If there is a relationship...it's probably the other way around. The Ainu/Jomon have a lineage that's traceable back at least 13,000 years, the Greeks don't.
Do you consider the Ainu Caucasian?
You take for granted that ainu are of jomon descedance. That is not proved. So we can't be sure of when the Ainu people appear in Japan. The fact that Ainu language relates to no other languages of the same area is bizzare! On the other hand Greeks seem to have been capable of making long journeys. Eratosthenes' world map (250BC) contains Britain and Ireland, labrys and potery of minoan origin are exposed in Bahamas museum, greek linguistic influence can be found in the Caribbean and Easter Island (Nors Josephson), extended use of 'meandros' schemes can be seen at the 'palacios de las Grecas' in Mexico, natives in Indonesia during a sacred ceremony paint their faces white, their hair red and wear crowns with leaves in order to honor the gods who came from the west and taught them agriculture technics and wine production, the names Indonesia and Polynesia are greek and were used by the natives long before the europeans arrived in 15th century. Not to mention the Greco-Budhism. Ainu don't seem to have had a naval technology similar to the greek one, so it must have been difficult for them to colonise Greece. If we look closer we can see that Ainu give unique names to their childs just like ancient Greeks did. The houses have a fire in the centre, remember the greek "estia". They offer wine to their gods, "spondi". They are taller than the asians, their hair are curly and dence and grow on their body too. They sing epic stories, they use intense geometrical schemes on their clothes and drawings.
I am not sure about Ainu's anchestry but it is my opinion that a long trip by some adventurus Greeks who never came back should be added in the current scenarios.