Surprised this is the top thread and unchallenged.
Buddhism arrived in China via the Greek Bactrian kingdom
{{citation-needed}}. Buddhism arrived in the geographic region currently encompassed by the modern Chinese state via multiple routes over more than a thousand years including innumerable exchanges through the overall Tarim Basin (Xinjiang) or "silk road" area via modern Tajikistan, Afghanistan and Pakistan but also via Tibet and by sea through Southeast Asia (Myanmar, Cambodia, Laos, Vietnam, and probably some sporadic oceanic contact with South Asia and Buddhist kingdoms in Indonesia). Tajikistan in particular is only slightly archaeologically investigated to date owing to its remote location, poor infrastructure, geopolitical challenges and IIRC the use of mines during 20th century military conflicts. In short, like any other set of ideas it probably arrived in waves of migration, monkly pilgrimages, exchanges, and instances of official diplomacy. There are records of Buddhism being preached to China by the geographically proximal Yuezhi people as early as 2BCE. Classical Chinese history teaches that Buddhism "came to China" in 67CE. Hellenistic artistic styles are, to my knowledge, never mentioned.
Buddha as depicted in East Asia is wearing Greek robes
The robes are not Greek, they are Buddhist. Buddhism was established in the sixth century BCE. We can read that in Buddhist tradition monks are only allowed a robe and a begging bowl, and that the historical Buddha also set forth a tradition that discouraged iconography as improper worship of likeness (idolatry). This may be because Buddha composed his philosophy in reaction to the prevailing religious environment, where he had previously partaken in such habits which were and remain deeply culturally entrenched.
In short, Greco-Buddhist art is influential because it is, in a sense, the first major period with any voluminous production of "Buddhist art" (a notion decidedly not Buddhist), and in any event, the Hellenistic style robes primarily only affects Mahayana art, and the modern geographical scope of China also encompasses an endemic Vajrayana philosophy, Theravada, and Tibetan. Even within Mahayana artistic tradition in China, I don't see how a cursory inspection of - for example - robes depicted at https://en.wikipedia.org/wiki/Chinese_Buddhist_sculpture or https://commons.wikimedia.org/wiki/Category:Buddhist_sculptu... could conclude they're all Greek-influenced. Such a claim is tenuous at best.
there are elements of Greek philosophy in Buddhism as practiced in East Asia.
I have never heard anyone make this claim. If we flip the claim, as Buddhist jataka tales are known to Christianity, the reverse could also be said: "elements of Indian philosophy in Christianity as practiced in Europe". Such a statement, outside of tangential academic arguments, is broadly untrue.
My apologies for the mistakes in my statement. You seen to know much, much about this topic than me; thank you for adding the nuances this complex topic. My comment was based on my recollection of what I saw at the San Francisco Museum of Asian Arts and some of the descriptions I read there about the various pieces. In all likelihood, my recollection was either incorrect and/or incomplete. I cited the Wikipedia article because it seems to reflect similar ideas as my recollection and also in case anyone else found the topic as fascinating as I did.
I'm past the window for editing my original comment but if the mods don't mind, it would be great to qualify my original comment with the response.
No worries happy to share. FYI I spent most of my 20s and early 30s traveling with an interest in this topic so have visited most museums out there (including SF) as well as many historic sites (including the white horse temple, most major Buddhist sculptural sites in China, remote border areas, various ancient sites in SEA, etc.). It is a true shame that the historic depth of cultural exchange that evidently occurred in these regions is seemingly being systemically oppressed from the popular and academic record due to modern-day geopolitical concerns.
Buddhism arrived in China via the Greek Bactrian kingdom
{{citation-needed}}. Buddhism arrived in the geographic region currently encompassed by the modern Chinese state via multiple routes over more than a thousand years including innumerable exchanges through the overall Tarim Basin (Xinjiang) or "silk road" area via modern Tajikistan, Afghanistan and Pakistan but also via Tibet and by sea through Southeast Asia (Myanmar, Cambodia, Laos, Vietnam, and probably some sporadic oceanic contact with South Asia and Buddhist kingdoms in Indonesia). Tajikistan in particular is only slightly archaeologically investigated to date owing to its remote location, poor infrastructure, geopolitical challenges and IIRC the use of mines during 20th century military conflicts. In short, like any other set of ideas it probably arrived in waves of migration, monkly pilgrimages, exchanges, and instances of official diplomacy. There are records of Buddhism being preached to China by the geographically proximal Yuezhi people as early as 2BCE. Classical Chinese history teaches that Buddhism "came to China" in 67CE. Hellenistic artistic styles are, to my knowledge, never mentioned.
Buddha as depicted in East Asia is wearing Greek robes
The robes are not Greek, they are Buddhist. Buddhism was established in the sixth century BCE. We can read that in Buddhist tradition monks are only allowed a robe and a begging bowl, and that the historical Buddha also set forth a tradition that discouraged iconography as improper worship of likeness (idolatry). This may be because Buddha composed his philosophy in reaction to the prevailing religious environment, where he had previously partaken in such habits which were and remain deeply culturally entrenched.
In short, Greco-Buddhist art is influential because it is, in a sense, the first major period with any voluminous production of "Buddhist art" (a notion decidedly not Buddhist), and in any event, the Hellenistic style robes primarily only affects Mahayana art, and the modern geographical scope of China also encompasses an endemic Vajrayana philosophy, Theravada, and Tibetan. Even within Mahayana artistic tradition in China, I don't see how a cursory inspection of - for example - robes depicted at https://en.wikipedia.org/wiki/Chinese_Buddhist_sculpture or https://commons.wikimedia.org/wiki/Category:Buddhist_sculptu... could conclude they're all Greek-influenced. Such a claim is tenuous at best.
there are elements of Greek philosophy in Buddhism as practiced in East Asia.
I have never heard anyone make this claim. If we flip the claim, as Buddhist jataka tales are known to Christianity, the reverse could also be said: "elements of Indian philosophy in Christianity as practiced in Europe". Such a statement, outside of tangential academic arguments, is broadly untrue.