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There can never be a "first" zero of Brownian motion. If $B_{t_{0}}=a$, the $B_{t}$ takes the value $a$ an infinite number of times on every interval containing $t_{0}$.
Your first integral is the total momentum $P$ carried by the B-O field. The second integral, with two terms in the integrand, is almost the corresponding energy $E$ carried by the field, but the relative sizes of the two terms in the integrand are wrong for that See, e.g., my paper arXiv:1902.04643
