One way communication with forward secrecy?
Is there a known protocol that can do this?


It's easy to see that forward security is impossible with the constraints you have.

Suppose an attacker breaks into B to create a B' with the exact same state.

Then, from now on, when A sends a message to B, the attacker takes that message and also submits it to B'.

B and B' are essentially identical; if B can understand the message, so can B'. 

This remains true even if B can generate some local entropy (e.g. has an entropy source whose output the attacker cannot predict); as B cannot send its entropy back to A, it has to understand the message from A no matter what the entropy is (and hence B' can as well, even if its entropy sample is different).