Every valid ECDSA public key corresponds to some valid ECDSA private key. Bitcoin addresses are not actually public keys though, they are a 160-bit hash of a particular binary representation of the public key.
Are there valid Bitcoin addresses for which it's possible to mathematically prove that there is no private key?
As far as I know, the answer to that is no. To construct such a proof, you would have to somehow consider the enormous set of all possible public keys, and prove that none of them have a binary representation whose 160-bit hash would match the Bitcoin address in question. The point of hash functions is that they cannot be easily inverted, so you can't just get a list of all the valid inputs that would result in the same hash.
The number of ECDSA public keys (approximately 2^256) is vastly greater than the number of Bitcoin addresses (2^160). So the average Bitcoin address actually corresponds to about 2^(256-160) = 2^(96) different public keys. So it seems like it would be hard to find an address that corresponds to 0 public keys.