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We present Feynman–Kac path integrals representations for scalar wave motions on variable medium. The main new points on them is about their rigorous mathematical validity on the space of continuous functions vanishing at infinity, besides of possessing intrinsically the physical property of causal wave field propagation, thus solving mathematically the long standing problem on the subject of automatically leading to causality wave propagation.

A note on Feynman–Kac path integral representations for scalar wave motions