Discrete Feynman-Kac formulas for branching random walks

Density-Dependent Analysis of Nonequilibrium Paths Improves Free Energy Estimates II. A Feynman-Kac Formalism

The nonequilibrium fluctuation theorems have paved the way for estimating equilibrium thermodynamic properties, such as free energy differences, using trajectories from driven nonequilibrium processes. While many statistical estimators may be derived from these identities, some are more efficient than others. It has recently been suggested that trajectories sampled using a particular time-dependent protocol for perturbing the Hamiltonian may be analyzed with another one. Choosing an analysis protocol based on the nonequilibrium density was empirically demonstrated to reduce the variance and bi…

Tempered Fractional Feynman-Kac Equation

Functionals of Brownian/non-Brownian motions have diverse applications and attracted a lot of interest of scientists. This paper focuses on deriving the forward and backward fractional Feynman-Kac equations describing the distribution of the functionals of the space and time tempered anomalous diffusion, belonging to the continuous time random walk class. Several examples of the functionals are explicitly treated, including the occupation time in half-space, the first passage time, the maximal displacement, the fluctuations of the occupation fraction, and the fluctuations of the time-averaged …

Feynman-Kac 理论中的非线性漂移：保留早期的概率洞察

Nonlinear Drift in Feynman-Kac Theory: Preserving Early Probabilistic Insights

本文表明 Feynman-Kac 理论可严格扩展至带漂移的非线性动力学，忠实其概率起源，从而为受限几何中的流动恢复确定性流线，并促进跨学科思想融合。

A fractional Feynman-Kac equation for weak ergodicity breaking

Continuous-time random walk (CTRW) is a model of anomalous sub-diffusion in which particles are immobilized for random times between successive jumps. A power-law distribution of the waiting times, $ψ(τ) τ^{-(1+α)}$, leads to sub-diffusion ($<x^2>~t^α$) for 0<α<1. In closed systems, the long stagnation periods cause time-averages to divert from the corresponding ensemble averages, which is a manifestation of weak ergodicity breaking. The time-average of a general observable $\bar{U} = \int_0^t U[x(τ)]dτ/ t$ is a functional of the path and is described by the well known Feynman-Kac equation if …

Functorial Statistical Physics: Feynman--Kac Formulae and Information Geometries

The main results of this paper comprise proofs of the following two related facts: (i) the Feynman--Kac formula is a functor $F_*$, namely, between a stochastic differential equation and a dynamical system on a statistical manifold, and (ii) a statistical manifold is a sheaf generated by this functor with a canonical gluing condition. Using a particular locality property for $F_*$, recognised from functorial quantum field theory as a `sewing law,' we then extend our results to the Chapman--Kolmogorov equation {\it via} a time-dependent generalisation of the principle of maximum entropy. This y…

具有生长波动的尺寸结构种群：费曼-凯克公式与解耦

Size-structured populations with growth fluctuations: Feynman--Kac formula and decoupling

本文研究了个体细胞生长速率由波动内部变量决定的尺寸结构种群模型，通过费曼-凯克公式建立了系谱动态与种群分布的关系，推导了解耦条件并探讨了不同集合中的弱解耦形式。

Feynman-Kac equation for anomalous processes with space- and time-dependent forces

Functionals of a stochastic process Y(t) model many physical time-extensive observables, e.g. particle positions, local and occupation times or accumulated mechanical work. When Y(t) is a normal diffusive process, their statistics are obtained as the solution of the Feynman-Kac equation. This equation provides the crucial link between the expected values of diffusion processes and the solutions of deterministic second-order partial differential equations. When Y(t) is an anomalous diffusive process, generalizations of the Feynman-Kac equation that incorporate power-law or more general waiting …

时间积分泛函的 Feynman-Kac 理论：Itô 微积分与泛函微积分的对比

Feynman-Kac theory of time-integrated functionals: Itô versus functional calculus

本文直接通过 Itô 微积分和泛函微积分推导了一般加性动力学泛函的 Feynman-Kac 理论，并利用 Dyson 级数重新概括了稳态方差结果，旨在统一该领域不同方法并提供更易理解的路径。

Feynman cycles in the Bose gas

We study the lengths of the cycles formed by trajectories in the Feynman-Kac representation of the Bose gas. We discuss the occurrence of infinite cycles and their relation to Bose-Einstein condensation.

驯化的Feynman-Kac扩散过程：杀死-分支交织

Tamed Feynman-Kac diffusion processes: Killing-branching intertwine

研究漂移布朗运动平衡松弛的Feynman-Kac核，通过引入杀死和分支速率解释负势区域的驯化效应，并用数值模拟验证一维非线性模型。

Feynman-Kac equation revisited

Functionals of particles' paths have diverse applications in physics, mathematics, hydrology, economics, and other fields. Under the framework of continuous time random walk (CTRW), the governing equations for the probability density functions (PDFs) of the functionals, including the ones of the paths of stochastic processes of normal diffusion, anomalous diffusion, and even the diffusion with reaction, have been derived. Sometimes, the stochastic processes in physics and chemistry are naturally described by Langevin equations. The Langevin picture has the advantages in studying the dynamics w…