Show that (p_{t_{1}, ldots, t_{n}}^{x}left(C_{1} times cdots times C_{n} ight)) of Remark 7.6 define measures on (mathscr{B}left(mathbb{R}^{d

Show that \(p_{t_{1}, \ldots, t_{n}}^{x}\left(C_{1} \times \cdots \times C_{n}\right)\) of Remark 7.6 define measures on \(\mathscr{B}\left(\mathbb{R}^{d \cdot n}\right)\).

Data From 7.6 Remark

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7.6 Remark (From semigroups to processes). So far we have considered semigroups given by a Brownian motion. It is an interesting question whether we can construct a Markov process starting from a semigroup of operators. If (T)to is a Markov semigroup where pt(x, C) := T+1c(x) is an everywhere defined kernel such that Cpt(x, C) is a probability measure for all x = Rd, (t, x)p(x, C) is measurable for all C B(Rd), this is indeed possible. We can use the formula from Theorem 7.5 to define for every Xe Rd P(Cxx Cn) := T [1c Tt-t [1c.. Tt-tn-11C]...](x),