Thermonuclear Explosions and Accretion-induced Collapses of White Dwarfs in Active Galactic Nucleus Accretion Disks

White dwarfs (WDs) embedded in the gaseous disks of active galactic nuclei (AGNs) can rapidly accrete materials from these disks and grow in mass to reach, or even exceed, the Chandrasekhar limit. Binary WD (BWD) mergers are also believed to occur in AGN accretion disks. We study observational signatures from these events. We suggest that mass-accreting WDs and BWD mergers in AGN disks can lead to thermonuclear explosions that drive an ejecta shock breakout from the disk surface and power a slow-rising, relatively dim Type Ia supernova (SN). It is possible that such SNe Ia may be outshone by the emission of the AGN disk around a supermassive black hole (BH) with a mass of M (SMBH) greater than or similar to 10(8) M (circle dot). In addition, accretion-induced collapses (AICs) of WDs in AGN disks may sometimes occur, which may form highly magnetized millisecond neutron stars (NSs). The subsequent spindown process of this nascent magnetar can deposit its rotational energy into the disk materials, resulting in a magnetar-driven shock breakout and a luminous magnetar-powered transient. We show that such an AIC event could power a rapidly evolving and luminous transient for a magnetic field of B similar to 10(15) G. The rising time and peak luminosity of the transient, powered by a magnetar with B similar to 10(14) G, are predicted to have similar properties to those of superluminous SNe. AIC events taking place in the inner parts of disks around relatively less massive supermassive BHs (M (SMBH) less than or similar to 10(8) M (circle dot)) are more likely to power transients that are much brighter than the AGN disk emission, and hence easily identified.