Brownian Motion, 2nd Edition

12 The growth of Brownian paths

In this chapter we study the question how fast a Brownian path grows as t → ∞. Since for a BM¹ (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 ≥ t₀ (є, ω). Lévy’s modulus of continuity, Theorem 10.6, gives a better bound. For all ∈ > 0 and almost ...