Additive Lévy Processes

Additive Lévy Processes Introduction Let X X N denote N independent Lévy processes on. We can construct an N-parameter stochastic process, indexed by N +, as follows: := X + + X N N for every := ( N ) N + We might also write := X X N And in this way, it follows that if N are independent additive Lévy processes, then N is an additive Lévy process as well, notation being more or less obvious. It is not hard to convince yourself that if Ψ Ψ N denote the respective Lévy exponents of X X N, then Ee ξ = e Ψ(ξ) for all N + and ξ where Ψ(ξ) := Ψ (ξ) Ψ N (ξ) And that Ψ determines uniquely the finite-ensional distributions of. Definition. The N-parameter stochastic process is called the additive Lévy process corresponding to X X N. The function Ψ is called the Lévy exponent of. I mention a simple example of additive Lévy process. Exercise 2 shows us how to create other types of additive Lévy processes from independent Lévy processes. 9

92 4. Additive Lévy Processes Example 2. Let X X N denote N independent -ensional Brownian motions. Then the N-parameter Gaussian process is called additive Brownian motion. More generally, if X X N are independent isotropic stable processes with the same index α (0 2], then is an additive stable process with index α. Note that Ψ(ξ) ξ α ( ) Let us define ( )() := E( + ) ( )() := N + e N = ( )() d [We could just as easily define λ for λ>0, or even λ N +, but there is no pressing need for doing this here.] You should check the following; it states that there are natural, and easy-to-understand, analogues of semigroups and resolvents in the present N-parameter setting. Lemma 3. If P P N denote the respective semigroups of X X N, then = P π() π() Pπ(N) π(n) for every permutation (π() π(n)) of ( ). And if N respectively denote the -resolvents of X X N, then = π() π(n). And, not surprisingly, we have also potential measures: Definition 4. The -potential measure of is defined as (A) := E e N = l A ( ) d for all A ( ). N + Lemma 5. If U UN denote the respective -potential measures of X X N, then = U UN. An addition theorem Theorem 6 (Khoshnevisan and Xiao, 2009; Khoshnevisan et al., 2003; Yang, 2007). Choose and fix a Borel set G +. N Then, E (+) N G > 0 if and only if there exists a Borel probability measure ρ on G such that = N +Ψ (ξ) ˆρ(ξ) 2 dξ< () I will prove the sufficiency of (); that is the easier half of Theorem 6;. You can find the details of the [much] more difficult half in Khoshnevisan and Xiao (2009).

An addition theorem 93 Define for all probability densities : and, (J)() := e N = ( + ) d N + Then, a careful computation, using Theorem 5 (page 86), reveals the following multiparameter analogue of Lemma 6 (page 88): Lemma 7. For all measurable probability densities : +, E (J)() d = E (J)() 2 d = (2π) N = +Ψ (ξ) ˆ(ξ) 2 dξ Proof of half of Theorem 6. If (J)() > 0 for some probability density, then certainly + X has hit the support of at some time. That is, P supp() X( + ) P {(J)() > 0} In particular, E supp() X( + ) P {(J)() > 0} d Lemma 7 and the Paley Zygmund inequality (page 89) together imply that E supp() X( + ) N (2π) ˆ(ξ) 2 dξ +Ψ (ξ) = where / := 0. Now we approximate G by the support of a probability density of the form := ρ, where ρ M (G) and is a bounded probability density with support in B(0 ). Since ˆ(ξ) ˆρ(ξ), the preceding shows that if there exists a probability measure ρ on G that satisfies (), then E X( + ) G = E G X( + ) > 0. Definition 8. We say that is absolutely continuous if (A) = A υ() d for some measurable υ. The function υ is called the -potential density of. It is not hard to see that υ can always be chosen to be a probability density. Moreover, if U U N have -potential densities N respectively, then υ = N. The following is proved similarly to Theorem 0 (page 82).

94 4. Additive Lévy Processes Theorem 9. Suppose is absolutely continuous with a -potential density υ such that υ(0) > 0. Then, for all G ( ), P{( N +) G = } > 0 if and only if there exists a probability measure ρ on G that satisfies (). Example 0. Let be an additive stable process on with N parameters and index α (0 2]. Then, P (+) N ˆρ(ξ) 2 G = > 0 iff +ξ Nα dξ < for some ρ M (G) When α =2, this says something about additive Brownian motion. A connection to Hausdorff ension Definition. The Hausdorff ension G of a Borel set G is defined as G := inf (0 ): ˆρ(ξ) 2 ρ M (G) such that dξ < +ξ [The preceding is well defined, provided that we set inf :=.] [This is not the usual definition, rather the consequence of a famous theorem called Frostman s theorem of classical potential theory.] In particular, Example 0 tells us the following. Proposition 2. If is an M-parameter additive stable process with index α (0 2], then for all G ( ): () If G> Mα, then P{( M + ) G = } > 0; whereas (2) If G< Mα, then P{( M + ) G = } =0. Now let be an N-parameter additive Lévy process on with Lévy exponent Ψ := (Ψ Ψ N ), independent from. Then, := is an (N + M)-parameter additive Lévy process on. It follows from Theorem 9 that P 0 (+ N+M ) > 0 iff N = +Ψ (ξ) dξ < +ξnα But 0 is in the closure of the range of if and only if the closures of the ranges of and intersect! Therefore, Proposition 2 implies the following: Theorem 3 (Khoshnevisan et al., 2003; Yang, 2007). With probability one, N ( +) N dξ = sup (0 ): +Ψ (ξ) ξ < =

An application to subordinators 95 where sup := 0. [Why can we replace ( + ξ ) by ξ +?] It is not hard to see that if C is at most countable, then (G C) = G for all G ( ). Therefore, one can use the fact that the X s are cadlag [hence have denumerably-many jumps] to prove that the closure sign of the preceding theorem can be removed. An application to subordinators Let us now apply Theorem 3 to the case where X := T is a subordinator with a Lévy exponent Φ [and Lévy exponent Ψ, still]. Theorem 4 (Horowitz, 968). With probability one, T( + ) = sup (0 ) : + Φ(λ) where sup := 0. 0 dλ λ < The following is a convenient method by which we can transform a Lévy exponent to a Laplace exponent. Proposition 5. For every λ>0, + Φ(λ) = πλ + Ψ() d +(/λ) 2 Proof. Define {C λ } λ>0 to be an independent linear Cauchy process, normalized so that E exp(c λ ) = exp( λ ) for and λ>0. By independence, e Φ(λ) = Ee λt = Ee C λt = Ee Ψ(C λ) But the probability density of C λ is () =(πλ) ( + (/λ) 2 ). Therefore, e Φ(λ) = πλ e Ψ() +(/λ) 2 d Multiply both sides by exp( ) and integrate [d] to finish. Proof of Theorem 4. Here is how we can apply Proposition 5. First, note that if 0 <θ<2 and, then 0 dλ λ +θ +(/λ) 2 θ

96 4. Additive Lévy Processes Therefore, if we multiply the equation in Proposition 5 by λ θ and integrate [dλ], then we obtain dλ 0 + Φ(λ) λ θ d + Ψ() θ This and Theorem 3 together prove the theorem. Let us conclude this section by applying Horowitz s theorem to the set of increase times of linear Brownian motion. Proposition 6 (Lévy XXX). If B denotes standard Brownian motion, then 0: B = sup B = a.s. [0] 2 Proof. Define T := inf { >0: B = } for all >0 Then, T is a stable subordinator with index α =/2; see Exercise 2 [page 5]. And Horowitz s theorem [ with Φ(λ) λ] tells us that the Hausdorff ension of the range of T is a.s. /2. On the other hand, it is a realvariable fact that T( + )= 0: B = sup B [0] (Check!) Therefore, the proposition follows. There is a theorem of Lévy which implies that { 0: B = sup [0] B} has the same law as the zero set { 0: B =0} of Brownian motion. Therefore, the preceding implies that the Hausdorff ension of the zero set of B is almost surely /2. ather than study this particular problem in greater depth, we study the zero set of a more general Lévy process in the next lecture. Problems for Chapter 4. Let {P } denote the two-sided semigroup of a two-sided Lévy process. () Is it true that P + = P P for all? [In other words, is {P } a semigroup of linear operators?] (2) Define linear operators P := P P for all. Prove that { P } is a semigroup of linear operators. 2. Let X and X 2 denote two independent Lévy processes on with respective exponents Ψ and Ψ 2.

Problems for Chapter 4 97 () Verify that := X Y defines a 2-parameter additive Lévy process on ; compute its Lévy exponent. (2) Verify that := (X Y ) defines a 2-parameter additive Lévy process on ( ) 2 ; compute its Lévy exponent. 3. Derive Lemma 7. 4. Let X denote an isotropic stable process on with index α (0 2]. Compute (X( + )). Indicate the changes to your formula if X( + ) is replaced by ( + ), where is an additive stable process on with index α (0 2] and N parameters. Or more generally still if := X X N, where X s are independent symmetric stable processes on the line with respective indices α α (0 2]. 5. Compute the Hausdorff ension of the range of Y := (B ), where B denotes -ensional Brownian motion. The range of Y is called the graph of Brownian motion. Indicate the changes made to your formula if we replace Brownian motion by isotropic stable process on with index α (0 2].