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Adrian is Walter's father. Walter tells the men in the Crown smoking room (and us) about his dad when he's explaining how he ended up in New Zealand.
It seems that, back in the motherland, Adrian had abandoned his wife and fled. When Walter finally found his father in New Zealand, he was shocked and appalled to realize the full extent of father's bad behavior (which now included bigamy) and scooted out of town almost immediately. That's how he ended up in Hokitika.
However, Frank Carver (who of course knew Adrian as well) ended up tipping Adrian off to his son's whereabouts, and so Adrian shows up at the end of the novel to beg forgiveness from his son.
He writes Walter a note that seems pretty genuine and contrite (unfortunately, it doesn't seem that the note reaches Walter, since he was leaving for a dig that day). After explaining some details that Walter didn't know that shed a slightly more positive light on his actions, he says:
I can do no more than to beg for your forgiveness, as I must also beg for Frederick's, and to promise that our next reunion, should you grant me one, will be conducted 'dry.' Good fortune, Walter. Know that I have confronted my true soul, and that I write this as a sober man. (IV.8.25)
Well, we'll just hope that's genuine, and that Walter goes to see his father again when he returns to Hokitika. But, maddeningly, we never get to find out. We smell a sequel.