An introduction to stochastic processes associated with resistance forms and their scaling limits

Scaling limits of loop-erased Markov chains on resistance spaces via a partial loop-erasing procedure

We introduce partial loop-erasing operators. We show that by applying a refinement sequence of partial loop-erasing operators to a finite Markov chain, we get a process equivalent to the chronological loop-erased Markov chain. As an application, we construct loop-erased random paths on bounded domains of resistance spaces as the weak limit of the loop erasure of the Markov chains on a sequence of finite sets approximating the space, and the limit is independent of the approximating sequences. The random paths we constructed are simple paths almost surely, and they can be viewed as the loop-era…

与电阻形式相关的随机过程局部时的收敛性

Convergence of local times of stochastic processes associated with resistance forms

本文在局部 Gromov-Hausdorff-vague 收敛及非爆炸、度量熵条件下，证明电阻度量空间上随机过程与局部时的收敛，并统一推出分形图、随机图及生成树等模型的新老结果。

Scaling limit for random walk on the range of random walk in four dimensions

We establish scaling limits for the random walk whose state space is the range of a simple random walk on the four-dimensional integer lattice. These concern the asymptotic behaviour of the graph distance from the origin and the spatial location of the random walk in question. The limiting processes are the analogues of those for higher-dimensional versions of the model, but additional logarithmic terms in the scaling factors are needed to see these. The proof applies recently developed machinery relating the scaling of resistance metric spaces and stochastic processes, with key inputs being n…

Resistance Scaling on $4N$-Carpets

The $4N$ carpets are a class of infinitely ramified self-similar fractals with a large group of symmetries. For a $4N$-carpet $F$, let $\{F_n\}_{n \geq 0}$ be the natural decreasing sequence of compact pre-fractal approximations with $\cap_nF_n=F$. On each $F_n$, let $\mathcal{E}(u, v) = \int_{F_N} \nabla u \cdot \nabla v \, dx$ be the classical Dirichlet form and $u_n$ be the unique harmonic function on $F_n$ satisfying a mixed boundary value problem corresponding to assigning a constant potential between two specific subsets of the boundary. Using a method introduced by Barlow and Bass (1990…

Scaling Limits for Stochastic SQG Equations and 2D Inviscid Critical Boussinesq Equations with Transport Noises

We study stochastic SQG equations on the torus $\mathbb{T}^2$ with multiplicative transport noise in the $L^2$-setting. Under a suitable scaling of the noise, we show that the solutions converge weakly to the unique solution to the deterministic dissipative SQG equation. A similar scaling limit result is proved also for the stochastic 2D inviscid critical Boussinesq equations.

Scaling limit for the random walk on the largest connected component of the critical random graph

A scaling limit for the simple random walk on the largest connected component of the Erdos-Renyi random graph in the critical window is deduced. The limiting diffusion is constructed using resistance form techniques, and is shown to satisfy the same quenched short-time heat kernel asymptotics as the Brownian motion on the continuum random tree.

具有余法噪声的随机波映射的抛物缩放：极限与涨落

Parabolic scaling of a stochastic wave map with co-normal noise: limit and fluctuations

本文研究受余法型乘性高斯噪声扰动的阻尼随机波映射，证明其抛物缩放极限为调和映射热流，并给出涨落的弱中心极限定理。

Scaling limit of stochastic 2D Euler equations with transport noises to the deterministic Navier-Stokes equations

We consider a family of stochastic 2D Euler equations in vorticity form on the torus, with transport type noises and $L^2$-initial data. Under a suitable scaling of the noises, we show that the solutions converge weakly to that of the deterministic 2D Navier--Stokes equations. Consequently, we deduce that the weak solutions of the stochastic 2D Euler equations are approximately unique and "weakly quenched exponential mixing".

弱关联整数晶格上步强化和随机Levy-Lorentz模型的标度极限

Scaling limit of the step-reinforced and stochastic Levy--Lorentz model on weakly entangled integer lattice

本文研究了弱关联整数晶格上步强化和随机Levy-Lorentz模型的标度极限，探讨了随机游走的记忆性及中心极限定理的获得。

Multi-scaling of moments in stochastic volatility models

We introduce a class of stochastic volatility models $(X_t)_{t \geq 0}$ for which the absolute moments of the increments exhibit anomalous scaling: $\E\left(|X_{t+h} - X_t|^q \right)$ scales as $h^{q/2}$ for $q < q^*$, but as $h^{A(q)}$ with $A(q) < q/2$ for $q > q^*$, for some threshold $q^*$. This multi-scaling phenomenon is observed in time series of financial assets. If the dynamics of the volatility is given by a mean-reverting equation driven by a Levy subordinator and the characteristic measure of the Levy process has power law tails, then multi-scaling occurs if and only if the mean re…

带输运噪声的三维全局修正随机Navier-Stokes方程的标度极限与大偏差

Scaling Limit and Large Deviation for 3D Globally Modified Stochastic Navier-Stokes Equations with Transport Noise

该研究针对带输运噪声的三维全局修正随机超粘性Navier-Stokes方程，证明了弱解的存在性与轨道唯一性，得到其标度极限下收敛于确定性方程解的结论，并建立了对应系统的大偏差原理。

流形上随机输运方程的标度极限

Scaling Limits of Stochastic Transport Equations on Manifolds

本文将环面上随机输运方程的标度极限结果推广到紧黎曼流形，研究有色时空噪声驱动下方程在不同初始数据下的收敛行为，并给出平方可积初值情形收敛速度的定量估计。

The Full Brownian Web as Scaling Limit of Stochastic Flows

In this paper we construct an object which we call the full Brownian web (FBW) and prove that the collection of all space-time trajectories of a class of one-dimensional stochastic flows converges weakly, under diffusive rescaling, to the FBW. The (forward) paths of the FBW include the coalescing Brownian motions of the ordinary Brownian web along with bifurcating paths. Convergence of rescaled stochastic flows to the FBW follows from general characterization and convergence theorems that we present here combined with earlier results of Piterbarg.

The random conductance model with heavy tails on nested fractal graphs

Recently, Kigami's resistance form framework has been applied to provide a general approach for deriving the scaling limits of random walks on graphs with a fractal scaling limit. As an illustrative example, this article describes an application to the random conductance model with heavy tails on nested fractal graphs.

与电阻形式相关的半群和热核的收敛率估计

Convergence rate estimates for semigroups and heat kernels associated with resistance forms

本文在测度正则性和下电阻估计假设下，建立了与电阻形式相关的随机过程的半群和热核的显式收敛率，并引入了一个便于评估的新度量，应用于Sierpinski垫片上布朗运动的随机游走逼近和Bouchaud陷阱模型。

Critical Scaling for the Simple SIS Stochastic Epidemic

We exhibit a scaling law for the critical SIS stochastic epidemic: If at time 0 the population consists of square root N infected and N - square root N susceptible individuals, then when time and number currently infected are both scaled by square root N, the resulting process converges, for large N, to a diffusion process related to the Feller diffusion by a change of drift. As a consequence, the rescaled size of the epidemic has a limit law that coincides with that of a first-passage time for the standard Ornstein- Uhlenbeck process. These results are the analogues for the SIS epidemic of re…

非平衡随机振子：标度极限与相关性估计

Stochastic oscillators out of equilibrium: scaling limits and correlation estimates

本文研究受随机噪声扰动的谐振振子链，在扩散标度下依据 Hamiltonian 动力学强度确定能量与体积的两类流体动力学方程，并刻画一般初始测度下的非平衡体积涨落。

Stochastic mSQG equations with multiplicative transport noises: white noise solutions and scaling limit

We consider the modified Surface Quasi-Geostrophic (mSQG) equation on the 2D torus $\mathbb{T}^2$, perturbed by multiplicative transport noise. The equation admits the white noise measure on $\mathbb{T}^2$ as the invariant measure. We first prove the existence of white noise solutions to the stochastic equation via the method of point vortex approximation, then, under a suitable scaling limit of the noise, we show that the solutions converge weakly to the unique stationary solution of the dissipative mSQG equation driven by space-time white noise. The weak uniqueness of the latter equation is …

Scaling limit of a kinetic inhomogeneous stochastic system in the quadratic potential

We consider a particle evolving in the quadratic potential and subject to a time-inhomogeneous frictional force and to a random force. The couple of its velocity and position is solution to a stochastic differential equation driven by an $α$-stable L{é}vy process with $α\in (1,2]$ and the frictional force is of the form $t^{-β}\text{sgn}(v)|v|^γ$. We identify three regimes for the behavior in long-time of the couple velocity-position with a suitable rescaling, depending on the balance between the frictional force and the index of stability $α$ of the noise.

Scaling limit for stochastic control problems in population dynamics

Going from a scaling approach for birth/death processes, we investigate the scaling limit of solutions to non-Markovian stochastic control problems by studying the convergence of solutions to BSDEs driven a sequence of converging martingales. In particular we manage to describe how the values and optimal controls of control problems converge when the models converge towards a continuous population model.

Scaling limit of the corrector in stochastic homogenization

In the homogenization of divergence-form equations with random coefficients, a central role is played by the corrector. We focus on a discrete space setting and on dimension 3 and more. Completing the argument started in previous work, we identify the scaling limit of the corrector, which is akin to a Gaussian free field.

度量图上的光滑各向同性协方差函数：基于多调和电阻距离

Smooth Isotropic Covariance Functions on Metric Graphs via Polyharmonic Resistance Distances

本文在度量图上定义多调和距离，统一扩展电阻与双调和距离，并构造各向同性高斯过程，实现任意有限阶均方可微且满足基尔霍夫条件。

Scaling limit of fluctuations in stochastic homogenization

We investigate the global fluctuations of solutions to elliptic equations with random coefficients in the discrete setting. In dimension $d\geq 3$ and for i.i.d.\ coefficients, we show that after a suitable scaling, these fluctuations converge to a Gaussian field that locally resembles a (generalized) Gaussian free field. The paper begins with a heuristic derivation of the result, which can be read independently and was obtained jointly with Scott Armstrong.

Stochastic Representations for the Wave Equation on Graphs and their Scaling Limits

This paper is devoted to an interacting particle system that provides probabilistic interpretation of the wave equation on graphs. A Feynman-Kac-type formula is established, connecting the expectation of the process with the wave equation on graphs. Non-asymptotic $L^2$ estimates are presented. It is then shown that the high-density hydrodynamic limit of the system is given by the wave equation in Euclidean space. The sharpness of scaling limit result is demonstrated by a phase transition phenomenon.

递归分布方程类的分析，包括串联-并联图的电阻

Analysis of a class of recursive distributional equations including the resistance of the series-parallel graph

本文分析了一类递归分布方程，包含串联-并联图的电阻对数，推导出类似准线性Fisher-KPP方程的离散时间演化方程，当偏置参数p=1/2时，推导出分布极限定理。

Scaling and Saturation in Infinite-Dimensional Control Problems with Applications to Stochastic Partial Differential Equations

We establish the dual notions of scaling and saturation from geometric control theory in an infinite-dimensional setting. This generalization is applied to the low-mode control problem in a number of concrete nonlinear partial differential equations. We also develop applications concerning associated classes of stochastic partial differential equations (SPDEs). In particular, we study the support properties of probability laws corresponding to these SPDEs as well as provide applications concerning the ergodic and mixing properties of invariant measures for these stochastic systems.

特殊随机标度极限下被动标量与矢量平流方程的背景Vlasov方程和Young测度

Background Vlasov equations and Young measures for passive scalar and vector advection equations under special stochastic scaling limits

该研究针对随机标度极限下被动标量与矢量平流方程，引入背景随机Vlasov方程，证明其收敛到满足带扩散项极限PDE的非平凡Young测度，为被动矢量场情形的随机平流方程提供了新的统计信息。

作为随机感应方程标度极限的平均场磁流体动力学模型

Mean-Field Magnetohydrodynamics Models as Scaling Limits of Stochastic Induction Equations

本文研究三维周期域中磁场感应方程随机模型的标度极限，证明各向同性湍流导致额外耗散并影响磁场衰减率，而沿优先方向发展的湍流可产生发电机效应。