6 problems

Let $D$ be the unit circle, let $P_N$ be the distribution of the discrete process on $N$ equally spaced points, and let $V:D\to\mathbb{R}$ be $C^3$. Define $W(x)=V(e^{ix})$. Let…

Let $\mathcal{C}$ be the circle of radius $r>0$ centered at $0$, let $T_y$ be the projection of $\mathbb{R}^2$ onto the tangent line to $\mathcal{C}$ at $y$, and let $P_N$ be the d…

Let $D\subset\mathbb{R}^n$ be a domain, let $V:D\to\mathbb{R}$ be $C^2$, and let $P_K$ be the distribution of the associated semi-discrete process. Let $(Y,S)$ solve … where $B$ is…

Let $D\subset\mathbb{R}^n$ be a domain, let $V:D\to\mathbb{R}$ be a $C^2$ potential, and let $P_K$ be the distribution of the semi-discrete process $(X,L)$ constructed from the lat…

Let $D\subset\mathbb{R}^n$ be a bounded connected domain, let $D_K$ be its connected square-lattice approximation with mesh $1/K$, and let $P_K$ be the distribution of the resultin…

Convergence conjecture. As $N\to\infty$, $P_N$ should converge to the distribution of reflected Brownian motion in $[0,1]$ with inert drift.