A Discovery Tour in Random Riemannian Geometry

We study random perturbations of Riemannian manifolds $(\mathsf{M},\mathsf{g})$ by means of so-called Fractional Gaussian Fields, which are defined intrinsically by the given manifold. The fields $h^\bullet: \omega\mapsto h^\omega$ will act on the manifolds via conformal transformation $\mathsf{g}\mapsto \mathsf{g}^\omega\colon\!\!= e^{2h^\omega}\,\mathsf{g}$. Our focus will be on the regular case with Hurst parameter $H>0$, the celebrated Liouville geometry in two dimensions being borderline. We want to understand how basic geometric and functional analytic quantities like diameter, volume, heat kernel, Brownian motion, spectral bound, or spectral gap will change under the influence of the noise. And if so, is it possible to quantify these dependencies in terms of key parameters of the noise. Another goal is to define and analyze in detail the Fractional Gaussian Fields on a general Riemannian manifold, a fascinating object of independent interest.

1.1.Random Riemannian Geometry.Given a Riemannian manifold (M, g) and a Gaussian random field h • : Ω → C(M), ω → h ω , we study random perturbations (M, g ω ) of the given manifold with conformally changed metric tensors g ω := e 2h ω g.For this Random Riemannian Geometry (M, g • ) with g • := e 2h • g we want to understand how basic geometric and functional analytic quantities like diameter, volume, heat kernel, Brownian motion, or spectral gap change under the influence of the noise.If possible, we want to quantify these dependencies in terms of key parameters of the noise.
Our main interest in the sequel will be in the case h • ∈ C 2 (M) a.s., where standard Riemannian calculus is not directly applicable and where no classical curvature concepts are at our disposal.Our approach to geometry, spectral analysis, and stochastic calculus on the randomly perturbed Riemannian manifolds (M, g • ) will be based on Dirichlet form techniques.
For convenience, we will assume throughout that the reference manifold (M, g) has bounded geometry.
The Brownian motion on (M, g ω ), defined as the reversible, Markov diffusion process B ω associated with the heat semigroup e t∆ ω /2 t>0 , allows for a more explicit construction if the conformal weight h ω is differentiable.
Proposition 1.3.If h ω ∈ C 1 (M) then B ω is obtained from the Brownian motion B on (M, g) by the combination of time change with weight e 2h ω and Girsanov transformation with weight (n − 2)h ω .
We will compare the random volume, random length, and random distance in the random Riemannian manifold (M, g • ) with analogous quantities in deterministic geometries obtained by suitable conformal weights.
Proposition 1.4.Put θ(x) := E[h • (x) 2 ] ≥ 0 and g n := e n θ g, g 1 := e θ g.Then for every measurable A ⊂ M, E[vol g • (A)] = vol g n (A) ≥ vol g (A) , and for every absolutely continuous curve γ : [0, 1] → M, Of particular interest is the rate of convergence to equilibrium for the random Brownian motion.
Let us emphasize that classical estimates for the spectral gap, based on Ricci curvature estimates, require that the metric tensor is of class C 2 , whereas our Theorem 1.5 -combined with Theorem 1.9 below -will apply whenever the random metric tensor is of class C 0 .1.2.Fractional Gaussian Field (FGF).In our approach to Random Riemannian Geometry, we will restrict ourselves to the case where the random field h • is a Fractional Gaussian Field, defined intrinsically by the given manifold.It is a fascinating object of independent interest.
Given a Riemannian manifold (M, g) of bounded geometry, for m > 0 and s ∈ R, we define the Sobolev spaces The scalar product u | v L 2 extends to a continuous bilinear pairing between H s m (M) and H −s m (M) as well as between D(M) and D (M).It follows, that the functional u → exp − Theorem 1.6.For every s ∈ R and m > 0, there exists a unique centered Gaussian field h • with called m-massive Fractional Gaussian Field on M of regularity s, briefly FGF M s,m .
For s = 0 this is the white noise on M. Note that, if h • is distributed according to FGF M s,m on some compact M, then Theorem 1.7.For s > 0, the Fractional Gaussian Field FGF M s,m is uniquely characterized as the centered Gaussian process h • with covariance where G s,m (x, y) := 1 Γ(s) ∞ 0 p t (x, y) e −m 2 t t s−1 dt.For s > n/2, this characterization simplifies to x, y ∈ M .(1.5) Indeed, for s > n/2, the Fractional Gaussian Field FGF M s,m is almost surely a continuous function.More precisely, Proposition 1.8.Assume M is compact and let h • ∼ FGF M s,m with s > n/2 + k, k ∈ N 0 .Then h ω ∈ C k (M) for a.e.ω.
A crucial role in our geometric estimates and functional inequalities for the Random Riemannian Geometry is played by estimates for the expected maximum of the random field.
Theorem 1.9.For every compact manifold M there exists a constant C = C(M) such that for h • ∼ FGF M s,m with any m > 0, If M is compact, then an analogous construction also works in the case m = 0 provided all function spaces H −s m are replaced by the subspaces H−s m obtained via the grounding map u → ů := u − 1 volg(M) udvol g .The F GF M s,m for s = 1, m = 0 is the celebrated Gaussian Free Field (GFF) on M. In the compact case, the Fractional Gaussian Field also admits a quite instructive series representation.
Theorem 1.10.Let (ϕ j ) j∈N0 be a complete orthonormal basis in L 2 consisting of eigenfunctions of −∆ with corresponding eigenvalues (λ j ) j∈N0 , and let a sequence ξ • j j∈N0 of independent, N (0, 1)distributed random variables be given.Then for s > n/2 and m ≥ 0, the series converges and provides a pointwise representation of h (a) For Euclidean spaces M = R n , the F GF M s,m is well studied with particular focus on the massless case m = 0.Here some additional effort is required to deal with the kernel of which is resolved by factoring out polynomials of degree ≤ s.The real white noise, the 1d Brownian motion, the Lévy Brownian motion, and the Gaussian Free Field on the Euclidean space are all instances of random fields in the larger family of Fractional Gaussian Fields.The article [37] by Lodhia, Sheffield, Sun, and Watson provides an excellent survey.
Despite the fact that it seems to be regarded as common knowledge (in particular in the physics literature), even in the most prominent case s = 1, the Riemannian context is addressed only occasionally, e.g.[10,22,28].In particular, Gelbaum [22] studies the existence on complete Riemannian manifolds of the fractional Brownian motions FGF M s,0 , s ∈ (n/2, n/2 + 1), and of the massive FGF M s,1 , with the same values of s.Fractional Brownian motions are also constructed on Sierpiński gaskets and related fractals in [6].
(b) The particular case of the FGF with s = 1 is the Gaussian Free Field, discussed and analyzed in detail in the landmark article [50] by Sheffield.The GFF arises as scaling limit of various discrete models of random (hyper-)surfaces over n-dimensional simplicial lattices, e.g.Discrete Gaussian Free Fields (DGFF) or harmonic crystals [50].The two-dimensional case is particularly relevant, for the GFF is then invariant under conformal transformations of D ⊂ R 2 ∼ = C, and constitutes therefore a useful tool in the study of conformally invariant random objects.For instance, the zero contour lines of the GFF (despite being random distributions, not functions) are well-defined SLE curves [49].
(c) Again in the two-dimensional case, the GFF gives rise to an impressive random geometry, the Liouville Quantum Gravity.It is a hot topic of current research with plenty of fascinating, deep results -despite the fact that many classical geometric quantities become meaningless, see e.g.[3,11,16,20,21,28,39,41].
In this paper, our focus will be on the Random Riemannian Geometry in the 'regular' case of Hurst parameter H := s − n/2 > 0 in arbitrary dimension.In general, this geometry is not conformally invariant, since neither the Laplace-Beltrami operator nor its powers are conformally covariant.For compact manifolds of arbitrary even dimension n, we shall address in [14] the conformally invariant case at the critical scale s = n/2, a high-dimensional Liouville Quantum Gravity.1.3.Higher Order Green Kernel.The regularity of the Fractional Gaussian Field h • and the quantitative geometric and functional analytic estimates for the Random Riemannian Geometry (M, g • ) will be determined by the Green kernel of order s, and, in the compact case, by its grounded counterpart The latter is also well-behaved in the massless case m = 0 whereas the application of the former is restricted to the case of positive mass parameter m.We analyze these Green kernels in detail and derive explicit formulas for model spaces, including Euclidean spaces, tori, hyperbolic spaces, and spheres.Theorem 1.12.For points x, y, let r := d(x, y).Then, (a) For the 1-dimensional torus .
(b) For the sphere in 2 and 3 dimensions, (c) For the hyperbolic space in three dimensions and m > 0, Of particular interest is the asymptotics of the Green kernel close to the diagonal.
Theorem 1.13.Let M be a compact manifold, m ≥ 0, and s > n/2.Then for every α ∈ (0, 1] with α < s − n/2 there exists a constant C = C(M) so that Gs,m (x, x) + Gs,m (y, y) − 2 Gs,m (x, y) Acknowledgements.The authors would like to thank Matthias Erbar and Ronan Herry for valuable discussions on this project.They are also grateful to Nathanaël Berestycki, and Fabrice Baudoin for respectively pointing out the references [7], and [6,22], and to Julien Fageot and Thomas Letendre for pointing out a mistake in a previous version of the proof of Proposition 3.10.The authors feel very much indebted to an anonymous reviewer for their careful reading and the many valuable suggestions that have significantly contributed to the improvement of the paper.
2. The Riemannian Manifold.Throughout this paper, (M, g) will be a complete connected ndimensional smooth Riemannian manifold without boundary, ∆ will denote its Laplace-Beltrami operator and p t (x, y) the associated heat kernel.The latter is symmetric in x, y, and as a function of t, x it solves the heat equation 1  2 ∆u = ∂ ∂t u.For convenience, we always assume that (M, g) is stochastically complete, i.e., which is a well-known consequence of uniform lower bounds for the Ricci curvature, see e. 2.1.Higher Order Green Operators.For m > 0, consider the positive self-adjoint operator on L 2 = L 2 (vol g ), and its powers A s m defined by means of the Spectral Theorem for all s ∈ R. On appropriate domains, A s m • A r m = A r+s m for all r, s ∈ R. For s > 0, the operator A −s m , called the Green operator of order s with mass parameter m, admits the representation (i) For s > 0, the Green operator of order s is an integral operator with density given by the Green kernel of order s with mass parameter m, where p t (x, y) is the heat kernel (i.e. the density for the operator e t∆/2 ).(ii) For each m > 0, the family (G s,m ) s>0 is a convolution semigroup of kernels, viz.
and the conclusion follows by Tonelli's Theorem and the definition (2.2) of G s,m .Assertions (ii) and (iii) are straightforward.

2.2.
The case of manifolds of bounded geometry.Let C ∞ c be the space of all smooth compactly supported functions on M. We recall some definitions of spaces of weakly differentiable functions on M.

2.2.1.
Bessel potential spaces.Fix m > 0, let p ∈ [1, ∞) and denote by p := p p−1 the Hölder conjugate of p ∈ (1, ∞).Following [52], we define the Bessel potential spaces L s,p m , s ≥ 0, as the space of all u ∈ L p so that u = A −s/2 m v for some v ∈ L p , endowed with the norm u L s,p m := v p .For s < 0, we define L s,p m as the space of all distributions u on M of the form u = A k m v, where v ∈ L 2k+s,p m and k is any integer so that 2k + s > 0, endowed with the norm u L s,p m := v L 2k+s,p m .As it turns out, the above definition is well-posed, i.e. independent of k, and we have the following result of R. S. Strichartz'.

Lemma 2.3 ([52], §4
).The spaces L s,p m , s ∈ R, are Banach spaces (Hilbert spaces for p = 2).The natural inclusion L s,p m ⊂ L r,p m , s > r, is bounded and dense for every r, s ∈ R and p ∈ (1, ∞).Furthermore, C ∞ c is dense in L s,p m for every s ∈ R, m > 0 and p ∈ (1, ∞).As a consequence, the , extends to a bounded bilinear form between L s,p m and L −s,p m , s > 0, thus establishing isometric isomorphisms between L s,p m and (L −s,p m ) , s ∈ R, p ∈ (1, ∞).For every m, s > 0, the space L s,p m coincides with the L p -domain of (−∆) s/2 , and the norm • L s,p m is equivalent to the graphnorm We note that, for m 1 , m 2 > 0, the spaces L s,p m1 = L s,p m2 coincide setwise, and the corresponding norms are bi-Lipschitz equivalent.For the sake of notational simplicity, we set H s m := L s,2 m for s ∈ R, m > 0.
2.2.2.Standard Sobolev spaces.For a given local chart on M let ∇ αi be the corresponding covariant derivatives.For smooth f : M → R and a non-negative integer k, we set ∇ 0 f := |f | and let ∇ k f be defined by For p ∈ (1, ∞), we denote by E k,p the space of all functions f ∈ C ∞ so that ∇ i f is in L p = L p (vol g ) for every 0 ≤ i ≤ k, and define the Sobolev space W k,p as the completion of E k,p with respect to the norm 2.2.3.Manifolds of bounded geometry.To simplify the presentation, at some places in the sequel we make the following assumption, corresponding to H ∞ in [4,Déf. 3].
Assumption 2.4.(M, g) has bounded geometry, i.e. the injectivity radius is bounded away from 0, and for every k ∈ N 0 there exists a constant Remark 2.5.It is the main result of [43] that, on an arbitrary smooth differential manifold, the conformal class [g] of any chosen Riemannian metric g contains a Riemannian metric g of bouded geometry.Thus, Assumption 2.4 poses no topological restriction on the class of manifolds we consider.
Our main interest lies in compact manifolds and in homogeneous spaces.All these spaces satisfy the above assumption.By Lemma 2.3 above and e.g.[56, §7.4.5], under Assumption 2.4, we have that W k,p * = W k,p and W k,p ∼ = L k,p m (bi-Lipschitz equivalence) for every integer k ≥ 0 and m > 0. Furthermore, L s,p m for s ∈ R may be equivalently defined via localization and pull-back onto R d , by using geodesic normal coordinates and corresponding fractional Sobolev spaces on R d , see [56, § §7.2.2, 7.4.5]or [25].In particular we have the following: Lemma 2.6.Under Assumption 2.4, all the standard Sobolev-Morrey and Rellich-Kondrashov embeddings hold for L s,p m .
We conclude this section with an auxiliary result.Proof.By duality, it suffices to show the statement for r, s > 0. In this case, by the definition of H t m , t > 0, and by the semigroup property of t → A t m , t > 0, The extension to H s m follows by the density of D in H s m , Lemma 2.3.Lemma 2.9.The space D embeds continuously into H s m for every s ∈ R and every m > 0.
Proof.A proof is standard in the case when s > 0 is a positive integer.The conclusion for general s follows since the identical inclusion H s m → H k m is continuous for every integer k ≤ s by the very definition of Bessel potential space.2.2.5.Heat-kernel estimates.We collect here some estimates for the heat kernel on (M, g), which we shall make use of throughout the rest of the work.We also provide estimates on its first and second derivatives, which we need for the Green kernel asymptotics in Section 6.These estimates are sharp.
Lemma 2.10.Let (M, g) be a Riemannian manifold of bounded geometry.Then: (i) there exists a constant C > 0, so that for all x, y ∈ M and every t > 0 (ii) there exists a constant C > 0 , so that for all x, y ∈ M and every t > 0 (2.4) (iii) there exists a constant C > 0, so that for all x, y ∈ M and every t > 0 (2.5) (iv) there exists a constant C > 0, so that for all x, y ∈ M and every t > 0 Proof.Throughout the proof C > 0 is a constant only depending on (M, g), possibly changing from line to line.(i) In light of the bounded geometry assumption we have the Gaussian heat kernel estimate for some 0 < r 0 < inj(M) and ν 0 > 0 [47,Thm. 4.2].The claim follows since vol g B r (x) ≥ Cr n for all r < inj(M) by virtue of [9,Prop. 14].
Then by [51,Thm. 1.1] we have where −K, K ≥ 0, is a lower bound of the Ricci curvature.By [47,Theorem 4.2] we have , where we used the volume-doubling property and consequently by (2.7) and (2.3) In order to estimate u(x, t) from below we use Corollary 1.2 in [36] and obtain for Kt ≤ 1 For Kt > 1 we use Corollary 1.7 in [36] |∇u for ν 0 and r 0 as in (i).We estimate the volume of the ball from below as in (i) by applying [9,Prop. 14].
Noting that ∂ t p t (x, y) = ∆p t (x, y) the result follows.
(iv) It follows from [34, Thm.2.1] that there exists a constant C > 3 depending on (M, g) so that, for all x, y ∈ M, Since p t ( • , y) is a solution to the heat equation, and using (2.5), we have for all t ∈ (0, 2] and every x, y ∈ M, for some constant C > 0 only depending on (M, g) and possibly changing from line to line.Combining this with the heat kernel estimate (2.3) yields the claim for t ≤ 2. For t ≥ 2 the claim follows from the bound for t ≤ 1 combined with the following inequalities for t ≥ 1: which concludes the proof.

The case of closed manifolds.
Let us now specialize our constructions to the case when M is additionally closed, i.e. compact and without boundary.
If M is closed, the operator (m 2 − 1 2 ∆) −1 is compact on L 2 (vol g ), and thus has discrete spectrum.We denote by (ϕ j ) j∈N0 the complete L 2 -orthonormal system consisting of eigenfunctions of −∆, each with corresponding eigenvalue λ j , so that (∆+λ j )ϕ j = 0 for every j.Since M is connected, we have 0 = λ 0 < λ 1 and ϕ 0 ≡ vol g (M) −1/2 .Weyl's asymptotic law implies that for some c > 0, Grounding.If M is closed, we further define the grounded Green operator of order s with mass parameter m as the (bounded) self-adjoint operator Å−s f dvol g .
We start by refining the heat-kernel estimates in Lemma 2.10 to the closed case.
Lemma 2.11 (Heat kernel estimates: compact case).Let (M, g) be a closed Riemannian manifold.Then, (i) there exists a constant C > 0, so that for all x, y ∈ M and every t > 0 (ii) for every ∈ N 0 there exists a constant C = C( ) > 0, so that for all x, y ∈ M and every t > 0 (2.12) (iii) there exists a constant C > 0, so that for all x, y ∈ M and every t > 0 Proof.(i) The estimate (2.10) was already shown in Lemma 2.10.We provide here an alternative proof which we subsequently adapt to the case of pt .For t ≥ 1, the estimate (2.10) immediately follows from the fact that by compactness of M the heat kernel is uniformly bounded on [1, ∞) × M × M. For t ≤ 1 it follows from the celebrated estimate of Li and Yau [35,Cor. 3.1], combined with the fact that vol g (B √ t (x)) ≥ 1 C t n/2 for each x ∈ M, which in turn follows from Bishop-Gromov volume comparison and compactness of M, see, e.g., [45,Lem. 9.1.36,p. 269].
Since −C ≤ pt (x, y) ≤ p t (x, y), the estimate (2.11) for t ≤ 2 follows immediately from the previous estimate.In order to prove (2.11) for t ≥ 2, note that, for t ≥ 1, uniformly in x, y ∈ M.Moreover, note that by the standard spectral calculus for ∆ and ultracontractivity of the heat semigroup, see e.g.[12, Thm.2.1.4],we may express the grounded heat kernel on M as the uniform limit of the series and with this we obtain This proves the claim.
(ii) It is shown in [53,Eqn. (1.1)] that for every x, y ∈ M for some constant C , henceforth possibly changing from line to line.As a consequence, In combination with the heat kernel estimate (2.10) from above this yields the claim for t ≤ 2. As in part (i), the claim for t ≥ 2 follows from the bound for t ≤ 1 together with the fact that, for t ≥ 1, according to the previous estimates (2.15), (2.10), and (2.14).
(iii) Let us first note that [34, Thm.2.1] holds with identical proof also in the case of closed M. Similarly to the proof of Lemma 2.10, it follows from [34, Thm.2.1] that there exists a constant C > 3 depending on (M, g) so that, for all x, y ∈ M, Since p t ( • , y) is a solution to the heat equation, and using (2.15), we have for all t ∈ (0, 2] and every x, y ∈ M, for some constant C > 0 depending on (M, g) and possibly changing from line to line.Combining this with the heat kernel estimate (2.3) yields the claim for t ≤ 2. Again, for t ≥ 2 the claim follows from the bound for t ≤ 1 combined with the following inequalities for t ≥ 1: ≤ C e −λ1t/2 .Lemma 2.12.If M is closed and s > 0, then Å−s m is an integral operator with density given by the grounded Green kernel of order s with mass parameter m ≥ 0, defined in terms of the grounded heat kernel, (2.17) Gs,m (x, y) .
For each m ≥ 0, the family ( Gs,m ) s>0 is a convolution semigroup of kernels, and Gs,m (x, • ) dvol g = 0 for all x ∈ M, s > 0.
Of particular interest will be Gs,0 , the massless grounded Green kernel of order s.
Proof of Lemma 2.12.Let us first observe that Gs,m (x, y) as defined above is finite for all x = y by virtue of (2.11).We claim that the integral is absolutely convergent for every f ∈ L 2 and a.e.x.Indeed, it defines an L 2 -function according to and since, due to (2.12), Thus, and moreover, (due to the absolute convergence of the integrals) by Fubini's Theorem, .
(b) For each s > 0, m ≥ 0 and x ∈ M, the distribution Gs,m (x, • )vol g is the unique distributional solution to Proof.As (a) is straightforward, we only prove (b).It is also standard that Gs,m (x, • )vol g is a distributional solution to (2.20), thus it suffices to show that the associated homogeneous equations A s m u = 0 and u | 1 = 0 admit a unique solution for every s ∈ R.
To this end, denote by D := {φ : ϕ ∈ D} the space of grounded test functions.Equivalently, we show that A s m : D → D is a bijection for every s ∈ R. The fact that A k m (D) ⊂ D for integer k holds by the standard Schauder estimates for elliptic operators (for closed manifolds see e.g.[38, Thm.III.5.2 (iii), p. 193]).This is readily extended to s ∈ R noting that the integral operator G s,m with kernel G s,m (x, • ) is a smoothing operator for s > n/2.
For m > 0, the injectivity on D (in fact on L 2 (vol g )) holds by Lemma 2.8, and the surjectivity by Lemma 2.2.For m = 0, the injectivity holds since ker A k 0 = ker(−∆ g ) k only consists of the constant functions for every non-negative integer k, and the surjectivity holds by Lemma 2.12.We omit the details.

Eigenfunction expansion.
We conclude the analysis of the closed case by discussing the expansion of the Green kernels G s,m and Gs,m in terms of eigenfunctions of the Laplace-Beltrami operator.
Lemma 2.14.Assume that M is closed.Then for all m > 0 and s > n/2, where the series is absolutely convergent for every x, y ∈ M.
Furthermore, for all m ≥ 0 and s > n/2, (Note that the summation now starts at j = 1.)In particular, Proof.By the spectral calculus (e.g.[12, Thm.2.1.4]),we may express the heat kernel on M as the uniform limit of the series  x,y (s) of the Laplace-Beltrami operator on M, introduced in [42].The massive grounded Green kernel Gs,m (x, y) is therefore the Hurwitz regularization of ζ ∆ with parameter m 2 .2.3.3.Sobolev spaces on compact manifolds.Again assume that M is closed, and let (ϕ j ) j∈N0 and (λ j ) j∈N0 be as above.Then for each m > 0 and s ∈ R, Definition 2.16.If M is closed we define the grounded Sobolev spaces for m ≥ 0 and s ∈ R by regarded as a subspace of H s m .
(i) For all m ≥ 0 and r, s ∈ R, is an isometry of Hilbert spaces.(ii) For all m > 0 and s ∈ R, (iii) For all m > 0 and s ∈ R, the spaces Hs m and Hs 0 coincide setwise, and the corresponding norms are bi-Lipschitz equivalent.
Similarly for s < 0, 2.4.The noise distance.Given any positive numbers s, m, a pseudo-distance ρ s,m on M, called noise distance (for reasons which become clear in Corollary 3.12), is defined by Indeed, symmetry and triangle inequality are immediate consequences of the fact that this is the L 2distance between p •/2 (x, •) and p •/2 (y, • ) w.r.t. a (possibly infinite) measure on R + × M. In the case of closed M, the analogous definition for p•/2 ( • , • ) results in ρs,m = ρ s,m .
Remark 2.18.Note that by the symmetry and the Chapman-Kolmogorov property of the heat kernel, Hence, for all s, m ∈ (0, ∞) and all x, y ∈ M with G s,m (x, y) < ∞, 3. The Fractional Gaussian Field.Let us now define Fractional Gaussian Fields.Theorem 3.1.For m > 0 and s ∈ R, there exists a unique Radon Gaussian measure µ m,s on D σ with characteristic functional Proof.Note that χ m,s (0) = 1 and that χ m,s is positive definite, e.g., [37, Prop.We omit the superscript M from the notation whenever apparent from context, and write h • ∼ FGF s,m to denote an m-massive Fractional Gaussian Field with regularity s.Here and henceforth, for random variables X • : ω → X ω on Ω the superscript • will indicate the ω-dependence.
The case h • ∼ FGF s,m with s = 0 is singled out in the scale of all FGF's on M as the only one independent of m.It corresponds to the Gaussian White Noise on M induced by the nuclear rigging D ⊂ L 2 (vol g ) ⊂ D , where we note that L 2 (vol g ) = H 0 m for all m > 0.
Remark 3.3.The White Noise W • on M is the D -valued, centered Gaussian random field uniquely characterized by either one of the following properties, see e.g. the monograph [32]: 3.1.Some characterizations.Let us now characterize the Fractional Gaussian Field h • ∼ FGF s,m in terms of the associated Gaussian Hilbert space.We recall that a Gaussian Hilbert space on (Ω, F , P) is a closed linear subspace of L 2 (Ω) consisting of centered Gaussian random variables, cf.e.g.[37,Dfn. 2.5].We say that a Gaussian Hilbert space Vice versa, every Gaussian Hilbert space on (Ω, F , P) linearly indexed by H −s m and satisfying is isomorphic to H s,m as a Hilbert space via the map The space H s,m is called the Gaussian Hilbert space of h • ∼ FGF s,m .
Corollary 3.5.Let h • ∼ FGF M s,m and Hs,m be any Gaussian Hilbert space linearly indexed by H −s m defined as in (3.4) and satisfying (3.5).Further suppose that there exists a D -valued Gaussian field X • on (Ω, F , P) so that X Remark 3.6 (Constructions with Schwartz functions).Suppose M = R n is a standard Euclidean space, and denote by S the space of Schwartz functions on M endowed with its canonical Fréchet topology, and by S σ the space of tempered distributions on M endowed with the weak topology σ(S , S ).Recall that S is a nuclear space, and embeds densely and continuously into H s m for every s ∈ R and m > 0. By the very same proof of Theorem 3.1, there exists a centered Gaussian field X • on Ω = S σ with characteristic functional satisfying (3.1) for every ϕ ∈ S .By comparison with the massless case, see e.g. the survey [37], the field X • too would deserve the name of massive Fractional Gaussian Field on M = R n .In fact, we have X • ∼ FGF M s,m in our sense.
Proof.Since the identical embedding D → S is continuous, the space S σ of tempered distributions on M embeds identically and continuously (in particular, measurably) into D σ .Thus, X • is in particular D -valued, and it may be regarded as defined on Ω = D σ .The conclusion follows in light of Corollary 3.5.f is linear, (3.5) shows that it is injective, and therefore an isomorphism of linear spaces.Analogously, f → h • | f is an isomorphism of linear spaces by (3.3).Thus, the map
In particular, we have the following: Corollary 3.7.For s > 0, h • ∼ FGF s,m is uniquely characterized as the centered Gaussian process with covariance Proposition 3.8.Let s ∈ R, m > 0, and h • ∼ FGF M s,m .Then, the following assertions hold: By Lemma 2.8, we have . Thus, similarly to the proof of the forward implication in Proposition 3.4, the equality in (3.7) extends from D to H , and m is a Gaussian Hilbert space on (Ω, F , P) linearly indexed by H . Furthermore, we conclude again from (3.7) and Lemma 2.8 that Again as in Proposition 3.4, the above equality extends from D to H , and we conclude that Hs−2k,m has covariance structure Corollary 3.9.The following assertions hold: where k is the only integer so that s − 2k ∈ [0, 2).(ii) if M is closed, then all the Fractional Gaussian Fields h • ∼ FGF M s,m for s ∈ R and m > 0 may be obtained from the White Noise W • on M as 3.2.Continuity of the FGF.The basic property concerning differentiability and Hölder continuity of FGF's is as follows.
Proposition 3.10.Let h • ∼ FGF M s,m .Then, the following assertions hold: In particular, the continuity of h • in the case s > n/2 will allow us to rewrite (3.6) in a more comprehensive and suggestive form.
Corollary 3.11.For each s > n/2 the centered Gaussian process h • ∼ FGF s,m is uniquely characterized by x, y ∈ M .(3.8) Corollary 3.12.For each s > n/2, the pseudo-distance ρ s,m is indeed a distance.It is given in terms of the process h x, y ∈ M .(3.9) s,m with s > n/2.Lemma 2.6 implies that H s m embeds continuously into a space of continuous functions on M by Morrey's inequality.As a consequence, δ x ∈ H −s m .Thus, Proposition 3.4 implies that h ω (x) := h ω | δ x is P-a.s.well-defined for every fixed x ∈ M. Together with Corollary 3.7, this proves the representation (3.9) in Corollary 3.12.
Combining (3.9) and Theorem 6.1 we have therefore that for some constant C α > 0. In particular, ω → h ω (x) − h ω (y) is a centered Gaussian random variable with covariance dominated by C α • d(x, y) α .Therefore, it has finite moments of all orders p > 1, and, for every such p, there exists a constant C α,p > 0 so that Since M is smooth, there exists an atlas of charts (U, Φ), with Φ : for some constant C U > 0 possibly depending on U .Define a random field on Φ(U ) by setting h By the standard Kolmogorov-Chentsov Theorem, e.g.[46, Thm.I.2.1], we conclude that, for every ε > 0 and every p > 1, the function h almost surely for all α ∈ (0, s−n/2).By arbitrariness of ε and p, and since α ranges in an open interval, we may conclude that h • Φ ∈ C 0,α (Φ(U )) almost surely for all α ∈ (0, s−n/2).Finally, since Φ is smooth, it follows that h • ∈ C 0,α (U ), and therefore that h • ∈ C 0,α loc (M) almost surely.(ii) Now assume that h • ∼ FGF M s,m with s > n/2+k +α with k ∈ N and α ∈ (0, 1).Note that (iii): Let K be a bounded convex subset of M with smooth boundary, and denote p K t the heat kernel with Neumann boundary conditions on K. Recall that a function f ∈ L 2 (M) belongs to W 1,2 (K)-the form domain for the Neumann heat semigroup on M-if and only if (3.12) lim by the very definition of the Neumann heat semigroup on K. Furthermore, the lim t→0 is in fact a monotone limit.In the case s > n/2 + 1, Theorem 6.1 below (applied with α = 1) implies that the continuous random function h where the last inequality follows from the Li-Yau estimate [35,Thm. 3.2] on the Neumann heat kernel.
Remark 3.13.The regularity of h • provided by Proposition 3.10 is sharp, in the sense that h

Series Expansions in the Compact
Case.If M is closed, Fractional Gaussian Fields may be approximated by their expansion in terms of eigenfunctions of the Laplace-Beltrami operator ∆.As before in §2.3.2, we denote by (ϕ j ) j∈N0 ⊂ D the complete L 2 -orthonormal system consisting of eigenfunctions of ∆, each with corresponding eigenvalue λ j , so that (∆+λ j )ϕ j = 0 for every j.Recall the representations of heat kernel (2.24), Green kernel (2.21), and grounded Green kernel (2.22) in terms of this eigenbasis.
Let now a sequence ξ • j j∈N0 of i.i.d.random variables on a common probability space (Ω, F , P) be given with ξ • j ∼ N (0, 1).For each > 0, define a random variable h Theorem 3.14.
(i) For every s ∈ R and f ∈ H −s m , the family ( h • | f ) ∈N is a centered, L 2bounded martingale on (Ω, F , P). (ii) As → ∞, it converges, both a.e. and in L 2 , to the random variable h | f • ∈ L 2 (Ω) given for a.e.ω by Proof.Assertion (i) and (ii) follow by standard arguments on centered Gaussian variables, e.g.[8, Thm.1.1.4].For (iii), observe that by definition, h | f • is a centered Gaussian random variable with variance where the first equality holds by orthogonality of (ϕ j ) j∈N0 and since ξ • j j∈N0 are i.i.d.∼ N (0, 1), the second equality since (ϕ j ) j∈N0 is a complete L 2 -orthonormal system of eigenfunctions of A m as well, and the third equality by the definition of the norm of H −s m .
Corollary 3.15.The family of random variables Theorem 3.16.For s > n/2, the series converges in L 2 (Ω) and almost surely on Ω for each x ∈ M. Moreover it converges on L 2 (Ω × M) and in L 2 (vol g ) almost surely.
Proof.The L 2 (Ω × M) as well as the L 2 (Ω) convergence follow by combining the identities and the fact that the terms on the right hand side of both equations converge to 0 as , → ∞ according to Weyl's asymptotics (2.9) and (2.21) respectively.The almost sure convergence for each x as well as the almost sure convergence for the L 2 (vol g ) sequence follow by Theorem 3.14 and Doob's Martingale Convergence Theorem.
3.4.The Grounded FGF.Assume now that M is closed.Then, the same arguments used to derive Theorem 3.1 also apply for the grounded norms, and in this case even for m ≥ 0.
In order to state the next result, let us set D := {ψ ∈ D : vol g | ψ = 0}, and denote by D the topological dual of D. We note that D is a nuclear space when endowed with the subspace topology inherited from D, since every linear subspace of a nuclear space is itself nuclear, e.g.[55, Prop.50.1, (50.3), p. 514].
Theorem 3.17.For m ≥ 0 and s ∈ R, there exists a unique Radon Gaussian measure μm,s on D with characteristic functional given by χm,s : Proof.Analogously to Theorem 3.1, it suffices to show that D embeds continuously into H−s m .In turn, this follows from the continuity of the embedding of D into H s m and Lemma 2.17(ii).
Definition 3.18.Let m ≥ 0 and s ∈ R. A grounded m-massive Fractional Gaussian Field on M with regularity s, in short: F GF M s,m , is any D -valued random field h • on Ω distributed according to μm,s .In the case m = 0, the field is called a grounded massless Fractional Gaussian Field on M with regularity s.
All results for the random fields FGF s,m have their natural counterparts for F GF s,m , now even admitting m = 0.In particular, we have the grounded versions of Corollary 3.7 and Theorem 3.16.
Corollary 3.19.For s > 0 and m ≥ 0, the random field h • ∼ F GF s,m is uniquely characterized as the centered Gaussian process with covariance Corollary 3.20.For s > n/2 and m ≥ 0, the series converges in L 2 (Ω) and almost surely on Ω for each x ∈ M. Moreover it converges on L 2 (Ω × M) and in L 2 (vol g ) almost surely.
In particular, h • ∼ F GF s,0 is given by h Lemma 3.21.For every s ∈ R and every m > 0, as an FGF M s,m , and thus it satisfies Proposition 3.10.Since ξ ω 1 ∈ D for every ω, the conclusion follows.
Remark 3.23.It is worth comparing the grounding of operators and fields presented above with the pinning for fractional Brownian motions in [22], where a Riesz field R s is defined as the centered Gaussian field with covariance for some fixed 'origin' o ∈ M. In particular, while grounding on a compact manifold (M, g) is canonical, the pinning of a Riesz field at o ∈ M, and hence the properties of the corresponding random Riemannian manifold (see §4 below), would depend on o.

Dudley's Estimate.
A crucial role in our geometric estimates and functional inequalities for the Random Riemannian Geometry is played by estimates for the expected maximum of the random field.The fundamental estimate of Dudley provides an estimate in terms of the covering number w.r.t. the pseudo-distance ρ s,m , introduced in (2.25).Notation 3.24.For any pseudo-distance ρ on M, we denote by N ρ (ε) the least number of ρ-balls of radius ε which are needed to cover M. When ρ = ρ s,m we write N s,m (ε) in place of N ρs,m (ε).Theorem 3.25 ([33,Thm. 11.17]).Fix s > n/2 and m ≥ 0 Then, for h ∼ FGF M s,m (and in the compact case also for h In Section 6 we will study in detail the asymptotics of the Green kernel close to the diagonal and in particular derive sharp estimates for the noise distance ρ in terms of the Riemannian distance d.This will lead to sharp estimates for the covering numbers N s,m (ε) and thus in turn to sharp estimates for the expected maximum of the random field.
4. Random Riemannian Geometry.Let a Riemannian manifold (M, g) be given together with a Fractional Gaussian Field h • ∼ FGF M s,m with s > n/2 and m > 0. If M is compact, we alternatively can choose h • ∼ F GF M s,m with s > n/2 and m ≥ 0. In the sequel, we assume that either M is closed or m > 0 and (M, g) has bounded geometry.
For almost every ω ∈ Ω, by Propositions 3.10 and 3.22, h ω is a continuous function on M. For each such ω, we consider the Riemannian manifold (M, g ω ) with g ω := e 2h ω g , (4.1) the new metric being the conformal change of the metric g by the conformal factor h ω .In other words, we consider the random Riemannian manifold with the random Riemannian metric g Assuming that M is closed, for a.e.ω, the Riemannian metric g ω is of class C k on M for k := s − n/2 − 1 ≥ 0, where we set a := min(Z ∩[a, ∞)).In particular, for s > n/2 + 2, it is almost surely of class C 2 , and the Riemannian manifolds M ω may be studied by smooth techniques.Our main interest in the sequel will be in the case s ∈ (n/2, n/2 + 2] where no such techniques are directly applicable and where we have no classical curvature concepts at our disposal.4.1.Random Dirichlet Forms and Random Brownian Motions.Our approach to geometry, spectral analysis, and stochastic calculus on the randomly perturbed Riemannian manifolds (M, g • ) will be based on Dirichlet-form techniques.Before going into details, let us recall some standard results on the canonical Dirichlet form on the 'un-perturbed' Riemannian manifold.Here g * denotes the inverse metric tensor obtained from g by musical isomorphism, d the differential on M, and ∇ the gradient; for functions in W 1,2 * , differentials and gradients have to be understood in the weak sense.In fact, however, C ∞ c is dense in the form domain F and thus in (4.3) we can restrict ourselves to ϕ, ψ ∈ C ∞ c .The form (E, F) is a regular, strongly local, conservative Dirichlet form properly associated with the standard Brownian motion B on (M, g), the Markov diffusion process with transition kernel p t introduced in §2.
The canonical Dirichlet form and the Laplace-Beltrami operator on (M, g) uniquely determine each other by Under conformal transformations with non-differentiable weights, however, the latter no longer admits a closed expression whereas the former still is easily representable.Theorem 4.3.Let h • ∼ FGF s,m with m > 0 and s > n/2.Then, (a) for P-a.e. ω ∈ Ω, the quadratic form is closable on L 2 (e nh ω vol g ); (b) its closure (E ω , F ω ) is a regular, irreducible, strongly local Dirichlet form, properly associated with an e nh ω vol g -symmetric Markov diffusion process B ω on M; (c) the generator of the closed bilinear form (E ω , F ω ), denoted by ∆ ω , is the unique self-adjoint operator on L 2 (e nh ω vol g ) with D(∆ ω ) ⊂ F ω and coincides with the Riemannian distance d ω on M given by Proof.(a) Let ω be given such that h ω is continuous.Then both σ := e nh ω and ρ := e (n−2)h ω are positive and in L 1 loc and so is 1/ρ.In particular, the weights thus satisfy the so-called Hamza condition.A proof of closability under this condition, in the case M = R n , is given in [40, §II.t>0 on M ω has an integral kernel p ω t (x, y) which is jointly locally Hölder continuous in t, x, y; (iii) for every starting point, the distribution of the Brownian motion on M ω is uniquely defined.(iv) For all x, y ∈ M, Lemma 4.6 ('Fukushima decomposition', see [19, §6.3]).
(a) For each continuous ψ ∈ W 1,2 * , there exist a unique martingale additive functional M [ψ] and a unique continuous additive functional N [ψ]  which is of zero energy such that The quadratic variation of M [ψ] is given by for any choice of a Borel version of the function loc , there exists a unique local martingale additive functional where, for every n ∈ N, we let M [ψn] be the martingale additive functional associated with a function such that ψ = ψ n a.e. on M n , for some exhausting sequence of relatively compact open sets M n M, and where τ n := inf {t ≥ 0 : loc , a super-martingale, multiplicative functional is defined by For the defining properties of 'martingale additive functionals' and of 'continuous additive functionals of zero energy' (as well as for the relevant equivalence relations that underlie the uniqueness statements) we refer to the monograph [19].
We are now able to provide an explicit construction of the Brownian motion on the randomly perturbed manifold (M, g • ) which previously was introduced by abstract Dirichlet form techniques.
Theorem 4.8.Let h • ∼ FGF s,m with m > 0 and s > n/2 + 1.Then for P-a.e. ω ∈ Ω, the process B ω is a time-changed Girsanov transform of the standard Brownian motion B on (M, g).More precisely: (a) For q.e.x ∈ M, the law P ω x is locally absolutely continuous up to life-time ζ ω w.r.t. the law P x of B on the natural filtration (F t ) t≥0 of B, viz.
Remark 4.9 (On conservativeness).It is not clear to the authors whether the Dirichlet form (E ω , F ω ) is P-a.s.conservative.In particular, the random Brownian motion (4.10) may in principle have finite lifetime ζ ω .
Proof of Theorem 4.8.By Proposition 3.10, the random field h • lies a.s. in W 1,2 loc ∩ C(M).Thus, also e (n−2)h ω /2 ∈ W 1,2 loc ∩ C(M), and we may consider the Girsanov transform (E φ , F φ ), e.g.[19, §6.3], of the canonical form (E, F) by the function φ = φ ω := e (n−2)h ω /2 , satisfying By standard results in the theory of Dirichlet forms, (E φ , F φ ) is a regular Dirichlet form on L 2 (φ 2 vol g ), properly associated with the Girsanov transform B φ of the standard Brownian motion B. Indeed, choosing G n := B n (o), n ∈ N, for some fixed o ∈ M yields a nondecreasing sequence of (quasi-)open sets with n G n = M such that φ, 1/φ and φ|∇φ| ∈ L 2 (G n , vol g ).Then, according to [18,Thm. 4.9], the Girsanovtransformed process is properly associated with the quasi-regular Dirichlet form obtained as the closure of E φ with pre-domain n∈N F Gn where as usual F Gn :={ψ ∈ F : ψ = 0 q.e. on M \ G n }.Since obviously C ∞ c ⊂ n∈N F Gn ⊂ F, this Dirichlet form is even regular.Now, let us denote by E φ,µ , F φ,µ the time-changed form, e.g.[19, §6.2], of (E φ , F φ ) with respect to the measure µ = µ ω := e 2h ω vol g .It is again standard that E φ,µ , F φ,µ is a regular Dirichlet form on L 2 (φ 2 µ), properly associated with the time change B φ,µ of B φ induced by µ.Since φ 2 µ = e nh ω vol g , the form E φ,µ coincides on C ∞ c with the form E ω defined in (4.4).By regularity of both forms we conclude that E φ,µ , F φ,µ = (E ω , F ω ) is the canonical form on the Riemannian manifold M ω = (M, g ω ), properly associated with the corresponding Brownian motion B ω = B φ,µ .
In order to characterize the law of B ω as in assertion (a), (b), it suffices to note the following.Since B is conservative, it is noted in e.g.[17, §5 a)] that the process where the functions log φ n are given as in Lemma 4.6(b) for log φ in place of ψ, and the stopping times τ n are defined as τ n := inf {t > 0 : X t / ∈ M n } with M n again as in Lemma 4.6(b).The conclusion follows by letting n to infinity, since B ω is a time change of B φ , and therefore: t with λ ω t as in Equation (4.12) for each t > 0, cf.[19, Eqn.(6.2.5)]; assertion (c) is [19, Exercise 6.2.1].

Geometric and Functional
Inequalities for RRG's.Given a Riemannian manifold (M, g) and the intrinsically defined FGF noise h • , we ask ourselves: how do basic geometric and spectral theoretic quantities of (M, g) change if we switch on the noise?For instance, will E vol g • (M) be smaller or larger than vol g (M)?How about λ • 0 , the random spectral bound, or λ • 1 , the random spectral gap?Can we estimate them in terms of the unperturbed spectral quantities?Can we estimate in average the rate of convergence to equilibrium on the random manifold?
In the following, let a Riemannian manifold (M, g) of bounded geometry be given and a random field h • ∼ FGF s,m with m > 0 and s > n/2.As before, put g • = e 2h • g.

5.
1. Volume, Length, and Distance.We will compare the random volume, random length, and random distance in the random Riemannian manifold (M, g • ) with analogous deterministic quantities in geometries obtained by suitable averages of the conformal weight.Recall that θ(x) := G s,m (x, x) = E[h • (x) 2 ] ≥ 0 and put g n := e n θ g, g 1 := e θ g .
Further, recall that for given ω with continuous h ω , the volume of a measurable subset A ⊂ M w.r.t. the Riemannian tensor g ω is given by Similarly, the length of an absolutely continuous curve γ : [0, 1] → M w.r.t. the Riemannian tensor g ω is given by In particular, Proof.It suffices to note that Proposition 5.2.For any absolutely continuous curve γ : Proof.It suffices to note that Proof.Given x and y, let γ be any absolutely continuous curve connecting them.Then This proves the upper bound.
For the lower bound, let us assume that inf z∈M h • (z) is finite for almost every ω.Otherwise, the lower bound is trivially satisfied.Then (M, g • ) is complete and locally compact so that there exists a constant speed geodesic γ ω : [0, 1] → M connecting x and y.Then Then, by Jensen's inequality and symmetry of the random field, 5.2.Spectral Bound.The L 2 -spectral bound for (M, g ω ) is defined by By the standard variational characterization of the spectrum via Rayleigh-Riesz quotients we have that Note that λ 0 is not necessarily 0, e.g.λ 0 = (n−1) 2

4
for the hyperbolic space of curvature −1.
Proof.Let C ∞ c be endowed with the C 1 -topology τ 1 , and note that (C ∞ c , τ 1 ) is separable.Further note that, P-almost surely, (C ∞ c , τ 1 ) embeds continuously into (F ω , (E ω ) 1 ), and that this embedding has dense image since (E ω , F ω ) is a regular Dirichlet form.Therefore, there exists a countable 1 -dense in F ω for P-a.e. ω.As a consequence, the variational characterization (5.1) holds as well when replacing C ∞ c by D. Since the integrals' quotient in this characterization is measurable as a function of ω, the corresponding infimum over D is as well a measurable function of ω, since D is countable and the infimum of any countable family of measurable functions is again measurable.|∇u| 2 e (n−2)h ω dvol g .
Integrating w.r.t.dP(ω) and applying Hölder's inequality yield Since this holds for all u we conclude that Remark 5.6.Following the argumentation from the proof of Theorem 5.10 below, we can also derive a two-sided, pointwise estimate for the spectral bound, valid for almost every ω: with α := 2(n − 1) if n ≥ 2 and α := 2 if n = 1.

Spectral Gap.
In the following we assume that M is closed, and we let vol ω g = vol g ω := e nh ω vol g .Then, the Laplacian ∆ g ω has compact resolvent and, in particular, it has discrete spectrum.The spectral gap is defined by λ ω 1 := inf spec(−∆ g ω ) \ {0} .Denoting by the mean value of f w.r.t. the measure vol ω g , the spectral gap has the variational representation Hence the spectral gap is the smallest non-zero eigenvalue of the Laplacian and the inverse of the smallest constant for which the Poincaré inequality holds.By the very same proof of the measurability of the random spectral bound (Lemma 5.4) we have as well the following: Lemma 5.7 (Measurability of the spectral gap).The function ω → λ ω 1 is measurable.
The function h • is P-a.s.continuous by Proposition 3.10, thus P-a.s.bounded by compactness of M. As a consequence, the L 2 (vol ω g )-norm is bi-Lipschitz equivalent to the L 2 (vol g )-norm.Thus, the spaces L 2 (vol g ) and L 2 (vol ω g ) coincide as sets.Again by boundedness of h ω , the form , and analogously for ω.By the equivalence of the L 2 -norms and forms established above, the norm E Since M is compact, both forms are regular, thus F ω too coincides with F as a set and the bi-Lipschitz equivalence of E and (E ω 1 ) 1/2 extends to F. Given ω with continuous h ω , let P ω t := e t∆ ω /2 , t > 0, denote the heat semigroup on L 2 (vol ω g ).For each f ∈ L 2 (vol g ), the functions P ω t f will converge as t → ∞ to π ω f .The rate of convergence is determined by λ ω 1 , viz.
Proof.Firstly, let us discuss some heuristics.For α > 0, set , and denote by (G ω α ) α≥0 the L 2 (vol ω g )-resolvent semigroup of (E ω , F ω ), satisfying (e.g.[40, Thm.I.2.8, p. 18]) We conclude the measurability in ω of the left-hand side from that of the right-hand side which is clear from the identifications of sets L 2 (vol ω g ) = L 2 (vol g ) and F ω = F.For fixed t, α > 0, writing the series expansion of e tα(αGα−1) we conclude that is measurable as a function of ω, since P ω t = lim α→∞ e tα(αGα −1) .The measurability of ω → π ω u | v may be concluded in a similar way, which would then show the assertion.
In order to make this argument rigorous, we resort to theory of direct integrals of quadratic forms in [13].In light of Corollary 3.15, we may assume with no loss of generality that (Ω, F , P) be the completion of a standard Borel space.Let D ⊂ C ∞ c be the countable Q-vector space simultaneously dense in (F ω , (E ω ) 1/2 1 ) for P-a.e. ω ∈ Ω constructed in the proof of Lemma 5.4.Now, let ω → F ω be the measurable field of Hilbert spaces with underlying linear space S := ω∈Ω F ω = F Ω in the sense of [15,§II.1.3,Dfn. 1,p. 164] with D as a fundamental sequence in the sense of [15, §II.1.3,Dfn.1(iii), p. 164].Further let ω → L 2 (vol ω g ) be the measurable field of Hilbert spaces with underlying space generated by S as above in the sense of [15,§II.1.3,Prop. 4,p. 167].In particular, for every f ∈ L 2 (vol g ), the constant field ω → f ∈ L 2 (vol ω g ) is a measurable vector field.Furthermore, since all constant fields are elements of the direct integral of Hilbert spaces It follows that ω → (P ω t − π ω ) is a measurable field of bounded operators.Now fix f ∈ L 2 (vol g ).Since the constant field ω → f is measurable as discussed above, ω → (P ω t − π ω )f too is a measurable vector field, by definition of measurable field of bounded operators.Thus, its norm ω → too is measurable, which concludes the assertion.
Proof.With Theorem 5.10 we estimate By the convexity we may apply Jensen's inequality and get the estimate Moreover, again by Jensen's inequality which yields the claim.
6.1.Green Kernel Asymptotics.The next Theorem illustrates the asymptotic behavior of the higherorder Green kernel G s,m (x, y) close to the diagonal in terms of the Riemannian distance d(x, y).The statement of the Theorem is sharp, as readily deduced by comparison with the analogous statement for Euclidean spaces, see Equation (6.7) below.Theorem 6.1.Let (M, g) be a Riemannian manifold with bounded geometry, and s > n/2.Then, for every α ∈ (0, 1] with α < s − n/2 there exists a constant C α,m > 0 so that for all m > 0 and all x, y ∈ M. If M is additionally closed, then additionally ρ s,m (x, y) = Gs,m (x, x) + Gs,m (y, y) − 2 Gs,m (x, y) for all m ≥ 0. In this case, the constant C α can be chosen such that with α * := α whenever α ∈ (0, 1/2] and α * := α − 1/2 whenever α ∈ (1/2, 1] and C > 0 is a constant only depending on M.
Proof.Note that Thus it suffices to prove the claim for Gs,m .Assume first that M is closed.Throughout the proof, C > 0 denotes a finite constant, only depending on M but possibly changing from line to line.For x, y ∈ M denote by ([x, y] r ) r∈[0,1] any constant speed distance-minimizing geodesic joining x to y.
Assume first that σ := 2α ∈ (0, 1].Then, By (2.12) For the last inequality, we used the fact that the function R → R (1−σ)/2 exp(−R/C) is uniformly bounded on (0, ∞), independently of σ ∈ (0, 1].Assume now that σ := 2α ∈ (1, 2].Then, similarly to the previous case, sup x,y∈M x =y Γ(s) d(x, y) σ Gs,m (x, x) + Gs,m (y, y) − 2 Gs,m (x, y) By (2.13), similarly Assume now that M has bounded geometry.The proof holds in a similar way to the case of closed M, having care to replace the application of (2.12) with (2.4) and (2.13) with (2.6).Corollary 6.2.Let M be a compact manifold.Then, there exists a constant C > 0 such that for all m ≥ 0 and all x, y ∈ M, The estimate in the third case is not sharp.The previous Theorem provides estimates ρ s,m ≤ C α d α for every α < s − n/2.(As α → s − n/2, however, the constant C α will diverge.) Proof.The eigenfunction representation (2.22) of Gs,m yields that Hence, ρ 2 s,m (x, y) ≤ ρ 2 s,0 (x, y) for all x, y, s, m under consideration.Moreover, for all x, y ∈ M the function Therefore, the first case s ≥ n 2 + 2 follows from the choice s = n 2 + 2 which is included in the second case.In the second case s ∈ ( n 2 +1, n 2 +2], with the choice α = 1 the previous Theorem provides the estimate In the third case s ∈ ( n 2 , n 2 + 1], with the choice α = 1 2 (s − n 2 ) ∈ (0, 1/2] the previous Theorem provides the estimate 6.2.Supremum estimates.Now let us combine Dudley's estimate, Theorem 3.25, for the supremum of the Gaussian field with our Hölder estimate, Corollary 6.2, for the noise distance.Theorem 6.3.For every compact manifold M there exists a constant C = C(M) such that for every h • ∼ F GF M s,m with any m ≥ 0, Proof.Recall the Notation 3.24 for the covering number of a pseudo-metric, and let ρ = ρ s,m be as in (2.25).For the Riemannian distance d on the compact manifold M, for some constant C > 0.
Proof.For convenience, we provide two proofs.The first one is based on direct calculations.(For the third equality above, we used the monotonicity of the integrand in r, and for the fifth, we used integration by parts.)In the case s = Proof.The first claim is an immediate consequence of the analogous formula for the heat kernel: The second claim follows from the first one combined with (6.5) according to   Observe that for all m > 0 as r → 0, GS 2  1,0 (r) G R The representations in (6.17) thus follow from the fact that the functions u 2 and u 3 given by the respective right-hand sides of (6.17
M), and is therefore the Fourier transform of a unique centered Gaussian field with variance u 2 H −s m by Bochner-Minlos Theorem applied to the nuclear space D (M).

Lemma 2
H s m is an isometry of Hilbert spaces for every r, s ∈ R and m > 0.

2. 2 . 4 .
Test functions.Denote by D := C ∞ c the space of smooth compactly supported functions on M endowed with its canonical LF topology.It is noted in the comments preceding [27, Ch.II, Thm. 10, p. 55] that D is a nuclear space.We denote by D the topological dual of D, and by • | • = D • | • D the canonical duality pairing, extending the L 2 (vol g )-scalar product.The weak topology σ(D , D) is the coarsest topology for which all functionals of the form • | ϕ , with ϕ ∈ D, are continuous.We write D σ for the space D endowed with the weak topology.Recall that a set B ⊂ D is bounded if for every neighborhood U ⊂ D of the origin in D there exists λ ≥ 0 such that B ⊂ λU .The strong topology β(D , D) on D is the topology of uniform convergence on bounded sets in D, e.g.[55, II.19,Example IV, p. 198].We write D β for the space D endowed with the strong topology.

Remark 2 . 15 .
The grounded Green kernel Gs,0 (x, y) coincides, up to the multiplicative factor 2 s , with the celebrated Minakshisundaram-Pleijel ζ-function ζ ∆ 2.4].Furthermore, χ m,s is additionally continuous on D, since D embeds continuously into H −s m for every s ∈ R and m > 0 by Lemma 2.9.Note that β(D , D) is finer than σ(D , D), hence every Radon probability measure on D β restricts to a Radon probability measure on D σ .Since D is nuclear, by Bochner-Minlos Theorem in the form [57, §VI.4.3,Thm.4.3, p. 410], there exists a Radon probability measure µ m,s on D β , and the conclusion follows by restricting this measure to a (non-relabeled) Radon measure on D σ .Everywhere in the following, (Ω, F , P) denotes a probability space supporting countably many i.i.d.Gaussian random variables.Definition 3.2.Let m > 0 and s ∈ R.An m-massive Fractional Gaussian Field on M with regularity s, in short: FGF M s,m , is any D -valued random field h • on Ω distributed according to µ m,s .

Proposition 3 . 4 .
For every ϕ ∈ D, the map t → χ m,s (tϕ) as in(3.1)  is analytic in t around t = 0. Differentiating it twice at t = 0 shows that the assignment Dϕ → h • | ϕ defines an isometry of D, • H −s m into L 2 (Ω).By density of D in H −sm , the latter extends to a linear isometry H −s m → L 2 (Ω).Thus, by construction, H s,m forms a closed linear subspace of L 2 (Ω).By the definition of χ m,s , the random variable h • | ϕ has centered Gaussian distribution with variance ϕ 2 H −s m for every ϕ ∈ D. By the H −s m -continuity in ϕ of the corresponding characteristic function, the latter distributional characterization extends to H −s m which yields (3.3).Vice versa, let Hs,m be as in (3.4) and (3.5).Since the indexing assignment ι : f → X • m for every r ∈ R. Proof.(i) Fix k ∈ N. Since A m : D → D, the operator A k m : D → D is well-defined on D by transposition.Thus, A k m h • is P-a.s. a well-defined element of D .By definition of A k m : D → D , we have (3.7)

m.
By the converse implication in Proposition 3.4, Hs−2k,m is isomorphic as a Hilbert space to the Gaussian Hilbert space H s−2k,m of an FGF M s−2k,m .Thus, A k m h • ∼ FGF M s−2k,m by Corollary 3.5.(ii) Since M is closed, A r m : D → D for every r ∈ R, thus A r m : D → D is well-defined by transposition.The rest of the proof follows exactly as in (i) replacing k by (s − r)/2.
m by Proposition 3.8, and A −k/2 m : C 0,α (M) → C k,α (M) for every k ∈ N. Thus the claim follows by the previous part (i).

1
allows us to easily switch between the random fields FGF M s,m and F GF M s,m , as in the next Lemma.