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In this paper, we study asymptotic behaviors of a subcritical branching killed Brownian motion with drift $-\rho$ and offspring distribution $\{p_k:k\ge 0\}$. Let $\widetilde{\zeta}^{-\rho}$ be the extinction time of this subcritical branching killed Brownian motion, $\widetilde{M}_t^{-\rho}$ the maximal position of all the particles alive at time $t$ and $\widetilde{M}^{-\rho}:=\max_{t\ge 0}\widetilde{M}_t^{-\rho}$ the all time maximal position. Let $\mathbb{P}_x$ be the law of this subcritical branching killed Brownian motion when the initial particle is located at $x\in (0,\infty)$. Under the assumption $\sum_{k=1}^\infty k (\log k) p_k <\infty$, we establish the decay rates of $\mathbb{P}_x(\widetilde{\zeta}^{-\rho}>t)$ and $\mathbb{P}_x(\widetilde{M}^{-\rho}>y)$ as $t$ and $y$ tend to $\infty$ respectively. We also establish the decay rate of $\mathbb{P}_x(\widetilde{M}_t^{-\rho}>z(t,\rho))$ as $t\to\infty$, where $z(t,\rho)=\sqrt{t}z-\rho t$ for $\rho\leq 0$ and $z(t,\rho)=z$ for $\rho>0$. As a consequence, we obtain a Yaglom-type limit theorem.

In this paper, we study branching Brownian motion with absorption, in which particles undergo Brownian motions with drift and are killed upon reaching the origin. We prove that the extremal process of this branching Brownian motion with absorption converges to a random shifted decorated Poisson point process. Furthermore, we show that the law of the right-most particle converges to the law of a random shifted Gumbel random variable., Comment: 23 pages

Precise segmentation of organs and tumors plays a crucial role in clinical applications. It is a challenging task due to the irregular shapes and various sizes of organs and tumors as well as the significant class imbalance between the anatomy of interest (AOI) and the background region. In addition, in most situation tumors and normal organs often overlap in medical images, but current approaches fail to delineate both tumors and organs accurately. To tackle such challenges, we propose a position-guided deformable UNet, namely PGD-UNet, which exploits the spatial deformation capabilities of deformable convolution to deal with the geometric transformation of both organs and tumors. Position information is explicitly encoded into the network to enhance the capabilities of deformation. Meanwhile, we introduce a new pooling module to preserve position information lost in conventional max-pooling operation. Besides, due to unclear boundaries between different structures as well as the subjectivity of annotations, labels are not necessarily accurate for medical image segmentation tasks. It may cause the overfitting of the trained network due to label noise. To address this issue, we formulate a novel loss function to suppress the influence of potential label noise on the training process. Our method was evaluated on two challenging segmentation tasks and achieved very promising segmentation accuracy in both tasks., Comment: Accepted by the 2020 International Joint Conference on Neural Networks (IJCNN 2020)