By Ciprian Tudor

Self-similar approaches are stochastic techniques which are invariant in distribution lower than compatible time scaling, and are a subject matter intensively studied within the previous couple of many years. This publication provides the elemental houses of those strategies and makes a speciality of the examine in their edition utilizing stochastic research. whereas self-similar techniques, and particularly fractional Brownian movement, were mentioned in different books, a few new sessions have lately emerged within the medical literature. a few of them are extensions of fractional Brownian movement (bifractional Brownian movement, subtractional Brownian movement, Hermite processes), whereas others are options to the partial differential equations pushed through fractional noises.

In this monograph the writer discusses the fundamental houses of those new sessions of self-similar procedures and their interrelationship. even as a brand new method (based on stochastic calculus, particularly Malliavin calculus) to learning the habit of the differences of self-similar approaches has been built over the past decade. This paintings surveys those fresh strategies and findings on restrict theorems and Malliavin calculus.

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The term B can easily be estimated. Indeed, by the change of variables u˜ = u v , v˜ = t−s , s − u, v˜ = v − s and then u˜ = t−s B = c(H )(t − s)2H − 2 . 22) Let us now consider the term A. By the change of variables u˜ = s − u, v˜ = v − s u v , v˜ = t−s we have and then u˜ = t−s s A= 0 s dudv|u − v|2H −2 (2t − 2s + u + v)− 2 d 0 − 2(t − s + u + v)− 2 + (u + v)− 2 d s t−s = (t − s)2H − 2 d s t−s 0 − (1 + u + v) 0 − d2 − (1 + u + v) Note that the integral − d2 (u + v) d + (u + v)− 2 d ∞ d dudv|u − v|2H −2 (2 + u + v)− 2 0 − d2 ≤ (t − s)2H − 2 d ∞ dudv|u − v|2H −2 (2 + u + v)− 2 d 0 + (u + v)− 2 .

This proves the upper bound. Next, we establish the lower bound. Suppose that t|ξ |2 ≤ 1. For any u ∈ [0, t], u|ξ |2 t|ξ |2 1 −x ≥ 1 − x for all x > 0, we conclude that: 2 ≤ 2 ≤ 2 . Using the fact that e exp − u|ξ |2 2 2 t ≥1− u|ξ |2 1 ≥ , 2 2 ∀u ∈ [0, t]. Hence Bt (ξ ) ≥ αH 1 2 t 0 0 1 1 1 |u − v|2H −2 dudv = t 2H ≥ t 2H 4 4 1 + |ξ |2 For the last inequality, we used the fact that 1 ≥ 2H . 1 . 1+|ξ |2 Suppose that t|ξ |2 ≥ 1. Using the change of variables u = u|ξ |2 /2, v = v|ξ |2 /2, we obtain: Bt (ξ ) = αH 22H |ξ |4H t|ξ |2 /2 0 t|ξ |2 /2 e−u e−v u − v 2H −2 du dv .

We will assume from now on that the spatial covariance of the noise W is given by the Riesz kernel. 1 and assume that the measure μ is dμ(ξ ) = |ξ |−d+β dξ with β ∈ (0, d). In this case the kernel f is given by f (ξ ) = cβ,d |ξ |−β with β ∈ (0, d). If f is as above, then t∧s Eu(t, x)u(s, x) = (2π)−d du 0 × Rd dξ sin((t − u)|ξ |) sin((s − u)|ξ |) −d+β dξ.