DAVAR KHOSHNEVISAN AND YIMIN XIAO

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1 HARMOIC AALYSIS OF ADDITIVE LÉVY PROCESSES DAVAR KHOSHEVISA AD YIMI XIAO Abstract. Let X 1,..., X denote independent d-dimensional Lévy processes, and consider the -parameter random field Xt := X 1 t X t. First we demonstrate that for all nonrandom Borel sets F, the Minkowski sum XR + F, of the range XR + of X with F, can have positive d-dimensional Lebesgue measure if and only if a certain capacity of F is positive. This improves our earlier joint effort with Yuquan Zhong 003 by removing a symmetry-type condition there. Moreover, we show that under mild regularity conditions, our necessary and sufficient condition can be recast in terms of one-potential densities. This rests on developing results in classical [non-probabilistic] harmonic analysis that might be of independent interest. As was shown in Khoshnevisan, Xiao, and Zhong 003, the potential theory of the type studied here has a large number of consequences in the theory of Lévy processes. We present a few new consequences here. 1. Introduction 1.1. Background. It is known that, for all integers d, the range of d-dimensional Brownian motion has zero Lebesgue measure. See Lévy 1940 for d =, Ville 194 for d = 3, and Kakutani 1944a for the remaining assertions. There is a perhaps betterknown, but equivalent, formulation of this theorem: When d, the range of d-dimensional Brownian motion does not hit points. Kakutani 1944b has generalized this by proving that, for all integers d 1, the range of d-dimensional Brownian motion can hit a nonrandom Borel set F if and only if capf > 0, where cap denotes, temporarily, the logarithmic capacity if d = and the Riesz capacity of index d if d 3; the case d = 1 is elementary. [Actually, Kakutani s paper discusses only the planar case. pointed out in Dvoretzky, Erdős, and Kakutani 1950.] The theorem, for d 3, is Kakutani s theorem is the starting point of a deep probabilistic potential theory initiated by Hunt 1958, 1957a, 1957b, The literature on this topic is rich and quite large; Date: June 4, Mathematics Subject Classification. 60G60, 60J55, 60J45. Key words and phrases. Additive Lévy processes, multiplicative Lévy processes, capacity, intersections of regenerative sets. Research supported in part by a grant from the ational Science Foundation. 1

2 KHOSHEVISA AD XIAO see, for example, the books by Blumenthal and Getoor 1968, Doob 001, Fukushima, Ōshima, and Takeda 1994, Getoor 1990, and Röckner 1993, together with their combined bibliography. One of the central assertions of probabilistic potential theory is that a nice Markov process will hit a nonrandom measurable set F if and only if capf > 0, where cap is a certain natural capacity in the sense of G. Choquet Dellacherie and Meyer, 1978, Chapter III, pp Moreover, that capacity is defined solely, and fairly explicitly, in terms of the Markov process itself. There are interesting examples where F is itself random. For instance, suppose X is d- dimensional standard Brownian motion, and F = Y 0, is the range minus the starting point of an independent standard Brownian motion Y on. In this particular case, it is well known that 1.1 P {Xs = Y t for some s, t > 0} > 0 if and only if d 3. This result was proved by Lévy 1940 for d =, Kakutani 1944a for d 5, and Dvoretzky, Erdős, and Kakutani 1950 for d = 3, 4. Peres 1996b, 1996a and Khoshnevisan 003 contain different elementary proofs of this fact. There are many generalizations of 1.1 in the literature. For example, Dvoretzky, Erdős, and Kakutani 1954 proved that the paths of an arbitrary number of independent planar Brownian motions can intersect. While LeGall 1987 proved that the trajectories of a planar Brownian motion can intersect itself countably many times. And Dvoretzky, Erdős, Kakutani, and Taylor 1957 showed that three independent Brownian-motion trajectories in can intersect if and only if d. For other results along these lines, see Hawkes 1978b, 1977, 1976/77, Hendricks 1979, 1973/74, Kahane 1985, Chapter 16, Section 6, Lawler 1989, 1985, 198, Pemantle, Peres, and Shapiro 1996, Le Gall 199, Peres 1999, 1996b, 1996a, Rogers 1989, Tongring 1988, and their combined bibliography. For a long time, a good deal of effort was concentrated on generalizing 1.1 to other concrete Markov processes than Brownian motion. But the problem of deciding when the paths of independent general but nice Markov processes can intersect remained elusive. It was finally settled, more than three decades later, by Fitzsimmons and Salisbury 1989, whose approach was to consider the said problem as one about a certain multiparameter Markov process. To be concrete, let us consider the case =, and let X := {Xt} t 0 and Y := {Y t} t 0 denote two independent nice Markov processes on a nice state space S. The starting point of the work of Fitzsimmons and Salisbury is the observation that P{Xs = Y t for some s, t > 0} > 0 if and only if the two-parameter Markov process X Y hits the

3 HARMOIC AALYSIS OF ADDITIVE LÉVY PROCESSES 3 diagonal diag S := {x x : x S} of S S, where Xs 1. X Y s, t := Y t for all s, t 0. In the special case that X and Y are Lévy processes, the Fitzsimmons Salisbury theory was used to solve the then-long-standing Hendricks Taylor conjecture Hendricks and Taylor, The said connection to multiparameter processes is of paramount importance in the Fitzsimmons Salisbury theory, and appears earlier in the works of Evans 1987a, 1987b. See also Le Gall, Rosen, and Shieh 1989 and Salisbury 1996, 199, Walsh 1986, pp discusses a connection between X Y and the Dirichlet problem for the bilaplacian on the bi-disc of. The Fitzsimmons Salisbury theory was refined and generalized in different directions by Hirsch 1995, Hirsch and Song 1999, 1996, 1995a, 1995b, 1995c, 1995d, 1994, and Khoshnevisan 00, See also Ren 1990 who derives an implicit-function theorem in classical Wiener space by studying a very closely-related problem. The two-parameter process X Y itself was introduced earlier in the works of Wolpert 1978, who used X Y to build a φ κ model of the Euclidean field theory. This too initiated a very large body of works. For some of the earlier examples, see the works by Aizenman 1985, Albeverio and Zhou 1996, Dynkin 1987, 1986, 1985, 1984a, 1984b, 1984c, 1983a, 1983b, 1981, 1980, Felder and Fröhlich 1985, Rosen 1983, 1984, and Westwater 1980, 1981, 198. [This is by no means an exhaustive list.] In the case that X and Y are Lévy processes on [i.e., have stationary independent increments], X Y is an example of the so-called additive Lévy processes. But as it turns out, it is important to maintain a broader perspective and consider more than two Lévy processes. With this in mind, let X 1,..., X denote independent Lévy processes on such that each X j is normalized via the Lévy Khintchine formula Bertoin, 1996; Sato, 1999: 1.3 E exp iξ X j t = exp tψ j ξ for all t 0 and ξ. The function Ψ j is called the characteristic exponent or Lévy exponent of X j, and its defining property is that Ψ j is a negative-definite function Schoenberg, 1938; Berg and Forst, The main results. The main object of this paper is to develop some basic probabilistic potential theory for the following -parameter random field X with values in : 1.4 Xt := X 1 t X t for all t := t 1,..., t R +.

4 4 KHOSHEVISA AD XIAO On a few occasions we might write X j t in place of Xt, as well. The random field X is a so-called additive Lévy process, and is characterized by its multiparameter Lévy Khintchine formula: 1.5 E exp iξ Xt = exp t Ψξ for all t R + and ξ ; where Ψξ := Ψ 1 ξ,..., Ψ ξ is the characteristic exponent of X. Our goal is to describe the potential-theoretic properties of X solely in terms of its characteristic exponent Ψ. Thus, it is likely that our harmonic-analytic viewpoint can be extended to study the potential theory of more general multiparameter Markov processes that are based on the Feller processes of Jacob 005, 00, 001. In order to describe our main results let us first consider the kernel K Ψ ξ := Re for all ξ. 1 + Ψ j ξ When = 1, this kernel plays a central role in the works of Orey 1967 and Kesten The kernel for general was introduced first by Evans 1987b; see also Khoshnevisan, Xiao, and Zhong 003. Based on the kernel K Ψ, we define, for all Schwartz distributions µ on, 1.7 I Ψ µ := 1 ˆµξ K π d Ψ ξ dξ. We are primarily interested in the case where µ is a real-valued locally integrable function, or a σ-finite Borel measure on. In either case, we refer to I Ψ µ as the energy of µ. Our notion of energy corresponds to a capacity cap Ψ, which is the following set function: For all Borel sets F, [ cap Ψ F := inf I Ψ µ], µ P cf where P c F denotes the collection of all compactly-supported Borel probability measures on F, inf :=, and 1/ := 0. The following is the first central result of this paper. Here and throughout, λ k denotes k-dimensional Lebesgue measure on R k for all integers k 1. Theorem 1.1. Let X be an -parameter additive Lévy process on with exponent Ψ. Then, for all Borel sets F, 1.9 E [ λ d XR + F ] > 0 if and only if cap Ψ F > 0.

5 Remark 1.. HARMOIC AALYSIS OF ADDITIVE LÉVY PROCESSES 5 1 Theorem 1.1 in the one-parameter setting is still very interesting, but much easier to derive. See Kesten 1969 for the case that F := {0} and Hawkes 1984 for general F. For a scholarly pedagogic account see Bertoin 1996, p. 60. One can view Theorem 1.1 as a contribution to the theory of Dirichlet forms for a class of infinite-dimensional Lévy processes. These Lévy processes are in general non-symmetric. Röckner 1993 describes a general theory of Dirichlet forms for nice infinite-dimensional Markov processes that are not necessarily symmetric. It would be interesting to know if the processes of the present paper lend themselves to the analysis of the general theory of Dirichlet forms. We have no conjectures along these lines. Our earlier collaborative effort with Yuquan Zhong 003 yielded the conclusion of Theorem 1.1 under an exogenous technical condition on X 1,..., X. A first aim of this paper is to establish the fact that Theorem 1.1 holds in complete generality. Also, we showed in our earlier works Khoshnevisan and Xiao, 005; 004, Khoshnevisan, Xiao, and Zhong ; 003 that such a theorem has a large number of consequences, many of them in the classical theory of Lévy processes itself. ext we describe a few such consequences that are nontrivial due to their intimate connections to harmonic analysis. Our next result provides a criterion for a Borel set F to contain intersection points of independent Lévy processes. It completes and complements the well-known results of Fitzsimmons and Salisbury See also Corollary 9.3 and Remark 9.4 below. Theorem 1.3. Let X 1,..., X be independent Lévy processes on, and assume that each X j has a one-potential density u j : R + such that u j 0 > 0. Then, for all nonempty Borel sets F, 1.10 P {X 1 t 1 = = X t F for some t 1,..., t > 0} > 0 if and only if there exists a compact-support Borel probability measure µ on F such that 1.11 ˆµ ξ ξ 1 Re dξ 1 dξ <. R 1 + Ψ d j ξ j Suppose, in addition, that every u j is continuous on, and finite on \ {0}. Then, another equivalent condition is that there exists a compact-support probability measure µ on F such that 1.1 uj x y + u j y x µdx µdy <.

6 6 KHOSHEVISA AD XIAO In order to describe our next contribution, let us recall that the one-potential measure U of a Lévy process X := {Xt} t 0 on is defined as 1.13 UA := 0 P {Xt A} e t dt, for all Borel sets A. ext we offer a two-parameter additive variant which requires fewer technical conditions than Theorem 1.3. Theorem 1.4. Suppose X 1 and X are independent Lévy processes on with respective one-potential measures U 1 and U. Suppose U 1 dx = u 1 x dx, where u 1 : R +, and u 1 U > 0 almost everywhere. Then, for all Borel sets F, 1.14 P {X 1 t 1 + X t F for some t 1, t > 0} > 0 if and only if there exists a compact-support Borel probability measure µ on F such that 1.15 ˆµξ 1 1 Re Re dξ <. R 1 + Ψ d 1 ξ 1 + Ψ ξ Suppose, in addition, that u 1 is continuous on, and finite on \ {0}. Then, 1.14 holds if and only if there exists a compact-support probability measure µ on F such that 1.16 Qx y µdx µdy <, where 1.17 Qx := for all x. [ ] u1 x + y + u 1 x y + u 1 x + y + u 1 x y U dy 4 Among other things, Theorem 1.4 confirms a conjecture of Bertoin 1999b and Bertoin 1999a, p. 49; see Remark 8.1 for details. Finally we mention a result on the Hausdorff dimension of the set of intersections of the sample paths of Lévy processes. Theorem 1.5. Let X 1,..., X be independent Lévy processes on, and assume that each X j has a one-potential density u j : R + such that u j 0 > 0. Then, almost surely on { k=1 X k R + }, 1.18 dim H k=1 X k R + { = sup s 0, d : Re Ψ j ξ j } dξ 1 + ξ ξ <, d s

7 HARMOIC AALYSIS OF ADDITIVE LÉVY PROCESSES 7 where sup := 0. Suppose, in addition, that the u j s are continuous on, and finite on \ {0}. Then, almost surely on { k=1 X kr + }, { uj z + u j z 1.19 dim H X k R + = sup s 0, d : k=1 } dz z <. s In the remainder of the paper we prove Theorem 1.1 and its stated corollaries in the order in which they are presented. Finally, we conclude by two zero-one laws for the Lebesgue measure and capacity of the range of an additive Lévy process, that, we believe, might have independent interest. We end this section with four problems and conjectures. Open problem 1. Throughout this paper, we impose continuity conditions on various onepotential densities. This is mainly because we are able to develop general harmonic-analytic results only for kernels that satisfy some regularity properties. Can the continuity conditions be dropped? We believe the answer is yes. This is motivated, in part, by the following fact, which follows from inspecting the proofs: The condition u is continuous on and finite on \ {0} is used only for proving the if portions in the second parts of Theorems 1.3 and 1.4. Open problem. Jacob 005; 00; 001 has constructed a very large class of Feller processes that behave locally like Lévy processes. Moreover, his construction is deeply connected to harmonic analysis. Because the results of the present paper involve mainly the local structure of Lévy processes, and are inextricably harmonic analytic, we ask: Is it possible to study the harmonic-analytic potential theory of several Jacob processes by somehow extending the methods of the present paper? Open problem 3. We ask: Is there a useful theory of excessive functions and/or measures for additive Lévy processes or more general multiparameter Markov processes? This question is intimately connected to Open Problem 1, but deserves to be asked on its own. In the one-parameter case, the answer is a decisive yes Getoor, But the one-dimensional theory does not appear to readily have a suitable extension to the multiparameter setting. Open problem 4. We conjecture that, under the conditions of Theorem 1.5, the following holds almost surely on { k=1 X kr + } : { 1.0 dim H X k R + = sup s 0, d : k=1 1,1 d uj z + u j z } dz z <. s [The difference between this and 1.19 is in the range of the integrals.] But in all but one case we have no proof; see Remark 9.6 below for the mentioned case. As we shall see in that

8 8 KHOSHEVISA AD XIAO remark, what we actually prove is the following harmonic-analytic fact: Suppose u is the one-potential density of a Lévy process, u0 > 0, u is continuous on, and u is finite on \ {0}. Then the local square-integrability of u implies the [global] square-integrability of u. We believe that the following more general result holds: If u 1,..., u are one-potential densities that share the stated properties for u, then uj + u j 1.1 L 1 loc If this is so, then the results of this paper imply Conjecture 1.0. uj + u j L 1.. The stationary additive Lévy random field Consider a classical Lévy process X := {Xt} t 0 on with characteristic exponent Ψ. Let us introduce an independent copy X of X, and extend the definition of X to a process indexed by R as follows: Xt if t 0,.1 Xt := X t if t < 0. This is the two-sided Lévy process with exponent Ψ in the sense that X := { Xt} t R has stationary and independent increments. Moreover, { Xt + s Xs} t 0 is a copy of X for all s R. We also define X := { Xt} t R all of R, whose values are in and are defined as as the corresponding -parameter process, indexed by. Xt := X1 t X t for all t := t 1,..., t R. We are assuming, of course, that X 1,..., X are independent two-sided extensions of the processes X 1,..., X, respectively. We intend to prove the following two-sided version of Theorem 1.1. Theorem.1. For all Borel sets F, [ ].3 E λ d XR F > 0 if and only if cap Ψ F > 0. This implies Theorem 1.1 effortlessly. Indeed, we know already from Remark 1. of Khoshnevisan, Xiao, and Zhong 003 that.4 cap Ψ F > 0 = E [ λ d XR + F ] > 0. Thus, we seek only to derive the converse implication. But that follows from Theorem.1, because XR + XR.

9 HARMOIC AALYSIS OF ADDITIVE LÉVY PROCESSES 9 Henceforth, we assume that the underlying probability space is the collection of all paths ω : R that have the form ωt = ω jt j for all t R, where each ω j maps R to such that ω j 0 = 0; and ω j D R, the Skorohod space of cadlag functions from R not [0, to. We can then assume that the stationary additive Lévy fields, described earlier in this section, are in canonical form. That is, Xtω := ωt for all t R and ω Ω. Because we are interested only in distributional results, this is a harmless assumption. Define P x to be the law of x + X, and E x the expectation operation with respect to P x, for every x. Thus, we are identify P with P 0, and E with E 0. We are interested primarily in the σ-finite measure.5 P λd := P x dx, and the corresponding expectation operator E λd, defined by.6 E λd f := fω P λd dω for all f L 1 P λd. Ω If A B := {a b : a A, b B}, then by the Fubini-Tonelli theorem, [ ] [ ] E λ d XR F = E 1 exr F x dx R d.7 = P x { Xt F for some t R } dx } = P λd { XR F. Thus, Theorem.1 is a potential-theoretic characterization of all polar sets for X under the σ-finite measure P λd. With this viewpoint in mind, we proceed to introduce some of the fundamental objects that are related to the process X. Define, for all t R, the linear operator P t as follows: ].8 P t fx := E x [f Xt for all x. This is well-defined, for example, if f : R + is Borel-measurable, or if f : R is Borel-measurable and P t f is finite at x. Also define the linear operator R by.9 Rfx := 1 P t fxe [t] dt for all x, R where.10 [t] := t t

10 10 KHOSHEVISA AD XIAO denotes the l 1 -norm of t R. [We will use this notation throughout.] The l -norm of t R will be denoted by t. Again, Rfx is well defined if f : R + is Borel-measurable, or if f : R is Borel-measurable and R f is finite at x. Our next result is a basic regularity lemma for R. It should be recognized as a multiparameter version of a very well-known property of Markovian semigroups and their resolvents. Lemma.. Each P t and R are contractions on L p, as long as 1 p. Proof. Choose and fix j {1,..., } and t R, and define µ j,t to be the distribution of the random variable X j t. If f : R + is Borel-measurable, then.11 P t f = f µ 1,t1 µ,t, where denotes convolution. This implies readily that P t is a contraction on L p for all p [1, ], and hence,.1 Rf L p 1 R P t f L p e [t] dt. Since P t is a contraction on L p, the preceding is bounded above by f L p. Henceforth, let denote the [Schwartz] Fourier transform on any and every Euclidean space R k. Our Fourier transform is normalized such that.13 ˆfξ := e R iξ x fx dx, k for all f L 1 R k, say. We then have the following. Lemma.3. If f L 1, then ˆR = K Ψ. That is, Rfξ = K Ψ ξ ˆfξ for all ξ. Proof. Recall µ j,t from the proof of Lemma.. Its Fourier transform is given by.14 µ j,t ξ = exp { t Ψ j sgntξ} for all ξ. Equations.11 and.14, and the Plancherel theorem together imply that.15 Pt fξ = ˆfξ exp t j Ψ j sgnt j ξ.

11 Consequently,.16 HARMOIC AALYSIS OF ADDITIVE LÉVY PROCESSES 11 Rfξ = 1 ˆfξ exp R = 1 ˆfξ 0 t j [ 1 + Ψ j sgnt j ξ ] dt e t[1+ψ j ξ] + e t[1+ψ j ξ] dt. A direct computation reveals that Rf = K Ψ ˆf, as asserted. The following is a functional-analytic consequence. Corollary.4. The operator R maps L into L, and is self-adjoint. Proof. By Lemma.3, if f L 1 L and g L, then.17 Rfxgx dx = 1 K R π d d Ψ ξ ˆfξ ĝξ dξ. Thanks to Lemma., the preceding holds for all f L. Duality then implies that R : L L. Moreover, since K Ψ is real, R is self-adjoint. The following lemma shows that for every t R, the distribution of Xt under P λ is λ d. This is the reason why we call X a stationary additive Lévy process. Lemma.5. If f : R + is Borel-measurable, then ].18 E λd [f Xt = P t fx dx = fy dy for all t R. Proof. We apply the Fubini-Tonelli theorem to find that ].19 E λd [f Xt = E x + Xt dx. A change of variables [y := x + Xt] proves that the preceding expression is equal to the integral of f. This implies half of the lemma. Another application of the Fubini Tonelli theorem implies the remaining half as well. f Let us choose and fix π {1,..., } and identify π with the partial order π, on R, which is defined as follows: For all s, t R, s i t i for all i π, and.0 s π t iff s i > t i for all i π. The collection of all π {1,..., } forms a collective total order on R in the sense that.1 for all s, t R there exists π {1,..., } such that s π t.

12 1 KHOSHEVISA AD XIAO For all π {1,..., }, we define the π-history of the random field X as the collection }. H π t := σ { Xs for all t R, s πt where σ denotes the σ-algebra generated by whatever is in the parentheses. Without loss of generality, we assume that each H π t is complete with respect to P x for all x ; else, we replace it with the said completion. Also, we assume without loss of generality that t H π t is π-right-continuous. More precisely, we assume that.3 H π t = H π s for all t R and π {1,..., }. s R : t πs If not, then we replace the left-hand side by the right-hand side everywhere. Proposition.6 A Markov-random-field property. Suppose π {1,..., } and s π t, both in R. Then, for all measurable functions f : R +, ].4 E λd [f Xt Hπ s = P t s f Xs P λd -a.s. Proof. Choose and fix Borel measurable functions g, φ 1,..., φ m : R +, and parameter time points s 1,..., s m R such that.5 s j π s π t for all j = 1,..., m. According to the Fubini-Tonelli theorem,.6 E λd [f Xt g Xs m = = φ j Xs ] j E E [ f x + Xt g x + Xs m φ j x + Xs ] j dx [ fa + y ] m φ j A j + y gy dy, where A := Xt Xs and A j := Xs j Xs for all j = 1,..., m. The independent-increments property of each of the Lévy processes X j implies that A is independent of {A j } m. This and the stationary-increments property of X 1,..., X together

13 imply that.7 E λd [f HARMOIC AALYSIS OF ADDITIVE LÉVY PROCESSES 13 Xt g Xs m φ j Xs ] j [ m ] = E [fa + y] E φ j A j + y gy dy. After a change of variables and an appeal to the stationary-independent property of the increments of X 1,..., X and X 1,..., X, we arrive at the following:.8 E λd [f Xt g Xs m φ j Xs ] j = E [ f [ ] m ] Xt s + y E y φ j A j gy dy [ m ] = P t s fye y φ j A j gy dy. By the monotone class theorem, for all nonnegative H π s-measurable random variables Y, ].9 E λd [f Xt g Xs Y = P t s fy ψy gy dy, where ψ : R + is a measurable function. This proves the proposition. Lemma.7. If f, g L and t, s R, then ].30 E λd [f Xs g Xt = fyp t s gy dy. Proof. We may consider, without loss of generality, measurable and nonnegative functions f, g L. Let π denote the collection of all i {1,..., } such that s i t i. Then s π t, and Proposition.6 implies that P λd -a.s., ].31 E λd [f Xt Hπ s = P t s f Xs. This and Lemma.5 together conclude the proof. 3. The sojourn operator Recall.10, and consider the sojourn operator, 3.1 Sf := 1 Xt e [t] dt. R f

14 14 KHOSHEVISA AD XIAO Our first lemma records the fact that S maps density functions to mean-one random variables [P λd ]. Lemma 3.1. If f is a probability density function on, then E λd [Sf] = 1. This follows readily from Lemma.5. Our next result shows that, under a mild condition on Ψ j, S embeds functions in L, quasi-isometrically, into the subcollection of all functions in L that have finite energy. amely, Proposition 3.. If f L, then 3. Sf L P λd I Ψ f. Suppose, in addition, that there exists a constant c 0,, such that the following sector condition holds for all j = 1,..., : 3.3 Im Ψ j ξ c 1 + Re Ψ j ξ for all ξ. Then, there exists a constant A 0, 1 such that 3.4 A I Ψ f Sf L P λd I Ψ f. Remark 3.3 Generalized Sobolev spaces. When = 1 and the Lévy process in question is symmetric, the following problem arises in the theory of Dirichlet forms: For what f in the class D, of Schwartz distributions on, can we define Sf as an element of L P λd say? This problem continues to make sense in the more general context of additive Lévy processes. And the answer is given by 3.4 in Proposition 3. as follows: Assume that the sector condition 3.3 holds for all j = 1,..., and some c 0,. Let S Ψ denote the completion of the collection of all members of L that have finite energy I Ψ, where the completion is made in the norm f Ψ := I 1/ Ψ f + f L R. Then, there d exists an a.s.-unique maximal extension S of S such that S : S Ψ SS Ψ is a quasi-isometry. The space S Ψ generalizes further some of the ψ-bessel potential spaces of Farkas, Jacob, and Schilling 001 and Farkas and Leopold 006; see also Jacob and Schilling 005, Masja and agel 1978, and Slobodeckiĭ The proof requires a technical lemma, which we develop first. Lemma 3.4. For all z C define 3.5 Λz := e t s t s σz; t s dt ds,

15 HARMOIC AALYSIS OF ADDITIVE LÉVY PROCESSES 15 where σz; r := z if r 0 and σz; r := z otherwise. Then for all z C with Re z 0, Λz 4 Re. 1 + z If, in addition, Im z c1 + Re z for some c 0,, then 3.7 Λz c 1 Re. 1 + z Proof. The double integral is computed by dividing the region of integration into four natural parts: i s, t 0; ii s, t 0; iii t 0 s; and iv s 0 t. Direct computation reveals that for all z C with Re z Similarly, one can compute Consequently, e t s t s σz; t s dt ds + e t s t s σz; t s dt ds Λz = Re e t s t s σz; t s dt ds e t s t s σz; t s dt ds Re z Im z, 1 + z 1 + z 4 for all z C with Re z 0. It follows that [ ] Λz Re 1 + Re 1 + z 1 + z = 1 = Re. 1 + z z z. for all z C with Re z 0. Whenever Re z 0, we have 0 Re1 + z 1 1, and hence 3.6 follows from On the other hand, if Im z c1 + Re z, then 3.10 yields Λz Re + 1 c [ ] 1 Re, 1 + z 1 + z from which the result follows readily, because 0 Re[1 + z 1 ] 1 when Re z 0. Proof of Proposition 3.. We apply Lemma.7 to deduce that 3.13 E λd Sf = 1 e [t] [s] fyp 4 t s fy dt ds dy. R R

16 16 KHOSHEVISA AD XIAO In accord with.15 and Parseval s identity, for all u R, fyp u fy dy = 1 ˆfξ Pu fξ dξ R π d d 3.14 = 1 π ˆfξ exp u d j Ψ j sgnu j ξ dξ. This and the Fubini-Tonelli theorem together reveal that 3.15 E λd Sf 1 = 4 π ˆfξ d ΛΨ j ξ dξ. Since Re Ψ j ξ 0 for all ξ and j = 1,...,, we apply Lemma 3.4 to this formula, and conclude the proof of the proposition. 4. Proof of Theorem.1 Thanks to the definition of cap Ψ, and to the countable additivity of P, it suffices to consider only the case that 4.1 F is a compact set. This condition is tacitly assumed throughout this section. We note, in particular, that P c F denotes merely the collection of all Borel probability measures that are supported on F. Proposition 5.7 of Khoshnevisan, Xiao, and Zhong 003 proves that for every compact set F, 4. cap Ψ F > 0 = E [ λ d X[0, r] F ] > 0 for all r > 0. It is clear the latter implies that E [ λ d XR F ] > 0, and one obtains half of the theorem. Since cap Ψ F = cap Ψ F, we may and will replace F by F throughout. In light of.7, we then assume that } 4.3 P λd { XR F > 0, and seek to deduce the existence of a probability measure µ on F such that I Ψ µ <. We will prove a little more. amely, that for all k > 0 there exists a constant A = Ak, 0, such that 4.4 P λd { X } [ k, k] F A cap Ψ F.

17 HARMOIC AALYSIS OF ADDITIVE LÉVY PROCESSES 17 In fact, we will prove that for all sufficiently large k > 0, 4.5 P λd { X } [ k, k] F e k 4 cap Ψ F. This would conclude our proof of the second half of the theorem. It is a standard fact that there exists a probability density function φ 1 in C with the following properties: P1. φ 1 x = 0 if x > 1; P. φ 1 x = φ 1 x for all x ; P3. ˆφ 1 x 0 for all x. This can be obtained, for example, readily from Plancherel s [duality] theorem of Fourier analysis. We recall also the following standard fact: φ 1 L 1 and P3 together imply that ˆφ 1 L 1 Hawkes, 1984, Lemma 1. ow we define an approximation to the identity {φ ɛ } ɛ>0 by setting 4.6 φ ɛ x := 1 x ɛ φ d 1 for all x and ɛ > 0. ɛ It follows readily from this that for every µ P c F : i µ φ ɛ is a uniformly continuous probability density; ii µ φ ɛ is supported on the closed ɛ-enlargement of F, which we denote by F ɛ, for all ɛ > 0; iii φ ɛ is symmetric; and iv lim ɛ 0 + ˆφɛ ξ = 1 for all ξ. As was done in Khoshnevisan, Xiao, and Zhong 003, we can find a random variable T with values in R { } such that: 1 {T = } is equal to the event that Xt F for all t R ; XT F on {T }. This can be accomplished pathwise. Consequently, 4.3 is equivalent to the condition that 4.7 P λd {T } > 0. Define for all Borel sets A and all integers k 1, 4.8 µ k A := P λd XT A T [ k, k].

18 18 KHOSHEVISA AD XIAO We claim that µ k is a probability measure on F for all k sufficiently large. In order to prove this claim we choose and fix l > 0, and consider } P λd { XT A, T [ k, k], X0 l 4.9 µ k,l A := } P λd {T [ k, k]., X0 l Because the P λd -distribution of X0 is λ d, µ k,l is a probability measure on F for all k and l sufficiently large; see 4.7. And µ k,l A converges to µ k A for all Borel sets A as l [monotone convergence theorem]. This proves the assertion that µ k P c F for all k large. Choose and fix such a large integer k. ow we define f ɛ := µ k φ ɛ, and observe that according to Proposition.6, for all nonrandom times τ R, [ E λd 4.10 π {1,...,} t πτ ] f ɛ Xt e [t] dt H πτ = π {1,...,} e [τ ] t πτ π {1,...,} P t τ f ɛ Xτ e [t] dt s π0 P s f ɛ Xτ e [s] ds. We can apply.1 to deduce then that for all nonrandom times τ R, E λd [Sf ɛ H π τ ] e [τ ] P s f ɛ Xτ e [s] ds 4.11 R = e [τ ] Rf ɛ Xτ. Because f ɛ is continuous and compactly supported, one can verify from.9 that Rf ɛ is continuous [this can also be shown by Lemma.3, the fact that ˆf ɛ L 1 and the Fourier inversion formula]. Also, the L P λd -norm of the left-most term in 4.11 is bounded above by Sf ɛ L P λd, and this is at most I Ψ f ɛ, in turn; see Proposition 3.. It is easy to see that 4.1 I Ψ f ɛ = I Ψ µ k φ ɛ 1 π ˆφ d ɛ ξ dξ <. Since ˆµ k ξ K Ψ ξ 1, we conclude that Sf ɛ L P λd is finite for all ɛ > 0.

19 HARMOIC AALYSIS OF ADDITIVE LÉVY PROCESSES 19 A simple adaptation of the proof of Proposition.6 proves that H π is a commuting filtration for all π {1,..., }. By this we mean that for all t R, 4.13 Hπ 1 t 1,..., Hπ t are conditionally independent [P λd ], given H π t, where Hπ j t j is defined as the following σ-algebra: σ Xj s; < s t j 4.14 Hπ j t j := σ Xj s; > s t j if j π, if j π. Here, σ denotes the σ-algebra generated by the random variables in the parenthesis. The stated commutation property readily implies that for all random variables Y L P λd and partial orders π {1,..., }: 1 τ E λd [Y H π τ ] has a version that is cadlag in each of its variables, uniformly in all other 1 variables; and The second moment of sup τ R E λd [Y H π τ ] is at most 4 times the second moment of Y [P λd ]. See Khoshnevisan 00, Theorem.3., p. 35 for the case where P λd is replaced by a probability measure. The details of the remaining changes are explained in a slightly different setting in Lemma 4. of Khoshnevisan, Xiao, and Zhong 003. In summary, 4.11 holds for all τ R, P λd -almost surely [note the order of the quantifiers]. It follows immediately from this that for all integers k 1, 4.15 sup E λd [Sf ɛ H π τ ] e k Rf ɛ XT τ R 1 {T [ k,k] } P λd -a.s. According to Item above, the second moment of the left-hand side is at most 4 times the second moment of Sf ɛ. As we noticed earlier, the latter is at most I Ψ f ɛ. Therefore, [ Rfɛ e k 4 I Ψ f ɛ E λd XT ; T [ k, k] ] 4.16 [ Rfɛ = E λd XT ] { T [ k, k] P } λd T [ k, k]. It follows from this and the Cauchy Schwarz inequality that ] 4.17 e k 4 I Ψ f ɛ E λd [Rf ɛ XT T [ k, k] { } Pλd T [ k, k].

20 0 KHOSHEVISA AD XIAO Using the definition of µ k, we can write the above as { e k 4 I Ψ f ɛ Rf ɛ dµ k P } λd T [ k, k] R 4.18 d = 1 π d ˆµ k ξ Rf { ɛ ξ dξ P } λd T [ k, k], where the equality follows from the Parseval identity. According to Lemma.3, the Fourier transform of Rf ɛ is K Ψ times the Fourier transform of f ɛ, and the latter is ˆµ k ˆφɛ. Hence 4.18 implies that 4.19 e k 4 I Ψ f ɛ 1 π d ˆµ k ξ ˆφɛ ξ K Ψ ξ dξ P λd { T [ k, k] }. ow we apply Property P3 to deduce that ˆφ ɛ 0. Because φ ɛ is also a probability density, it follows that ˆφ ɛ ˆφ ɛ. Consequently, e k 4 I Ψ f ɛ π d ˆf ɛ ξ K Ψ ξ dξ = I Ψ f ɛ P λd { T [ k, k] }. P λd { T [ k, k] } We have seen in 4.1 that I Ψ f ɛ is finite for each ɛ > 0. If it were zero for arbitrary small ɛ > 0, then we apply Fatou s lemma to deduce that I Ψ µ k lim inf ɛ 0 I Ψ f ɛ = 0. This and the fact that K Ψ ξ > 0 for all ξ would imply ˆµ k ξ 0 for all ξ, which is a contradiction. Hence we can deduce that for all ɛ > 0 small enough, 4.1 e k 4 I Ψ f ɛ P λ d { T [ k, k] }, and this is positive for k large; see 4.7. The right-hand side of 4.1 is independent of ɛ > 0. Hence, we can let ɛ 0 and appeal to Fatou s lemma to deduce that I Ψ µ k <. Thus, in any event, we have produced a probability measure µ k on F whose energy I Ψ µ k is finite. This concludes the proof, and also implies 4.5, thanks to the defining properties of the function T. 5. On kernels of positive type In this section we study kernels of positive type; they are recalled next. Here and throughout, R+ := [0, ] is defined to be the usual one-point compactification of R + := [0,, and is endowed with the corresponding Borel sigma-algebra. Definition 5.1. A kernel [on ] is a Borel measurable function κ : R + such that κ L 1 loc Rd. If, in addition, ˆκξ 0 for all ξ, then we say that κ is a kernel of positive type.

21 HARMOIC AALYSIS OF ADDITIVE LÉVY PROCESSES 1 Clearly, every kernel κ can be redefined on a Lebesgue-null set so that the resulting modification κ maps into R +. However, we might lose some of the nice properties of κ by doing this. A notable property is that κ might be continuous; that is, κx κy in R + as x converges to y in. In this case, κ might not be continuous. From this perspective, it is sometimes advantageous to work with the R + -valued function κ. examples, we have in mind Riesz kernels. They are defined as follows: Choose and fix some number α 0, d, and then let the Riesz kernel κ α of index α be x α d if x 0, 5.1 κ α x := if x = 0. It is easy to check that κ α is a continuous kernel for each α 0, d. In fact, every κ α is a kernel of positive type, as can be seen via the following standard fact: 5. ˆκ α ξ = c d,α κ d α ξ for all ξ. Here c d,α is a universal constant that depends only on d and α; see Kahane 1985, p. 134 or Mattila 1995, eq. 1.10, p. 161, for example. It is true but still harder to prove that for all Borel probability measures µ on, µdx µdy 5.3 x y = c d,α ˆµξ dξ d α π d R ξ. d α See Mattila 1995, p. Lemma 1.1, p. 16. The utility of 5.3 is in the fact that it shows that Riesz-type energies of the left-hand side are equal to Pólya Szegő energies of the right-hand side. This is a probabilistically significant fact. For example, consider the case that α 0, ]. Then, µ x y d+α µdx µdy is the energy functional associated to continuous additive functionals of various stable processes of index α. At the same time, µ π d ˆµξ ξ α dξ is a Fourier-analytic energy form of the type that appears more generally in the earlier parts of the present paper. Roughly speaking, π d ˆµξ ξ α dξ I Ψ µ, where Ψξ = ξ α defines the Lévy exponent of an isotropic stable process of index α. [Analytically speaking, this is the Sobolev norm of µ that corresponds to the fractional Laplacian operator α/.] Thus, we seek to find a useful generalization of 5.3 that goes beyond one-parameter stable processes. We define for all finite Borel measures µ and ν on, and all kernels κ on, κx y + κy x 5.4 E κ µ, ν := µdx νdy. For

22 KHOSHEVISA AD XIAO This is called the mutual energy between µ and ν in gauge κ, and defines a quadratic form with pseudo-norm 5.5 E κ µ := E κ µ, µ. This is the κ-energy of the measure µ. There is a corresponding capacity defined as 5.6 C κ F := 1 inf E κ µ, where the infimum is taken over all compactly supported probability measures µ on F, inf :=, and 1/ := 0. We can recognize the left-hand side of 5.3 to be E κd α µ. Because κ d α x < if and only if x 0, the following is a nontrivial generalization of 5.3. Theorem 5.. Suppose κ is a continuous kernel of positive type on which satisfies one of the following two conditions: 1 κx < if and only if x 0; ˆκ L, and κx < when x 0. Then for all Borel probability measures µ and ν on, 5.7 E κ ν µ = 1 ˆκξ Re ˆνξ ˆµξ dξ. π d Remark 5.3. If κ L 1, then ˆκ can be define by the usual Fourier transform, ˆκξ = expix ξκx dx. That is, the condition ˆκ L is automatically verified in this case. In fact, ˆκ is bounded in this case, as can be seen from sup ξ ˆκξ = κ L 1 R. d Our proof of Theorem 5. proceeds in four steps; the first three are stated as lemmas. The folklore of harmonic analysis contains precise versions of the loose assertion that typically, kernels of positive type achieve their supremum at the origin. The first step in the proof of Theorem 5. is to verify a suitable form of this statement. Lemma 5.4. Suppose κ is a kernel of positive type such that ˆκ L. Suppose also that κ is continuous on all of, and finite on \ {0}. Then, κ0 = sup x κx. Proof. Recall the functions φ ɛ from the proof of Theorem.1; see 4.6. Then, κ φ ɛ L 1 and κ φ ɛ = ˆκ ˆφ ɛ 0. But ess sup ξ ˆκξ < and sup ξ ˆφ ɛ ξ φ ɛ L 1 = 1. Moreover, the construction of φ ɛ ensures that ˆφ ɛ is integrable. Therefore, κ φ ɛ L 1 L, and for Lebesgue-almost all x, 5.8 κ φ ɛ x = 1 e ix ξ ˆκξ π ˆφ d ɛ ξ dξ,

23 HARMOIC AALYSIS OF ADDITIVE LÉVY PROCESSES 3 thanks to the inversion formula for Fourier transforms. Since both sides of 5.8 are continuous functions of x, that equation is valid for all x. Furthermore, because κ is a kernel of positive type, it follows that κ φ ɛ x π d ˆκξ dξ. But if x 0, then lim ɛ 0 κ φ ɛ x = κx, and hence, 5.9 sup κx 1 x \{0} π d It suffices to prove that 5.10 ˆκξ dξ. 1 ˆκξ dξ κ0. π d This holds trivially if κ0 is infinite. Therefore, we may assume without loss of generality that κ0 <. The continuity of κ ensures that it is uniformly continuous in a neighborhood of the origin, thence we have lim ɛ 0 κ φ ɛ 0 = κ0 by the classical Fejér theorem. Also, we recall that lim ɛ 0 ˆφɛ ξ = 1 for all ξ. We use these facts in conjunction with 5.8 and Fatou s lemma to deduce 5.10, and hence the lemma. ext we present the second step in the proof of Theorem 5.. This is another folklore fact from harmonic analysis. We say that a kernel κ is lower semicontinuous if there exist a sequence of continuous functions κ 1, κ,... : R + such that κ n x κ n+1 x for all n 1 and x, such that κ n x κx for all x, as n. Because κ n x is assumed to be in R + [and not R + ], our definition of lower semicontinuity is slightly different from the usual one. onetheless, the following is a consequence of Lemma 1.11 of Mattila 1995, p Lemma 5.5. Suppose κ is a lower semicontinuous kernel of positive type on. Then, for all Borel probability measures µ on, 5.11 E κ µ 1 π d ˆκξ ˆµξ dξ. Our next lemma constitutes the third step of our proof of Theorem 5.. Lemma 5.6. Suppose κ is a continuous kernel of positive type on, which satisfies one of the following two conditions: 1 κx < if and only if x 0; ˆκ L, and κx < when x 0. Then, for all Borel probability measures µ on, 5.1 E κ µ = 1 ˆκξ ˆµξ dξ. π d

24 4 KHOSHEVISA AD XIAO A weaker version of this result is stated in Kahane 1985, p. 134 without proof. According to Kahane loc. cit., functions κ that satisfy the conditions of Lemma 5.6 are called potential kernels. They can be defined, in equivalent terms, as kernels of positive types that are continuous on \ {0} and lim x 0 κx =. We will not use this terminology: the term potential kernel is reserved for another object. Proof of Lemma 5.6. Regardless of whether κ satisfies condition 1 or, it is lower semicontinuous. Therefore, in light of Lemma 5.5, it suffices to prove that 5.13 E κ µ 1 ˆκξ ˆµξ dξ. π d From here on, our proof considers two separate cases: Case 1. First, let us suppose κ satisfies condition 1 of the lemma. Without loss of generality, we may assume that E κ µ <. Since lim x 0 κx =, this implies that µ does not charge singletons. According the Lusin s theorem, for every η 0, 1 we can find a compact set K η \{0} with µk c η η such that κ µ is continuous and hence uniformly continuous on K η. Define for all Borel sets A, 5.14 µ η A = µa K η. 1 η Then µ η is supported by the compact set K η, and η µ ηk η = µk η 1 η 1. ow lim r 0 µ η Bx, r = 0 for all x, where Bx, r denotes the l -ball of radius r > 0 about x. Therefore, a compactness argument reveals that for all η 0, 1, 5.16 lim r 0 sup µ η Bx, r = 0. x K η For otherwise we can find δ > 0 and x r K η such that for all r > 0, µ η Bx r, r δ. By compactness we can extract a subsequence r 0 and x K η such that x r x. It follows easily then µ η Bx, ɛ δ for all ɛ > 0, whence µ η {x} δ, which contradicts the fact that µ η does not charge singletons. ext we choose and fix y K η and η > 0. We claim that 5.17 sup x K η κy z µ η dz 0 as ɛ 0. Bx,ɛ

25 HARMOIC AALYSIS OF ADDITIVE LÉVY PROCESSES 5 Indeed, by 5.16 and the fact that κ µ η y <, we see that for all ρ > 0 there exists θ > 0 such that 5.18 lim ɛ 0 sup κy z µ η dz ρ. x K η: x y θ/ Bx,ɛ On the other hand, by the continuity of κ on \{0} and 5.16, we have 5.19 lim κy z µ η dz = 0. ɛ 0 sup x K η: x y >θ/ Bx,ɛ By combining 5.18 and 5.19, we find that 5.0 lim κy z µ η dz ρ. ɛ 0 sup x K η Bx,ɛ Thus 5.17 follows from 5.0, because we can choose ρ as small as we want. By 5.17 and another appeal to compactness, we obtain 5.1 sup y K η sup x K η κy z µ η dz 0 as ɛ 0. Bx,ɛ Consequently, for all k 1, we can find ɛ k 0 such that 5. sup sup κ µ ηy κy z µ η dz 1 y K η x K η Bx,ɛ k k. c Let y K η and let {y n } n=1 be an arbitrary sequence in K η such that lim n y n = y. Because z κy z is uniformly continuous on By, ɛ k c K η, we have 5.3 lim κy n z µ η dz = κy z µ η dz. n By,ɛ k c By,ɛ k c This and 5. together imply that for all k 1, 5.4 lim n κ µη y n κ µ η y k. Let k to deduce that κ µ η is continuous, and hence uniformly continuous, on K η. On the other hand, it can be verified directly that κ µ η is continuous on K c η. Hence, we have shown that κ µ η is continuous on. If 0 < ɛ, η < 1, then we can appeal to the Fubini-Tonelli theorem, a few times in succession, to deduce that 5.5 E κ ψɛ µ η = κ z a ɛ,η b ɛ,η z dz,

26 6 KHOSHEVISA AD XIAO where a ɛ,η := φ ɛ µ η, b ɛ,η := φ ɛ µ η, and µ η is the Borel probability measure on that is defined by 5.6 µ η A := µ η A for all A. We observe the following elementary facts: 1 Both a ɛ,η and b ɛ,η are infinitely differentiable functions of compact support; because φ ɛ is of positive type, the Fourier transform of a ɛ,η b ɛ,η is ˆφ ɛ ˆµ η ; 3 the Fourier transform of z κ z is the same as that of κ because κ is of positive type; Therefore, we can combine the preceding with the Parseval identity, and deduce that E κ ψɛ µ η = 1 ˆκξ â π d ɛ,η ξ ˆb ɛ,η ξ dξ R 5.7 d = 1 ˆκξ π ˆφ d ɛ ξ ˆµ η ξ dξ. We can apply the Fubini-Tonelli theorem to write the left-most term in another way, as well. amely, 5.8 E κ ψɛ µ η = κ ψ ɛ µ η dµ η. The continuity of κ µ η, and Fejér s theorem, together imply that κ µ η ψ ɛ converges to κ µ η uniformly on K η as ɛ 0, and hence E κ ψɛ µ η converges to E κ µ η as ɛ 0. This and 5.7 together imply that E κ µ η = 1 π lim ˆκξ ˆφ d ɛ ξ ˆµ η ξ dξ ɛ 0 R 5.9 d 1 ˆκξ ˆµ π d η ξ dξ, owing to Fatou s lemma. Since µ η is 1 η 1 times a restriction of µ, it follows that η E κµ 1 ˆκξ ˆµ π d η ξ dξ. Because for all ξ, 5.31 ˆµ ηξ ˆµξ 1 η µkc η 1 η η 1 η, we deduce that lim η 0 ˆµ η = ˆµ pointwise. An appeal to Fatou s lemma justifies 5.13, and this completes our proof in the first case.

27 HARMOIC AALYSIS OF ADDITIVE LÉVY PROCESSES 7 Case. ext, we consider the case that κ satisfies condition of the lemma. If κ0 =, then condition 1 is satisfied, and therefore the proof is complete. Thus, we may assume that κ0 <. According to Lemma 5.4, κ is a bounded and continuous function from into R +. For all Borel sets A define 5.3 µ n A := µ n A [ n, n] d, χ n where χ n := µ [ n, n] d. If n > 0 is sufficiently large, then µ n is a well-defined Borel probability measure on [ n, n] d, and since κ is uniformly continuous on [ n, n] d, χ n E κ µ E κ µ n = lim E κ ψɛ µ n ɛ 0 = 1 π lim ˆκξ ˆψ d ɛ ξ ˆµ n ξ dξ. ɛ 0 The first equality holds because κ ψ ɛ converges uniformly to κ on [ n, n] d, as ɛ 0. The second is a consequence of the Parseval identity. Thanks to Fatou s lemma, we have proved that for all n > 0 sufficiently large, χ n E κ µ 1 π d ˆκξ ˆµ n ξ dξ. As n tends to infinity, χ n 1 and ˆµ n ˆµ pointwise. Therefore, another appeal to Fatou s lemma implies 5.13, and hence the lemma. ow we derive Theorem 5.. Proof of Theorem 5.. Let us observe that by the Fubini-Tonelli theorem, 5.35 E κ ν µ = E κ µ, ν µ. In fact, both sides are equal to E[κX X Y ], where X, X, Y are independent, X and X are distributed as µ, and Y is distributed as ν. Lemma 5.6, and polarization, together imply that the following holds for every Borel probability measure σ on : 5.36 E κ µ, σ = 1 π d Indeed, we first notice that 5.37 E κ µ + σ ˆκξ ˆσξ ˆµξ dξ. = 1 4 E κµ E κσ + 1 E κµ, σ.

28 8 KHOSHEVISA AD XIAO Thus, we solve for E κ µ, σ and apply Lemma 5.6 to deduce that E κ µ, σ = µ + σ ˆκξ ξ dξ π d R 5.38 d 1 ˆκξ ˆµξ dξ 1 ˆκξ ˆσξ dξ. π d R π d d We solve to obtain Thus we define σ := ν µ, observe that σ is a Borel probability measure on, and E κ ν µ = E κ µ, σ. Thus, we may apply 5.35 and 5.36 in this order to find that E κ ν µ = 1 ˆκξ ˆσξ ˆµξ dξ π d R 5.39 d = 1 ˆκξ ˆν ξ ˆµξ dξ. π d In the last line, we have only used the fact that ˆσξ = ˆν ξˆµξ for all ξ. Because the left-most term in 5.39 is real-valued, so is the right-most term. Therefore, we may consider only the real part of the right-most item in 5.39, and this proves the result. 6. Absolute continuity considerations Suppose E is a Borel measurable subset of R that has positive Lebesgue measure, and Y := {Y t} t E is an -valued random field that is indexed by E. Definition 6.1. We say that Y has a one-potential density v if v L 1 is nonnegative and satisfies the following for all Borel measurable functions f : R + : [ ] E e [s] ds E fy te [t] dt = fxvx dx. E In particular, X has a one-potential density v if for all Borel measurable functions f : R +, 6. Rfx = fx + yvy dy := f vx for all x, where ğz := g z for all functions g. It follows from Lemma.1 in Hawkes 1979 that X has a one-potential density if and only if the operator efined by.9 is strong Feller. That is, if Rf is continuous whenever f is Borel measurable and has compact support. In order to be concrete, we choose a nice version of the one-potential density v, when it exists. Before we proceed further, let us observe that when v exists it is a probability

29 HARMOIC AALYSIS OF ADDITIVE LÉVY PROCESSES 9 density. In particular, the Lebesgue density theorem tell us that vx = lim vy dy for almost all x [λ d ]. ɛ 0 λ d B0, ɛ Bx,ɛ We can recognize the integral as R1 B0,ɛ x. Moreover, we can alter v on a Lebesgue-null set and still obtain a one-potential density for X. Therefore, from now on, we always choose the following version of v: 6.4 vx := lim inf ɛ 0 R10,ɛ x λ d B0, ɛ for all x. Lemma.3 states that the Fourier multiplier of R is K Ψ. Therefore, the Fourier transform of v is also K Ψ ; confer with 6.. Because K Ψ is real-valued, this proves the following: Lemma 6.. If X has a one-potential density v, then ˆv = K Ψ in the sense of Schwartz. In particular, v is an integrable kernel of positive type on. Remark 6.3. Let ɛ 1,..., ɛ denote random variables, all independent of one another, as well as {X j } and {X j}, with P{ɛ 1 = ±1} = 1/. We can re-organize the order of integration a few times to find that [ 1 ] 6.5 Xt E e [t] dt R f = E [ R + ] f ɛ j X j s j e [s] ds. Thus, in particular, X has a one-potential density if and only if the R +-indexed random field t 1,..., t ɛ jx j t j does [interpreted in the obvious sense]. Interestingly enough, the latter random field appears earlier though for quite different reasons as ours in the works of Marcus and Rosen 1999b, 1999a. Remark 6.4. It follows from the definitions that a random field {Y t} t E has a one-potential density if and only if the Borel probability measure [ ] 6.6 A E 1 A Y te [t] dt ow suppose f 0 is Borel measurable. Then 6.7 Xt e [t] dt E R f R + is absolutely continuous. fxte [t] dt. We take expectations of both sides to deduce from 6.6 that if X has a one-potential density, then so does X. When = 1, it can be verified that the converse it also true. However, the previous remark can be used to show that, when, the converse is not necessarily true. Thus, we can conclude that the existence of a one-potential density for X is a more stringent

30 30 KHOSHEVISA AD XIAO condition than the existence of a one-potential density for X. Another consequence of the preceding is the following: If the one-potential density of X exists and the potential density of X is a.e.-positive, then the one-potential density of X is per force also a.e.-positive. We have, and will, encounter these conditions several times. Our next theorem is the main result of this subsection. From a technical point of view, it is also a key result in this paper. In order to describe it properly, we introduce some notation first. If T is a nonempty subset of {1,..., }, then we define T to be the cardinality of T, and X T to be the subprocess associated to the index T. That is, X T is the following T -parameter, -valued random field: 6.8 X T t := j T X j t j for all t R T +. In order to obtain nice formulas, we define X to be the constant 0. In this way, it follows that, regardless of whether or not T > 0, each X T is itself an additive Lévy process, and the Lévy exponent of X T is the function Ψ j j T : C T. Thus, we can talk about the stationary field X T for all T {1,..., }, etc. Despite this new notation, we continue to write X and X in place of the more cumbersome X {1,...,} and X {1,...,}. Theorem 6.5. Suppose there exists a nonrandom and nonempty subset T {1,..., }, such that X T has a one-potential density v T : R + that is continuous on, and finite on \ {0}. Then, X has a one-potential density v on, and for all Borel sets F, 6.9 E [ λ d XR + F ] > 0 C v F > 0. Proof. First, consider the case that T = {1,..., }. In this case, X has a one-potential density v := v {1,...,} that is continuous on, and finite away from the origin. In light of Remark 5.3, Lemma 6. shows us that κ := v satisfies condition of Theorem 5.. Theorem 5., in turn, implies that I Ψ µ = E v µ for all Borel probability measures µ on, and thence C v F = cap Ψ F. This and Theorem 1.1 together imply Theorem 6.5 in the case that X has a continuous one-potential density that is finite on \ {0}. ext, consider the remaining case that 1 T 1. For all Borel sets A define [ ] 6.10 V T ca := E XT cs e [s] ds. R T 1 A