Mathematics Problem Archive

Let $X_t$ be $d$-dimensional Brownian motion starting at the origin, let $D$ be an open subset of $\mathbb{R}^d$ containing the origin, and let $\tau=...

Let $D\subset\mathbb{R}^d$, $d\ge2$, be bounded, connected, and open. Suppose there are coupled reflected Brownian motions $X_t,Y_t$ in $D$ and $\vare...

For the exclusion process on $\mathbb{Z}$ with $p(x,y)=p(y-x)$, assume $\sum_x|x|p(x)<\infty$, $\sum_xxp(x)>0$, and $\sum_{x<0}x^2p(x)=\infty$. Does t...

On $\mathbb{Z}^2$, take nearest-neighbor jump probabilities $p_1,q_1,p_2,q_2$ in directions $\pm e_1,\pm e_2$, with $p_1>q_1$ and $p_2>q_2$. If the an...

For the exclusion process on $\mathbb{Z}^d$ with translation-invariant kernel $p(x,y)=p(y-x)$ and zero mean $\sum_xxp(x)=0$, prove that every stationa...

For nearest-neighbor asymmetric exclusion on $\mathbb{Z}$ with $p(1)=p>q=p(-1)$, start from the deterministic configuration $\cdots11110000\cdots$. Is...

For Aldous's self-similar random partition of the plane, determine its topological properties: in particular, do region boundaries have fractal dimens...

Does every unimodular random countable locally finite rooted graph arise as a local weak limit of finite graphs?...

Let $S$ and $S'$ be random walks on $\mathbb{R}$ with the same i.i.d. step-length distribution, starting at $0$ and $x$. Suppose the step lengths are ...

Let $(X,\xi)$ consist of a random element and a diffuse random measure on a locally compact second countable Abelian group. Is mass-stationarity of $(...

Does the invariant-transport characterization of mass-stationarity remain valid if the bounded jointly invariant preserving kernels are restricted to ...

For every infinite vertex-transitive graph $G$, prove that $p_c(G)<p_u(G)$ if and only if $h(G)>0$....

Give general conditions implying $p_u(G)<1$. In particular, prove or disprove that every one-ended transitive graph has $p_u(G)<1$....

For nearest-neighbor independent bond percolation on $\mathbb{Z}^d$, $d\ge2$, let $p_c(d)$ be the critical edge-retention probability. Prove that at $...