6 Brownian motion as a Markov process

We have seen in 2.13 that for a d-dimensional Brownian motion (Bt)t≥0 and any s > 0 the shifted process Wt := Bt+s - Bs, t ≥ 0, is again a BMd which is independent of (Bt)0≤t≤s. Since Bt+s = Wt + Bs, we can interpret this as a renewal property: Rather than going from 0 = B0 straight away to x = Bt+s in (t + s) units of time, we stop after time s at x′ = Bs and move, using a further Brownian motion Wt for t units of time, from x’ to x = Bt+s. This situation is shown in Figure 6.1:

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Fig. 6.1. Wt := Bt+s - Bs, t ≥ 0, is a Brownian motion in the new coordinate system with origin (s, Bs).

6.1 The Markov property ...

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