= Solution
{c}
The increment $B_t-B_s$ is independent of $\mathcal F_s$, with <characteristic function> $e^{-\xi^2(t-s)/2}$. Since $B_s$ is <measurable> at time $s$,
$$
\mathbb E(e^{i\xi B_t}\mid\mathcal F_s)
=e^{i\xi B_s}e^{-\xi^2(t-s)/2}.
$$
The <heat kernel> identity in the hint gives $(P_uf)(x)=e^{i\xi x}e^{-\xi^2u/2}$, so this is exactly $(P_{t-s}f)(B_s)$. Multiplying by the deterministic factor $e^{\xi^2t/2}$ gives
$$
\boxed{\mathbb E(M_t\mid\mathcal F_s)=M_s.}
$$
The process is <adapted>, and $|M_t|=e^{\xi^2t/2}$ is finite at each fixed time. Thus it is a complex <martingale>, meaning that its real and imaginary parts are integrable <martingales>.
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