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By Girsanov's thoerem and using the existing log-Harnack inequality fordistribution independent SDEs, the log-Harnack inequality is derived forpath-distribution dependent stochastic Hamiltonian systems. As an application,the exponential ergodicity in relative entropy is obtained by combining withtransportation cost inequality. In addition, the quantitative propagation ofchaos in the sense of Wasserstein distance, which together with the coupling bychange of measure implies the quantitative propagation of chaos in totalvariation norm as well as relative entropy are obtained.

Exponential Ergodicity and Propagation of Chaos for Path-Distribution Dependent Stochastic Hamiltonian System