Do landlords who live in California have a specific time limit to midigate damages?

Mitigation of damages with respect to landlords would normally concern lessees who have moved out before the lease is up. The have a duty to try to find a new tenant, if they are going to sue the old tenant for the amount due over the life of the lease. Is this what you are asking about?

If so, are you asking about the statute of limitations for suing the tenant for lost rent?