In this chapter we study the question how fast a Brownian path grows as t → ∞. Since for a BM^{1} (B(t))_{t≥0}, the process (tB(t^{-1}))_{t>0}, is again a Brownian motion, we get from the growth at infinity automatically local results for small t and vice versa. A first estimate follows from the strong law of large numbers. Write for no ≥ 1

and observe that the increments of a Brownian motion are stationary and independent random variables. By the strong law of large numbers we get that B(t, ω) ≤ єt for any ∈ > 0 and all t ≥ t_{0} (є, ω). Lévy’s modulus of continuity, Theorem 10.6, gives a better bound. For all ∈ > 0 and almost ...

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