Existence of the zero range process and a deposition model with superlinear growth rates

Euler simulation of interacting particle systems and McKean-Vlasov SDEs with fully superlinear growth drifts in space and interaction

We consider in this work the convergence of a split-step Euler type scheme (SSM) for the numerical simulation of interacting particle Stochastic Differential Equation (SDE) systems and McKean-Vlasov Stochastic Differential Equations (MV-SDEs) with full super-linear growth in the spatial and the interaction component in the drift, and non-constant Lipschitz diffusion coefficient. The super-linear growth in the interaction (or measure) component stems from convolution operations with super-linear growth functions allowing in particular application to the granular media equation with multi-well c…

具有超线性增长系数和单边Lipschitz漂移的SDE的指数Euler格式的强收敛性

Strong convergence of the exponential Euler scheme for SDEs with superlinear growth coefficients and one-sided Lipschitz drift

本文研究具有超线性增长系数和单边Lipschitz漂移的一维SDE的指数Euler格式强收敛性，证明了连续漂移下1/2阶收敛速率，并分析了不连续漂移及扩散系数在零点消失情形下的收敛行为与稳定性。

\(\mathbb{R}^d\) 上由超线性噪声驱动的 McKean-Vlasov 随机分数阶 \((α,p)\)-拉普拉斯方程的一致大偏差

Uniform Large Deviations of Mckean-Vlasov Stochastic Fractional $(α,p)$-Laplacian Equations Driven by Superlinear Noise on $\mathbb{R}^d$

研究 \(\mathbb{R}^d\) 上由超线性噪声驱动的 McKean-Vlasov 随机分数阶 \((α,p)\)-拉普拉斯方程，用单调方法和区域扩张论证建立全局适定性，利用广义弱收敛方法等建立一致大偏差原理，克服相关困难并控制扩散项。

On the Growth of a Ballistic Deposition Model on Finite Graphs

We revisit a ballistic deposition process introduced by Atar, Athreya and Kang. Let $\mathcal{G}=(V,E)$ be a finite connected graph. We choose independently and uniformly vertices in $\mathcal{G}$. If a vertex $x$ is chosen and the previous height configuration is given by $h=(h_y)_{y \in V} \in \mathbb{N}_0^V$, the height $h_x$ is replaced by \[ \tilde{h}_x := 1 + \max_{y \sim x} h_y. \] We study asymptotic properties of this growth model. We determine the asymptotic growth parameter $γ(\mathcal{G} )$ for some graphs and prove a central limit theorem for the fluctuations around $γ( \mathcal{G…

Growth fluctuations in a class of deposition models

We compute the growth fluctuations in equilibrium of a wide class of deposition models. These models also serve as general frame to several nearest-neighbor particle jump processes, e.g. the simple exclusion or the zero range process, where our result turns to current fluctuations of the particles. We use martingale technique and coupling methods to show that, rescaled by time, the variance of the growth as seen by a deterministic moving observer has the form |V-C|*D, where V and C is the speed of the observer and the second class particle, respectively, and D is a constant connected to the eq…

Existence, uniqueness and approximation of solutions of SDEs with superlinear coefficients in the presence of discontinuities of the drift coefficient

Existence, uniqueness, and $L_p$-approximation results are presented for scalar stochastic differential equations (SDEs) by considering the case where, the drift coefficient has finitely many spatial discontinuities while both coefficients can grow superlinearly (in the space variable). These discontinuities are described by a piecewise local Lipschitz continuity and a piecewise monotone-type condition while the diffusion coefficient is assumed to be locally Lipschitz continuous and non-degenerate at the discontinuity points of the drift coefficient. Moreover, the superlinear nature of the coe…