奇异路径空间及其应用

Sample-path large deviation principle for a 2-D stochastic interacting vortex dynamics with singular kernel

We consider a stochastic interacting vortex system of $N$ particles, approximating the vorticity formulation of 2-D Navier-Stokes equation on torus. The singular interaction kernel is given by the Biot-Savart law. We only require the initial state to have finite energy, and obtain a sample-path large deviation principle for the empirical measure when the number of vortices goes to infinity. The rate function is characterized by an explicit formula supporting on sample paths with finite energy and finite integral of $L^2$ norms over time. The proof utilizes a symmetrization technique for the re…

Convergence rate for the hedging error of a path-dependent example

We consider a Brownian functional $F=g\bigl(\int_0^T η(s) dW_s\bigr)$ with $g \in L_2(γ)$ and a singular deterministic $η$. We deduce the $L_2$-convergence rate for the approximation $F^{(n)} = E F + \int_0^T ϕ^{(n)}(s) dW_s$ for a class of piecewise constant predictable integrands $ϕ^{(n)}$ from the fractional smoothness of $g$ quantified by Besov spaces and the rate of singularity of $η$.

Central limit theorem for Gibbs measures on path spaces including long range and singular interactions and homogenization of the stochastic heat equation

We consider a class of Gibbs measures defined with respect to increments $\{ω(t)-ω(s)\}_{s<t}$ of $d$-dimensional Wiener measure, with the underlying Hamiltonian carrying interactions of the form $H(t-s,ω(t)-ω(s))$ that are invariant under uniform translations of paths. In such interactions we allow {\it{long-range}} dependence in the time variable (including power law decay up to $t\mapsto (1+t)^{-(2+\varepsilon)}$ for $\varepsilon>0$) and unbounded (singular) interactions (including singularities of the form $x\mapsto 1/|x|^p$ in $d\geq 3$ or $x\mapsto δ_0(x)$ in $d=1$) attached to the space…

On a class of path-dependent singular stochastic control problems

This paper studies a class of non$-$Markovian singular stochastic control problems, for which we provide a novel probabilistic representation. The solution of such control problem is proved to identify with the solution of a $Z-$constrained BSDE, with dynamics associated to a non singular underlying forward process. Due to the non$-$Markovian environment, our main argumentation relies on the use of comparison arguments for path dependent PDEs. Our representation allows in particular to quantify the regularity of the solution to the singular stochastic control problem in terms of the space and …

奇异相互作用粒子系统及其平均场极限的路径正则化

Pathwise Regularisation of Singular Interacting Particle Systems and their Mean Field Limits

本文研究奇异相互作用粒子系统的噪声正则化，证明加入适当不规则的公共噪声路径可在宽泛 Besov 核条件下恢复 McKean–Vlasov 平均场极限。

Backward Stochastic Differential Equations with Nonmarkovian Singular Terminal Values

We solve a class of BSDE with a power function $f(y) = y^q$, $q > 1$, driving its drift and with the terminal boundary condition $ ξ= \infty \cdot \mathbf{1}_{B(m,r)^c}$ (for which $q > 2$ is assumed) or $ ξ= \infty \cdot \mathbf{1}_{B(m,r)}$, where $B(m,r)$ is the ball in the path space $C([0,T])$ of the underlying Brownian motion centered at the constant function $m$ and radius $r$. The solution involves the derivation and solution of a related heat equation in which $f$ serves as a reaction term and which is accompanied by singular and discontinuous Dirichlet boundary conditions. Although t…

由Rosenblatt过程驱动的随机微分方程在路径空间上的参数估计与律的奇异性

Parameter estimation and singularity of laws on the path space for SDEs driven by Rosenblatt processes

本文针对Rosenblatt过程驱动的SDE，提出联合估计漂移、扩散和Hurst指数的方法，证明强相合性，并揭示不同漂移下测度的奇异性，导致Girsanov型定理失效。

Ornstein-Uhlenbeck processes with singular drifts: integral estimates and Girsanov densities

We consider a perturbation of a Hilbert space-valued Ornstein--Uhlenbeck process by a class of singular nonlinear non-autonomous maximal monotone time-dependent drifts. The only further assumption on the drift is that it is bounded on balls in the Hilbert space uniformly in time. First we introduce a new notion of generalized solutions for such equations which we call pseudo-weak solutions and prove that they always exist and obtain pathwise estimates in terms of the data of the equation. Then we prove that their laws are absolutely continuous with respect to the law of the original Ornstein--…

由分数布朗运动驱动的奇异随机延迟方程：动力学、长时间行为与路径稳定性

Singular stochastic delay equations driven by fractional Brownian motion: Dynamics, longtime behaviour, and pathwise stability

本文研究由分数布朗运动驱动的奇异随机延迟方程，证明其适定性，建立Lyapunov谱与局部流形，并给出路径局部指数稳定性条件。

路径空间上的熵及其在奇异扩散和平均场模型中的应用

Entropy on the path space and application to singular diffusions and mean-field models

本文基于路径空间相对熵提出奇异相互作用粒子系统及其McKean-Vlasov极限的统一处理方法，推导了多类奇异扩散的存在唯一性，证明了粒子数趋于无穷时的收敛性与混沌传播，给出已知结果的新证明、拓展及全新结论。

BPHZ Renormalization in Gaussian Rough Paths

We construct a procedure for Bogoliubov-Parasiuk-Hepp-Zimmermann (BPHZ) renormalization of a rough path in view of the relation between rough path theory and regularity structure. We also provide a plain expression of the BPHZ-renormalized model in a rough path. BPHZ renormalization plays a central role in the theory of singular stochastic partial differential equations and assures the convergence of the model in the regularity structure. Here we demonstrate that the renormalization is also effective in a rough path setting by considering its application to rough path theory and mathematical f…

Volterra Equations Driven by Rough Signals

This article is devoted to the extension of the theory of rough paths in the context of Volterra equations with possibly singular kernels. We begin to describe a class of two parameter functions defined on the simplex called Volterra paths. These paths are used to construct a so-called Volterra-signature, analogously to the signature used in Lyon's theory of rough paths. We provide a detailed algebraic and analytic description of this object. Interestingly, the Volterra signature does not have a multiplicative property similar to the classical signature, and we introduce an integral product be…

On the Construction and Malliavin Differentiability of Levy Noise Driven SDEs with Singular Coefficients

In this paper we introduce a new technique to construct unique strong solutions of SDEs with singular coefficients driven by certain Levy processes. Our method which is based on Malliavin calculus does not rely on a pathwise uniqueness argument. Furthermore, the approach, which provides a direct construction principle, grants the additional insight that the obtained solutions are Malliavin differentiable.

Geometric singular perturbation theory for stochastic differential equations

We consider slow-fast systems of differential equations, in which both the slow and fast variables are perturbed by noise. When the deterministic system admits a uniformly asymptotically stable slow manifold, we show that the sample paths of the stochastic system are concentrated in a neighbourhood of the slow manifold, which we construct explicitly. Depending on the dynamics of the reduced system, the results cover time spans which can be exponentially long in the noise intensity squared (that is, up to Kramers' time). We obtain exponentially small upper and lower bounds on the probability of…

具有块马尔可夫依赖的稠密随机矩阵的奇异值分布

Singular value distribution of dense random matrices with block Markovian dependence

本文研究具有块马尔可夫依赖的稠密随机矩阵，引入具有方差剖面的近似不相关随机矩阵，通过矩方法和耦合论证，建立了样本路径长度为 $Θ(n^2)$ 时经验转移矩阵与经验频率矩阵奇异值分布的极限定律。