On motion after the death of a person who commenced an action or proceeding, the court shall allow a pending action or proceeding that does not abate to be continued by the decedent’s personal representative or, if none, by the decedent’s successor in interest.
Source:
Section 377.31, https://leginfo.legislature.ca.gov/faces/codes_displaySection.xhtml?lawCode=CCP§ionNum=377.31..