Consistency of a family of probabilities in the construction of a pre-brownian motion

Solution 1:

Say $J = \{t_1,\dots,t_n\}$ and $J' = \{t_{k_1},\dots,t_{k_m}\}$. Fix $A_{k_1},\dots,A_{k_m} \in \mathcal{B}(\mathbb{R})$. Then, $$\pi_J(\pi_{J'}^{-1}(A_{k_1}\times \dots \times A_{k_m})) = \mathbb{R}\times\dots\times\mathbb{R}\times A_{k_1}\times\mathbb{R}\times\dots\times\mathbb{R}\times A_{k_2}\times \mathbb{R}\times\dots.$$ So, with $A := A_{k_1}\times\dots\times A_{k_m}$, we have $$P_J(\pi_J(\pi_{J'}^{-1}(A))) = \int \dots \int p(x_1,t_1)p(x_2-x_1,t_2-t_1)\dots p(x_n-x_{n-1},t_n-t_{n-1})dx_1\dots dx_n.$$ The integrals over $x_1,\dots,x_{k_1-1}$ are over $\mathbb{R}$, as are the integrals over $x_{k_1+1},\dots,x_{k_2-1}$, etc, while the integrals over $x_{k_i}$ are over $A_{k_i}$. So, thinking through the notation for a minute, it suffices to prove the following two general identities: $$ \int_{\mathbb{R}} p(t_1,x_1)p(t_2-t_1,x_2-x_1)dx_1 = p(t_2,x_2)$$ $$\int_{\mathbb{R}} p(t_2-t_1,x_2-x_1)p(t_2-t_2,x_3-x_2)dx_2 = p(t_3-t_1,x_3-x_1).$$ But these are just standard gaussian integrals which can be easily computed via standard formulas.