work, we characterize all the point processes $θ=\sum_{i\in \mathbb{N}} δ_{x_i}$ on $\mathbb{R}$ which are left invariant under branching Brownian motions with critical drift $-\sqrt{2}$. Our characterization holds under the only assumption that $θ(\mathbb{R}_+)<\infty$ almost surely.

math.PR↗

arXiv2020

Maximal local time of randomly biased random walks on a Galton-Watson tree

We consider a recurrent random walk on a rooted tree in random environment given by a branching random walk. Up to the first return to the root, its edge local times form a Multi-type Galton-Watson tree with countably infinitely many types. When the walk is the diffusive or sub-diffusive, by studying the maximal type of this Galton-Watson tree, we establish the asymptotic behaviour of the largest local times of this walk during n excursions, under the annealed law.

math.PR↗

arXiv2020

Heavy range of the randomly biased walk on Galton-Watson trees in the slow movement regime

We consider the randomly biased random walk on trees in the slow movement regime as in [HS16], whose potential is given by a branching random walk in the boundary case. We study the heavy range up to the $n$-th return to the root, i.e., the number of edges visited more than $k_n$ times. For $k_n=n^θ$ with $θ\in(0,1)$, we obtain the convergence in probability of the rescaled heavy range, which improves one result of [AD20].

math.PR↗

arXiv2020

Branching Brownian motion conditioned on small maximum

We consider a standard binary branching Brownian motion on the real line. It is known that the maximal position $M_t$ among all particles alive at time $t$, shifted by $m_t = \sqrt{2} t - \frac{3}{2\sqrt{2}} \log t$ converges in law to a randomly shifted Gumbel variable. Derrida and Shi (2017) conjectured the precise asymptotic behaviour of the corresponding lower deviation probability $\mathbb{P}(M_t \leq \sqrt{2}αt)$ for $α< 1$. We verify their conjecture, and describe the law of the branching Brownian motion conditioned on having a small maximum.

math.PR↗

arXiv2019

Doob equivalence and non-commutative peaking for Markov chains

In this paper we show how questions about operator algebras constructed from stochastic matrices motivate new results in the study of harmonic functions on Markov chains. More precisely, we characterize coincidence of conditional probabilities in terms of (generalized) Doob transforms, which then leads to a stronger classification result for the associated operator algebras in terms of spectral radius and strong Liouville property. Furthermore, we characterize the non-commutative peak points of the associated operator algebra in a way that allows one to determine them from inspecting the matrix. This leads to a concrete analogue of the maximum modulus principle for computing the norm of operators in the ampliated operator algebras.

math.OA↗

arXiv2019

Branching processes in correlated random environment

We consider the critical branching processes in correlated random environment which is positively associated and study the probability of survival up to the n-th generation. Moreover, when the environment is given by fractional Brownian motion, we estimate also the tail of progeny as well as the tail of width.

math.PR↗

arXiv2019

On the tail distribution of the solution to some law equation

We consider a distribution equation which was initially studied by Bertoin \cite{Bertoin}: \[M \stackrel{d}{=} \max\{\widetildeν, \max_{1\leq k\leq ν}M_k\}.\] where $\{M_k\}_{k\geq 1}$ are i.i.d. copies of $M$ and independent of $(\widetildeν, ν)\in\mathbb{R}_+\times\mathbb{N}$. We obtain the tail behaviour of the solution of a generalised equation in a different but direct method by considering the joint tail of $(\widetildeν, ν)$.

math.PR↗

arXiv2018

On the trajectory of an individual chosen according to supercritical Gibbs measure in the branching random walk

Consider a branching random walk on the real line. Madaule showed the renormalized trajectory of an individual selected according to the critical Gibbs measure converges in law to a Brownian meander. Besides, Chen proved that the renormalized trajectory leading to the leftmost individual at time $n$ converges in law to a standard Brownian excursion. In this article, we prove that the renormalized trajectory of an individual selected according to a supercritical Gibbs measure also converges in law toward the Brownian excursion. Moreover, refinements of this results enables to express the probability for the trajectories of two individuals selected according to the Gibbs measure to have split before time $t$, partially answering a question of Derrida and Spohn.

math.PR↗

arXiv2018

Lower deviation and moderate deviation probabilities for maximum of a branching random walk

Given a super-critical branching random walk on $\mathbb R$ started from the origin, let $M_n$ be the maximal position of individuals at the $n$-th generation. Under some mild conditions, it is known from \cite{A13} that as $n\rightarrow\infty$, $M_n-x^*n+\frac{3}{2θ^*}\log n$ converges in law for some suitable constants $x^*$ and $θ^*$. In this work, we investigate its moderate deviation, in other words, the convergence rates of $$\mathbb{P}\left(M_n\leq x^*n-\frac{3}{2θ^*}\log n-\ell_n\right),$$ for any positive sequence $(\ell_n)$ such that $\ell_n=O(n)$ and $\ell_n\uparrow\infty$. As a by-product, we also obtain lower deviation of $M_n$; i.e., the convergence rate of \[ \mathbb{P}(M_n\leq xn), \] for $x<x^*$ in Böttcher case where the offspring number is at least two. Finally, we apply our techniques to study the small ball probability of limit of derivative martingale.

math.PR↗

arXiv2017

On large deviation probabilities for empirical distribution of branching random walks: Schr{ö}der case and B{ö}ttcher case

Given a super-critical branching random walk on $\mathbb{R}$ started from the origin, let $Z\_n(\cdot)$ be the counting measure which counts the number of individuals at the $n$-th generation located in a given set. Under some mild conditions, it is known in \cite{B90} that for any interval $A\subset \mathbb{R}$, $\frac{Z\_n(\sqrt{n}A)}{Z\_n(\mathbb{R})}$ converges a.s. to $ν(A)$, where $ν$ is the standard Gaussian measure. In this work, we investigate the convergence rates of $$\mathbb{P}\left(\frac{Z\_n(\sqrt{n}A)}{Z\_n(\mathbb{R})}-ν(A)>Δ\right),$$ for $Δ\in (0, 1-ν(A))$, in both Schr{ö}der case and B{ö}ttcher case.

math.PR↗

arXiv2016

Speed of Vertex reinforced jump process on Galton-Watson trees

We give an alternative proof of the fact that the vertex reinforced jump process on Galton- Watson tree has a phase transition between recurrence and transience as a function of c, the initial local time, see [3]. Further, applying the techniques in [1], we show a phase transition between positive speed and null speed for the discrete time process associated in the transient regime.

math.PR↗

arXiv2016

Long Brownian bridges in hyperbolic spaces converge to Brownian trees

We show that the range of a long Brownian bridge in the hyperbolic space converges after suitable renormalisation to the Brownian continuum random tree. This result is a relatively elementary consequence of $\bullet$ A theorem by Bougerol and Jeulin, stating that the rescaled radial process converges to the normalized Brownian excursion, $\bullet$ A property of invariance under re-rooting, $\bullet$ The hyperbolicity of the ambient space in the sense of Gromov. A similar result is obtained for the rescaled infinite Brownian loop in hyperbolic space.

math.PR↗

arXiv2016

Range and critical generations of a random walk on Galton-Watson trees

In this paper we consider a random walk in random environment on a tree and focus on the boundary case for the underlying branching potential. We study the range $R\_n$ of this walk up to time $n$ and obtain its correct asymptotic in probability which is of order $n/\log n$. Thisresult is a consequence of the asymptotical behavior of the number of visited sites at generations of order $(\log n)^2$,which turn out to be the most visited generations. Our proof which involves a quenched analysisgives a description of the typical environments responsible for the behavior of $R\_n$.

math.PR↗