# Small fluctuations for time-dependent
spherical random fields

Marco Carfagnini† Department of Mathematics
University of California, San Diego
La Jolla, CA 92093-0112, U.S.A<EMAIL_ADDRESS>and  Anna Paola Todino⋆
Dipartimento di Scienze e Innovazione Tecnologica
Università del Piemonte Orientale
Italy<EMAIL_ADDRESS>
###### Abstract.

We consider time-dependent space isotropic and time stationary spherical
Gaussian random fields. We establish Chung’s law of the iterated logarithm and
solve the small probabilities problem. Our results depend on the high-
frequency behaviour of the angular power spectrum: the speed of decay of the
small ball probability is faster as either the memory parameter or the space-
parameter decreases.

###### Key words and phrases:

Time-dependent spherical random fields, spherical harmonics, small ball
probabilities, Chung’s law of the iterated logarithm.

###### 1991 Mathematics Subject Classification:

60G60, 60G17, 60G15, 42C05.

11footnotemark: 1${\dagger}$ Research was supported in part by an AMS-Simons
Travel Grant

22footnotemark: 2$\star$ Research was supported in part by “INdAM - GNAMPA
Project”, codice CUP E53C23001670001.

###### Contents

  1. 1 Introduction
  2. 2 Background and main results
    1. 2.1 Upper and lower bounds for Gaussian random fields
    2. 2.2 Time-dependent spherical random fields
    3. 2.3 Main Results
    4. 2.4 Isotropic Gaussian spherical random fields (non-time-dependent)
  3. 3 Preliminary results
  4. 4 Chung’s law of the iterated logarithm
  5. 5 Small probabilities
  6. A Proof of Proposition 2.5
  7. B Technical Lemmas

##  1\. Introduction

Time-dependent spherical random fields have recently aroused lots of interest
as mathematical models in many applications. In particular atmospheric and
climate sciences, medical imaging, geophysics, astrophysics and cosmology, see
for instance [21, 4, 5, 6, 15, 12, 46] for recent contributions. In this paper
we focus on sample path properties of time-dependent spherical random fields,
i.e. defined on $\mathbb{S}^{2}\times\mathbb{R}$. Fluctuations of geometric
functionals associated to such fields have been studied recently in [38, 39].
In particular we establish Chung’s law of the iterated logarithm and small
deviations.

We say that a real-valued random field $\\{X_{t},t\in\mathbb{R}^{n}\\}$
satisfies Chung’s law of the iterated logarithm (LIL) at
$t_{0}\in\mathbb{R}^{n}$ with rate function $\phi$ if there exists a constant
$C$ such that

(1.1) $\displaystyle\liminf_{r\rightarrow 0}\phi(r)\max_{t\in
B_{r}(t_{0})}|X_{t}-X_{t_{0}}|=C\;\;\;a.s.$

where $B_{r}(t_{0})$ is an appropriate ball of radius $r>0$ centered at
$t_{0}$. When $X_{t}$ is a Brownian motion and $t_{0}=0$, it was proven in a
famous paper by K.-L. Chung in 1948 [11] that (1.1) holds with
$\phi(r)=\sqrt{\frac{\log|\log r|}{r}}$ and $C=\frac{\pi}{\sqrt{8}}$.

Chung’s LIL has many applications and we refer to the survey [30] for more
details. In this paper we are mostly interested in connections to small
probabilities. The small ball problem (SBP) consists in finding a function
$\theta$ and a constant $c$ such that

(1.2) $\lim_{\varepsilon\rightarrow
0}\theta(\varepsilon)\log\mathbb{P}\left(\max_{t\in
B_{r}(t_{0})}|X_{t}-X_{t_{0}}|<\varepsilon\right)=c.$

The small ball problem and Chung’s LIL for stochastic processes have been
extensively studied, and the literature on the subject is vast. In [3] the
authors considered the case of a one-dimensional Brownian motion and Hölder
norms, in [20] a Brownian sheet in Hölder norms has been considered, [19]
studied the integrated Brownian motion in the uniform norm, and [10] the
$m$-fold integrated Brownian motion in both the uniform and $L^{2}$-norm, and
in [7] the $n$-step Kolmogorov diffusion. In [41] and [8] a small deviation
problem and Chung’s law of iterated logarithm is proven for a class of
stochastic integrals and for the hypoelliptic Brownian motion on the
Heisenberg group respectively.

As far as random fields $\\{X_{t},t\in\mathbb{R}^{n}\\}$ $n\geq 1$ on
Euclidean spaces are concerned, sample path properties have been studied in
both the isotropic and anisotropic case by many authors, see for instance [31,
34, 33, 40, 43, 48, 44, 49, 45, 27, 29]. In particular, Chung’s LIL for
Gaussian random fields with stationary increments has been proven in [33].
Estimates on small ball probabilities of random fields have been extensively
studied as well, see [43].

In recent years, sample paths properties of random fields defined on manifolds
have also been investigated. In particular, in the case of the sphere
$\mathbb{S}^{2}$, differentiability and Hölder continuity properties of sample
paths have been studied in [18, 26]; Hölder conditions of local times and
exact moduli of non-differentiability for spherical Gaussian fields in [24]; a
strong local non determinism of an isotropic spherical Gaussian field and the
exact uniform modulus of continuity have been investigated in [23]. These last
results have been then generalized to isotropic Gaussian random fields on
compact two-point homogeneous space in [32]. Finally we refer to [25] for
regularity properties of the solution to a stochastic wave equation given by a
fractional Gaussian noise on $\mathbb{S}^{2}$.

In the current paper we consider a time-dependent spherical Gaussian random
field $\\{T(x,t),\;(x,t)\in\mathbb{S}^{2}\times[0,T]\\}$ which is stationary
in time and isotropic in space. Our main results are collected in Theorem 2.7
and Theorem 2.8. The former provides the rate function for Chung’s LIL, and
the latter describes the small ball probability asymptotics. Both the rate
function in (1.1) and the asymptotics in (1.2) depend on the high frequency
behaviour of the angular power spectrum of the field
$\\{T(x,t),\;(x,t)\in\mathbb{S}^{2}\times[0,T]\\}$. The asymptotic behaviour
of the spectral density depends on a space-parameter $\alpha$ and a time-
parameter $\beta$. The former describes the high-energy behaviour of the field
and the latter can be thought as a ”memory” parameter. The small ball
probability dacays to zero like
$\exp\left(\frac{-c}{\varepsilon^{\frac{2}{\beta}+\frac{4}{\alpha-2}}}\right)$,
for some positive constant $c$. In particular, for a fixed $\alpha$, the
smaller the time-memory parameter $\beta$ the faster the speed of the decay.
We refer to (2.22) and (2.23) for a more explicit rate and asymptotics
function.

Let us now explain the strategy of the proof. Generally, the first step is to
show that the LHS of (1.1) is measurable with respect to a tail $\sigma$-field
and hence constant. Our approach is to use Theorem 2.8 to prove that the LHS
of (1.1) is constant. More precisely, we first show that is almost surely
positive and finite. Then by means of stationarity and isotropy of the field
we are able to connect (1.1) to (1.2). Lastly, we show that $C$ in (1.1) is
constant by giving an explicit expression in terms of $c$ in (1.2).

The paper is organized as follows. In Section 2 we recall Talagrand’s bounds
for random fields, time-spherical Gaussian random fields, and then state the
main results of this paper, namely Theorem 2.7 and Theorem 2.8. Section 3
contains some preliminary results which are needed to prove Chung’s LIL in
Section 4 and small probabilities in Section 5. Lastly, in Appendix A and
Appendix B we collect proofs of some technical results.

##  2\. Background and main results

###  2.1. Upper and lower bounds for Gaussian random fields

Small ball probabilities of random fields have been extensively studied, see
[40, 28, 49, 47] and references therein. In this sections we recall results
from [43] that we will use throughout the paper. Let
$\\{X_{t},\,t\in\mathbb{R}^{n}\\}$ be a real-valued random field, and $d_{X}$
be its canonical metric, that is,

(2.1) $d_{X}(t,s):=\sqrt{\mathbb{E}\left[(X_{t}-X_{s})^{2}\right]}.$

One of the main ingredients in our proofs is Talagrand’s bound on small ball
probabilities for Gaussian processes.

######  Lemma 2.1.

[43, Lemma 2.1 and Lemma 2.2] Let $\\{X_{t},\,t\in K\\}$ be a separable, real-
valued, mean zero Gaussian process indexed by a compact set $K$ with canonical
metric $d_{X}$. Let $N(K,d_{X},\varepsilon)$ be the smallest number of
$d_{X}$-balls of radius $\varepsilon$ needed to cover the set $K$. Suppose
that there is a function $\Psi:(0,\delta)\rightarrow(0,\infty)$ such that
$N(K,d_{X},\varepsilon)\leqslant\Psi(\varepsilon)$ for all
$\varepsilon\in(0,\delta)$ and there are constants $1<a_{1}\leqslant a_{2}$
such that for all $\varepsilon\in(0,\delta)$

(2.2) $\displaystyle
a_{1}\Psi(\varepsilon)\leqslant\Psi\left(\frac{\varepsilon}{2}\right)\leqslant
a_{2}\Psi(\varepsilon).$

Then, there is a finite constant $k$ depending only on $a_{1}$ and $a_{2}$
such that for all $u\in(0,\delta)$

(2.3) $\displaystyle\mathbb{P}\left(\sup_{s,t\in K}|X_{t}-X_{s}|\leqslant
u\right)\geqslant\exp\left(-k\Psi(u)\right).$

Moreover, there exists a positive constant $c$ such that for any $u>0$

(2.4) $\mathbb{P}\left(\max_{s,t\in K}|X_{s}-X_{t}|\geqslant
c\left(u+\int_{0}^{D}\sqrt{\log
N(K,d_{X},\varepsilon)}d\varepsilon\right)\right)\leqslant\exp\left(-\frac{u^{2}}{D^{2}}\right),$

where $D$ is the diameter of $K$ with respect to $d_{X}$.

###  2.2. Time-dependent spherical random fields

Let $\mathbb{S}^{2}$ be the unit 2-dimensional sphere and
$\Delta_{\mathbb{S}^{2}}$ be the Laplace-Beltrami operator on $\mathbb{S}^{2}$
defined as

$\dfrac{1}{\sin\theta}\dfrac{\partial}{\partial\theta}\bigg{\\{}\sin\theta\dfrac{\partial}{\partial\theta}\bigg{\\}}+\dfrac{1}{\sin^{2}\theta}\dfrac{\partial^{2}}{\partial\varphi^{2}},\mbox{
}0\leq\theta\leq\pi,\mbox{ }0\leq\varphi<2\pi.$

It is well-known that the spectrum of $\Delta_{\mathbb{S}^{2}}$ is purely
discrete, its eigenvalues are of the form $-\lambda_{\ell}=-\ell(\ell+1)$
where $\ell\in\mathbb{N}$, and the eigenspace corresponding to
$\lambda_{\ell}$ is the $(2\ell+1)$-dimensional space
$\mathcal{S}_{\ell}:=\text{span}\\{Y_{\ell m}(\cdot):m=-\ell,\dots,\ell\\}$ of
degree-$\ell$ spherical harmonics. Note that
$\\{\mathcal{S}_{\ell},\,\ell\in\mathbb{N}\\}$ is an
$L^{2}(\mathbb{S}^{2},dx)-$orthonormal basis, where $dx$ denotes the Lebesgue
measure on $\mathbb{S}^{2}$. In particular, the functions $Y_{\ell m}$ satisfy

$\Delta_{\mathbb{S}^{2}}Y_{\ell m}+\lambda_{\ell}Y_{\ell m}=0,$

and the addition and duplication formulae

(2.5) $\sum_{m=-\ell}^{\ell}Y_{\ell m}(x){Y}_{\ell
m}(y)=\frac{2\ell+1}{4\pi}P_{\ell}(\left\langle x,y\right\rangle)\text{ ,}$
(2.6) $\int_{\mathbb{S}^{2}}\frac{2\ell+1}{4\pi}P_{\ell}(\left\langle
x,z\right\rangle)\frac{2\ell+1}{4\pi}P_{\ell}(\left\langle
z,y\right\rangle)dz=\frac{2\ell+1}{4\pi}P_{\ell}(\left\langle
x,y\right\rangle)\text{ ,}$

for all $x,y\in\mathbb{S}^{2},$ and $P_{\ell}:[-1,1]\to\mathbb{R}$, denotes
the Legendre polynomial of degree $\ell$:

$P_{\ell}(t)=\frac{1}{2^{\ell}\ell!}\frac{d^{\ell}}{dt^{\ell}}(t^{2}-1)^{\ell}\quad\ell=1,2,\dots$

and $P_{\ell}(1)=1.$ We refer to [36, Eq. (3.42) and Sec. 13.1.2] for more
details.

A space-time real-valued spherical random field $T$ is a collection of random
variables $\\{T(x,t),\,(x,t)\in\mathbb{S}^{2}\times\mathbb{R}\\}$ defined on a
common probability space $(\Omega,\mathcal{F},\mathbb{P})$ such that
$T:\Omega\times\mathbb{S}^{2}\times\mathbb{R}\longrightarrow\mathbb{R}$ is a
$\mathcal{F}\times\mathcal{F}(\mathbb{S}^{2}\times\mathbb{R})$-measurable map,
where $\mathcal{F}(\mathbb{S}^{2}\times\mathbb{R})$ denotes the Borel sigma-
algebra of $\mathbb{S}^{2}\times\mathbb{R}$. We are interested in a space-time
real-valued spherical random field $T$ that is

(1) Gaussian, i.e. its finite-dimensional distributions are Gaussian;

(2) centered, i.e. $\mathbb{E}\left[T(x,t)\right]=0$ for every
$(x,t)\in\mathbb{S}^{2}\times\mathbb{R}$;

(3) isotropic in space and stationary in time, that is, $T(x,t)$ and
$T(g.x,\tau+t)$ have the same distribution for every
$(x,t)\in\mathbb{S}^{2}\times\mathbb{R}$, $g\in SO(3)$, and
$\tau\in\mathbb{R}$;

(4) mean-square continuous.

Condition (3) implies that, for every
$(x,t),(y,s)\in\mathbb{S}^{2}\times\mathbb{R}$

$\mathbb{E}\left[T(x,t)T(y,s)\right]=\Gamma(\langle x,y\rangle,t-s),$

where $\langle\cdot,\cdot\rangle$ denotes the standard inner product in
$\mathbb{R}^{3}$, and $\Gamma:[-1,1]\times\mathbb{R}\rightarrow\mathbb{R}$ is
a positive-semidefinite function. Under the above conditions, it is well known
that [4, Theorem 3.3] [35, Theorem 3] $\Gamma$ can be written as a uniformly
convergent series

(2.7)
$\Gamma(\eta,\tau)=\sum_{\ell=0}^{\infty}\frac{2\ell+1}{4\pi}C_{\ell}(\tau)P_{\ell}(\eta),\quad(\eta,\tau)\in[-1,1]\times\mathbb{R},$

where $\\{C_{\ell}(\cdot)\\}_{\ell=0}^{\infty}$ is a sequence of continuous
positive semi-definite functions on $\mathbb{R}$, and
$\\{P_{\ell}\\}_{\ell=0}^{\infty}$ denotes the Legendre polynomial. Let

$\displaystyle a_{\ell m}(t):=\int_{\mathbb{S}^{2}}T(x,t)Y_{\ell m}(x)dx,\quad
t\in\mathbb{R},$

where $Y_{\ell m}$ denotes the $(\ell,m)-$th spherical harmonic. By our
assumptions it follows that $\\{a_{\ell
m},\,m=-\ell,\ldots,\ell\\}_{\ell=0}^{\infty}$ is a family of independent,
stationary, centered Gaussian processes on $\mathbb{R}$ such that

(2.8) $E[a_{\ell
m}(t)a_{\ell^{\prime}m^{\prime}}(s)]=C_{\ell}(t-s)\delta_{\ell}^{\ell^{\prime}}\delta_{m}^{m^{\prime}}.$

From the spectral representation for $\Gamma$ one can deduce the following
Karhunen–Loève expansion

(2.9) $T(x,t)=\sum_{\ell=0}^{\infty}\sum_{m=-\ell}^{\ell}a_{\ell m}(t)Y_{\ell
m}(x),$

where the series converges in
$L^{2}\left(\Omega\times\mathbb{S}^{2}\times[0,T]\right)$ for any $T>0$.

Throughout the paper we will assume the following.

Assumption I There exists $0<\beta<2$ such that the family of functions
$\\{C_{\ell}(\cdot)\\}_{\ell=0}^{\infty}$ satisfies

(2.10) $C_{\ell}(\tau)=C_{\ell}(0)(1-|\tau|^{\beta}),$

for all $|\tau|<1$. Moreover, there exists an $\alpha$ with $2+\beta<\alpha<4$
such that

(2.11) $\displaystyle
C_{\ell}(0)=G(\ell)\ell^{-\alpha}>0,\;\;\ell=1,2,\ldots,,$ (2.12)
$\displaystyle C_{0}(0)=c_{1},$

and we assume that

$\displaystyle c_{0}^{-1}\leqslant G(\ell)\leqslant c_{0},$

for some finite constants $c_{0}\geqslant 1$, $c_{1}>0$.

######  Remark 2.2.

The isotropic Gaussian spherical random field (not time-dependent)
$\\{Z(x),\,x\in\mathbb{S}^{2}\\}$ with zero mean, can be represented by

(2.13) $\displaystyle Z(x)$ $\displaystyle=$
$\displaystyle\sum_{\ell=1}^{\infty}\sqrt{C_{\ell}}T_{\ell}(x)=\sum_{\ell=1}^{\infty}\sqrt{C_{\ell}}\sum_{m=-\ell}^{\ell}a_{\ell
m}Y_{\ell m}(x),\quad x\in\mathbb{S}^{2},$ (2.14) $\displaystyle Z_{\ell}(x)$
$\displaystyle:=$ $\displaystyle\sum_{m=-\ell}^{\ell}a_{\ell m}Y_{\ell
m}(x),\quad x\in\mathbb{S}^{2},$

where the random variables $\\{a_{\ell m},m=-\ell,\dots,\ell\\}$ are standard
Gaussian and independent, the sequence $\\{C_{\ell}\\}_{\ell=0,1,\dots}$ is
called angular power spectrum of $Z(x)$ and it is non-negative and such that
$\sum_{\ell}C_{\ell}\frac{2\ell+1}{4\pi}=\mathbb{E}Z^{2}<\infty$ (see [36,
37]). The series (2.13) converges in $L^{2}(\Omega\times\mathbb{S}^{2})$, and
in $L^{2}(\Omega)$ at every fixed $x$.

The Assumption I on the power spectrum translates here simply as the
following.

Condition (A): The angular power spectrum $C_{\ell}$ is such that

(2.15) $\displaystyle C_{\ell}=G(\ell)\ell^{-\alpha}>0,\;\;\ell=1,2,\ldots,$
$\displaystyle C_{0}(0)=c_{1},$

where $\alpha>2$ is a constant, and we assume that

$\displaystyle c_{0}^{-1}\leqslant G(\ell)\leqslant c_{0},$

for some finite constant $c_{0}\geqslant 1$, $c_{1}>0$.

We recall that (2.15) has been used in [23] and it does not require stronger
regularity assumptions such as the ones used in [2, 22, 16, 42] among others.
Moreover, Condition (A) also cover the Matérn class of covariance functions
[17], which is the prototype of covariance models in the applications. The
Condition (2.15) with $2<\alpha<4$ is suggested by many models for data
analysis of the Cosmic Microwave Background radiation [14, 9]. For the time-
dependent case, the restriction $2+\beta<\alpha<4$ will be needed in
Proposition 4.2.

######  Remark 2.3.

We remark that there is a fractal behaviour for $2<\alpha<4$. For these values
of $\alpha$ the modulus of continuity decays slower than linearly with respect
to the angular distance, and hence making the sample function of $Z(x)$ non-
differentiable, see [23, Theorem 2]. Many applied fields do satisfy
$2<\alpha<4$. For example, the value of $\alpha$ for Cosmic Microwave
Background data is known to be very close to $2$, see [13]. For this reason,
the estimate (2.21) is not believed to hold true for $\alpha>4$, but it is
conjectured for the critical case $\alpha=4$, and we refer to [23] for more
details.

Let us consider the function
$\mu_{\alpha,\beta}:\mathbb{S}^{2}\times\mathbb{R}\times\mathbb{S}^{2}\times\mathbb{R}\longrightarrow[0,\infty)$
given by

(2.16)
$\mu^{2}_{\alpha,\beta}((x,t),(y,s)):=|t-s|^{\beta}+(1-|t-s|^{\beta})\rho_{\alpha}^{2}(d_{\mathbb{S}^{2}}(x,y)),$

where

(2.17) $\rho_{\alpha}(t):=t^{\frac{\alpha-2}{2}}.$

In [23] it was shown that the function $\rho_{\alpha}$ is equivalent to the
canonical metric on $\mathbb{S}^{2}$ induced by a spherical random field
satisfying (2.15) with $2<\alpha<4$. This function was introduced in [23] to
prove the strong non-local determinism for a spherical Gaussian random field
$Z$ on the sphere. More precisely, if (2.15) holds for some $2<\alpha<4$, then
there exist two positive and finite constants $C$ and $\varepsilon_{0}$ such
that for all integers $n\geqslant 1$, and all
$x_{0},\ldots,x_{n}\in\mathbb{S}^{2}$ with $\min_{1\leqslant k\leqslant
n}d_{\mathbb{S}^{2}}(x_{0},x_{k})\leqslant\varepsilon_{0}$ we have that

(2.18) $\text{Var}\left(Z(x_{0})\,|\,Z(x_{1}),\ldots,Z(x_{n})\right)\geqslant
C\min_{1\leqslant k\leqslant
n}\rho_{\alpha}\left(d_{\mathbb{S}^{2}}(x_{0},x_{k})\right)^{2},$

where $\rho_{\alpha}$ is given by (2.17).

One of our main ingredients is Proposition 2.5, where we prove a strong non-
local determinism similar to (2.21) for the time-space random field $T(x,t)$.
We end this section by showing that the function $\mu_{\alpha,\beta}$ is
equivalent to the canonical distance on $\mathbb{S}^{2}\times\mathbb{R}$
induced by $T(x,t)$.

######  Proposition 2.4.

Let $\\{T(x,t),\,(x,t)\in\mathbb{S}^{2}\times\mathbb{R}\\}$ be a Gaussian
random field satisfying Assumption I. Then there exist constants $1\leqslant
c_{\alpha}<\infty$ and $0<\varepsilon<1$ such that for all
$x,y\in\mathbb{S}^{2}$ with $d_{\mathbb{S}^{2}}(x,y)<\varepsilon$ and all
$s,t\in\mathbb{R}$ with $|t-s|<1$ one has that

(2.19) $\displaystyle
c_{\alpha}^{-1}\mu^{2}_{\alpha,\beta}((x,t),(y,s))\leqslant
d_{T}^{2}((x,t),(y,s))\leqslant
c_{\alpha}\mu^{2}_{\alpha,\beta}((x,t),(y,s)).$

###### Proof.

By (2.10) we have that

$\displaystyle
d^{2}_{T}((x,t),(y,s))=\mathbb{E}\left[T(x,t)^{2}\right]+\mathbb{E}\left[T(y,s)^{2}\right]-2\mathbb{E}\left[T(x,t)T(y,s)\right]$
$\displaystyle=2\sum_{\ell=1}^{\infty}\frac{2\ell+1}{4\pi}C_{\ell}(0)-2\sum_{\ell=1}^{\infty}\frac{2\ell+1}{4\pi}C_{\ell}(t-s)P_{\ell}(\cos\theta)$
$\displaystyle=\sum_{\ell=1}^{\infty}\frac{2\ell+1}{2\pi}(C_{\ell}(0)-C_{\ell}(t-s)P_{\ell}(\cos\theta))$
$\displaystyle=\sum_{\ell=1}^{\infty}\frac{2\ell+1}{2\pi}\left(C_{\ell}(0)-C_{\ell}(t-s)\right)+\sum_{\ell=1}^{\infty}\frac{2\ell+1}{2\pi}C_{\ell}(t-s)\left(1-P_{\ell}(\cos\theta)\right)$
$\displaystyle=|t-s|^{\beta}\sum_{\ell=1}^{\infty}\frac{2\ell+1}{2\pi}C_{\ell}(0)+(1-|t-s|^{\beta})\sum_{\ell=1}^{\infty}\frac{2\ell+1}{2\pi}C_{\ell}(0)\left(1-P_{\ell}(\cos\theta)\right),$

and the result then follows by [23, Lemma 4]. ∎

###  2.3. Main Results

We want to study the asymptotic behaviour of the maximum of a Gaussian random
field $\\{T(x,t),\,(x,t)\in\mathbb{S}^{2}\times\mathbb{R}\\}$ in a small time-
space regime. Thus, we assume $(x,t)\in\mathbb{S}^{2}\times[0,T]$, for some
fixed $T>0$. On the product metric space $\mathbb{S}^{2}\times[0,T]$ one has
the natural $q$-product metrics given by

(2.20) $\displaystyle
d_{q}((x,t),(y,s)):=\left(|t-s|^{q}+d_{\mathbb{S}^{2}}(x,y)^{q}\right)^{\frac{1}{q}}$

for any $1\leqslant q\leqslant\infty$. These metrics are all equivalent and
induce the same topology. It is easy to see that our results are independent
of the choice of $q$, and for this reason we fix $q=2$. Let

$B_{r}(x,t):=\left\\{(y,s)\in\mathbb{S}^{2}\times[0,T]\,:\,d_{2}((x,t),(y,s))<r\right\\}.$

First, let us state an auxiliary result: the strong non-local determinism for
space-time real-valued spherical random fields, whose proof can be found in
Appendix A.

######  Proposition 2.5.

Let $\\{T(x,t),\,(x,t)\in\mathbb{S}^{2}\times[0,T]\\}$ be a Gaussian random
field satisfying Assumption I. Then there exist two positive and finite
constants $C$ and $\varepsilon_{0}$ such that for all integers $n\geqslant 1$,
and all $(x_{0},t_{0}),\ldots,(x_{n},t_{n})\in\mathbb{S}^{2}\times[0,T]$ with
$\min_{1\leqslant k\leqslant
n}d_{2}\left((x_{0},t_{0}),(x_{k},t_{k})\right)\geqslant\varepsilon_{0}$, we
have that

(2.21)
$\text{Var}\left(T(x_{0},t_{0})\,|\,T(x_{1},t_{1}),\ldots,T(x_{n},t_{n})\right)\geqslant
C\min_{1\leq k\leq n}\rho_{\alpha}^{2}(d_{\mathbb{S}^{2}}(x,x_{k}))).$

Let us introduce the following notation, and state our main results.

######  Notation 2.6.

For any $(x,t)\in\mathbb{S}^{2}\times[0,T]$, and $r>0$ let us set

$\displaystyle M_{r}(x,t):=\max_{(y,s)\in B_{r}(x,t)}|T(y,s)-T(x,t)|,$

where $B_{r}(x,t)$ is defined above. Moreover, for any $r>0$ and
$\varepsilon>0$ we define the following functions

(2.22) $\displaystyle\phi(r)=\phi_{\alpha,\beta}(r):=\left(\frac{\log|\log
r|}{r^{3}}\right)^{p},$ (2.23)
$\displaystyle\psi(r,\varepsilon)=\psi_{\alpha,\beta}(r,\varepsilon):=r^{3}\varepsilon^{-\frac{1}{p}},$

where

(2.24)
$p=p(\alpha,\beta):=\left(\frac{2}{\beta}+\frac{4}{\alpha-2}\right)^{-1}.$

######  Theorem 2.7.

Let $\\{T(x,t),\,(x,t)\in\mathbb{S}^{2}\times[0,T]\\}$ be a time-spherical
Gaussian random field satisfying Assumption I. Then there exists a finite
positive constant $\kappa_{\alpha,\beta}$ such that, for all
$(x,t)\in\mathbb{S}^{2}\times[0,T]$

(2.25) $\liminf_{r\rightarrow
0}\phi(r)M_{r}(x,t)=\kappa_{\alpha,\beta},\;\;\text{a.s.}$

where $\phi=\phi_{\alpha,\beta}$ is given by (2.22).

As an application of Theorem 2.7 we solve the small ball problem for the
random field $T(x,t)$.

######  Theorem 2.8.

Let $\\{T(x,t),\,(x,t)\in\mathbb{S}^{2}\times[0,T]\\}$ be a Gaussian random
field satisfying Assumption I, and $\kappa_{\alpha,\beta}$ be given by (2.25).
Then, for every $r>0$

(2.26) $\lim_{\varepsilon\rightarrow
0}-\varepsilon^{\frac{1}{p}}\log\mathbb{P}\left(M_{r}(x,t)<\varepsilon\right)=r^{3}\kappa_{\alpha,\beta}^{\frac{1}{p}},$

where $p=p(\alpha,\beta)$ is given by (2.24).

We remark that the constant $\kappa_{\alpha,\beta}$ only depends on $\alpha$
and $\beta$ and not on the point $(x,t)\in\mathbb{S}^{2}\times[0,T]$. Chung’s
LIL (2.25) describes sample paths properties of the random field $T(x,t)$
which do not depend on the fixed point $(x,t)\in\mathbb{S}^{2}\times[0,T]$ due
to isotropy and stationarity of the field.

###  2.4. Isotropic Gaussian spherical random fields (non-time-dependent)

Chung’s LIL for Gaussian isotropic spherical random fields (non-time-
dependent) can be derived from [33] and exploiting the strong non-local
determinism in [32]. However, it can also be seen as a special case of time-
dependent spherical Gaussian fields. Moreover, it is possible here to solve
also the small ball probabilities for Gaussian random fields as a special case
of Theorem 2.8.

More precisely, let us consider the field in 2.13. For any
$x\in\mathbb{S}^{2}$, and $r>0$ let

$\displaystyle M_{r}(x):=\max_{y\in B_{r}(x)}|Z(y)-Z(x)|,$

where $B_{r}(x)$ denotes the geodesic ball in $\mathbb{S}^{2}$ centered in
$x\in\mathbb{S}^{2}$; and for any $r>0$ and $2<\alpha<4$

$\displaystyle\phi_{\alpha}(r):=\left(\frac{\log|\log
r|}{r^{2}}\right)^{\frac{\alpha-2}{4}},$
$\displaystyle\rho_{\alpha}(t):=t^{\frac{\alpha-2}{2}}.$

It was shown in [23] that the function $\rho_{\alpha}$ is equivalent to the
canonical metric $d_{T}$ on $\mathbb{S}^{2}$ induced by the random field
$\\{Z(x),x\in\mathbb{S}^{2}\\}$.

######  Remark 2.9.

Note that the induced distance $\mu^{2}_{\alpha,\beta}((x,t),(y,s))$ in (2.16)
reduces to $\rho_{\alpha}^{2}(d_{\mathbb{S}^{2}}(x,y))$ if we set $t=s$.

Chung’s LIL and small ball probabilities are given by the following two
results respectively.

######  Theorem 2.10.

Let $\\{Z(x),\,x\in\mathbb{S}^{2}\\}$ be an isotropic Gaussian field
satisfying Condition (A) for some $2<\alpha<4$. Then there exists a finite
positive constant $p_{\alpha}$ such that

$\liminf_{r\rightarrow 0}\phi_{\alpha}(r)M_{r}(x)=p_{\alpha},\;\text{a.s.}$

for all $x\in\mathbb{S}^{2}$.

######  Theorem 2.11.

Let $\\{Z(x),x\in\mathbb{S}^{2}\\}$ be an isotropic Gaussian field satisfying
Condition (A) for some $2<\alpha<4$ and $p_{\alpha}$ be given by Theorem 2.10.
Then, for every $r>0$

$\lim_{\varepsilon\rightarrow
0}-\varepsilon^{\frac{4}{\alpha-2}}\log\mathbb{P}\left(M_{r}(x)<\varepsilon\right)=r^{2}p_{\alpha}^{\frac{4}{\alpha-2}}.$

######  Remark 2.12.

As we also mentioned for the time-dependent spherical Gaussian field, a key
tool to prove Theorem 2.10 (and then Theorem 2.11) is the strong non-local
determinism given in (2.21) and proved in [23]. This result holds if
$2<\alpha<4$ and it is conjectured to hold even for $\alpha=4$. If we assume
its validity for $\alpha=4$, under the assumption that the random field
$\\{Z(x),\,x\in\mathbb{S}^{2}\\}$ satisfies Condition (A) with $\alpha=4$ and
(2.21) holds with $\alpha=4$, Chung’s LIL and small probabilities can be
proven even in this case, with the functions

$\displaystyle\phi_{4}(r):=\sqrt{\frac{\log|\log r|}{r^{2}|\log r|}},$
$\displaystyle\rho_{4}(t):=t\sqrt{|\ln t|}.$

However, we leave the proof of the strong non-local determinism for spherical
random field for $\alpha=4$ as a future research.

##  3\. Preliminary results

We first prove a weaker version of Theorem 2.8, that is, we show that

$\displaystyle\liminf_{\varepsilon\rightarrow
0}-\varepsilon^{\frac{1}{p}}\log\mathbb{P}\left(M_{r}(x,t)<\varepsilon\right)>0,$
$\displaystyle\limsup_{\varepsilon\rightarrow
0}-\varepsilon^{\frac{1}{p}}\log\mathbb{P}\left(M_{r}(x,t)<\varepsilon\right)<\infty.$

This is the content of Proposition 3.1 and Proposition 3.2 below.

######  Proposition 3.1.

Let $\\{T(x,t),\,(x,t)\in\mathbb{S}^{2}\times[0,T]\\}$ be a time-spherical
Gaussian random field satisfying Assumption I. Then there exists a positive
finite constant $A_{1}=A_{1}(\alpha,\beta)$ only depending on $\alpha$ and
$\beta$ such that for all $\varepsilon>0$ and $r>\varepsilon^{\frac{1}{3p}}$
we have that

(3.1)
$\displaystyle\mathbb{P}\left(M_{r}(x,t)<\varepsilon\right)\leqslant\exp\left(-A_{1}\psi(r,\varepsilon)\right),$

where $\psi=\psi_{\alpha,\beta}$ is given by (2.23).

###### Proof.

The proof is based on the strong non-local determinism for the field $T(x,t)$
Propositon 2.5, and on a conditioning argument similar to [28, Proposition
4.2]. Let us recall that

$B_{r}(x,t):=\left\\{(y,s)\in\mathbb{S}^{2}\times[0,T]\,:\,d_{\mathbb{S}^{2}}(x,y)^{2}+|t-s|^{2}<r^{2}\right\\}.$

Let us consider the sets

$\displaystyle
B_{\frac{r}{\sqrt{2}}}(x):=\left\\{y\in\mathbb{S}^{2}\,:\,d_{\mathbb{S}^{2}}(x,y)<\frac{r}{\sqrt{2}}\right\\},$
$\displaystyle
I_{\frac{r}{\sqrt{2}}}(t):=\left[0,t+\frac{r}{\sqrt{2}}\right),$

and note that $B_{\frac{r}{\sqrt{2}}}(x)\times
I_{\frac{r}{\sqrt{2}}}(t)\subset B_{r}(x,t)$, so that

$\displaystyle\mathbb{P}\left(M_{r}(x,t)<\varepsilon\right)\leqslant\mathbb{P}\left(\max_{y\in
B_{\frac{r}{\sqrt{2}}}(x)}\max_{s\in
I_{\frac{r}{\sqrt{2}}}(t)}\left|T(y,s)-T(x,t)\right|<\varepsilon\right).$

Let $F:A_{r}\rightarrow B_{\frac{r}{\sqrt{2}}}(x)$ be a local diffeomorphism,
where $A_{r}\subset\mathbb{R}^{2}$ is a rectangle and $F(0,0)=x$, and for $r$
small enough such that $[0,r]\times[0,r]\subset A_{r}$. We can choose $F$ in
such a way that $d_{\mathbb{S}^{2}}(x_{i},x_{j})\geqslant
d_{\mathbb{R}^{2}}\left((\theta_{i},\varphi_{i}),(\theta_{j},\varphi_{j})\right)$.

Let $G:[0,r]\rightarrow[0,t+r/\sqrt{2}]$ be the bijection given by
$G(\eta)=(t/r+1/\sqrt{2})\eta$ so that $G(0)=0$ and $G(r)=t+r/\sqrt{2}$. We
then have a local diffeomorphism

$H:[0,r]\times[0,r]\times[0,r]\longrightarrow B_{\frac{r}{\sqrt{2}}}(x)\times
I_{\frac{r}{\sqrt{2}}}(t).$

Let us divide $[0,r]\times[0,r]$ into sub-rectangles of side-length

$\frac{r}{\bigg{\lfloor}\frac{r}{\varepsilon^{\frac{1}{2p}}}\bigg{\rfloor}+1},$

and $[0,r]$ into sub-rectangles of side-length $r/(\lfloor r\rfloor+1)$. We
then get a division of $[0,r]\times[0,r]\times[0,r]$ into $N$ sub-rectangles,
where

$N=\left(\bigg{\lfloor}\frac{r}{\varepsilon^{\frac{1}{2p}}}\bigg{\rfloor}+1\right)^{2}(\lfloor
r\rfloor+1)\geqslant\frac{r^{3}}{\varepsilon^{\frac{1}{p}}}.$

Note that $N>1$ since $r>\varepsilon^{\frac{1}{3p}}$. Now let
$(\theta_{i},\varphi_{i})$ for $i=1,\ldots,N$ denote the upper right-most
vertex in the subdivision of $[0,r]\times[0,r]$, and $\xi_{i}$ for
$i=1,\ldots,N$ be the right-most endpoint in the subdivision of $[0,r]$, and
set $p_{i}:=(x_{i},t_{i})$, where $x_{i}:=F(\theta_{i},\varphi_{i})$,
$t_{i}:=G(\xi_{i})$. Note that $p_{i}\in B_{\frac{r}{\sqrt{2}}}(x)\times
I_{\frac{r}{\sqrt{2}}}(t)\subset B_{r}(x,t)$, and by construction we have that

$\displaystyle
d_{\mathbb{R}^{2}}\left((\theta_{i},\varphi_{i}),(\theta_{j},\varphi_{j})\right)\geqslant
c\,\frac{r}{\bigg{\lfloor}\frac{r}{\varepsilon^{\frac{1}{2p}}}\bigg{\rfloor}+1}\quad\text{
for all }i,j=1,\ldots,N$

for some constant $c>0$. Set

$\displaystyle
A_{j}:=\left\\{\max_{i=1,\ldots,j}|T(p_{i})-T(x,t)|<\varepsilon\right\\},$

and write

$\displaystyle\mathbb{P}(A_{j})=\mathbb{E}\left[1_{A_{j-1}}\,\mathbb{P}\left(|T(p_{i})-T(x,t)|<\varepsilon\,\left.\right|T(p_{i})\,:\,i=0,\ldots,j-1\right)\right].$

By Proposition 2.5 we have that

$\displaystyle\text{Var}\left(T(p_{j})\,|\,T(p_{i})\,:\,i=0,\ldots,j-1\right)$
$\displaystyle\geqslant c_{2}\min_{1\leqslant i\leqslant
j-1}\rho^{2}_{\alpha}(d_{\mathbb{S}^{2}}(x_{j},x_{i}))=c_{2}\min_{1\leqslant
i\leqslant j-1}d_{\mathbb{S}^{2}}(x_{j},x_{i})^{\alpha-2}$
$\displaystyle\geqslant c_{2}\min_{1\leqslant i\leqslant
j-1}d_{\mathbb{R}^{2}}\left((\theta_{j},\varphi_{j}),(\theta_{i},\varphi_{i})\right)^{\alpha-2}$
$\displaystyle\geqslant
c_{2}c^{\alpha-2}\left(\frac{r}{\bigg{\lfloor}\frac{r}{\varepsilon^{\frac{1}{2p}}}\bigg{\rfloor}+1}\right)^{\alpha-2}=:\gamma.$

By the estimate $x\leqslant\lfloor x\rfloor+1\leqslant 2x$, for $x\geqslant
1$, it follows that

(3.2) $\displaystyle
c_{2}\left(\frac{c}{2}\right)^{\alpha-2}\leqslant\frac{\gamma}{\varepsilon^{\frac{\alpha-2}{2p}}}\leqslant
c_{2}c^{\alpha-2}.$

The conditional distribution of $T(p_{j})$ given all the $T(p_{i})$ is
Gaussian with conditional variance
$\text{Var}\left(T(p_{j})\,|\,T(p_{i})\,:\,i=0,\ldots,j-1\right)$ since the
field $T(p)$ is Gaussian. Then by the above lower bound on the conditional
variance and Anderson’s inequality [1] it follows that

$\displaystyle\mathbb{P}\left(|T(p_{j})-T(x,t)|<\varepsilon\,\left.\right|T(p_{i})\,:\,i=0,\ldots,j-1\right)\leqslant\mathbb{P}\left(|Z|\leqslant\frac{\varepsilon^{\frac{\alpha-2}{2p}}}{\sqrt{\gamma}}\right)\leqslant\exp(-C),$

where $Z$ is a standard Gaussian random variable and the last inequality holds
for some constant $C=C(\alpha,\beta)>0$ thanks to (3.2). Thus,

$\displaystyle\mathbb{P}(A_{N})\leqslant\exp(-C)\mathbb{P}(A_{N-1})\leqslant\cdots\leqslant\exp(-NC),$

and hence

$\displaystyle\mathbb{P}\left(M_{r}(x,t)<\varepsilon\right)\leqslant\mathbb{P}\left(\max_{i=1,\ldots,N}|T(p_{i})-T(x,t))|<\varepsilon\right)=\mathbb{P}(A_{N})$
$\displaystyle\leqslant\exp(-NC)\leqslant\exp\left(-C\,\frac{r^{3}}{\varepsilon^{\frac{1}{p}}}\right).$

∎

######  Proposition 3.2.

Let $\\{T(x,t),\,(x,t)\in\mathbb{S}^{2}\times[0,T]\\}$ be a time-spherical
Gaussian random field satisfying Assumption I. Then there exist a positive
finite constant $A_{2}=A_{2}(\alpha,\beta)$ such that for all $\varepsilon>0$
we have that

(3.3)
$\displaystyle\mathbb{P}\left(M_{r}(x,t)<\varepsilon\right)\geqslant\exp\left(-A_{2}\psi(r,\varepsilon)\right),$

where $\psi=\psi_{\alpha,\beta}$ is given by (2.23).

###### Proof.

Let $N(B_{r}(x,t),d_{T},\varepsilon)$ be the smallest number of $d_{T}$-balls
of radius $\varepsilon$ needed to cover $B_{r}(x,t)$. By Proposition 2.4 we
know that the canonical metric $d_{T}$ is equivalent to $\mu_{\alpha,\beta}$,
and hence it is enough to consider
$N(B_{r}(x,t),\mu_{\alpha,\beta},\varepsilon)$. The $\mu_{\alpha,\beta}$-ball
of radius $\varepsilon$ is given by

$\displaystyle
B^{\mu_{\alpha,\beta}}_{\varepsilon}(x,t)=\left\\{(y,s)\in\mathbb{S}^{2}\times[0,T]\,:\,\mu_{\alpha,\beta}((x,t),(y,s))<\varepsilon\right\\}$
$\displaystyle=\left\\{(y,s)\in\mathbb{S}^{2}\times[0,T]\,:\,|t-s|^{\beta}+(1-|t-s|^{\beta})\rho_{\alpha}^{2}(d_{\mathbb{S}^{2}}(x,y))<\varepsilon^{2}\right\\}.$

Due to isotropy and stationary of the field, we can assume $t=0$ and $x=N$,
the north pole. Then $d(N,y)=\theta$ and in spherical coordinates we get

$\displaystyle\text{Vol}(B^{\mu_{\alpha,\beta}}_{\varepsilon}(N,0))=\int_{B^{\mu_{\alpha,\beta}}_{\varepsilon}(N,0)}dyds=\int_{0}^{2\pi}d\varphi\int_{\left\\{|s|^{\beta}\left(1-\theta^{\alpha-2}\right)+\theta^{\alpha-2}<\varepsilon^{2}\right\\}}\sin\theta
d\theta ds.$

Note that

$\displaystyle\left\\{|s|^{\beta}\left(1-\theta^{\alpha-2}\right)+\theta^{\alpha-2}<\varepsilon^{2}\right\\}$
$\displaystyle=\left\\{(s,\theta)\,:\,0\leqslant\theta\leqslant\varepsilon^{\frac{2}{\alpha-2}}\,,\,|s|<\left(\frac{\varepsilon^{2}-\theta^{\alpha-2}}{1-\theta^{\alpha-2}}\right)^{\frac{1}{\beta}}\right\\}\cup\left\\{(s,\theta)\,:\,1<\theta<\pi\,,\,\left(\frac{\theta^{\alpha-2}-\varepsilon^{2}}{\theta^{\alpha-2}-1}\right)^{\frac{1}{\beta}}<s<T\right\\},$

and since we are interested in the asymptotics near $(x,t)=(N,0)$, we can rule
oiut the second set since $d(N,y)=\theta>1$. We then have that

$\displaystyle\text{Vol}(B^{\mu_{\alpha,\beta}}_{\varepsilon}(N,0))=2\pi\int_{0}^{\varepsilon^{\frac{2}{\alpha-2}}}\int_{-\left(\frac{\varepsilon^{2}-\theta^{\alpha-2}}{1-\theta^{\alpha-2}}\right)^{\frac{1}{\beta}}}^{\left(\frac{\varepsilon^{2}-\theta^{\alpha-2}}{1-\theta^{\alpha-2}}\right)^{\frac{1}{\beta}}}\sin\theta
dsd\theta$
$\displaystyle=2\pi\int_{0}^{\varepsilon^{\frac{2}{\alpha-2}}}2\left(\frac{\varepsilon^{2}-\theta^{\alpha-2}}{1-\theta^{\alpha-2}}\right)^{\frac{1}{\beta}}\sin\theta
d\theta=4\pi\varepsilon^{\frac{2}{\beta}+\frac{2}{\alpha-2}}\int_{0}^{1}\left(\frac{1-u^{\alpha-2}}{1-\varepsilon^{2}u^{\alpha-2}}\right)^{\frac{1}{\beta}}\sin\left(\varepsilon^{\frac{2}{\alpha-2}}u\right)du$
$\displaystyle=4\pi\varepsilon^{\frac{2}{\beta}+\frac{4}{\alpha-2}}\int_{0}^{1}u\left(\frac{1-u^{\alpha-2}}{1-\varepsilon^{2}u^{\alpha-2}}\right)^{\frac{1}{\beta}}\frac{\sin\left(\varepsilon^{\frac{2}{\alpha-2}}u\right)}{\varepsilon^{\frac{2}{\alpha-2}}u}du=:\varepsilon^{\frac{1}{p}}F_{\alpha,\beta}(\varepsilon),$

where the function $F_{\alpha,\beta}(\varepsilon)$ is bounded uniformly in
$\varepsilon$ above and below by a positive constant. Similarly, one can show
that

$\displaystyle\text{Vol}(B_{r}(N,0))=2\pi
r^{3}\int_{0}^{1}u(1-ru)^{\frac{1}{2}}\frac{\sin(ru)}{ru}du=:r^{3}H(r),$

where the function $H(r)$ is bounded uniformly in $r$ above and below by a
positive constant. Thus,

(3.4) $\displaystyle
N(B_{r}(N,0),\mu_{\alpha,\beta},\varepsilon)\leqslant\frac{\text{Vol}\left(B_{r}(N,0)\right)}{\text{Vol}(B^{\mu_{\alpha,\beta}}_{\varepsilon}(N,0))}=\frac{r^{3}H(r)}{\varepsilon^{\frac{1}{p}}F_{\alpha,\beta}(\varepsilon)},$

and hence we can take

$\displaystyle\psi_{\alpha,\beta}(r,\varepsilon):=\frac{r^{3}}{\varepsilon^{\frac{1}{p}}},$

which satisfied the assumption of Lemma 2.1. ∎

######  Remark 3.3.

The proof of Proposition 3.2 does not rely on the strong non-local determinism
property from Proposition 2.5. In particular, it holds for any
$\alpha\geqslant 4$ as well.

##  4\. Chung’s law of the iterated logarithm

The goal of this section is to show that the random variable

$\displaystyle\kappa_{\alpha,\beta}:=\liminf_{r\rightarrow
0}\phi(r)M_{r}(x,t),$

where

$\displaystyle M_{r}(x,t):=\max_{(y,s)\in B_{r}(x,t)}|T(y,s)-T(x,t)|,$

satisfies

$\displaystyle\mathbb{P}\left(0<\kappa_{\alpha,\beta}<\infty\right)=1.$

In the next section we will show that $\kappa_{\alpha,\beta}$ is constant a.s.
as a byproduct of Theorem 2.8. Let us recall that

$\displaystyle
p=p(\alpha,\beta)=\left(\frac{2}{\beta}+\frac{4}{\alpha-2}\right)^{-1}$
$\displaystyle\phi(r)=\phi_{\alpha,\beta}(r):=\left(\frac{\log|\log
r|}{r^{3}}\right)^{p}.$

Firs, let us prove that $\kappa_{\alpha,\beta}>0$ a.s.

######  Proposition 4.1.

Let $\\{T(x,t),\,(x,t)\in\mathbb{S}^{2}\times[0,T]\\}$ be a time-spherical
Gaussian random field satisfying Assumption I. Then

(4.1) $\kappa_{\alpha,\beta}\geqslant A_{1}^{p}\;\;\;a.s.$

where $\phi(r)$, $p$, and $A_{1}$ are given by (2.22), (2.24), and (3.1)
respectively.

###### Proof.

For $n\geqslant 1$ and $R>1$ let us set

$\displaystyle r_{n}:=R^{-n},\quad
0<\gamma<\left(\frac{A_{1}}{R^{3}}\right)^{p},$ $\displaystyle
A_{n}:=\left\\{\phi(r_{n-1})M_{r_{n}}(x,t)\leqslant\gamma\right\\}.$

Then by (3.1) we have that

$\displaystyle\mathbb{P}(A_{n})\leqslant\exp\left(-A_{1}\psi\left(r_{n},\frac{\gamma}{\phi(r_{n-1})}\right)\right)=\exp\left(-A_{1}\frac{r_{n}^{3}}{\gamma^{\frac{1}{p}}}\phi(r_{n-1})^{\frac{1}{p}}\right),$
$\displaystyle=\exp\left(-A_{1}\frac{r_{n}^{3}}{r_{n-1}^{3}}\log|\log
r_{n-1}|\gamma^{-\frac{1}{p}}\right)=\exp\left(-\frac{A_{1}}{R^{3}}\gamma^{-\frac{1}{p}}\log((n-1)\log
R)\right)$ $\displaystyle=\left(\frac{1}{(n-1)\log
R}\right)^{\frac{A_{1}}{R^{3}}\gamma^{-\frac{1}{p}}}$

and hence $\sum_{n=1}^{\infty}\mathbb{P}(A_{n})<\infty$ since
$A_{1}>\gamma^{\frac{1}{p}}R^{3}$. Note that $\phi(s)>\phi(t)$ for $0<s<t\ll
1$ and then

$\displaystyle\inf_{r_{n}\leqslant r\leqslant r_{n-1}}\phi(r)\max_{(y,s)\in
B_{r}(x,t)}|T(y,s)-T(x,t)|\geqslant\phi(r_{n-1})\max_{(y,s)\in
B_{r_{n}}(x,t)}|T(y,s)-T(x,t)|.$

Note that, for any $0<\gamma<\frac{A_{1}^{p}}{R^{3}}$

$\displaystyle\mathbb{P}\left(\liminf_{r\rightarrow 0}\phi(r)\max_{(y,s)\in
B_{r}(x,t)}|T(y,s)-T(x,t)|<\gamma\right)$
$\displaystyle\leqslant\mathbb{P}\left(\bigcup_{k\geqslant
1}\bigcap_{n\geqslant k}\left\\{\inf_{r_{n}\leqslant r\leqslant
r_{n-1}}\phi(r)\max_{(y,s)\in
B_{r}(x,t)}|T(y,s)-T(x,t)|<\gamma\right\\}\right)$
$\displaystyle\leqslant\mathbb{P}\left(\bigcup_{k\geqslant
1}\bigcap_{n\geqslant k}\left\\{\phi(r_{n-1})\max_{(y,s)\in
B_{r_{n}}(x,t)}|T(y,s)-T(x,t)|<\gamma\right\\}\right)=\mathbb{P}\left(\liminf_{n\rightarrow\infty}A_{n}\right)=0,$

where the latter is zero by the first Borel-Cantelli Lemma. Thus, we have that

$\displaystyle\liminf_{r\rightarrow 0}\phi(r)\max_{(y,s)\in
B_{r}(x,t)}|T(y,s)-T(x,t)|\geqslant\gamma\;\;a.s.\quad\text{ for every
}0<\gamma<\left(\frac{A_{1}}{R^{3}}\right)^{p},$

and the result follows by letting $R$ go to 1 and taking the supremum over
$\gamma$ on both sides. ∎

We now prove that $\kappa_{\alpha,\beta}<\infty$ a.s.

######  Proposition 4.2.

Let $\\{T(x,t),\,(x,t)\in\mathbb{S}^{2}\times[0,T]\\}$ be a time-spherical
Gaussian random field satisfying Assumption I. Then

(4.2) $\kappa_{\alpha,\beta}\leqslant A_{2,}^{p}\;\;\;a.s.$

where $\phi(r)$, $p$, and $A_{2}$ are given by (2.22), (2.24), and (3.3)
respectively.

###### Proof.

Let us set

$\displaystyle r_{n}:=e^{-n}\lfloor e^{n^{2}}\rfloor^{-1},$ $\displaystyle
d_{n}:=\lfloor e^{n^{2}}\rfloor,$

and note that

$\displaystyle r_{n}d_{n}=e^{-n}$ $\displaystyle r_{n}d_{n+1}>e^{n}.$

Let us consider the random fields

$\displaystyle\Xi_{n}(x,t):=\sum_{\ell=d_{n}+1}^{d_{n+1}}\sum_{m=-\ell}^{\ell}a_{\ell
m}(t)Y_{\ell m}(x),$ $\displaystyle\Theta_{n}(x):=T(x,t)-\Xi_{n}(x,t)$

that is,

$\displaystyle\Theta_{n}(x,t)=\sum_{\ell=1}^{d_{n}}\sum_{m=-\ell}^{\ell}a_{\ell
m}(t)Y_{\ell m}(x)+\sum_{\ell=d_{n+1}+1}^{\infty}\sum_{m=-\ell}^{\ell}a_{\ell
m}(t)Y_{\ell m}(x).$

For a fixed $(x,t)\in\mathbb{S}^{2}\times[0,T]$, the sequence of random fields
$\\{\Xi_{n}(x,t),(x,t)\in\mathbb{S}^{2}\times[0,T]\\}_{n=1}^{\infty}$ is
independent. Indeed, if $n<m$ then $d_{n}+1<d_{n+1}<d_{m}$ and the sets

$\displaystyle\\{a_{\ell
m}(t),\,m=-\ell,\ldots,\ell,\;\;\ell=d_{n}+1,\ldots,d_{n+1}\\},$
$\displaystyle\\{a_{\ell
m}(t),\,m=-\ell,\ldots,\ell,\;\;\ell=d_{m}+1,\ldots,d_{m+1}\\}$

are uncorrelated by (2.8) and hence independent because they are Gaussian.
Moreover, for each $n$ and $(x,t),\,(y,s)\in S^{2}\times[0,T]$, $\Xi_{n}(x,t)$
is independent of $\Theta_{n}(y,s)$ since they are Gaussian and

$\displaystyle\mathbb{E}\left[\Xi_{n}(x,t)\Theta_{n}(y,s)\right]=0,$

by (2.8). The proof is completed once we show that

(4.3)
$\displaystyle\sum_{n=1}^{\infty}\mathbb{P}\left(\phi(r_{n})\max_{(y,s)\in
B_{r_{n}}(x,t)}|\Xi_{n}(y,s)-\Xi_{n}(x,t)|\leqslant\gamma\right)=\infty,\quad\text{
for any }\gamma>A_{2}^{p},$ (4.4)
$\displaystyle\sum_{n=1}^{\infty}\mathbb{P}\left(\phi(r_{n})\max_{(y,s)\in
B_{r_{n}}(x,t)}|\Theta_{n}(y,s)-\Theta_{n}(x,t)|>\varepsilon\right)<\infty,\quad\text{
for any }\varepsilon>0.$

Indeed, by Borel-Cantelli Lemmas and (4.3) (4.4) it follows that

$\displaystyle\liminf_{r\rightarrow
0}\phi(r)M_{r}(x,t)\leqslant\liminf_{n\rightarrow\infty}\phi(r_{n})M_{r_{n}}(x,t)\leqslant\liminf_{n\rightarrow\infty}\phi(r_{n})\max_{(y,s)\in
B_{r_{n}}(x,t)}|\Xi_{n}(y,s)-\Xi_{n}(x,t)|$
$\displaystyle+\liminf_{n\rightarrow\infty}\phi(r_{n})\max_{(y,s)\in
B_{r_{n}}(x,t)}|\Theta_{n}(y,s)-\Theta_{n}(x,t)|\leqslant\gamma+\varepsilon,$

for any $\gamma>A_{2}^{p}$ and $\varepsilon>0$. The proof is then completed by
letting $\varepsilon$ go to zero and $\gamma$ to $A_{2}^{p}$.

Let us first prove (4.3). By Anderson inequality [1] and Proposition 3.2 it
follows that

$\displaystyle\mathbb{P}\left(\phi(r_{n})\max_{(y,s)\in
B_{r_{n}}(x,t)}|\Xi_{n}(y,s)-\Xi_{n}(x,t)|\leqslant\gamma\right)$
$\displaystyle\geqslant\mathbb{P}\left(\phi(r_{n})\max_{(y,s)\in
B_{r_{n}}(x,t)}|\Xi_{n}(y,s)-\Xi_{n}(x,t)+\Theta_{n}(y,s)-\Theta_{n}(x,t)|\leqslant\gamma\right)$
$\displaystyle=\mathbb{P}\left(\phi(r_{n})M_{r_{n}}(x)\leqslant\gamma\right)\geqslant\exp\left(-A_{2}\psi\left(r_{n},\frac{\gamma}{\phi(r_{n})}\right)\right)$
$\displaystyle=\exp\left(-A_{2}\gamma^{-\frac{1}{p}}\log|\log
r_{n}|\right)=\left(\frac{1}{|\log
r_{n}|}\right)^{A_{2}\gamma^{-\frac{1}{p}}},$

and (4.3) follows by Borel-Cantelli Lemma for independent events since
$\gamma>A_{2}^{p}$. We now prove (4.4) by means of (2.4) with $K=B_{r}(x,t)$
and $X=\Theta_{n}$. First, let us bound the diameter $D$ of $B_{r}(x,t)$ with
respect to the canonical metric $d_{\Theta_{n}}$ induced by the field
$\Theta_{n}$. We have that

$\displaystyle\mathbb{E}\left[(\Theta_{n}(y,s)-\Theta_{n}(x,t))^{2}\right]$
$\displaystyle=\sum_{\ell=1}^{d_{n}}\frac{2\ell+1}{2\pi}\left(C_{\ell}(0)-C_{\ell}(t-s)P_{\ell}(\cos\theta)\right)+\sum_{\ell=d_{n+1}+1}^{\infty}\frac{2\ell+1}{2\pi}\left(C_{\ell}(0)-C_{\ell}(t-s)P_{\ell}(\cos\theta)\right)$
$\displaystyle=|t-s|^{\beta}\sum_{\ell=1}^{d_{n}}\frac{2\ell+1}{2\pi}C_{\ell}(0)+(1-|t-s|^{\beta})\sum_{\ell=1}^{d_{n}}\frac{2\ell+1}{2\pi}C_{\ell}(0)\left(1-P_{\ell}(\cos\theta)\right)$
$\displaystyle+|t-s|^{\beta}\sum_{\ell=d_{n+1}+1}^{\infty}\frac{2\ell+1}{2\pi}C_{\ell}(0)+(1-|t-s|^{\beta})\sum_{\ell=d_{n+1}+1}^{\infty}\frac{2\ell+1}{2\pi}C_{\ell}(0)\left(1-P_{\ell}(\cos\theta)\right),$

where $\theta:=d_{\mathbb{S}^{2}}(x,y)$. Let us set

$\displaystyle S_{1}:=\max_{(y,s)\in
B_{r}(x,t)}|t-s|^{\beta}\sum_{\ell=1}^{d_{n}}\frac{2\ell+1}{2\pi}C_{\ell}(0),$
$\displaystyle S_{2}:=\max_{(y,s)\in
B_{r}(x,t)}(1-|t-s|^{\beta})\sum_{\ell=1}^{d_{n}}\frac{2\ell+1}{2\pi}C_{\ell}(0)\left(1-P_{\ell}(\cos\theta)\right),$
$\displaystyle S_{3}:=\max_{(y,s)\in
B_{r}(x,t)}|t-s|^{\beta}\sum_{\ell=d_{n+1}+1}^{\infty}\frac{2\ell+1}{2\pi}C_{\ell}(0),$
$\displaystyle S_{4}:=\max_{(y,s)\in
B_{r}(x,t)}(1-|t-s|^{\beta})\sum_{\ell=d_{n+1}+1}^{\infty}\frac{2\ell+1}{2\pi}C_{\ell}(0)\left(1-P_{\ell}(\cos\theta)\right),$

so that

$\displaystyle\max_{(y,s)\in
B_{r}(x,t)}\mathbb{E}\left[(\Theta_{n}(y,s)-\Theta_{n}(x,t))^{2}\right]\leqslant
S_{1}+S_{2}+S_{3}+S_{4}.$

Note that $|P_{\ell}(\cos\theta)|\leqslant 1$ and hence, by Assumption I and
Lemma B.2

$\displaystyle S_{1}\leqslant
c\,r_{n}^{\beta}\sum_{\ell=1}^{d_{n}}\ell^{1-\alpha}\leqslant
c\,r_{n}^{\beta}\sum_{\ell=1}^{d_{n+1}}\ell^{1-\alpha}\leqslant
c\,r_{n}^{\beta}d_{n+1}^{2-\alpha}=c\,r_{n}^{\beta}\left(\frac{1}{d_{n+1}}\right)^{\alpha-2}\leqslant
c\,r_{n}^{\alpha-2+\beta}e^{-n(\alpha-2)},$ $\displaystyle S_{2}\leqslant
c\,\max_{|s-t|<r_{n}}(1-|t-s|^{\beta})\sum_{\ell=1}^{d_{n}}\ell^{1-\alpha}\max_{y\in
B_{r}(x)}(1-P_{\ell}(\cos\theta))\leqslant
c\,\sum_{\ell=1}^{d_{n}}\ell^{1-\alpha}(\ell^{2}r_{n}^{2}+\ell^{4}r_{n}^{4})$
$\displaystyle=c\,r_{n}^{2}\sum_{\ell=1}^{d_{n}}\ell^{3-\alpha}+c\,r_{n}^{4}\sum_{\ell=1}^{d_{n}}\ell^{5-\alpha}\leqslant
c\,(r_{n}^{2}d_{n}^{4-\alpha}+r_{n}^{4}d_{n}^{6-\alpha})=c\,r_{n}^{\alpha-2}\left((r_{n}d_{n})^{4-\alpha}+(r_{n}d_{n})^{6-\alpha}\right)$
$\displaystyle=c\,r_{n}^{\alpha-2}\left(e^{-n(4-\alpha)}+e^{-n(6-\alpha)}\right)\leqslant
c\,r_{n}^{\alpha-2}e^{-n(4-\alpha)},$ $\displaystyle S_{3}\leqslant
c\,r_{n}^{\beta}\sum_{d_{n+1}+1}^{\infty}\ell^{1-\alpha}\leqslant
c_{\alpha}r_{n}^{\beta}\left(\frac{1}{d_{n+1}}\right)^{\alpha-2}\leqslant
c_{\alpha}r_{n}^{\alpha-2+\beta}e^{-n(\alpha-2)},$ $\displaystyle
S_{4}\leqslant
c\,\max_{|s-t|<r_{n}}(1-|t-s|^{\beta})\sum_{d_{n+1}+1}^{\infty}\ell^{1-\alpha}\leqslant
c_{\alpha}\,r_{n}^{\alpha-2}e^{-n(\alpha-2)},$

for some finite constant $c_{\alpha}>0$. Combining everything together it
follows that

$\displaystyle D^{2}\leqslant S_{1}+S_{2}+S_{3}+S_{4}\leqslant
c_{\alpha}r_{n}^{\alpha-2}e^{-n\lambda},$

where $\lambda=\lambda(\alpha):=\min(\alpha-2,4-\alpha)$. If $N_{\varepsilon}$
denotes the number of $d_{T}$-balls of radius $\varepsilon$ needed to cover
$B_{r_{n}}(x,t)$, then proceeding as in the proof of Proposition 3.2 we have
that

$N_{\varepsilon}\leqslant c\frac{r^{3}}{\varepsilon^{\frac{1}{p}}},$

for some finite constant $c>0$. Thus,

$\displaystyle\int_{0}^{D}\sqrt{\log
N_{\varepsilon}}d\varepsilon\leqslant\int_{0}^{c_{\alpha}r_{n}^{\frac{\alpha-2}{2}}e^{-\frac{n}{2}\lambda}}\sqrt{\log
c\frac{r_{n}^{3}}{\varepsilon^{\frac{1}{p}}}}d\varepsilon$
$\displaystyle=c_{\alpha,\beta}r_{n}^{3p}\int_{0}^{a}w^{-q}\sqrt{-\log w}dw,$

where
$a:=d_{\alpha,\beta}e^{-\frac{n}{2p}\lambda}r_{n}^{\frac{\alpha-2}{2p}-3}$,
for some constant $d_{\alpha,\beta}$, and $q=1-p<1$. In order to apply Lemma
B.3 we need $a<1$, which is satisfied if $\frac{\alpha-2}{2p}-3>0$, that is,
if $\alpha>2+\beta$. By Lemma B.3 it then follows that

$\displaystyle\int_{0}^{D}\sqrt{\log N_{\varepsilon}}d\varepsilon\leqslant
c_{\alpha,\beta}r_{n}^{3p}a^{1-q}\sqrt{-\log a}$
$\displaystyle=c_{\alpha,\beta}d_{\alpha\beta}^{p}r_{n}^{3p}e^{-\frac{n}{2}\lambda}r_{n}^{\frac{\alpha-2}{2}-3p}\sqrt{\frac{n}{2}\lambda\frac{1}{p}-\log
d_{\alpha,\beta}+(n+n^{2})\left(\frac{\alpha-2}{2p}-3\right)}$ (4.5)
$\displaystyle\leqslant
c_{\alpha\beta}r_{n}^{\frac{\alpha-2}{2}}e^{-\frac{n}{2}\lambda}n.$

Set

$\displaystyle
u:=\frac{\varepsilon}{c\phi(r_{n})}-c_{\alpha\beta}r_{n}^{\frac{\alpha-2}{2}}e^{-\frac{n}{2}\lambda}n$

where $c$ and $c_{\alpha\beta}$ are the constants given in (2.4) and (4.5)
respectively. Note that $u>0$ for all $n$ large enough since

$\displaystyle u=\frac{\varepsilon}{c}\frac{r_{n}^{3p}}{(\log|\log
r_{n}|)^{p}}\left(1-\frac{c}{\varepsilon}\left(r_{n}^{\frac{\alpha-2}{\beta}-1}\log|\log
r_{n}|\right)^{p}e^{-\frac{n}{2}\lambda}n\right)>0.$

Then by (2.4)

$\displaystyle\mathbb{P}\left(\phi(r_{n})\max_{(y,s)\in
B_{r_{n}}(x,t)}|\Theta_{n}(y,s)-\Theta_{n}(x,t)|>\varepsilon\right)$
$\displaystyle\leqslant\mathbb{P}\left(\max_{(z,v),(y,s)\in
B_{r_{n}}(x,t)}|\Theta_{n}(y,s)-\Theta_{n}(z,v)|>\frac{\varepsilon}{\phi(r_{n})}\right)$
$\displaystyle=\mathbb{P}\left(\max_{(z,v),(y,s)\in
B_{r_{n}}(x,t)}|\Theta_{n}(y,s)-\Theta_{n}(z,v)|>c\left(u+c_{\alpha\beta}r_{n}^{\frac{\alpha-2}{2}}e^{-\frac{n}{2}\lambda}n\right)\right)$
$\displaystyle\leqslant\mathbb{P}\left(\max_{(z,v),(y,s)\in
B_{r_{n}}(x,t)}|\Theta_{n}(y,s)-\Theta_{n}(z,v)|>c\left(u+\int_{0}^{D}\sqrt{\log
N_{\varepsilon}}d\varepsilon\right)\right)\leqslant\exp\left(-\frac{u^{2}}{D^{2}}\right).$

Note that for $n$ large

$u=\frac{1}{\phi(r_{n})}\left(\frac{\varepsilon}{c}-\phi(r_{n})c_{\alpha\beta}r_{n}^{\frac{\alpha-2}{2}}e^{-\frac{n}{2}\lambda}n\right)\geqslant\frac{b}{\phi(r_{n})},$

for some constant $b=b(\alpha,\beta)$ since
$\phi(r_{n})c_{\alpha\beta}r_{n}^{\frac{\alpha-2}{2}}e^{-\frac{n}{2}\lambda}n\rightarrow
0$. Thus

$\displaystyle\mathbb{P}\left(\phi(r_{n})\max_{(y,s)\in
B_{r_{n}}(x,t)}|\Theta_{n}(y,s)-\Theta_{n}(x,t)|>\varepsilon\right)\leqslant\exp\left(-\frac{b^{2}}{\phi(r_{n})^{2}}\frac{1}{D^{2}}\right)$
$\displaystyle\leqslant\left(-C\frac{e^{n\lambda}}{(\log|\log
r_{n}|)^{p}\,r_{n}^{\alpha-2-{6p}}}\right),$

for some finite constant $C>0$, and the proof of (4.4) is then complete. ∎

##  5\. Small probabilities

In this section we prove Theorem 2.8, the small ball principle for the random
field $\\{T(x,t),(x,t)\in\mathbb{S}^{2}\times[0,T]\\}$, and we show that the
random variable $\kappa_{\alpha,\beta}$ is constant a.s. By Proposition 4.1
and Proposition 4.2 we know that, for any $(x,t)\in\mathbb{S}^{2}\times[0,T]$

$\displaystyle\kappa_{\alpha,\beta}=\liminf_{r\rightarrow
0}\phi_{\alpha,\beta}(r)M_{r}(x,t),\;\;\;a.s.$ $\displaystyle
A_{1}^{p}\leqslant\kappa_{\alpha,\beta}\leqslant A_{2}^{p}\;\;\;a.s.$

where $A_{1}$, $A_{2}$, and $p$ are given by (3.1), (3.3), and (2.24)
respectively.

###### Proof of Theorem 2.8.

Let

$\displaystyle c_{+}:=\limsup_{\varepsilon\rightarrow
0}-\frac{1}{\psi(r,\varepsilon)}\log\mathbb{P}\left(M_{r}(x,t)<\varepsilon\right),$
$\displaystyle c_{-}:=\liminf_{\varepsilon\rightarrow
0}-\frac{1}{\psi(r,\varepsilon)}\log\mathbb{P}\left(M_{r}(x,t)<\varepsilon\right),$

where $\psi=\psi_{\alpha,\beta}$ is given by (2.23). By Proposition 3.2 and
Proposition 3.1 we know that

$0<A_{1}\leqslant c_{-}\leqslant c_{+}\leqslant A_{2}<\infty.$

The strategy of the proof is to show that

$\displaystyle c_{+}\leqslant\kappa_{\alpha,\beta}^{\frac{1}{p}}\leqslant
c_{-}\;\;\;a.s.$

and in particular

$\displaystyle\kappa_{\alpha,\beta}^{\frac{1}{p}}=c_{-}=c_{+}=\lim_{\varepsilon\rightarrow
0}-\frac{1}{\psi(r,\varepsilon)}\log\mathbb{P}\left(M_{r}(x,t)<\varepsilon\right)$

is constant. We first prove that
$c_{+}\leqslant\kappa_{\alpha,\beta}^{\frac{1}{p}}$ a.s. Let us fix
$\delta\in(0,c_{+})$. Then there exists an $\varepsilon(\delta)$ such that

(5.1)
$\mathbb{P}\left(M_{r}(x,t)<\varepsilon\right)\leqslant\exp\left(-\delta\psi(r,\varepsilon)\right),$

for any $\varepsilon\leqslant\varepsilon(\delta)$. Let $r_{n}:=R^{-n}$ for
some $R>1$ and let us choose $\gamma$ such that $0<R^{3}\gamma<\delta$. Then

$\displaystyle\mathbb{P}\left(M_{r_{n}}(x,t)<\frac{\gamma^{p}}{\phi(r_{n-1})}\right)=\mathbb{P}\left(\max_{(y,s)\in
B_{r_{n}}(x,t)}|T(y,s)-T(x,t)|<\frac{\gamma^{p}}{\phi(r_{n-1})}\right)$
$\displaystyle\leqslant\exp\left(-\delta\psi\left(r_{n},\frac{\gamma^{p}}{\phi(r_{n-1})}\right)\right)=\exp\left(-\frac{\delta}{R^{3}\gamma}\log((n-1)\log
R)\right)=\left(\frac{1}{(n-1)\log R}\right)^{\frac{\delta}{R^{3}\gamma}},$

and hence

$\displaystyle\sum_{n=1}^{\infty}\mathbb{P}\left(M_{r_{n}}(x,t)<\frac{\gamma^{p}}{\phi(r_{n-1})}\right)<\infty,$

since $0<R^{3}\gamma<\delta$. Thus by Borel-Cantelli Lemma we have that

$\displaystyle\mathbb{P}\left(\bigcap_{k\geqslant 1}\bigcup_{n\geqslant
k}\left\\{M_{r_{n}}(x,t)<\frac{\gamma^{p}}{\phi(r_{n-1})}\right\\}\right)=0,\text{
i.e.}$ $\displaystyle\mathbb{P}\left(\bigcup_{k\geqslant 1}\bigcap_{n\geqslant
k}\left\\{M_{r_{n}}(x,t)\geqslant\frac{\gamma^{p}}{\phi(r_{n-1})}\right\\}\right)=1,$

and hence almost surely for all large $n$

$M_{r_{n}}(x,t)\geqslant\frac{\gamma^{p}}{\phi(r_{n-1})}.$

Note that $\phi(r)>\phi(r_{n-1})$ for $r_{n}\leqslant r\leqslant r_{n-1}$, and
it then follows that

$\displaystyle M_{r}(x,t)\geqslant
M_{r_{n}}(x,t)\geqslant\frac{\gamma^{p}}{\phi(r_{n-1})}\geqslant\frac{\gamma^{p}}{\phi(r)},$

which yields

$\displaystyle\kappa_{\alpha,\beta}:=\liminf_{r\rightarrow
0}\phi(r)M_{r}(x,t)\geqslant\gamma^{p}\;a.s.$

for any $\gamma<\frac{\delta}{R^{3}}<\frac{c_{+}}{R^{3}}$, and hence
$c_{+}\leqslant\kappa_{\alpha,\beta}^{\frac{1}{p}}$ a.s. by letting first $R$
go to 1, and then $\delta$ to $c_{+}$.

Let us now prove that $\kappa_{\alpha,\beta}^{\frac{1}{p}}\leqslant c_{-}$
a.s. Let $\delta>c_{-}$ be fixed. Then there exists an $\varepsilon(\delta)$
such that

$\displaystyle\mathbb{P}\left(M_{r}(x,t)<\varepsilon\right)\geqslant\exp\left(-\delta\psi(r,\varepsilon)\right),$

for any $\varepsilon\leqslant\varepsilon(\delta)$. Let $r_{n}$ ,
$\Xi_{n}(x,t)$, and $\Theta_{n}(x,t)$ be defined as in the proof of
Proposition 4.2. Then

$\displaystyle\mathbb{P}\left(\phi(r_{n})\max_{(y,s)\in
B_{r_{n}}(x,t)}|\Xi_{n}(y,s)-\Xi_{n}(x,t)|\leqslant\delta\,c_{-}^{p-1}\right)$
$\displaystyle\geqslant\mathbb{P}\left(\max_{(y,s)\in
B_{r_{n}}(x,t)}|\Xi_{n}(y,s)-\Xi_{n}(x,t)|\leqslant\frac{\delta\,c_{-}^{p-1}}{\phi(r_{n})}\right)$
$\displaystyle\geqslant\exp\left(-\delta\psi\left(r_{n},\frac{\delta\,c_{-}^{p-1}}{\phi(r_{n})}\right)\right)=\exp\left(-\left(\frac{c_{-}}{\delta}\right)^{\frac{1-p}{p}}\log|\log
r_{n}|\right)=\left(\frac{1}{|\log r_{n}|}\right)^{a},$

where $a:=\left(\frac{c_{-}}{\delta}\right)^{\frac{1-p}{p}}<1$ since
$\delta>c_{-}$. The events

$\left\\{\phi(r_{n})\max_{(y,s)\in
B_{r_{n}}(x,t)}|\Xi_{n}(y,s)-\Xi_{n}(x,t)|\leqslant\delta\,c_{-}^{p-1}\right\\}$

are independent, and hence by Borel-Cantelli Lemma for independent events it
follows that

$\liminf_{n\rightarrow\infty}\phi(r_{n})\max_{(y,s)\in
B_{r_{n}}(x,t)}|\Xi_{n}(y,s)-\Xi_{n}(x,t)|\leqslant\delta\,c_{-}^{p-1}\,a.s.\,$

for any $\delta>c_{-}$, and thus,

$\liminf_{n\rightarrow\infty}\phi(r_{n})\max_{(y,s)\in
B_{r_{n}}(x,t)}|\Xi_{n}(y,s)-\Xi_{n}(x,t)|\leqslant\delta\,c_{-}^{p-1}\leqslant
c_{-}^{p}\;a.s.$

Note that by (4.4) we have that

$\displaystyle\kappa_{\alpha,\beta}:=\liminf_{r\rightarrow
0}\phi(r)M_{r}(x,t)\leqslant\liminf_{n\rightarrow\infty}\phi(r_{n})M_{r_{n}}(x,t)$
$\displaystyle\leqslant\liminf_{n\rightarrow\infty}\phi(r_{n})\max_{(y,s)\in
B_{r_{n}}(x,t)}|\Xi_{n}(y,s)-\Xi_{n}(x,t)|+\liminf_{n\rightarrow\infty}\phi(r_{n})\max_{(y,s)\in
B_{r_{n}}(x,t)}|\Theta_{n}(y,s)-\Theta_{n}(x,t)|$ $\displaystyle\leqslant
c_{-}^{p}+\varepsilon$

for any $\varepsilon>0$, and hence the result follows by letting $\varepsilon$
go to zero. ∎

##  Appendix A Proof of Proposition 2.5

In this section we prove Proposition 2.5, i.e. the strong non-local
determinism for this field. We work in spherical coordinates
$(\theta,\varphi)$ and we take without loss of generality $x_{0}=(0,0)$ to be
the North Pole and $x_{j}=(\theta_{j},\varphi_{j})$ so that
$d_{\mathbb{S}^{2}}(x_{0},x_{j})=\theta_{j}$. Since $T(x,t)$ is a Gaussian
random field, it is known that

$Var(T(x_{0},t_{0})|T(x_{1},t_{1}),...T(x_{n},t_{n}))=\inf\\{\mathbb{E}[(T(x_{0},t_{0})-\sum_{j=1}^{n}\gamma_{j}T(x_{j},t_{j}))^{2}]:\gamma_{1},\dots,\gamma_{n}\in\mathbb{R}\\}.$

We observe that

$\displaystyle\mathbb{E}\left[\left(T(x,t)-\sum_{j=1}^{n}\gamma_{j}T(x_{j},t_{j})\right)^{2}\right]=\Gamma(x,t,x,t)-2\sum_{j=1}^{n}\gamma_{j}\Gamma(x,t,x_{j},t_{j})+\sum_{j,i=1}^{n}\gamma_{i}\gamma_{j}\Gamma(x_{j},t_{j},x_{i},t_{i}).$

Indeed,

$\displaystyle\mathbb{E}\left[\left(T(x,t)-\sum_{j=1}^{n}\gamma_{j}T(x_{j},t_{j})\right)^{2}\right]=\mathbb{E}\left[\left(\sum_{\ell=1}^{\infty}\sum_{m=-\ell}^{\ell}a_{\ell
m}(t)Y_{\ell
m}(x)-\sum_{\ell=1}^{\infty}\sum_{m=-\ell}^{\ell}\sum_{j=1}^{n}\gamma_{j}a_{\ell
m}(t_{j})Y_{\ell m}(x_{j})\right)^{2}\right]$
$\displaystyle=\sum_{\ell,\ell^{\prime}=1}^{\infty}\sum_{m=-\ell}^{\ell}\sum_{m^{\prime}=-\ell^{\prime}}^{\ell^{\prime}}\mathbb{E}[a_{\ell
m}(t)a_{\ell^{\prime}m^{\prime}}(t)]Y_{\ell
m}(x)Y_{\ell^{\prime}m^{\prime}}(x)$
$\displaystyle\qquad-2\sum_{\ell,\ell^{\prime}=1}^{\infty}\sum_{m=-\ell}^{\ell}\sum_{m^{\prime}=-\ell^{\prime}}^{\ell^{\prime}}\sum_{j=1}^{n}\gamma_{j}\mathbb{E}[a_{\ell
m}(t)a_{\ell^{\prime}m^{\prime}}(t_{j})]Y_{\ell
m}(x)Y_{\ell^{\prime}m^{\prime}}(x_{j})$
$\displaystyle\qquad+\sum_{\ell,\ell^{\prime}=1}^{\infty}\sum_{m=-\ell}^{\ell}\sum_{m^{\prime}=-\ell^{\prime}}^{\ell^{\prime}}\sum_{j,j^{\prime}=1}^{n}\gamma_{j}\gamma_{j^{\prime}}\mathbb{E}[a_{\ell
m}(t_{j})a_{\ell^{\prime}m^{\prime}}(t_{j^{\prime}})]Y_{\ell
m}(x_{j})Y_{\ell^{\prime}m^{\prime}}(x_{j^{\prime}})$

In view of (2.8) we obtain

$\displaystyle\mathbb{E}\left[\left(T(x,t)-\sum_{j=1}^{n}\gamma_{j}T(x_{j},t_{j})\right)^{2}\right]$
$\displaystyle=\sum_{\ell=1}^{\infty}\sum_{m=-\ell}^{\ell}C_{\ell}(0)Y_{\ell
m}(x)Y_{\ell
m}(x)-2\sum_{\ell=1}^{\infty}\sum_{m=-\ell}^{\ell}\sum_{j=1}^{n}\gamma_{j}C_{\ell}(t-t_{j})Y_{\ell
m}(x)Y_{\ell^{\prime}m^{\prime}}(x_{j})$
$\displaystyle\qquad+\sum_{\ell=1}^{\infty}\sum_{m=-\ell}^{\ell}\sum_{j,j^{\prime}=1}^{n}\gamma_{j}\gamma_{j^{\prime}}C_{\ell}(t_{j}-t_{j^{\prime}})Y_{\ell
m}(x_{j})Y_{\ell^{\prime}m^{\prime}}(x_{j^{\prime}})$
$\displaystyle=\sum_{\ell=1}^{\infty}C_{\ell}(0)\frac{2\ell+1}{4\pi}P_{\ell}(0)-2\sum_{\ell=1}^{\infty}\sum_{j=1}^{n}\gamma_{j}C_{\ell}(t-t_{j})\frac{2\ell+1}{4\pi}P_{\ell}(d_{\mathbb{S}^{2}}(x,x_{j}))$
$\displaystyle\qquad+\sum_{\ell=1}^{\infty}\sum_{j,j^{\prime}=1}^{n}\gamma_{j}\gamma_{j^{\prime}}C_{\ell}(t_{j}-t_{j^{\prime}})\frac{2\ell+1}{4\pi}P_{\ell}(d_{\mathbb{S}^{2}}(x_{j},x_{j^{\prime}})).$

Then we write

$\displaystyle\mathbb{E}\left[\left(T(x,t)-\sum_{j=1}^{n}\gamma_{j}T(x_{j},t_{j})\right)^{2}\right]=\sum_{\ell=0}^{\infty}\frac{2\ell+1}{4\pi}C_{\ell}(0)\bigg{[}P_{\ell}(0)-2\sum_{j=1}^{n}\gamma_{j}\frac{C_{\ell}(t-t_{j})}{C_{\ell}(0)}P_{\ell}(d(x,x_{j}))$
$\displaystyle\quad+\sum_{j,j^{\prime}=1}^{n}\gamma_{j}\gamma_{j^{\prime}}\frac{C_{\ell}(t_{j}-t_{j^{\prime}})}{C_{\ell}(0)}P_{\ell}(d(x_{j^{\prime}},x_{j}))\bigg{]}$
$\displaystyle=\sum_{\ell=0}^{\infty}\frac{2\ell+1}{4\pi}C_{\ell}(0)\bigg{[}\left(1-\sum_{j=1}^{n}\gamma_{j}\frac{C_{\ell}(t-t_{j})}{C_{\ell}(0)}P_{\ell}(d(x,x_{j}))\right)^{2}$
$\displaystyle\quad+\sum_{j,j^{\prime}=1}^{n}\gamma_{j}\gamma_{j^{\prime}}\bigg{(}\frac{C_{\ell}(t_{j}-t_{j^{\prime}})}{C_{\ell}(0)}P_{\ell}(d(x_{j^{\prime}},x_{j}))-\frac{C_{\ell}(t_{j}-t)C_{\ell}(t-t_{j^{\prime}})}{C_{\ell}(0)C_{\ell}(0)}P_{\ell}(d(x,x_{j}))P_{\ell}(d(x,x_{j^{\prime}}))\bigg{)}\bigg{]}.$

From Lemma B.1 it follows that

$\displaystyle\mathbb{E}\left[\left(T(x,t)-\sum_{j=1}^{n}\gamma_{j}T(x_{j},t_{j})\right)^{2}\right]\geq\sum_{\ell=0}^{\infty}\frac{2\ell+1}{4\pi}C_{\ell}(0)\left(1-\sum_{j=1}^{n}\gamma_{j}\frac{C_{\ell}(t-t_{j})}{C_{\ell}(0)}P_{\ell}(d(x,x_{j}))\right)^{2}.$

The thesis follows by proving the proposition below.

######  Proposition A.1.

Assume condition I holds. For all $\varepsilon\in(0,\varepsilon_{0}/2]$ there
exists a constant $C>0$ such that for all choices of $n\in\mathbb{N}$, all
$(\theta_{j},\varphi_{j}):\theta_{j}>\varepsilon$ and
$\gamma_{j}\in\mathbb{R}$, $j=1,\dots,n$ we have

$\displaystyle\sum_{\ell=0}^{\infty}\frac{2\ell+1}{4\pi}C_{\ell}(0)\left(1-\sum_{j=1}^{n}\gamma_{j}\frac{C_{\ell}(t-t_{j})}{C_{\ell}(0)}P_{\ell}(d(x,x_{j}))\right)^{2}\geq
C\min_{1\leq k\leq n}\rho_{\alpha}^{2}(d_{\mathbb{S}^{2}}(x,x_{k}))).$

###### Proof.

For any fixed $\varepsilon>0$, let $\delta_{\varepsilon}(\cdot,\cdot)$ the
spherical bump function constructed in [23, Section 2.2] with the
corresponding coefficients $\\{b_{\ell}(\varepsilon)\\}$. We follow the proof
of [23, Proposition 7]. We define

$I:=\sum_{\ell=0}^{\infty}\sqrt{\frac{2\ell+1}{4\pi
C_{\ell}(0)}}b_{\ell}(\varepsilon)\left[\sqrt{\frac{2\ell+1}{4\pi}C_{\ell}(0)}\left(1-\sum_{j=1}^{n}\gamma_{j}\frac{C_{\ell}(t-t_{j})}{C_{\ell}(0)}P_{\ell}(d(x,x_{j}))\right)\right].$

By Cauchy-Schwarz

$I^{2}\leq\sum_{\ell=0}^{\infty}\frac{2\ell+1}{4\pi}\frac{b_{\ell}(\varepsilon)^{2}}{C_{\ell}(0)}\sum_{\ell=0}^{\infty}\frac{2\ell+1}{4\pi}C_{\ell}(0)\left(1-\sum_{j=1}^{n}\gamma_{j}\frac{C_{\ell}(t-t_{j})}{C_{\ell}(0)}P_{\ell}(d(x,x_{j}))\right)^{2}$

From which it follows that

$\sum_{\ell=0}^{\infty}\frac{2\ell+1}{4\pi}C_{\ell}(0)\left(1-\sum_{j=1}^{n}\gamma_{j}\frac{C_{\ell}(t-t_{j})}{C_{\ell}(0)}P_{\ell}(d(x,x_{j}))\right)^{2}\geq\frac{I^{2}}{\sum_{\ell=0}^{\infty}\frac{b_{\ell}(\varepsilon)^{2}}{C_{\ell}(0)}\frac{2\ell+1}{4\pi}}.$

Let us compute I now. We have that

$I=\sum_{\ell-0}^{\infty}b_{\ell}(\varepsilon)\frac{2\ell+1}{4\pi}-\sum_{\ell=0}^{\infty}b_{\ell}(\varepsilon)\frac{2\ell+1}{4\pi}\sum_{j=1}^{n}\gamma_{j}\frac{C_{\ell}(t-t_{j})}{C_{\ell}(0)}P_{\ell}(d(x,x_{j})).$

From [23] we know that

$\sum_{\ell=0}^{\infty}b_{\ell}(\varepsilon)\frac{2\ell+1}{4\pi}=\delta_{\varepsilon}(0,0)\geq\frac{c^{\prime}}{2\varepsilon^{2}}$

and $C_{\ell}(t-t_{j})=(1-|t-t_{j}|^{\beta})C_{\ell}(0)$ since $|t-t_{j}|<1$,
and hence

$\displaystyle\sum_{\ell=0}^{\infty}b_{\ell}(\varepsilon)\frac{2\ell+1}{4\pi}\sum_{j=1}^{n}\gamma_{j}\frac{C_{\ell}(t-t_{j})}{C_{\ell}(0)}P_{\ell}(d(x,x_{j}))$
$\displaystyle\quad=\sum_{j=1}^{n}\gamma_{j}(1-|t-t_{j}|^{\beta})\sum_{\ell=0}^{\infty}b_{\ell}(\varepsilon)\frac{2\ell+1}{4\pi}P_{\ell}(d(x,x_{j}))$
$\displaystyle\quad=\sum_{j=1}^{n}\gamma_{j}(1-|t-t_{j}|^{\beta})\delta_{\varepsilon}(\theta_{j},\phi_{j})=0.$

Then we have $I\geq\frac{c^{\prime}}{2}\varepsilon^{-2}$. It remains to prove
that

$\sum_{\ell=0}^{\infty}\frac{b_{\ell}(\varepsilon)^{2}}{C_{\ell}(0)}\frac{2\ell+1}{4\pi}=O(\varepsilon^{-(\alpha+2)}),$

which follows from the proof of [23, Proposition 7] in view of Assumption I. ∎

##  Appendix B Technical Lemmas

We collect here other technical lemmas exploited along the paper.

######  Lemma B.1.

Assume condition I. For any $(x_{j},t_{j})$ and
$(x_{j^{\prime}},t_{j^{\prime}})$ points on $\mathbb{S}^{2}\times[0,T]$ and
for any $\gamma_{j}\in\mathbb{R}$, $j=1,\dots,n$ we have that

$\displaystyle\sum_{\ell=0}^{\infty}$
$\displaystyle\frac{2\ell+1}{4\pi}C_{\ell}(0)$
$\displaystyle\times\sum_{j,j^{\prime}=1}^{n}\gamma_{j}\gamma_{j^{\prime}}\bigg{(}\frac{C_{\ell}(t_{j}-t_{j^{\prime}})}{C_{\ell}(0)}P_{\ell}(d(x_{j^{\prime}},x_{j}))-\frac{C_{\ell}(t_{j}-t)C_{\ell}(t-t_{j^{\prime}})}{C_{\ell}(0)C_{\ell}(0)}P_{\ell}(d(x,x_{j}))P_{\ell}(d(x,x_{j^{\prime}})\bigg{)}\geq
0$

###### Proof.

Note that

$\displaystyle\frac{2\ell+1}{4\pi}C_{\ell}(0)\bigg{(}\frac{C_{\ell}(t_{j}-t_{j^{\prime}})}{C_{\ell}(0)}P_{\ell}(d(x_{j^{\prime}},x_{j}))-\frac{C_{\ell}(t_{j}-t)C_{\ell}(t-t_{j^{\prime}})}{C_{\ell}(0)C_{\ell}(0)}P_{\ell}(d(x,x_{j}))P_{\ell}(d(x,x_{j^{\prime}})\bigg{)}$
$\displaystyle=\frac{4\pi}{(2\ell+1)C_{\ell}(0)}\bigg{(}C_{\ell}(0)\frac{2\ell+1}{4\pi}P_{\ell}(0)C_{\ell}(t_{j}-t_{j^{\prime}})\frac{2\ell+1}{4\pi}P_{\ell}(d(x_{j^{\prime}},x_{j}))$
$\displaystyle-
C_{\ell}(t_{j}-t)P_{\ell}(d(x,x_{j}))\frac{2\ell+1}{4\pi}C_{\ell}(t-t_{j^{\prime}})\frac{2\ell+1}{4\pi}P_{\ell}(d(x,x_{j^{\prime}}))\bigg{)}$

Let us set
$C_{\ell}(x,y,t,s):=C_{\ell}(t-s)\frac{2\ell+1}{4\pi}P_{\ell}(d(x,y))$. Then
$C_{\ell}(x,y,t,s)$ is the covariance function of the random field
$T_{\ell;t}(x):=\sum_{m=-\ell}^{\ell}a_{\ell m}(t)Y_{\ell m}(x)$. By
Proposition 1 [32] the function
$C_{\ell}(x,x,t,t)C_{\ell}(x_{1},x_{2},t_{1},t_{2})-C_{\ell}(x,x_{1},t,t_{1})C_{\ell}(x,x_{2},t,t_{2})$
is still a covariance function, and as such it is positive definite. Thus,

$\displaystyle\sum_{\ell=0}^{\infty}\frac{2\ell+1}{4\pi}C_{\ell}(0)$
$\displaystyle\quad\times\sum_{j,j^{\prime}=1}^{n}\gamma_{j}\gamma_{j^{\prime}}\bigg{(}\frac{C_{\ell}(t_{j}-t_{j^{\prime}})}{C_{\ell}(0)}P_{\ell}(d(x_{j^{\prime}},x_{j}))-\frac{C_{\ell}(t_{j}-t)C_{\ell}(t-t_{j^{\prime}})}{C_{\ell}(0)C_{\ell}(0)}P_{\ell}(d(x,x_{j}))P_{\ell}(d(x,x_{j^{\prime}})\bigg{)}$
$\displaystyle=\sum_{\ell=0}^{\infty}\frac{4\pi}{(2\ell+1)C_{\ell}(0)}\left(C_{\ell}(x,x,t,t,)C_{\ell}(x_{j},x_{j^{\prime}},t_{j},t_{j^{\prime}})-C_{\ell}(x,x_{j},t,t_{j})C_{\ell}(x,x_{j^{\prime}},t,t_{j^{\prime}})\right)\geq
0$

since $C_{\ell}(0)$ is also positive. ∎

######  Lemma B.2.

Let $P_{\ell}$ be the $\ell$-th Legendre polynomial. Then there exists a
constant $c$ such that for any $\theta$ small enough

(B.1) $0<1-P_{\ell}(\cos\theta)\leqslant
c(\ell^{2}\theta^{2}+\ell^{4}\theta^{4}).$

###### Proof.

The Taylor expansion of $P_{\ell}(t)$ at $t=1$ reads

$\displaystyle
P_{\ell}(t)=1+P^{\prime}_{\ell}(t)(t-1)+R_{\ell}(t)=1+\frac{\ell(\ell+1)}{2}(t)(t-1)+R_{\ell}(t).$

Note that

$\displaystyle P^{\prime\prime}_{\ell}(t)\leqslant
P^{\prime\prime}_{\ell}(1)=\frac{(\ell+2)!}{8(\ell-2)!}\leqslant c\ell^{4},$

and

$\displaystyle|R_{\ell}(t)|=\frac{1}{2}|P^{\prime\prime}_{\ell}(\xi)|(t-1)^{2}\leqslant
c\ell^{4}(t-1)^{2},$

for some $\xi\in[t,1]$. Thus,

$\displaystyle
1-P_{\ell}(\cos\theta)\leqslant\frac{\ell(\ell+1)}{2}(1-\cos\theta)+c\ell^{4}(1-\cos\theta)^{2}$
$\displaystyle=\ell(\ell+1)\sin^{2}\frac{\theta}{2}+c\ell^{4}\sin^{4}\frac{\theta}{2}\leqslant
c\left(\ell^{2}\theta^{2}+\ell^{4}\theta^{4}\right).$

∎

######  Lemma B.3.

For any $q<1$ and $0<\delta<1$ there exists an explicit constant
$c_{q,\delta}$ such that for any $0<a<\delta<1$

(B.2) $\displaystyle\int_{0}^{a}u^{-q}\sqrt{-\log u}du\leqslant
c_{q,\delta}a^{1-q}\sqrt{-\log a}.$

###### Proof.

A simple integration by parts yields

$\displaystyle\int_{0}^{a}u^{-q}\sqrt{-\log
u}du=\frac{1}{1-q}a^{1-q}\sqrt{-\log
a}+\frac{1}{2(1-q)}\int_{0}^{a}u^{-q}(-\log u)^{-\frac{1}{2}}du$
$\displaystyle\leqslant\frac{1}{1-q}a^{1-q}\sqrt{-\log
a}+\frac{1}{2(1-q)}(-\log a)^{-\frac{1}{2}}\int_{0}^{a}u^{-q}du$
$\displaystyle=\sqrt{|\log
a|}a^{1-q}\left(\frac{1}{1-q}+\frac{1}{2(1-q)^{2}}\frac{1}{|\log
a|}\right)\leqslant c_{q,\delta}\sqrt{|\log a|}a^{1-q},$

where

$\displaystyle
c_{q,\delta}:=\frac{1}{1-q}+\frac{1}{2(1-q)^{2}}\frac{1}{|\log\delta|}.$

∎

######  Acknowledgement.

The authors want to thank Domenico Marinucci for helpful discussions during
the preparation of this work. This article is dedicated to the memory of
Orietta di Biagio.

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