The Local Equilibrium State of a Crystal Surface Jump Process in the Rough Scaling Regime

具有任意正则性的跳伊藤型公式

Jump Itô-type formula with arbitrary regularity

本文提出一种适用于具有有限p-变差且存在跳跃的路径的伊藤型公式，区分了粗糙积分与跳跃修正项，并在连续和 càdlàg 情况下分别推导了相应公式，同时给出了随机微分方程的链式法则和路径上的对数财富分解。

Existence of weak martingale solution of Nematic Liquid Crystals driven by Pure Jump Noise

In this work we consider a stochastic evolution equation which describes the system governing the nematic liquid crystals driven by a pure jump noise. The existence of a martingale solution is proved for both 2D and 3D cases. The construction of the solution is based on the classical Faedo-Galerkin approximation, compactness method and the Jakubowski's version of the Skorokhod representation theorem for non-metric spaces. We prove the solution is pathwise unique and further establish the existence of a strong solution in the 2D case.

On the equivalence between some jumping SDEs with rough coefficients and some non-local PDEs

We study some jumping SDE and the corresponding Fokker-Planck (or Kolmogorov forward) equation, which is a non-local PDE. We assume only some measurability and growth conditions on the coefficients. We prove that for any weak solution $(f_t)_{t\in [0,T]}$ of the PDE, there exists a weak solution to the SDE of which the time marginals are given by $(f_t)_{t\in[0,T]}$. As a corollary, we deduce that for any given initial condition, existence for the PDE is equivalent to weak existence for the SDE and uniqueness in law for the SDE implies uniqueness for the PDE. This extends some ideas of Figalli…

莱维驱动随机模型的鲁棒滤波

Robust Filtering of Lévy-driven Stochastic Models

本文研究了由莱维过程驱动的随机模型的鲁棒非线性滤波问题，重点在于信号与观测过程通过共同的布朗运动和跳跃噪声耦合。通过构造滤波器并证明其在两种不同情形下的连续性，提出了一种基于Stratonovich和Marcus流分解的方法，而非传统的Itô方法。

粗糙Hawkes-Heston模型的微观结构基础

Microstructural Foundation for the Rough Hawkes--Heston Model

本文构建泊松嵌入标记Hawkes订单流模型，为含共同价格-波动率跳跃的粗糙Hawkes-Heston模型提供微观结构基础，推导其收敛性并经数值实验验证。

The rough Hawkes Heston stochastic volatility model

We study an extension of the Heston stochastic volatility model that incorporates rough volatility and jump clustering phenomena. In our model, named the rough Hawkes Heston stochastic volatility model, the spot variance is a rough Hawkes-type process proportional to the intensity process of the jump component appearing in the dynamics of the spot variance itself and the log returns. The model belongs to the class of affine Volterra models. In particular, the Fourier-Laplace transform of the log returns and the square of the volatility index can be computed explicitly in terms of solutions of …

From local equilibrium to numerical PDE: Metropolis crystal surface dynamics in the rough scaling limit

We derive the PDE governing the hydrodynamic limit of a Metropolis rate crystal surface height process in the "rough scaling" regime introduced by Marzuola and Weare. The PDE takes the form of a continuity equation, and the expression for the current involves a numerically computed multiplicative correction term similar to a mobility. The correction accounts for the fact that, unusually, the local equilibrium distribution of the process is not a local Gibbs measure even though the global equilibrium distribution is Gibbs. We give definitive numerical evidence of this fact, originally suggested…

粗糙数据下薛定谔密度的分形性质

Fractality of the Schrödinger Density for Rough Data

本文证明了有界变差且含至少一个跳跃的粗糙数据对应的环面自由薛定谔演化密度几乎处处为分形，其临界索伯列夫质量的发散速率可探测色散关系，相关预测有待实验验证。

与晶体生长模型有关的过程的正重现性

Positive recurrence of processes associated to crystal growth models

我们发现，某些马尔可夫跳跃过程与晶体生长模型是积极的经常性时，参数满足一个相当自然的条件。

CTRW approximations for fractional equations with variable order

The standard diffusion processes are known to be obtained as the limits of appropriate random walks. These prelimiting random walks can be quite different however. The diffusion coefficient can be made responsible for the size of jumps or for the intensity of jumps. The "rough" diffusion limit does not feel the difference. The situation changes, if we model jump-type approximations via CTRW with non-exponential waiting times. If we make the diffusion coefficient responsible for the size of jumps and take waiting times from the domain of attraction of an $α$-stable law with a constant intensity…

General contact processes: inhomogeneous models, models on graphs and on manifolds

The contact process is a particular case of birth-and-death processes on infinite particle configurations. We consider the contact models on locally compact separable metric spaces. We prove the existence of a one-parameter set of invariant measures in the critical regime under the condition imposed on the associated Markov jump process. This condition, roughly speaking, requires the separation of any pair of trajectories of this jump process. The general scheme can be applied to the contact process on the lattice in a heterogeneous and random environments as well as to the contact process on …