Hölder estimates for magnetic Schrödinger semigroups in $${\mathbb {R}}^{d}$$ from mirror coupling

We use the mirror coupling of Brownian motion to show that under a $\beta\in (0,1)$-dependent Kato type assumption (which is satisfied suitable $L^q$-assumption on electro-magnetic potential, where $q$ depends $\beta$ and dimension $d$) possibly nonsmooth corresponding magnetic Schr\"odinger semigroup in $\mathbb{R}$ has global $L^{p}$-to-$C^{0,\beta}$ H\"older smoothing property for all $p\in [1,\infty]$, particular eigenfunctions are uniformly $\beta$-H\"older continuous. This result shows Hamilton operator molecule field continuous weak $L^q$-assumptions potential.