Future-path Markov property

Fix distinct times $0\le t_1<\cdots <t_k$ and a bounded Borel $G:{\mathbb{R}}^k\to \mathbb{R}$, and let $\mathrm{\Phi }$ be the cylinder functional $\mathrm{\Phi }(u)=G(u(t_1),\dots ,u(t_k))$. Then $Y:=(B_{s+t_i}-B_s{)}_{i\le k}$ is a random element of ${\mathbb{R}}^k$ independent of ${\mathcal{F}}_s^0$ with the law $\nu$ of $(W_{t_i}{)}_{i\le k}$ under Wiener measure: independence follows as in the finite-cylinder argument by deleting repetitions, expressing $(B_s,B_{s+t_1},\dots ,B_{s+t_k})$ through the independent increments of [F1] and applying [F2]; for any finite past times $r_j\le s$, include the $r_j$, $s$ and the $s+t_i$ in a common ordered grid. The increments before and after $s$ are independent, while $B_0=0$ almost surely causes no change in event probabilities. This proves independence from every finite past cylinder, and [F2] extends it to their generated sigma-algebra ${\mathcal{F}}_s^0$. The law identity holds because both $Y$ and $(W_{t_i})$ are obtained from independent $N(0,t_i-t_{i-1})$ increments (with $t_0=0$) by the same cumulative-sum map, the Wiener marginal being [F4].

With $\mathrm{\Phi }$ and $Y$ as in step 1.1 apply [F3] to $X=B_s$, the sigma-algebra ${\mathcal{F}}_s^0$, the noise $Y$, and $h(x,y):=G(x+y_1,\dots ,x+y_k)$: $H(x)=\int G(x+y_1,\dots ,x+y_k)\nu (dy)$ is Borel and $E[\mathrm{\Phi }((B_{s+t}{)}_{t\ge 0})\mathrm{\mid }{\mathcal{F}}_s^0]=H(B_s)$ almost surely. By step 1.1's law identity, $H(x)=\int \mathrm{\Phi }(x+w)\mu (dw)={\mathrm{\Psi }}_{\mathrm{\Phi }}(x)$ for every $x$, because $\mathrm{\Phi }(x+w)=G(x+w(t_1),\dots ,x+w(t_k))$ and the marginal of $(w(t_i))$ under $\mu$ is $\nu$.

Let $\mathcal{D}$ be the class of product-measurable sets $\mathrm{\Gamma }\subseteq {\mathbb{R}}^{[0,\mathrm{\infty })}$ such that $E[1_{\mathrm{\Gamma }}((B_{s+t}{)}_{t\ge 0})\mathrm{\mid }{\mathcal{F}}_s^0]={\mu }_{\mathrm{\Gamma }}(B_s)$ almost surely, where ${\mu }_{\mathrm{\Gamma }}(x):=\mu (x+W\in \mathrm{\Gamma })$. Each ${\mu }_{\mathrm{\Gamma }}$ is Borel by [F7] applied to the constant kernel $\mu$, since $(x,w)\mapsto x+w$ is product measurable. Finite cylinder sets lie in $\mathcal{D}$ by step 2.1, and $\mathcal{D}$ is a lambda-system: it contains the whole space because ${\mu }_{{\mathbb{R}}^{[0,\mathrm{\infty })}}\equiv 1$; it is closed under complements because ${\mu }_{{\mathrm{\Gamma }}^c}=1-{\mu }_{\mathrm{\Gamma }}$ and conditional expectations are additive on bounded inputs; and for disjoint sets in the class, countable additivity of the kernel and monotone convergence of the nonnegative finite sums give the event-integral identity for their union. Thus it is closed under disjoint countable unions, as required for a lambda-system.

The finite cylinder sets are a pi-system containing the whole space and generate the product sigma-algebra, so [F6] gives $\mathcal{D}=$ all product-measurable sets: for every product-measurable $\mathrm{\Gamma }$, $E[1_{\mathrm{\Gamma }}((B_{s+t}{)}_{t\ge 0})\mathrm{\mid }{\mathcal{F}}_s^0]={\mu }_{\mathrm{\Gamma }}(B_s)$ almost surely.

For bounded nonnegative $\mathrm{\Phi }$, choose simple functionals $s_n\uparrow \mathrm{\Phi }$ and use step 4.1 together with linearity of the integral to get ${\int }_A{s}_n((B_{s+t}))dP={\int }_A{\mathrm{\Psi }}_{s_n}(B_s)dP$ for every $A\in {\mathcal{F}}_s^0$; monotone convergence [F7] applied to both sides gives ${\int }_A\mathrm{\Phi }((B_{s+t}))dP={\int }_A{\mathrm{\Psi }}_{\mathrm{\Phi }}(B_s)dP$. Since ${\mathrm{\Psi }}_{\mathrm{\Phi }}(B_s)$ is bounded and ${\mathcal{F}}_s^0$-measurable, [F5] gives $E[\mathrm{\Phi }((B_{s+t}))\mathrm{\mid }{\mathcal{F}}_s^0]={\mathrm{\Psi }}_{\mathrm{\Phi }}(B_s)$ almost surely, and splitting a bounded real $\mathrm{\Phi }$ into positive and negative parts extends this to all bounded Borel $\mathrm{\Phi }$. This is the raw-filtration half of assertion 2.

We next prove the identity for the usual augmentation, first for a bounded continuous cylinder $\mathrm{\Phi }(u)=G(u(t_1),\dots ,u(t_k))$. Set $u_n=s+1\mathrm{/}n$. For $A\in {\mathcal{F}}_s\subseteq \bar{\mathcal{F}}{}_{u_n}^0$, the raw identity of step 5.1 at time $u_n$ extends from ${\mathcal{F}}_{u_n}^0$ to its completion by [F8], and gives ${\int }_A\mathrm{\Phi }((B_{u_n+t}{)}_{t\ge 0})dP={\int }_A{\mathrm{\Psi }}_{\mathrm{\Phi }}(B_{u_n})dP$. Brownian continuity makes the left integrand converge almost surely to $\mathrm{\Phi }((B_{s+t}{)}_{t\ge 0})$. Also $B_{u_n}\to B_s$ almost surely and ${\mathrm{\Psi }}_{\mathrm{\Phi }}$ is continuous, because bounded convergence under Wiener measure applies to $\mathrm{\Phi }(x_n+w)\to \mathrm{\Phi }(x+w)$ for a continuous cylinder. Dominated convergence therefore yields ${\int }_A\mathrm{\Phi }((B_{s+t}))dP={\int }_A{\mathrm{\Psi }}_{\mathrm{\Phi }}(B_s)dP$.

Approximate each half-line coordinate-cylinder indicator by decreasing bounded continuous cylinder functions using $g_{m,c}(x)=\max (0,1-m\max (x-c,0))$ and finite products of these functions; dominated convergence in step 6.1 gives the same event-integral identity for all half-line cylinders. The class of product-measurable sets for which that identity holds for every $A\in {\mathcal{F}}_s$ is a lambda-system by the same event-integral and disjoint-union calculation as step 3.1; half-line cylinders form a generating pi-system, so it is the whole product sigma-algebra by [F6]. Increasing simple approximation and monotone convergence then extend the identity to every bounded nonnegative Borel $\mathrm{\Phi }$, and positive-minus-negative decomposition to every bounded real $\mathrm{\Phi }$. Since ${\mathrm{\Psi }}_{\mathrm{\Phi }}(B_s)$ is bounded and ${\mathcal{F}}_s$-measurable by [F7], [F5] identifies it as $E[\mathrm{\Phi }((B_{s+t}))\mid {\mathcal{F}}_s]$. This completes assertion 2 without reversing the tower property.

For assertion 1, fix a finite cylinder $\mathrm{\Gamma }=\{z:z(t_1)\in B_1,\dots ,z(t_k)\in B_k\}$ and let ${\mathrm{\Phi }}_{\mathrm{\Gamma }}(u):=1_{\mathrm{\Gamma }}((u(t)-u(0){)}_{t\ge 0})$, a bounded Borel functional. For every $x$, ${\mathrm{\Phi }}_{\mathrm{\Gamma }}(x+w)=1_{\mathrm{\Gamma }}((w(t)-w(0){)}_{t\ge 0})$ is independent of $x$, so ${\mathrm{\Psi }}_{{\mathrm{\Phi }}_{\mathrm{\Gamma }}}(x)={\mu }_{\mathrm{c}\mathrm{y}\mathrm{l}}(\mathrm{\Gamma })$ is the constant $q_{\mathrm{\Gamma }}$; step 7.1 therefore gives $P(A\cap \{Z\in \mathrm{\Gamma }\})=q_{\mathrm{\Gamma }}P(A)$ for every $A\in {\mathcal{F}}_s$, and $q_{\mathrm{\Gamma }}=P(Z\in \mathrm{\Gamma })$. Hence $\sigma (Z)$ is independent of ${\mathcal{F}}_s$: the class of product-measurable $\mathrm{\Gamma }$ satisfying $P(A\cap \{Z\in \mathrm{\Gamma }\})=P(A)P(Z\in \mathrm{\Gamma })$ for all $A\in {\mathcal{F}}_s$ is a lambda-system containing the cylinder pi-system, so equals the product sigma-algebra by [F6]. Its finite-dimensional marginals are those of $\mu$ by step 1.1's law identity, so the law of $Z$ is the pushforward $j_{\ast }\mu$ on the product sigma-algebra; this is the assertion that the increment process is a Brownian motion independent of the past.

For the continuous-path formulation, every coordinate of $V^s$ is ambient-measurable and every value is in continuous path space, so [F4] makes $V^s$ a Borel random element. For a continuous-path cylinder functional its evaluation at $V^s$ agrees almost surely with the corresponding product cylinder evaluated on the original future, simultaneously in every time on $A_{\ast }$. Thus steps 2.1 and 6.1 give its raw and augmented event-integral identities. The translation map $(x,w)\mapsto x+w$ into continuous path space is measurable because all its coordinates are measurable and [F4] gives the target sigma-algebra. Use the half-line approximation of step 7.1 for the augmented identity, then apply the lambda-system calculation of step 3.1 to cylinder sets in continuous path space, then [F6] and the simple approximation of step 5.1, to obtain both identities for every bounded Borel $\varphi$. This argument asserts no adaptedness of the normalized coordinates: the right side is measurable with respect to the past because it is a Borel function of the original $B_s$. Two choices of $A_{\ast }$ give identical paths on their probability-one intersection, proving the asserted independence of the convention.

The degenerate and endpoint cases are covered and consistent: if $k=0$ or $\mathrm{\Phi }$ is constant, both sides equal that constant; if $t_1=0$ then $\mathrm{\Phi }(u)=G(u(0),u(t_2),\dots )$ and the coordinate $B_{s+0}-B_s=0$, so $Y$ has a degenerate first coordinate and $\nu$ charges it correspondingly, while $\mu$'s marginal at time $0$ is the point mass at $w(0)$ — the identity of step 1.1 remains valid because both laws are the same law of a vector with a deterministic coordinate; for $s=0$ one has $B_0=0$ almost surely, ${\mathrm{\Psi }}_{\mathrm{\Phi }}(B_0)={\mathrm{\Psi }}_{\mathrm{\Phi }}(0)$, and the past sigma-algebras ${\mathcal{F}}_0^0=\sigma (B_0)$ and ${\mathcal{F}}_0={\bigcap }_{u>0}\bar{\mathcal{F}}{}_u^0$ are handled by steps 5.1 and 7.1 unchanged. AC is inherited from the Brownian and Wiener interfaces and from [F8], including ambient completion and the conditional-expectation interface. The time sequence is explicit and no pathwise selection is made.