Vector Levy characterization

First establish true coordinate martingales, without intersecting localizers. Apply [F3] to each $M^i$, since $[M^i{]}_t=t$. For $A\in {\mathcal{F}}_s$, let ${\mu }_A^i(C)=P(A\cap \{M_t^i-M_s^i\in C\})$. This finite positive Borel measure has transform $P(A)e^{-{\theta }^2(t-s)\mathrm{/}2}$ by componentwise conditional event testing. Finite-measure Fourier uniqueness [F4] in dimension one identifies it with $P(A)N_1(0,t-s)$, including when $P(A)=0$, without normalization. With $A=\mathrm{\Omega }$, this proves integrability and zero mean of each increment. At $s=0$, $M_0^i=0$ almost surely gives integrability of $M_t^i$; $M_0^i$ is itself integrable. Integrating the identity function against ${\mu }_A^i=P(A)N_1(0,t-s)$ gives $E[1_A(M_t^i-M_s^i)]=0$. The pushforward integral identity here follows first for indicators from the definition of ${\mu }_A^i$, then for simple functions and nonnegative increasing limits, and finally for integrable signed functions. Thus every coordinate is a true all-pairs martingale by its defining event tests.

Fix $\lambda \in {\mathbb{R}}^d$ and put $X_t^{\lambda }={\sum }_i{\lambda }_i{M}_t^i$. It is an integrable adapted martingale by step 1.1 and [F1], hence a local martingale, and has continuous paths on the finite intersection of the coordinate continuity events. It starts at zero almost surely. For every deterministic partition its square sums are exactly ${\sum }_{i,j}{\lambda }_i{\lambda }_j$ times the respective cross sums. Their uniform error is bounded by the sum of the finitely many absolute coefficients times the corresponding uniform errors. The union bound therefore proves existence, not merely a formal use of bilinearity, of $[X^{\lambda }{]}_t=\mathrm{\mid }\lambda {\mathrm{\mid }}^{2}t$ along every permitted sequence.

For $\lambda \mathrm{\ne }0$, $N^{\lambda }=X^{\lambda }\mathrm{/}\mathrm{\mid }\lambda \mathrm{\mid }$ satisfies the hypotheses of [F3]. Apply its componentwise identity at frequency $\theta =\mathrm{\mid }\lambda \mathrm{\mid }$. This gives $E[e^{i\lambda \cdot (M_t-M_s)}\mid {\mathcal{F}}_s]=e^{-(t-s)\mathrm{\mid }\lambda {\mathrm{\mid }}^2\mathrm{/}2}$. For $\lambda =0$ both sides are $1$. No common exceptional set for all frequencies is needed: each fixed frequency identity gives a numerical equality of event integrals.

Conditional law of the vector increment: for each $A\in {\mathcal{F}}_s$ define the finite Borel measure $Q_A(\mathrm{\Gamma }):=E[1_A{1}_{\{M_t-M_s\in \mathrm{\Gamma }\}}]$ on ${\mathbb{R}}^d$; its Fourier transform is $\int e^{i\lambda \cdot x}{Q}_A(dx)=E[1_A{e}^{i\lambda \cdot (M_t-M_s)}]=P(A)e^{-(t-s)\mathrm{\mid }\lambda {\mathrm{\mid }}^2\mathrm{/}2}$ by step 3.1 and [F5], which is $P(A)$ times the Fourier transform of $N_d(0,(t-s)I_d)$ by [F4]. By multivariate Fourier uniqueness [F4], $Q_A=P(A)N_d(0,(t-s)I_d)$ for every $A\in {\mathcal{F}}_s$; in particular, with $A=\mathrm{\Omega }$, the increment has law $N_d(0,(t-s)I_d)$, and with general $A$ the identity is exactly the independence of the increment from ${\mathcal{F}}_s$.

Finite lists of increments: for $0=t_0<t_1<\cdots <t_n$ the increments $M_{t_j}-M_{t_{j-1}}$ are independent with laws $N_d(0,(t_j-t_{j-1})I_d)$. Induction on $n$: the case $n=1$ is step 4.1; given the claim for $n$ increments, the increment at $t_{n+1}$ has conditional law $N_d(0,(t_{n+1}-t_n)I_d)$ given ${\mathcal{F}}_{t_n}$ and is independent of ${\mathcal{F}}_{t_n}$ by step 4.1 applied with $s=t_n$, hence independent of the sigma-algebra generated by the earlier increments, and [F5] multiplies the joint law.

Conclusion and boundary cases: $M$ is adapted, continuous, starts at $0$ almost surely, and its finite-dimensional increment laws are those of a standard $d$-dimensional Brownian motion by step 5.1; this is precisely the defining increment condition, so $M$ is a standard $d$-dimensional Brownian motion with the stated filtration property. For $d=1$ the same Fourier event-test argument gives the scalar characterization; for $\lambda =0$ the linear combination is the zero process and the identity is trivial; the coordinate increments at $s=t$ are zero with law $N(0,0)$; the hypothesis ${\delta }_{ij}t$ excludes degenerate covariance matrices, and no independence of the coordinates is assumed in the proof — the Brownian definition derives it from the verified vector increment laws; and AC has the uses declared in [F6].