How a specialized court to handle claims arising from asbestos and other injurious dusts has operated since 1989 in Australia's largest state, New South Wales (official LawLink site and FAQ; 2007 regulation, Law Society site; Wikipedia page). Note that the tribunal hears civil claims and is an entirely separate entity from what is known as the Dust Diseases Board, which compensates industrial dust ailments on a no-fault workers' comp model. In the United States, by contrast, neither the courts nor the workers' comp system have developed specialized branches devoted to this type of claim.

P.S. Ted writes to point out, "Depends on your definition of 'specialized.' There's no formal 'Asbestos Court,' but most jurisdictions (including the federal courts) consolidate their asbestos dockets and place them under a single judge -- Madison County notably stopped being a magnet jurisdiction for asbestos cases when they switched who the asbestos judge was."