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 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 and Gaussian Free Fields The theory of (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 E, a non- degenerate centered Gaussian W on E is a Borel such that for every x∗ ∈ E∗\{0}, the functional x ∈ E 7→ hx, x∗i ∈ 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∗ ∈ E∗ on E. Further assume H is a real separable 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 ∈ E hx∗ ∈ H hh, x i = (h, hx∗ )H all h ∈ H. Under this setting if the W on E has the following Fourier transform:

2 ! kh ∗ k W [exp (i h·, x∗i)] = exp − x H for all x∗ ∈ E∗, E 2 or equivalently, if h·, x∗i under W is a centered Gaussian random variable with 2 for every ∗ ∗, then the triple is called an Abstract khx∗ kH x ∈ E (H,E, W) ∗ ∗ Wiener Space. Moreover, since {hx∗ : x ∈ E } is dense in H, the mapping ∗ 2 I : hx∗ ∈ H 7→ I (hx∗ ) := h·, x i ∈ 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 {I (h): h ∈ H}, known as the Paley-Wiener , form a centered Gaussian family whose covariance is given by W for all E [I (h) I (g)] = (h, g)H h, g ∈ 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 §8 of [5]. We now apply the general theory of AWS to study Gaussian measures on function or spaces. To be specic, given s ∈ R and ν ∈ N, ν ≥ 2, consider the following inner product on ∞ ν , the space of compactly supported Cc (R ) smooth functions on ν : for every ∞ ν , R φ, ψ ∈ Cc (R ) s (φ, ψ) := ((I − ∆) φ, ψ) 2 ν s L (R ) s 1  2 ˆ ˆ = ν 1 + |ξ| φ (ξ) ψ (ξ)dξ, (2π) ˆ ν R where denotes the Fourier transform. The closure of ∞ ν under is the ˆ· Cc (R ) (·, ·)s 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 |θ (x)| Θ 2 := θ ∈ C (R ) : lim = 0 , |x|→∞ log (e + |x|) equipped with the norm

θ |θ (x)| kθk ν+1 := = sup . Θ 2 ν log (e + |·|) u x∈R log (e + |x|) So the dim- order- ν+1 GFF actually consists of continuous functions. By the Riesz ν 2  ν+1 ? representation theorem, every λ ∈ Θ 2 is characterized as a on Rν with the property that log (e + |x|) |λ| (dx) < ∞. ˆ ν R ν+1 − ν+1 In addition, if we identify the 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 ˆ kλk− ν+1 = ν 1 + |ξ| λ (ξ) dξ 2 (2π) ˆ ν R 1−ν π 2 = ν+1  exp (− |x − y|) 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 (− |x − y|) λ (dy) , ˆ ν Γ 2 R SPHERICAL AVERAGES OF GAUSSIAN FREE FIELDS 3

ν+1  ν+1 ∗ gives the unique element hλ ∈ H 2 such that the action of λ ∈ 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 ∈ 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 ∈ 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 ∈ H 2 . Finally, for general , s2 is the isometric image of s1 under the s1, s2 ∈ 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 ∈ 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 ∈ R t > 0 σ¯t ∈ H (R ) is given by ν (2π) 2 2−ν x i(x,ξ) ν 2 σ¯ct (ξ) = e R · (t |ξ|) J ν−2 (t |ξ|) αν 2 for every ν , where is the standard Bessel function of the rst kind. ξ ∈ R Jµ Since x −s ν , we can apply x to the GFF element in the sense of σ¯t ∈ H (R ) σ¯t θ x  the Paley-Wiener . Namely X (θ) := I h x (θ) is well-dened for W- t σ¯t a.e. θ ∈ Θs as a Gaussian random variable, which, heuristically speaking, gives SPHERICAL AVERAGES OF GAUSSIAN FREE FIELDS 4 the spherical average of θ. Based on the previous lemma, we make the following observation.

Lemma 2. x ν is a two-parameter centered Gaussian family un- {Xt : x ∈ R , t > 0} der Ws with covariance given by

s Ws x y x y C (x, t; y, r) := E [Xt Xr ] = (¯σt , σ¯r )−s −s 1  2 x y = ν 1 + |ξ| σ¯ct (ξ) σc¯r (ξ)dξ (2π) ˆ ν R 2− ν A ∞ τ 2 J ν−2 (tτ) J ν−2 (rτ) J ν−2 (|x − y| τ) (2.1) ν 2 2 2 = ν−2 2 s dτ, (tr |x − y|) 2 ˆ0 (1 + τ )

ν Γ2 1+ ν where 2  2 ( 2 ). In particular, when , i.e., in the concentric case, Aν := π ν2 x = y ∞ 1 2−ν τJ ν−2 (tτ) J ν−2 (rτ) (2.2) s s 2 2 2 C (t, r) := C (x, t; x, r) = (tr) 2 s dτ. αν ˆ0 (1 + τ ) These results follow from Lemma 1, computations in polar coordinates as well as integral representations of Bessel functions [6]. Details are omitted. Since the kernel of (I − ∆)s is stationary, i.e., invariant under translation, when studying the behavior of the concentric spherical averages x's, we may assume Xt x is the origin and omit x in the expressions. In fact, we have rather explicit formulas for the covariance function of the concentric family when s ∈ N, s ≥ 1. We now discuss the cases s = 1 and s ≥ 2 separately.

2.1. When s = 1. The rst result of this subsection is the following: Theorem 3. Following the same notation as above, let C1 (t, r) be as in (2.2) with s = 1, then for every t, r > 0,

1 2−ν 1 2 C (t, r) = (tr) I ν−2 (t ∧ r) K ν−2 (t ∨ r) , αν 2 2 where Iµ and Kµ are modied Bessel functions (with pure imaginary argument). Proof. This result can be derived from the formula (5) in §13.53 in [6]. For the same of completeness, we hereby give another proof. In fact, we are going to prove the general result for the operator p2I − ∆ with p ∈ R, which will be useful in the 2 −1 next subsection. Set ζt := p I − ∆ σ¯t and

−1  2 2 ζbt (ξ) = p + |ξ| σ¯bt (ξ) ν (2π) 2 2−ν  −1 2 2 2 = (t |ξ|) p + |ξ| J ν−2 (t |ξ|) . αν 2 Notice that is actually a function in ν and ζt (u) u ∈ R

1 −i(u,ξ) ν ζt (u) = ν e R · ζbt (ξ) dξ (2π) ˆ ν R 2−ν 2 1 −i(u,ξ) ν (t |ξ|) = ν e R · 2 J ν−2 (t |ξ|) dξ. 2 ˆ ν 2 2 αν (2π) R p + |ξ| SPHERICAL AVERAGES OF GAUSSIAN FREE FIELDS 5

Switching to the polar coordinates and making use of the fact that

ν−2 2 π αν−1 (|u| τ) i|u|τ cos ϕ ν−2 J ν−2 (|u| τ) = ν e sin ϕdϕ, 2 (2π) 2 ˆ0 we have that ∞ 1 2−ν τJ ν−2 (|u| τ) J ν−2 (tτ) 2 2 2 ζt (u) = (t |u|) 2 2 dτ. αν ˆ0 p + τ If we replace (I − ∆) by p2I − ∆ in deriving (2.2) and set s = 1, then expres- sion for the covariance will become ∞ 1 2−ν τJ ν−2 (tτ) J ν−2 (rτ) (2.3) 1 2 2 2 Cp (t, r) = (tr) 2 2 dτ. αν ˆ0 p + τ It follows from above that 1 for every ν , and hence ζt (u) = Cp (t, |u|) u ∈ R 2  1 2  p − ∆ Cp (t, |·|) = p − ∆ ζt =σ ¯t in the sense of tempered distribution. Since σ¯t is a measure supported on the sphere and 1 is radially symmetric function on ν then in the radial S (0, t) Cp (t, |·|) R , component we get the following equation:  ν − 1  p2 − ∂2 − ∂ C1 (t, r) = 0, r r r p wherever 0 < r < t with xed t > 0. The solution to this equation is of the form

K ν−2 (pr) I ν−2 (pr) 2 2 C1 (t) ν−2 + C2 (t) ν−2 , r 2 r 2 where C1 and C2 are functions only depending on t. To determine C1 and C2, let's examine (2.3) more carefully. First by the basic properties of Bessel functions,

 − 1  sup J ν−2 (u) ≤ Kν · min 1, u 2 for some Kν > 0 u>0 2

ν−2 and lim J ν−2 (u) = 0. Therefore, (2.3) implies that for any xed t > 0, r 2 · u→0+ 2 1 as . However, and exists. Thus Cp (t, r) → 0 r ↓ 0 limK ν−2 (pr) = ∞ limI ν−2 (pr) r→0 2 r→0 2 I ν−2 (pr) we can conclude that and 1 2 . C1 (t) ≡ 0 Cp (t, r) = C2 (t) ν−2 r 2 Next, since 1 1 , we apply exactly the same argument to get the Cp (t, r) = Cp (r, t) equation in t, i.e.,  ν − 1  p2 − ∂2 − ∂ C1 (t, r) = 0 t t t p for all t > r with r > 0 xed, whence C2 (t) must be of the form

K ν−2 (pt) I ν−2 (pt) 2 2 for some constants C2 (t) = C3 ν−2 + C4 ν−2 C3,C4. t 2 t 2 ν−2 This time we notice that 2 1 but and supt>0 t Cp (t, r) < ∞ lim I ν−2 (pt) = ∞ t→∞ 2 lim K ν−2 (pt) = 0, which implies that C4 = 0 and t→∞ 2

K ν−2 (pt) I ν−2 (pr) (2.4) 1 2 2 Cp (t, r) = C3 ν−2 ν−2 , t 2 r 2 SPHERICAL AVERAGES OF GAUSSIAN FREE FIELDS 6 for all . Since 1 is clearly a in and , we have t > r > 0 Cp (t, r) t r that

2−ν 1 1 2 Cp (t, r) = Cp (r, t) = C3 (tr) K ν−2 (p (t ∨ r)) I ν−2 (p (t ∧ r)) 2 2 for any t, r > 0. The only thing left is to determine C3. To this end, we consider 1 given by (2.3) notice that Cp (r, r)

2 ∞ J ν−2 (u) ν−2 1 1 2 1 1 lim r · Cp (r, r) = du = · . r→0+ αν ˆ0 u αν ν − 2 The second equation is due to a well known integral identity of Bessel function (for- mula (1), §13.42, [6]), which can be easily veried by invoking the series expansion 1 of J ν−2 . On the other hand, lim K ν−2 (pr) I ν−2 (pr) = by a simple analysis 2 r→0+ 2 2 ν−2 1 of the asymptotics of K ν−2 and I ν−2 near zero. Therefore, we have that C3 = . 2 2 αν This completes the proof. 

One direct result from the previous theorem is the following.

Corollary 4. Let  O be as introduced above, then Xt := Xt : t > 0

 2 W1 ν−1 ν−1 (2.5) 2 2 for every E t Xt − r Xr ≤ |t − r| t, r > 0.

1 In particular, {Xt : t > 0} is W -a.e. continuous. Proof. The second assertion follows directly from (2.5) and Kolmogorov's continuity   1 ν−1 ν−1 2 theorem. To prove (2.5), let's assume . Then W 2 2 t ≥ r > 0 E t Xt − r Xr can be written as

ν−1 1 ν−1 1 ν−1 ν−1 1 t C (t, t) + r C (r, r) − 2t 2 r 2 C (t, r) √ √ √  √ √ √  = tK ν−2 (t) tI ν−2 (t) − rI ν−2 (r) + rI ν−2 (r) rK ν−2 (r) − tK ν−2 (t) 2 2 2 2 2 2 t h√   √   i = tK ν−2 (t) · DI ν−2 (u) − rI ν−2 (r) · DK ν−2 (u) du, ˆr 2 2 2 2 where √ √ I ν−2 (u)   d   2 u   DI ν−2 (u) := uI ν−2 (u) = √ + I ν−4 (u) + I ν (u) 2 du 2 2 u 2 2 2 and √ √ K ν−2 (u)   d   2 u   DK ν−2 (u) := uK ν−2 (u) = √ − K ν−4 (u) + K ν (u) . 2 du 2 2 u 2 2 2 √ One can easily verify that as u ∈ (0, ∞) increases, uK ν−2 (u) decreases and √ √ 2 √ uI ν−2 (u) increases. Therefore, DI ν−2 ≥ 0, DK ν−2 ≤ 0, tK ν−2 (t) ≤ uK ν−2 (u) √2 √ 2 2 2 2 and rI ν−2 (r) ≤ uI ν−2 (u) whenever r ≤ u ≤ t. Putting everything together, 2 2 SPHERICAL AVERAGES OF GAUSSIAN FREE FIELDS 7 we have that  2 W1 ν−1 ν−1 2 2 E t Xt − r Xr t h√   √   i ≤ uK ν−2 (u) DI ν−2 (u) − uI ν−2 (u) DK ν−2 (u) du ˆr 2 2 2 2 1 t   = u I ν−4 (u) K ν−2 (u) + I ν−2 (u) K ν−4 (u) + I ν (u) K ν−2 (u) + I ν−2 (u) K ν (u) du 2 2 2 ˆr 2 2 2 2 2 2 = t − r. The last equality is due to the fact that for every µ ∈ R, 1 (2.6) I (u) K (u) + I (u) K (u) = . µ−1 µ µ µ−1 u  With all these preparations, the following theorem becomes an almost immediate result.

1 Theorem 5. {Xt : t > 0} is a W -a.e. continuous Gaussian Markov process, and its transition probability density is given by ( ) 1 [y − m (r, t) x]2 p (x, r; y, t) = exp − , p2πσ2 (r, t) 2σ2 (r, t)

2−ν K ν−2 (t) where 0 < r ≤ t, x, y ∈ , m (r, t) := t  2 2 , and R r K ν−2 (r) 2

I ν−2 (r) 2 ν−2 ν−2 2 2 σ (r, t) := t K ν−2 (t) I ν−2 (t) − t K ν−2 (t) . 2 2 2 K ν−2 (r) 2

The Kolmogorov backward equation associated with Xt's is given by

ν−3 K ν (r) r 2 2 ∂rp (x, r; y, t) + ∂xp (x, r; y, t) + ∂xp (x, r; y, t) = 0. 2 K ν−2 (r) 2

Proof. To complete the rst assertion, we only need to show that {Xt : t > 0} is Markovian. Xt is a , and it's well known that a Gaussian process is Markovian if and only if its covariance function has the following property (): (2.7) W1 W1 W1 W1 E [XtXw] E [XrXr] = E [XtXr] E [XrXw] whenever t ≥ r ≥ w > 0. But this is a direct consequence of Theorem 3. The second statement follows from the standard property of conditional expectation of Gaussian random variables. Namely, W1 is again a Gaussian random E [Xt|Xr] variable with value C1(t,r) and variance m (r, t) Xr = C1(r,r) Xr 2 C1 (t, t) C1 (r, r) − C1 (t, r) σ2 (r, t) = . C1 (r, r) One can directly verify that p (x, r; y, t) satises the given equation. In general (§3.3 in [2]), the associated Kolmogorov backward equation is given by ∂ σ2 (r, t) ∂ m (r, t) ∂ p (x, r; y, t) = r ∂2p (x, r; y, t) + r ∂ p (x, r; y, t) . r 2m2 (r, t) x m (r, t) x SPHERICAL AVERAGES OF GAUSSIAN FREE FIELDS 8

The rest follows from the observations that

K ν (r) 2 ν−3 2 2 ∂rσ (r, t) = −r m (r, t) and ∂rm (r, t) = − m (r, t) . K ν−2 (r) 2  So far we have been focusing on the spherical averages of the GFF. One would naturally wonder whether the properties we derived above also hold for the averages of θ over concentric solid balls {B (t) := B (O, t): t > 0}. To this end, let µt be the tempered distribution on ν such that for any ∞ ν , R φ ∈ Cc (R ) 1 hφ, µti = φ (x) dx, βν (t) ˆB(t)

ν where π 2 ν is the volume of dim- ball with radius . Set βν (t) := ν t ν t > 0 Γ( 2 +1) βν := βν (1). We have that

ν (2π) 2 ν − 2 µt (ξ) = (t |ξ|) J ν (t |ξ|) . b 2 βν It's easy to check that −1 ν for every , and in fact µt ∈ H (R ) t > 0 d (β (t) µ )(ξ) = α (t) σ¯ (ξ) for every ξ ∈ ν . dt v bt ν bt R Dene the Gaussian process for , which corresponds to the Yt := I (hµt ) t > 0 averages over concentric balls.

In order to study the behavior of Yt, we will need to know the covariance of the family {Yt : t > 0}. In fact, we will compute Yt's covariance through the bigger family which consists of and . To this end, rst consider W1 . Xt Yt F (r, t) := E [YrXt] Clearly, F (r, t) is continuous in t and C1 in r. Moreover, based on the observations above, one can verify that d d (β (r) F (r, t)) = β (r)(µ , σ¯ )  = α (r) (¯σ , σ¯ ) . dr ν dr ν r t −1 ν r t −1 On one hand,

∞ J ν (rτ) J ν−2 (tτ) 2−ν ν 2 (2.8) 2 2 2 βν (r) F (r, t) = t r 2 dτ, ˆ0 1 + τ and on the other hand,

ν 2−ν (2.9) αν (r) (¯σr, σ¯t) = r 2 t 2 I ν−2 (t ∧ r) K ν−2 (t ∨ r) . −1 2 2

Let's x t > 0 and rst assume 0 < r ≤ t. (2.8) implies that lim βν (r) F (r, t) = 0. r→0+ Combining with (2.9), we have that, when 0 < r ≤ t,

r  ν  2−ν βν (r) F (r, t) = ρ 2 I ν−2 (ρ) dρ t 2 K ν−2 (t) ˆ0 2 2 ν 2−ν = r 2 I ν (r) t 2 K ν−2 (t) . 2 2 SPHERICAL AVERAGES OF GAUSSIAN FREE FIELDS 9

When r ≥ t, making use of the continuity of F (r, t) in r, we have that r  ν  2−ν β (r) F (r, t) = tI ν (t) K ν−2 (t) + ρ 2 K ν−2 (ρ) dρ t 2 I ν−2 (t) . ν 2 2 ˆt 2 2  ν 2−ν  2 2 r = tI ν (t) K ν−2 (t) + −ρ K ν (r) t I ν−2 (t) |t 2 2 2 2 ν ν−2 = 1 − r 2 K ν (r) t 2 I ν−2 (t) . 2 2 The last equality again makes use of (2.6). Therefore,

 ν 2−ν 1 − 2 2 ν if  β · r t I (r) K ν−2 (t) , 0 < r ≤ t, (2.10) F (r, t) = ν 2 2 1  −ν − ν 2−ν  r − r 2 t 2 K ν (r) I ν−2 (t) , if r > t.  βν 2 2

Next, set W1 , and we will just repeat the procedure above to G (r, t) := E [YrYt] get the explicit formula for . This time, d , G (r, t) dt (βν (t) G (r, t)) = αν (t) F (r, t) and with xed r > 0, lim βν (t) G (r, t) = 0. Hence, when 0 < t ≤ r, t→0+

t 1 αν  ν−1 −ν − ν ν  G (r, t) = ρ r − r 2 K ν (r) ρ 2 I ν−2 (ρ) dρ 2 βν (t) βν ˆ0 2   1 1 − ν − ν = − νr 2 K ν (r) t 2 I ν (t) . ν 2 2 βν r Similarly, since G (r, t) is continuous in t, we have that   1 1 − ν − ν G (r, t) = − νr 2 I ν (r) t 2 K ν (t) ν 2 2 βν t for t > r > 0. We are now ready to state the following theorem.

Theorem 6. Let {Xt : t > 0} and {Yt : t > 0} be the Gaussian processes as dened above, then W1-a.e. 1 t Yt = αν (τ) Xτ dτ. βν (t) ˆ0

Moreover, {Yt : t > 0} has the covariance function   W1 ν 1 − ν G (r, t) = [Y Y ] = − (tr) 2 I ν (t ∧ r) K ν (t ∨ r) , E r t ν 2 2 βν ν (t ∨ r) and {Yt : t > 0} is not Markovian. However, if Vt := (Xt,Yt), then {Vt : t > 0} under W1 is a 2-vector valued a.e. continuous Gaussian Markov process.

Proof. Recall that is continuous 1-a.e., so t is well {Xt : t > 0} W 0 αν (τ) Xτ dτ dened as a Gaussian random variable. With all the computational´ results above, one can easily conclude the rst assertion by verifying that

" t 2# W1 1 E Yt − αν (τ) Xτ dτ = 0. βν (t) ˆ0

Next, G(r, t), the covariance of {Yt : t > 0}, clearly violates the condition (2.7), so {Yt : t > 0} is not Markovian. SPHERICAL AVERAGES OF GAUSSIAN FREE FIELDS 10

As for vector valued Gaussian process {Vt : t > 0}, it's also known that to prove the Markov property for {Vt : t > 0}, we only need to show that the corresponding  1 h i covariance matrix W has the following property: C2×2 (t, r) := E (Vt)i (Vr)j (2.11) −1 C2×2 (t, w) = C2×2 (t, r) C2×2 (r, r) C2×2 (r, w) whenever t ≥ r ≥ w > 0. Furthermore, it would be sucient if we can show that C2×2 (t, r) = A2×2 (t ∨ r) B2×2 (t ∧ r) for some non-degenerate 2×2 matrices A2×2 and B2×2. Let's assume that t ≥ r > 0. In fact we have that  C1 (t, r) F (r, t)  C (t, r) = 2×2 F (t, r) G (r, t) 1 2−ν 2−ν − ν 2−ν ! 1 r 2 t 2 I ν−2 (r) K ν−2 (t) r 2 t 2 I ν (r) K ν−2 (t) ν 2 2 2 2 = −ν − ν 2−ν −ν − ν − ν βν t − t 2 r 2 K ν (t) I ν−2 (r) , t − νr 2 I ν (r) t 2 K ν (t) 2 2 2 2 2−ν ! 1 2−ν − ν ! 1 t 2 K ν−2 (t) 0 r 2 I ν−2 (r) r 2 I ν (r) 2 ν 2 2 = − ν −ν . βν −νt 2 K ν (t) t 1 1 2

The continuity of Vt simply follows from the continuity of Xt and the rst assertion. 

2.2. When s ≥ 2 and s ∈ N. In this subsection, we are going to discuss the spherical averages of dim-ν order-s GFF for general s ∈ N and s ≥ 2. First, we make the remark that the condition (2.11) applies to any k-vector valued Gaussian process with . Namely, if is a -vector k ∈ N {Wt = (Wt,1,Wt,2,...,Wt,k): t > 0} k s s valued Gaussian process on Θ under W , with the covariance matrix Ck×k (t, r) :=  s h i W . Then, is Markovian if and only if for any E (Wt)i (Wr)j {Wt : t > 0} t ≥ r ≥ w > 0, (2.12) −1 Ck×k (t, w) = Ck×k (t, r) Ck×k (r, r) Ck×k (r, w) . Again we consider the spherical average . When and , we no- σ¯t s ≥ 2 s ∈ N tice that that (k) d k with , as tempered distri- σ¯t := dt σ¯t k ∈ {0, 1, . . . , s − 1} butions, are all elements of H−s. Consequently, for every k ∈ {0, 1, . . . , s − 1}, n   o (k) forms a Gaussian family under s. We identify (0) Xt := I h (k) : t > 0 W Xt σ¯t as Xt. The following theorem says that although Xt fails to be Markovian when , by putting (1) (s−1) together, we can restore the Markov prop- s ≥ 2 Xt,Xt ,...,Xt erty.   Theorem 7. Set (1) (s−1) , then is a -vector Wt := Xt,Xt ,...,Xt {Wt : t > 0} s valued a.e. continuous Gaussian Markov process on Θs under Ws. Proof. Recall the notations

s  s h i s W and W C (t, r) = E [XtXr] Cs×s (t, r) = E (Wt)i (Wr)j .

It's immediate from the denition of (k) for that Xt k ∈ {0, 1, . . . , s − 1}  d i−1  d j−1 (C (t, r)) = Cs (t, r) . s×s ij dt dr SPHERICAL AVERAGES OF GAUSSIAN FREE FIELDS 11

s Therefore, to understand Cs×s (t, r), it comes down to computing C (t, r). Recall from (2.2) , (2.3) and (2.4) that

∞ 1 2−ν τJ ν−2 (tτ) J ν−2 (rτ) s 2 2 2 C (t, r) = (tr) 2 s dτ, αν ˆ0 (1 + τ )

∞ 1 2−ν τJ ν−2 (tτ) J ν−2 (rτ) 1 2 2 2 Cp (t, r) = (tr) 2 2 dτ αν ˆ0 p + τ 1 2−ν 2 = (tr) K ν−2 (p (t ∨ r)) I ν−2 (p (t ∧ r)) , αν 2 2 from which it follows that  s−1 s 1 1 d 1 C (t, r) = − Cp (t, r) s! 2p dp p=1 s−1 1  1 d   1 2−ν  2 = − (tr) K ν−2 (p (t ∨ r)) I ν−2 (p (t ∧ r)) . 2 2 s! 2p dp p=1 αν Now let's assume that t ≥ r > 0. Then it's obvious that Cs (t, r) must be of the form s−1 s X C (t, r) = ak (t) bk (r) , k=0 where ak and bk are functions only depending on t and r respectively. Therefore, Cs×s (t, r) = As×s (t) Bs×s (r), where As×s (t) and Bs×s (r) are s × s matrices, and d i−1 and d j−1 . Hence, (As×s (t))ij = dt aj−1 (t) (Bs×s (r))ij = dr bi−1 (r) Cs×s (t, r) satised the condition (2.12). Finally, to establish the continuity of X(k), we notice that the covariance of n o t (k) is in fact Xt : t > 0

 d k  d k Cs,(k) (t, r) := Cs (t, r) dt dr ν−1+2k  k   k  ∞ τ D J ν−2 (tτ) D J ν−2 (rτ) 1 2 2 = 2 s dτ αν ˆ0 (1 + τ ) where  k  k  d  2−ν  D J ν−2 (u) = u 2 · J ν−2 (u) . 2 du 2

 k  By the basic identities of the of J ν−2 , we recognize that D J ν−2 (u) 2 2 can be written as a nite sum of the terms in the form of

2−ν −l+j cj,l · u 2 · J ν−2 (u) where 0 ≤ l ≤ k, 0 ≤ j ≤ l and cj,l ∈ . 2 +l R

2 h k   k  i We will need an estimate on D J ν−2 (tτ) − D J ν−2 (rτ) in terms of |t − r|. 2 2 By invoking the series expansion of J ν−2 , it's not hard to see that the worst 2 +l SPHERICAL AVERAGES OF GAUSSIAN FREE FIELDS 12 term in that dierence will occur when j = 0, and even in that situation (assuming 0 < r ≤ t) we have that 2 ! t 2 J ν−2 (tτ) J ν−2 (rτ)  2−ν  2 2 +l 2 +l 2 2−ν−2l −l c − = c τ ρ 2 J ν +l (τρ) dρ 0,l ν−2 +l ν−2 +l 0,l 2 (tτ) 2 (rτ) 2 ˆr  1  ≤ Cτ 2−ν−2l · rν−2+2l · (t − r) · min 1, √ . tτ for every 0 ≤ l ≤ k. Finally we see that 2 ν−1+2k h k   k  i   ∞ τ D J ν−2 (tτ) − D J ν−2 (rτ) s 2 1 W X(k) − X(k) = 2 2 dτ E t r 2 s αν ˆ0 (1 + τ ) ν−2+2k − 1 is always bounded by C r t 2 (t − r) so long as 0 ≤ k ≤ s − 1. It follows immediately from Kolmogorov's continuity theorem that (k) is s-a.e. continuous Xt W on t ∈ (0, ∞).  2.3. Non-Concentric Spherical Averages. For the Gaussian family consisting of non-concentric spherical averages, it is also possible to obtain the explicit formu- las for the covariance in some situations. However, the technicality is rather heavy. Again, we need various integral formulas for Bessel functions, for which we refer to pp 429-430 of [6] for technical details. The main results are the following:

Theorem 8. Let , and x ν the Gaussian family as dened s = 1 {Xt : x ∈ R , t > 0} at the beginning of §2 with covariance

1 W1 x y C (x, t; y, r) = E [Xt Xr ] . If t ≥ |x − y| + r, then

1 Aν C (x, t; y, r) = ν−2 K ν−2 (t) I ν−2 (r) I ν−2 (|x − y|) , (tr |x − y|) 2 2 2 2 and if |x − y| > t + r, then

1 Aν C (x, t; y, r) = ν−2 K ν−2 (|x − y|) I ν−2 (r) I ν−2 (t) , (tr |x − y|) 2 2 2 2

ν Γ2 1+ ν where 2  2 ( 2 ). Aν := π ν2 For the general s ≥ 2 and s ∈ N, we also have a result that is analogous to the concentric case. We will need the following remark.

Remark 9. Dene 1 similarly as 1 , i.e., consider the operator Cp (x, t; y, r) Cp (t, r) p2I − ∆ instead of (I − ∆). Then, if t ≥ |x − y| + r, then

1 Aν Cp (x, t; y, r) = ν−2 K ν−2 (pt) I ν−2 (pr) I ν−2 (p |x − y|); (ptr |x − y|) 2 2 2 2 and if |x − y| > t + r, then

1 Aν Cp (x, t; y, r) = ν−2 K ν−2 (p |x − y|) I ν−2 (pr) I ν−2 (pt) . (ptr |x − y|) 2 2 2 2 SPHERICAL AVERAGES OF GAUSSIAN FREE FIELDS 13

Theorem 10. Let s ≥ 2 and s ∈ N. The covariance of the Gaussian family x ν is given by {Xt : x ∈ R , t > 0}  s−1 s 1 1 d 1 C (x, t; y, r) = − Cp (x, t; y, r) . s! 2p dp p=1 Therefore, either when t > |x − y| + r or when |x − y| > t + r, one can derive the explicit formulas for Cs (x, t; y, r) based on the previous remark. In particular, in either case, there exist vector-valued functions V : (0, ∞) → Rs and s, and matrix-valued function s s such U : (0, ∞) → R As×s : (0, ∞) → R × R that Cs (x, t; y, r) = V> (t) A (|x − y|) U (r) . Set x,(k) d (k) x. Then x,(k) is well-dened as a Gaussian random Xt := dt Xt Xt variable on Θs under Ws for every k = 0, 1, ··· , s − 1. Identify Xx,(0) as Xx,   t t and denote x x,(0) x,(s−1) . Then x ν forms a Wt := Xt , ··· ,Xt {Wt : x ∈ R , t > 0} vector-valued two-parameter Gaussian family under Ws with the covariance matrix  Ws h x y i Cs×s (x, t; y, r) = E (Wt )i (Wr )j . Then, either when t > |x − y| + r or when |x − y| > t + r, we have that

Cs×s (x, t; y, r) = Vs×s (t) A (|x − y|) Us×s (r) where Vs×s (t) and Us×s (r) are s × s matrices such that  d i−1 (V (t)) = (V (t)) s×s ij dt j and  d y−1 (U (r)) = (U (r)) . s×s ij dr i A detailed treatement in the case when s = 2 and ν = 4 can be found in the appendix of [1].

References

4 [1] L. Chen and D. Jakobson. Gaussian free elds and kpz relation in R . Annales Henri Poincaré, 15(7):12451283, 2014. [2] J.L. Doob. Stochastic processes. Wiley-Interscience, 1990. [3] L. Gross. Abstract wiener spaces. Proc. 5th Berkeley Symp. Math. Stat. and Probab., 2(1):31 42, 1965. [4] D. Stroock. Abstract wiener space, revisited. Comm. on Stoch. Anal., 2(1):145151, 2008. [5] D. Stroock. Probability, an analytic view, 2nd edition. Cambridge Univ. Press, 2011. [6] G.N. Watson. A treatise on the theory of bessel functions, 2nd edition. Cambridge Univ. Press, 1995.

Department of Mathematics and , McGill University, Montreal, QC, Canada. E-mail address: [email protected]