A Brownian-Motion Approach to the Second Main Theorem for Meromorphic Mappings and Hypersurfaces with Truncated Counting Functions

By using Brownian motion and stochastic calculus, we establish a second main theorem for holomorphic curves into a projective subvariety $V\subset {\mathbb{P}}^n(\mathbb{C})$ with an arbitrary family $\mathcal{Q}$ of $q$ hypersurfaces $Q_1,\dots ,Q_q$ concerning its distributive constant ${\mathrm{\Delta }}_{\mathcal{Q},V}$. In our result, the counting functions are truncated to level $H_V(d)-1$, where $d=lcd(\mathrm{°}{Q}_1,\dots ,\mathrm{°}{Q}_d)$ and $H_V(d)$ is the Hilbert function of $V$. As an application of the second main theorem, we give a uniqueness theorem for holomorphic curves from $\mathbb{C}$ into $V$ sharing an arbitrary family of hypersurfaces regardless of multiplicity.