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Abstract Wiener Space and Gaussian Free Fields

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Abstract Wiener Space and Gaussian Free Fields SPHERICAL AVERAGES OF GAUSSIAN FREE FIELDS LINAN CHEN Abstract. In this note we study the properties of the spherical average (circle average if dim=2) of Gaussian free elds as a stochastic process parametrized by the radius. In particular, we give explicit formulas for the covariance of the Gaussian family which consists of the spherical averages as well as certain functionals of the spherical averages. We further prove the Markov property for various processes involving the spherical averages. These results are useful in the study of point-wise approximation of generic element of Gaussian free elds. 1. Introduction: Abstract Wiener Space and Gaussian Free Fields The theory of Abstract Wiener Space (AWS), rst introduced by Gross [3], provides an analytical foundation to construct and study Gaussian measures in innite dimensions. To be specic, given a real separable Banach space E, a non- degenerate centered Gaussian measure W on E is a Borel probability measure such that for every x∗ 2 E∗nf0g, the functional x 2 E 7! hx; x∗i 2 R has non-degenerate centered Gaussian distribution under W, where E∗ is the space of bounded linear functionals on E, and h·; x∗i is the action of x∗ 2 E∗ on E. Further assume H is a real separable Hilbert space which is continuously embedded in E as a dense subspace. Then E∗ can also be continuously and densely embedded into H, and for any ∗ ∗, there exists a unique such that ∗ for x 2 E hx∗ 2 H hh; x i = (h; hx∗ )H all h 2 H. Under this setting if the Gaussian measure W on E has the following Fourier transform: 2 ! kh ∗ k W [exp (i h·; x∗i)] = exp − x H for all x∗ 2 E∗; E 2 or equivalently, if h·; x∗i under W is a centered Gaussian random variable with variance 2 for every ∗ ∗, then the triple is called an Abstract khx∗ kH x 2 E (H; E; W) ∗ ∗ Wiener Space. Moreover, since fhx∗ : x 2 E g is dense in H, the mapping ∗ 2 I : hx∗ 2 H 7! I (hx∗ ) := h·; x i 2 L (W) can be uniquely extended as a linear isometry between H and L2 (W). The ex- tended isometry, also denoted by I, is called the Paley-Wiener map and its images fI (h): h 2 Hg, known as the Paley-Wiener integrals, form a centered Gaussian family whose covariance is given by W for all E [I (h) I (g)] = (h; g)H h; g 2 H: Date: September 2014. Key words and phrases. Abstract Wiener Space, Gaussian Free Fields, Spherical Averages, Bessel Functions. 1 SPHERICAL AVERAGES OF GAUSSIAN FREE FIELDS 2 It is clear that although W is a measure on E, it is the inner product of H that fully determines the covariance structure of W. H is known as the Cameron-Martin space. In fact, the theory of AWS says that given any separable Hilbert space H, one can always nd E and W such that the triple (H; E; W) forms an AWS. On the other hand, given a separable Banach space E, a non-degenerate centered Gaussian measure W on E must exist in the form of an AWS. For further discussions on the construction and the properties of AWS, we refer to [3], [4] and x8 of [5]. We now apply the general theory of AWS to study Gaussian measures on function or generalized function spaces. To be specic, given s 2 R and ν 2 N, ν ≥ 2, consider the following inner product on 1 ν , the space of compactly supported Cc (R ) smooth functions on ν : for every 1 ν , R φ, 2 Cc (R ) s (φ, ) := ((I − ∆) φ, ) 2 ν s L (R ) s 1 2 ^ ^ = ν 1 + jξj φ (ξ) (ξ)dξ; (2π) ˆ ν R where denotes the Fourier transform. The closure of 1 ν under is the ^· Cc (R ) (·; ·)s Sobolev space Hs := Hs (Rν ) which will be taken as the Cameron-Martin space. According to above, there exists a separable Banach space Θs := Θs (Rν ) and the Gaussian measure Ws := Ws (Rν ) on Θs such that the triple (Hs; Θs; Ws) forms an AWS, to which we refer as the dim-ν order-s Gaussian free eld (GFF). It's clear that the covariance of such a GFF is characterized by the kernel of the operator s (I − ∆) on Rν . For some special values of (ν; s) we have rather explicit formulations of the abstract theory introduced above. For example, when ν+1 , it's proven that s = 2 ν+1 Θ 2 can be taken as ν+1 ν jθ (x)j Θ 2 := θ 2 C (R ) : lim = 0 ; jxj!1 log (e + jxj) equipped with the norm θ jθ (x)j kθk ν+1 := = sup : Θ 2 ν log (e + |·|) u x2R log (e + jxj) So the dim- order- ν+1 GFF actually consists of continuous functions. By the Riesz ν 2 ν+1 ? representation theorem, every λ 2 Θ 2 is characterized as a Borel measure on Rν with the property that log (e + jxj) jλj (dx) < 1: ˆ ν R ν+1 − ν+1 In addition, if we identify the dual space of H 2 with H 2 , then λ can also be − ν+1 treated as an element in H 2 and ν+1 1 − 2 2 2 2 ^ kλk− ν+1 = ν 1 + jξj λ (ξ) dξ 2 (2π) ˆ ν R 1−ν π 2 = ν+1 exp (− jx − yj) λ (dx) λ (dy) : ¨ ν ν Γ 2 R ×R This further implies that the map 1−ν − ν+1 π 2 (1.1) 2 λ 7! hλ := (I − ∆) λ = ν+1 exp (− jx − yj) λ (dy) ; ˆ ν Γ 2 R SPHERICAL AVERAGES OF GAUSSIAN FREE FIELDS 3 ν+1 ν+1 ∗ gives the unique element hλ 2 H 2 such that the action of λ 2 Θ 2 restricted ν+1 ν+1 on H 2 , denoted by h·; λi ν+1 , coincides with (·; hλ) ν+1 on H 2 . Moreover, H 2 2 − ν+1 ν+1 being an isometry between H 2 and H 2 , the map (1.1) naturally extends to − ν+1 λ 2 H 2 and we still denote the image by hλ. Therefore, all the Paley-Wiener n − ν+1 o integrals, written as I (hλ): λ 2 H 2 , forms a centered Gaussian family with the covariance ν+1 W 2 E [I (hλ) I (hη)] = (hλ; hη) ν+1 = (λ, η) ν+1 2 − 2 − ν+1 for every λ, η 2 H 2 : Finally, for general , s2 is the isometric image of s1 under the s1; s2 2 R H H s1−s2 Bessel-type operator (I − ∆) 2 . Therefore, if s ν+1 − s ν+1 Θ := (I − ∆) 4 2 Θ 2 ; and s − ν+1 + s ν+1 W := (I − ∆) 4 2 W 2 ; ? then the triple (Hs; Θs; Ws) forms the dim-ν order-s GFF. The formulations above ν+1 − s will follow accordingly by the action of (I − ∆) 4 2 . It's also clear from this aspect that with xed dimension, the larger the order s is, the more regular the GFF is; on the other hands, when s is xed, the higher the dimension is, the more singular the GFF becomes. In most of the cases that of interest to us, a generic element of GFF is only a tempered distribution which is not dened pointwise. 2. Spherical Averages of Gaussian Free Fields Throughout this note, we assume s ≥ 1 and ν ≥ 2. As shown in the previous section, for most (ν; s), if θ is a generic element of the dim-ν order-s GFF, i.e., θ is sampled from Θs under Ws, θ is only a generalized function and θ (x) may not be dened. Hence, to study the behavior of θ, we will need some proper approximations of pointwise value of θ. To achieve this, we consider the average of θ over a sphere (circle in the case when ν = 2) centered at x with radius t > 0. To make this precise, we need to introduce some notations. Let S (x; t) be the ν −1 dimensional sphere centered at ν with radius , x the surface measure x 2 t > 0 σt ν R on , νπ 2 ν−1 the surface area of with , and S (x; t) αν (t) := ν t S (x; t) αν := αν (1) Γ( 2 +1) σx nally σ¯x := t the average over S (x; t). We simply omit x when x is the t αν (t) origin. Under this setting, we have the following simple fact about x (proof is σ¯t omitted). Lemma 1. For every ν and , x −s ν and its Fourier transform x 2 R t > 0 σ¯t 2 H (R ) is given by ν (2π) 2 2−ν x i(x,ξ) ν 2 σ¯ct (ξ) = e R · (t jξj) J ν−2 (t jξj) αν 2 for every ν , where is the standard Bessel function of the rst kind. ξ 2 R Jµ Since x −s ν , we can apply x to the GFF element in the sense of σ¯t 2 H (R ) σ¯t θ x the Paley-Wiener integral. Namely X (θ) := I h x (θ) is well-dened for W- t σ¯t a.e. θ 2 Θs as a Gaussian random variable, which, heuristically speaking, gives SPHERICAL AVERAGES OF GAUSSIAN FREE FIELDS 4 the spherical average of θ.
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