5. Examples
5.4. Other examples
In this section, we briefly discuss two further examples illustrating possible ways of con- structing cluster distributionsηx.
5.4.1. Manifolds of non-positive curvature. Let X be a complete, path-connected man- ifold with non-positive sectional curvature (Cartan–Hadamard manifold). In this case, for every two pointsx, y ∈Xthere is a unique geodesicgx,y(t),t∈[0,1], such thatgx,y(0) =x,
gx,y(1) = y. Assume in addition that X is simply connected. It follows from the Cartan–
Hadamard theorem that the exponential mapexpx: TxX →Xis a diffeomorphism for every
x∈X (see, e.g., [13, 21]). Choosex0 ∈X, and let
dgx0,x :Tx0X →TxX
be the parallel translation along the geodesicgx0,x. To deploy the construction of Section
3.4, we setW :=Tx0Xand
ϕx := expx◦dgx0,x : W →X.
For a given probability measureQ=Qx0on(Tx0X) ∞
0 , consider the corresponding translated
(push-forward) measures on(TxX)∞0 ,
Qx = dgx∗0,xQx0 x∈X. (5.21)
According to a general formula (3.40), we can now define a family of probability distribu- tions on the spaceXby
ηx:= ¯ϕ∗xQx0 = exp ∗
xQx, x∈X. (5.22)
Remark 5.2. In fact, X is essentially the Euclidean space Rd (d = dimX) with a non- constant metric which defines a family of exponential mapsexpx: Rd →
Rd(x ∈ Rd). In
this interpretation, we haveW =Rdandϕ
x = expx: W →X.
Remark5.3. Consider the diffeomorphism
ix := expx◦dgx0,x◦exp −1
x0 : X→X, (5.23)
From (5.21), (5.22) and (5.23) it follows that the family of distributions{ηx}x∈X is transla-
5.4.2. Metric spaces. Let (X, ρ) be a metric space, endowed with the natural topology generated by the open ballsBr0(x) :={x0 ∈X : ρ(x, x0)< r}(x∈X,r >0) and equipped with a (locally finite) reference measureϑ.
In this section, we construct an example of a family of probability measures{ηx(d¯y)}x∈X
on X = F
nX
n, based on a different idea that avoids using any family of maps ϕ x as in
Sections 5.1–5.3. To this end, note that by a radial-angular decomposition (based on Fubini’s theorem) we can represent theϑ-volume of a (closed) ballBr(x) :={x0 ∈X : ρ(x, x0)≤r}
(x∈X) as ϑ(Br(x)) = Z r 0 Z ∂Bs(x) ϑangx (dy|s) ϑradx (ds), (5.24) where ∂Br(x) = Br(x)\Br0(x) is the sphere of radius r centred at x, ϑangx (dy|r) is the
uniform “surface” measure on∂Br(x) induced by the measure ϑ(dy), andϑradx (dr) is the
radial component ofϑ as seen fromx. According to formula (5.24), the measureϑ can be symbolically expressed as a skew product
ϑ(dy) = ϑangx (dy|r)ϑradx (dr)
r=ρ(x,y).
Forx∈X andy¯∈X, setρ¯(x,y¯) := (ρ(x, yi))yi∈y¯ ∈X. As usual, the measureϑcan be
lifted to the spaceXby setting
¯
ϑ(d¯y) := N
yi∈¯y
ϑ(dyi), y¯∈X. (5.25)
Similarly, for eachx∈Xdefine
¯ ϑradx (d¯r) := N ri∈¯r ϑradx (dri), (5.26) ¯ ϑangx (d¯y|r¯) := N yi∈y¯ ϑangx (dyi|ri). (5.27)
Let us now fix a pointx0 ∈ X, and letf: R∞0 → R+ be such that
R
Xf(¯r) ¯ϑ x0
rad(d¯r) = 1.
Then we can construct a family of cluster distributions by setting, for eachx∈X,
ηx(d¯y) :=f(¯r) ¯ϑradx0 (d¯r)· ¯ ϑangx (d¯y|r¯) ¯ ϑx ang(∂B¯r¯(x)|r¯) ¯ r= ¯ρ(x,¯y) , y¯∈X. (5.28)
That is to say, under the measureηxa random vectory¯is sampled in two stages: first, a vector
¯
rof the distances fromx toy¯is sampled with the probability densityf(¯r)(with respect to the measureϑ¯x0
rad), and then the componentsyiofy¯are chosen, independently of each other,
with the uniform distribution over the corresponding spheres∂Bri(x), respectively.
Remark5.4. By the definition (5.28), the measureηxmay be considered as a “translation” of
the pattern measureηx0 fromx0 tox; however, this is not being done by a push-forward of
ηx0 under some mapping ϕx of the spaceX, as prescribed by the general recipe of Section
3.4; instead, we compensate the lack of such a mapping by using the same statistics of the distances at each pointx ∈ X (prescribed by the pattern distribution ϑx0
rad) and by taking
advantage of the uniform distribution on the corresponding spheres, which does not require any further angular information.
Remark5.5. If there is a groupGof isometries ofX acting transitively, then we can use the same method as for homogeneous spaces (see Section 5.3).
Appendix
A. On a definition of the skew-product measureµˆ
In order to verify that the measureµˆis well defined by the expression (2.17) (which requires the internal integral in (2.18) to be measurable as a function ofγ ∈ ΓX), we shall construct
an auxiliary measureµ˜onΓX ×X∞and show thatµˆis its image under a certain measurable
map.
Let us fix an indexationi={iγ, γ ∈ΓX}inΓX, whereiγ:γ →Nis a bijection for each
γ ∈ΓX. Define
ΓX,1 :={(γ, x)∈ΓX ×X: x∈γ}.
The indexationidefines a natural bijection
ΓX,1 3(γ, x)7→(γ,iγ(x))∈ΓX ×N. (A.1)
Moreover, the indexation i can be constructed so that the bijection (A.1) is measurable (see [31]). This ensures that the map
ΓX 3γ 7→jk(γ) :=i−γ1(k)∈X (A.2)
is measurable for eachk ∈N. Consider a family{νγ, γ ∈Γ X}of measures onX∞defined by νγ(d¯y) := N k∈N ηjk(γ)(d¯y), y¯∈X ∞ .
IfA∈ B(X∞)is a cylinder set,A=A1× · · · ×An×X× · · ·, then
νγ(A) =
n
N
k=1
ηjk(γ)(Ak).
The function ΓX 3 γ 7→ νγ(A) ∈ Ris measurable due to the measurability of jk(γ)and
Condition 2.1. Hence, the measure
˜
µ(dγ×d¯y) := νγ(dy)µ(dγ), (γ,y¯)∈ΓX ×X∞,
is well defined.
Finally, a direct calculation shows that the measureµˆdefined by (2.17) can be represented asµˆ=I∗µ˜, whereI:Γ
X×X∞→ΓX×Xγis a measurable map defined by(γ,(yk)k∈N)7→
(γ,(yjk(γ))k∈N). This proves the result.
B. Correlation functions
For a more systematic exposition and further details, see the classical books [16, 28, 29]; more recent useful references include [4, 22, 23].
Definition B.1. Letµbe a probability measure on the generalised configuration space ΓX], and letθbe a (locally finite) measure onX. Then the correlation functionκn
µ: Xn→R+of
for any functionφ ∈ M+(Xn)symmetric with respect to permutations of its arguments, it holds Z ΓX] X {x1,...,xn}⊂γ φ(x1, . . . , xn)µ(dγ) = 1 n! Z Xn φ(x1, . . . , xn)κnµ(x1, . . . , xn)θ(dx1)· · ·θ(dxn). (B.1)
RemarkB.1. Note that possible multiple points on the configurationγ ∈ ΓX] will lead cor- respondingly to some coinciding points among {x1, . . . , xn} ⊂ γ on the left-hand side of
formula (B.1) (cf. our convention on the use of set-theoretic notation, see Section 2.1). By a standard approximation argument, equation (B.1) can be extended to any (symmet- ric) functionsφ∈L1(Xn, θ⊗n).
Condition B.1. Correlation functionsκmµ(x1, . . . , xm)up to the n-th order (n ∈ N) of the
measureµwith respect toθexist and are bounded.
Remark B.2. Formula (B.1) with n = 1 and φ(x) = 1B(x) for B ∈ B(X) shows that
Condition B.1 automatically implies thatµ-a.a. configurationsγ are locally finite.
Lemma B.1. Assume that ConditionB.1is satisfied with somen ∈N. Thenµ∈ Mn θ(ΓX).
Proof. Similarly as in the proof of Lemma 3.6, we obtain (cf. (3.28))
Z ΓX |hf, γi|nµ(dγ)≤ Z ΓX X x∈γ |f(x)| !n µ(dγ) = n X m=1 Z ΓX X {x1,...,xm}⊂γ φn(x1, . . . , xm)µ(dγ), (B.2)
whereφn(x1, . . . , xm)is a (symmetric) function given by the expression (3.29). Note that,
by definition of the correlation functions (see (B.1)), the integral on the right-hand side of (B.2) is reduced to 1 m! Z Xm φn(x1, . . . , xm)κmµ(x1, . . . , xm)θ(dx1)· · ·θ(dxm). (B.3)
By Condition B.1, κmµ ≤ cm (m = 1, . . . , n) with some constant cm < ∞. Hence, the
integral in (B.3) is bounded by X i1,...,im≥1 i1+···+im=n cmn! i1!· · ·im! m Y j=1 Z Z |f(xj)|ijθ(dxj)<∞, (B.4)
since each integral in (B.4) is finite owing to the assumption f ∈ T
1≤q≤nLq(X, θ). As a
C. Integration-by-parts formula for push-forward measures
For any Riemannian manifoldsW andY, denote byC2
b(W,Y)the space of twice differen-
tiable mapsφ: W → Y with globally bounded derivatives dφ, d2φ. In particular, for any
¯
w∈ W, the first derivativedφ( ¯w)is a bounded linear operator from the tangent spaceTw¯W
to the tangent spaceTφ( ¯w)Y. In what follows, we fix φ ∈ Cb2(W,Y). Note that the adjoint
operatordφ( ¯w)∗: Tφ∗( ¯w)Y → Tw∗¯W can be identified with a bounded operator fromTφ( ¯w)Y
toTw¯W via the scalar products in the tangent spacesTw¯W andTφ( ¯w)Y (defined by the Rie-
mannian structure of the manifoldsW andY, respectively). Furthermore, defineVect1b(W)
as the space of differentiable vector fields onW with a globally bounded first derivative.
Definition C.1. We say that a probability measureQ(d ¯w)onW satisfies anintegration-by- parts(IBP)formulaif for any vector fieldV ∈Vect1b(W)there is a functionβV
Q ∈L1(W, Q)
(logarithmic derivativeofQin the directionV) such that, for anyg ∈C1
b(W), the following identity holds Z W (∇g( ¯w), V( ¯w))Tw¯WQ(d ¯w) =− Z W g( ¯w)βQV( ¯w)Q(d ¯w). (C.1)
Whenever it exists, the functionβQV can be represented in the form
βQV( ¯w) = (βQ( ¯w), V( ¯w))Tw¯W+ divV( ¯w), w¯∈ W, (C.2)
whereβQis a vector field onW(called thevector logarithmic derivativeofQ) satisfying
Z
W
|βQ( ¯w)|Tw¯WQ(d ¯w)<∞.
Consider the push-forward measureη:=φ∗QonY, and denote byIφthe operator acting
on functionsf: Y →Rby the formula
Iφf =f◦φ.
Because of the definition of the measureη, the operatorIφis an isometry fromLr(Y, η)to
Lr(W, Q), for anyr ∈[1,∞]. Hence, the adjoint operator defines an isometry between the corresponding dual spaces,
Iφ∗ :Lr(W, Q)0 →Lr(Y, η)0.
Furthermore, for any r ∈ (1,∞)we have the isomorphisms Lr(W, Q)0 ∼
= Ln(W, Q)and
Lr(Y, η)0 ∼
=Ln(Y, η), wheren = r/(r−1)(see, e.g., [30, Ch. II,§2, p. 43]). Sincer > 1
is arbitrary, this means thatI∗
φcan be treated as an isometry fromL
n(W, Q)toLn(Y, η)for
anyn > 1. Moreover, repeating the arguments used in the proof of Lemma 4.3, it can be shown that the same also holds forn= 1. To summarise, for anyn≥1the operator
Iφ∗ :Ln(W, Q)→Ln(Y, η)
Theorem C.1. Letφ ∈C2
b(W,Y)be such that the operator
dφ( ¯w)∗:Tφ( ¯w)Y → Tw¯W, w¯∈ W,
is an isometry, and suppose that the measure Qsatisfies the IBP formula (C.1). Then the push-forward measure η = φ∗Q satisfies an IBP formula with the logarithmic derivative
βU η =I
∗
φβQV, whereV =VU is a vector field onW given by
V( ¯w) = dφ( ¯w)∗U(φ( ¯w)), U ∈Vect1b(Y).
Proof. Note that V ∈ Vect1b(W). Applying the IBP formula (C.1), making the change of measure η = φ∗Q and taking into account that dφ( ¯w)dφ( ¯w)∗ is the identity operator in
Tφ( ¯w)Y, we see that (C.1) holds forη with the corresponding logarithmic derivative βηU =
I∗
φβQV. Finally, theη-integrability ofβηU follows by the isometry ofI
∗
. RemarkC.1. All of the above remains true in the case whereW =F∞
i=0WiandY =
F∞
i=0Yi
are countable disjoint unions of Riemannian manifolds(Wi)and(Yi)respectively, and the
mappingφacts component-wise, that is,φ: Wi → Yi. Although the spacesW andYdo not
possess a proper Riemannian manifold structure, all notions introduced above (including the IBP formula (C.1)) can be understood component-wise, and we use the analogous notation without further explanations.
Acknowledgements
The research of L. Bogachev was supported in part by a Leverhulme Research Fellowship. Partial support from SFB 701, ZiF (Bielefeld) and DFG Grant 436 RUS 113/722 is gratefully acknowledged. The authors would like to thank Sergio Albeverio, Yuri Kondratiev, Eugene Lytvynov, Stanislav Molchanov and Chris Wood for helpful discussions.
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