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This is mainly for reference, I have not found a simple answer outside of the diffusion approximation. Assume $\alpha\gg 1$. The random walk has unit step size and the angle $\phi$ of a step with th …

Without the reflecting boundary condition, the diffusion coefficient follows from the mean square displacement, $$E[x(t)^2]=2Dt.$$ Reflections at the origin have no effect on this expectation value, …

Because of the central-limit-theorem, for large $N$ the absolute distance $d_N$ converges in distribution as $$P_N(d_N/\sqrt N)\to p(|X|),$$ where $X$ is a Gaussian random variable with mean zero and …

Indeed, the Lévy flight is a random walk where the step increments $\nu$ are i.i.d. with a Lévy distribution, $p(\nu)\rightarrow 1/\nu^{1+\alpha}$ for $\nu\rightarrow\infty$, with exponent $0<\alpha< …

as requested, an explicit calculation: we seek the mean square displacement in the long-time limit for a random walker on an $L\times L$ square; the position of the walker for $N\gg L^2$ is uniformly …

The non-differentiability of the stochastic process is needed for the Markov assumption. If you relax the Markov assumption and allow for non-delta-function correlated noise, the process can be differ …

This random walk is known in the literature as the "geodesic random walk". For a manifold with positive curvature, theorems 1 and 4 of arXiv:1609.02901 prove that the uniform measure on the manifold i …

The region $Y$ introduces a delay time $\tau$, the average time between entry and exit. Let me first consider the case that the dynamics in the two-dimensional region $Y$ is a Brownian motion (diffusi …

An alternative to the Hurwitz decomposition of the Haar measure on $SO(n)$ has been developed by Julie Mitchell, "Sampling rotation groups by successive orthogonal images", SIAM J. Sci. Comput. 30, 52 …

source Both striped sites and black sites belong to the percolation cluster, but only the black sites are part of the backbone. The backbone can be defined as the set of current carrying paths from …

Example 2 of arXiv:0704.2826 considers the analogous problem for the continuous-time random walk, in the more general case that the curve has the form $g(t)=a+b\sqrt{T-t}$ with $a+b\sqrt T\geq 0$. The …

Q: Is there some application where $\mu$ plays a role in some kind of "connectedness" which would excuse the name? A: The application is to crystalline structure. The name originates from Hammersley, …

The probability $\mathbb P[\max_{t\leq \tau}|X_t|\geq a]$ is the probability to reach a point at a distance $a>0$ from the origin before returning to the origin, which is just $1/a$, see https://math. …

Smooth random functions, random ODEs, and Gaussian processes (2018) describes an approach that takes a finite Fourier series on the interval $(0,1)$ with randomly chosen coefficients. The integral of …

To carry out the limit, it helps to start from an integral representation of the Bessel function, $$P_n(T)=e^{-T}I_n(T)=\frac{1}{2\pi}\int_{-\pi}^\pi \exp [i k n+T \cos k-T]\,dk.$$ For $T\gg 1$ this m …