Gibbs-non-Gibbs dynamical transitions for mean-field interacting Brownian motions

We consider a system of real-valued spins interacting with each other through a mean-field Hamiltonian that depends on the empirical magnetisation of the spins. The system is subjected to a stochastic dynamics where the spins perform independent Brownian motions. Using large deviation theory we show that there exists an explicitly computable crossover time $t_c\in [0,\infty ]$ from Gibbs to non-Gibbs. We give examples of immediate loss of Gibbsianness ($t_c=0$), short-time conservation and large-time loss of Gibbsianness ($t_c\in (0,\infty )$), and preservation of Gibbsianness ($t_c=\infty$). Depending on the potential, the system can be Gibbs or non-Gibbs at the crossover time $t=t_c$.     

On purely discontinuous additive functionals of subordinate Brownian motions

Let $A_t={\sum }_{s\le t}F(X_{s?},X_s)$ be a purely discontinuous additive functional of a subordinate Brownian motion $X=(X_t,{\mathbb{P}}_x)$. We give a sufficient condition on the non-negative function $F$ that guarantees that finiteness of $A_{\infty }$ implies finiteness of its expectation. This result is then applied to study the relative entropy of ${\mathbb{P}}_x$ and the probability measure induced by a purely discontinuous Girsanov transform of the process $X$. We prove these results under the weak global scaling condition on the Laplace exponent of the underlying subordinator.     

Lp + Lq and Lp ∩ Lq are not isomorphic for all 1 ≤ p,q ≤ ∞, p ≠ q

We prove that if $1\le p,q\le \infty$, then the spaces $L_p+L_q$ and $L_p\cap L_q$ are isomorphic if and only if $p=q$. In particular, $L_2+L_{\infty }$ and $L_2\cap L_{\infty }$ are not isomorphic, which is an answer to a question formulated in [2].     

Some multicolor bipartite Ramsey numbers involving cycles and a small number of colors

For bipartite graphs $G_1,G_2,\dots ,G_k$, the bipartite Ramsey number $b(G_1,G_2,\dots ,G_k)$ is the least positive integer $b$ so that any coloring of the edges of $K_{b,b}$ with $k$ colors will result in a copy of $G_i$ in the $i$th color for some $i$. In this paper, our main focus will be to bound the following numbers: $b(C_{2t_1},C_{2t_2})$ and $b(C_{2t_1},C_{2t_2},C_{2t_3})$ for all $t_i\ge 3,$$b(C_{2t_1},C_{2t_2},C_{2t_3},C_{2t_4})$ for $3\le t_i\le 9,$ and $b(C_{2t_1},C_{2t_2},C_{2t_3},C_{2t_4},C_{2t_5})$ for $3\le t_i\le 5.$ Furthermore, we will also show that these mentioned bounds are generally better than the bounds obtained by using the best known Zarankiewicz-type result.