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I am looking to work out an analytical solution (if it is known) for the autocovariance $Cov[X_s,X_t]$ of a particle which behaves according to the Langevin equation for a Harmonic Oscillator in a ...

We consider a harmonic function $u:\mathbb{R}^d \to \mathbb{R}$ $(\Delta u=0).$ Suppose that $$\exists \alpha,\beta \in \mathbb{R},\forall x\in \mathbb{R}^d,u(x)\leq \alpha |x|+\beta.$$ Therefore $u-u(...

I have a question which relates to an argument appearing in this paper 1. Let $D$ be a domain of $\mathbb{R}^d$ and $X=(X_t, P_x)$ be a diffusion process on $D$. Let $h : D \to [0,\infty]$ be a ...

I have a question about harmonic functions and hitting probabilities. Let $d \ge 3$ be an integer. Let $D=\mathbb{R}^d \times (-1,1)$ and denote points $z \in D$ by $z=(r,\theta,y),$ where $(r,\...

A harmonic function on a graph is a function on its vertices such that the value at every vertex is the average of the values at its neighbors. Consider the following method for approximating a ...