The best replacement for asics is nike.

The best alternatives to asics are rated as following:
★★★★★  nike
★★★★★  new balance
★★★★★  brooks
★★★★  saucony
★★★  mizuno
★★  adidas
★  reebok
★  new balance running shoes
★  fpga
★  brooks running shoes
★  puma
★  nike size
★  mizuno running shoes
★  brooks sizing
★  nike running shoes
★  nike free
★  new balance running
★  mizuno volleyball shoes
  new balance size
  new balance sizing
  saucony sizing
  free runs
  mizuno running
  brooks running shoe
  mizuno sizing
  nike for running
  nike running
  avia
  new balance for running
  newton
  under armour
  sketcher
  ryka
  fpgas
  saucony shoes
  saucony running shoes

Your alternative for asics is not listed? Suggest!
Speaking about asics, one probably shouldn't forget about nike+.

Other search terms related to asics and their ratings are:
★★★★★ nike+
You have in mind a companion for asics and it is not yet here? Propose!



Google Trends

Scale is based on the average worldwide traffic of asics in all years. Click on graph for more.
asics
nike
new balance