If you have the V2 version of the hub (The current model), it supports a very limited amount of local processing. The V1 version of the hub does not support local processing at all.
The only app that can run locally at present is the official smartlighting feature, and then only if that Smart lighting automation is using devices that have a device type handler which is eligible to run locally and not using any device type handlers which are not eligible to run locally. See the list at the top of this thread for the ones that are eligible to run locally.
Also, some of the smart home monitor features run locally, but not all of them.
Nothing else runs locally. No custom code, no custom device type handlers, no other Smartapps.
https://support.smartthings.com/hc/en-us/articles/209979766-Local-processing
If you set up a smart lighting automation that uses only devices using the device type handlers mentioned at the top of this thread, it should run locally. That’s it.
To keep from cluttering up this FAQ thread, I suggest taking any follow-up questions (other than the specifics of which device type handlers run locally) to the discussion topic for local processing: