S. Albeverio, A. Daletskii, E. Lytvynov
Spaces of differential forms over configuration spaces with Poisson measures are constructed. The corresponding Laplacians (of Bochner and de Rham type) on forms and associated semigroups are considered. Their probabilistic interpretation is given.
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We consider a rough differential equation indexed by a small parameter $\varepsilon>0$. When the rough differential equation is driven by fractional Brownian motion with Hurst parameter $H$ ($1/4<H<1/2$), we prove the Laplace-type asymptotics for the solution as the parameter $\varepsilon$ tends to zero.
Neural Laplace for learning Stochastic Differential Equations
本文从理论和实践角度回顾了神经拉普拉斯框架在学习多种随机微分方程中的潜在应用，扩展其统一学习能力至含随机性的时空动态建模。
This work focuses on the Laplace approximation for the rough differential equation (RDE) driven by mixed rough path with as . Firstly, based on geometric rough path lifted from mixed fractional Brownian motion (fBm), the Schilder-type large deviation principle (LDP) for the law of the first level path of the solution to the RDE is given. Due to the particularity of mixed rough path, the main difficulty in carrying out the Laplace approximation is to prove the Hilbert-Schmidt property for the Hessian matrix of the Itô map restricted on the Cameron-Martin space of the mixed fBm. To this end, we …
We present the idea of intertwining of two diffusions by Feynman-Kac operators. We present some variations and implications of the method and give examples of its applications. Among others, it turns out to be a very useful tool for finding the expectations of some functionals of diffusions, especially for computing the Laplace transforms of stochastic processes. The examples give new results on marginal distributions of many stochastic processes including a generalized squared Bessel processes and joint distribution for squared Bessel bridge and its integral - the close formulae of the Laplac…
Let $Γ_X$ denote the space of all locally finite configurations in a complete, stochastically complete, connected, oriented Riemannian manifold $X$, whose volume measure $m$ is infinite. In this paper, we construct and study spaces $L^2_μΩ^n$ of differential $n$-forms over $Γ_X$ that are square integrable with respect to a probability measure $μ$ on $Γ_X$. The measure $μ$ is supposed to satisfy the condition $Σ_m'$ (generalized Mecke identity) well known in the theory of point processes. On $L^2_μΩ^n$, we introduce bilinear forms of Bochner and deRham type. We prove their closabilty and call t…
Given a bounded $\mathcaligr{C}^2$ domain $G\subset{\mathbb{R}}^m$, functions $g\in\mathcaligr{C}(\partial G,{\mathbb{R}})$ and $h\in\mathcaligr {C}(\bar{G},{\mathbb{R}}\setminus\{0\})$, let $u$ denote the unique viscosity solution to the equation $-2Δ_{\infty}u=h$ in $G$ with boundary data $g$. We provide a representation for $u$ as the value of a two-player zero-sum stochastic differential game.
We formulate a stochastic differential game in continuous time that represents the unique viscosity solution to a terminal value problem for a parabolic partial differential equation involving the normalized $p(x,t)$-Laplace operator. Our game is formulated in a way that covers the full range $1<p(x,t)<\infty$. Furthermore, we prove the uniqueness of viscosity solutions to our equation in the whole space under suitable assumptions.
Transition probabilities for stochastic differential equations using the Laplace approximation: Analysis of the continuous-time limit
本文提出连续时间极限下随机微分方程转移概率的拉普拉斯近似方法，通过控制视角和Girsanov变换实现，并证明Strang分裂优于Euler离散化，数值验证了噪声和时间区间的影响。
Inference in stochastic differential equations using the Laplace approximation: Demonstration and examples
本文展示如何正确使用拉普拉斯逼近来估计随机微分方程中的状态和参数，特别关注非线性动力学、状态依赖噪声和非高斯测量误差，并通过模拟案例验证其计算可行性和灵活性。
In this paper we develop a method to solve evolution equations on Gelfand triples with time-fractional derivative based on monotonicity techniques. Applications include deterministic and stochastic quasi-linear partial differential equations with time-fractional derivatives, including time-fractional (stochastic) porous media equations (including the case where the Laplace operator is also fractional) and $p$-Laplace equations as special cases.
On the stationary distribution of reflected Brownian motion in a wedge: differential properties
研究二维楔形中半鞅反射布朗运动平稳分布的Laplace变换的代数与微分性质，推导其成为有理式、代数式、微分有限或微分代数式的充要条件，并利用Tutte不变量方法和Galois理论给出简单表达式与证明。
Spaces of differential forms over configuration spaces with Poisson measures are constructed. The corresponding Laplacians (of Bochner and de Rham type) on 1-forms and associated semigroups are considered. Their probabilistic interpretation is given.
Multilevel Monte Carlo for the Feynman-Kac Formula for the Laplace Equation
本文探讨了利用费曼-科茨公式将偏微分方程转化为随机过程，并通过Walk on Sphere算法生成随机路径，研究多级蒙特卡洛方法在拉普拉斯方程中的应用及扩展可能性。
We study a nonlinear stochastic partial differential equation whose solution is the conditional log-Laplace functional of a superprocess in a random environment. We establish its existence and uniqueness by smoothing out the nonlinear term and making use of the particle system representation developed by Kurtz and Xiong [Stochastic Process. Appl. 83 (1999) 103-126]. We also derive the Wong-Zakai type approximation for this equation. As an application, we give a direct proof of the moment formulas of Skoulakis and Adler [Ann. Appl. Probab. 11 (2001) 488-543].
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