[a,b]Kari J. Eskola

# Exclusive quarkonium photoproduction in $A$+$A$ UPCs at the LHC in NLO pQCD

Christopher A. Flett  Vadim Guzey  Topi Löytäinen  Hannu Paukkunen

###### Abstract

We present the first study of coherent exclusive quarkonium ($J/\psi$,
$\Upsilon$) photoproduction in ultraperipheral nucleus-nucleus collisions
(UPCs) at the LHC in the framework of collinear factorization and next-to-
leading order (NLO) perturbative QCD (pQCD). We make NLO predictions for the
$J/\psi$ and $\Upsilon$ rapidity distributions for lead (Pb) and oxygen (O)
beams, and quantify their dependence on the factorization/renormalization
scale, nuclear parton distribution functions (PDFs) and their uncertainties,
and on differences between nuclear PDFs and generalized parton distribution
functions (GPDs). We show that within the PDF-originating uncertainties our
approach provides a good description of the available $J/\psi$ photoproduction
data in Pb+Pb UPCs at the LHC but that the scale uncertainty is significant.
We demonstrate that at NLO pQCD the quark contributions are important in the
$J/\psi$ case but that gluons clearly dominate the $\Upsilon$ cross sections.
We also study how the scale dependence could be tamed by considering O+O/Pb+Pb
ratios of the exclusive $J/\psi$ UPC cross sections, and how HERA and p+p/Pb
LHC data can help in obtaining better-controlled NLO predictions in the
$\Upsilon$ case.

##  1 Motivation for NLO studies

It was originally proposed by Ryskin [1] that exclusive coherent
photoproduction of $J/\psi$ at HERA, $\gamma+\text{p}\rightarrow
J/\psi+\text{p}$, probes very efficiently collinearly factorized small-$x$
gluon distributions: The leading-order (LO) cross section is directly
proportional to $[xg(x,Q)]^{2}$ at momentum fractions $x={\cal
O}(M^{2}/W^{2})$ and scales $Q={\cal O}(M)$, where $M$ is the the vector meson
mass and $W$ is the center-of-mass system energy of the $\gamma$+p system. Our
NLO studies of coherent photoproduction of $J/\psi$ [2, 3] and $\Upsilon$ [4]
in $A$+$A$ UPCs at the LHC aim at studying the role of these processes as
gluon probes, and at understanding whether one could use these as further
constraints in the global analyses of nuclear PDFs. For this, we study the
gluon and quark contributions, and uncertainties arising from the scale
choices, PDFs and GPD modeling. In particular, it is interesting to see
whether the NLO calculation could simultaneously fit the mid- and
forward/backward-rapidity measurements better than the previous LO
calculations – see the comparison in Ref. [5].

##  2 Theoretical framework

The theoretical framework here is that of collinear factorization. We compute
the rapidity-differential cross sections for coherent photoproduction of heavy
vector mesons $V$ in $A_{1}+A_{2}$ UPCs as products of the photon fluxes and
photon-nucleus cross sections, accounting for the fact that either one of the
colliding nuclei can be the photon emitter,

$\frac{d\sigma^{A_{1}A_{2}\rightarrow
A_{1}VA_{2}}}{dy}=\left[k\frac{dN_{\gamma}^{A_{1}}}{dk}\sigma^{\gamma
A_{2}\rightarrow VA_{2}}\right]_{k=k^{+}}\\\
+\left[k\frac{dN_{\gamma}^{A_{2}}}{dk}\sigma^{A_{1}\gamma\rightarrow
A_{1}V}\right]_{k=k^{-}}$ (1)

where $k^{\pm}\approx Me^{\pm y}/2$ are the energies of the photons emitted by
the nuclei $A_{1,2}$. The photon fluxes are obtained as impact-parameter
integrals of the number of equivalent Weizsäcker-Williams photons,
$N_{\gamma}^{A}(k,b)$, at an energy $k$ and impact parameter $\vec{b}$,
multiplied by the Poissonian probability $\Gamma_{AA}(\vec{b})$ of having no
hadronic interactions,

$k\frac{dN_{\gamma}^{A}}{dk}=\int
d^{2}bN_{\gamma}^{A}(k,b)\Gamma_{AA}(\vec{b}),$ (2)

where $N_{\gamma}^{A}(k,b)$ is from Ref. [6], and
$\Gamma_{AA}(\vec{b})=\exp[-\sigma_{NN}T_{AA}(b)]$ with $\sigma_{NN}$ being
the total nucleon-nucleon cross section and $T_{AA}(b)$ the standard nuclear
overlap function computed from the Woods-Saxon nuclear density distribution.
Note that there is no cut-off for $b$ in the above integral. The photon-
nucleus cross section for $\gamma+A_{2}\rightarrow V+A_{2}$ (and
correspondingly for $\sigma^{A_{1}\gamma\rightarrow A_{1}V}$) is

$\sigma^{\gamma A\rightarrow VA}=\frac{d\sigma_{A}^{\gamma N\rightarrow
VN}}{dt}\bigg{|}_{t=0}\int\limits_{t_{\rm
min}}^{\infty}dt^{\prime}|F_{A}(-t^{\prime})|^{2},$ (3)

where the nuclear form factor $F_{A}$, originating from the nuclear GPDs, is a
Fourier transform of the Woods-Saxon nuclear density distribution, and $t$ is
the Mandelstam variable. The per-nucleon cross section, computed here in the
$t\rightarrow 0$ limit, is ${d\sigma_{A}^{\gamma N\rightarrow
VN}}/{dt}={|\mathcal{M}^{\gamma N\rightarrow VN}_{A}|^{2}}/(16\pi W^{4})$.
Collinear factorization is done here at the amplitude level [7],

$\mathcal{M}^{\gamma N\rightarrow VN}_{A}\propto\sqrt{{\langle
O_{1}\rangle_{V}}}\int\limits_{-1}^{1}dx[T_{g}(x,\xi,\mu_{F},\mu_{R})F^{g}_{A}(x,\xi,t,\mu_{F})+T_{q}(x,\xi,\mu_{F},\mu_{R})F^{q,S}_{A}(x,\xi,t,\mu_{F})],$
(4)

where $T_{g}$ and $T_{q}$ are the pQCD coefficient functions for gluons and
quarks, and $F^{g}_{A}$ and $F^{q,S}_{A}$ are the nuclear GPDs (nGPDs),
correspondingly ($S$ for flavor singlet). These depend on the momentum
fraction $x$, skewness $\xi$, factorization scale $\mu_{F}$, and
renormalization scale $\mu_{R}$. The $T_{g}$ and $T_{q}$ were computed in NLO
pQCD (${\cal O}(\alpha_{s}^{2})$) in Ref. [8], and we apply their results
here. Note that quarks start to contribute only from the NLO level on. We
obtain the non-perturbative NRQCD element $\langle O_{1}\rangle_{V}$ by
matching the measured leptonic decay width of the vector meson with the one
calculated at NLO. In the forward limit ($t=0$, $\xi=0$), the nGPDs become the
usual nuclear PDFs (nPDFs),

$\displaystyle F^{g}_{A}(x,0,0,\mu_{F})$ $\displaystyle=$ $\displaystyle
F^{g}_{A}(-x,0,0,\mu_{F})=|x|g_{A}(|x|,\mu_{F})\,,$ $\displaystyle
F^{q,S}_{A}(x,0,0,\mu_{F})$ $\displaystyle=$
$\displaystyle\sum_{q}\left[\theta(x)q_{A}(x,\mu_{F})-\theta(-x)\bar{q}_{A}(-x,\mu_{F})\right],$
(5)

where the sum is over massless quarks, and $\theta(x)$ is the step function.
Here, we use the EPPS21 [9], nNNPDF3.0 [10] and nCTEQWZSIH [11] nPDFs.

Figure 1: Rapidity-differential coherent $J/\psi$ photoproduction cross
section vs. rapidity in Pb+Pb UPCs at $\sqrt{s_{NN}}=5.02$ TeV (left) and 2.76
TeV (right), obtained with EPPS21 and scales
$\mu=M_{J/\psi}/2,\,0.77M_{J/\psi}$ and $M_{J/\psi}$. For the LHC data
references, see [3]. Figure from [3].

##  3 NLO results for $J/\psi$ and $\Upsilon$ in $A$+$A$ UPCs at the LHC

Taking the nGPDs in the forward limit and equating the scales
$\mu_{F}=\mu_{R}=\mu$, we chart the scale dependence of the NLO results by
varying $\mu$ from $M_{J/\psi}/2$ to $M_{J/\psi}$. As seen in Fig. 1, the
scale uncertainty is significant, a factor of 55 (22) at $y=0$ for the left
(right) panel results. With an “optimal” scale $\mu\approx 0.77M_{J/\psi}$ we
can, however, fit both the Run 2 and Run 1 LHC ALICE data at $y=0$, although
at forward/backward rapidities the NLO calculation has difficulties in
reproducing the Run 2 ALICE and latest LHCb data. This, and the HERA data (see
comparison in Ref. [2]) suggests that there is still room for next-to-NLO
(NNLO) and NRQCD corrections, as well as effects from GPD modeling.

Figure 2: Breakdown of the NLO cross section into contributions from quarks
only, gluons only, and their interference, for $J/\psi$ (left, figure from
[2]) and $\Upsilon$ (right, figure from [4]).

Somewhat surprisingly, see the left panel of Fig. 2, quarks actually dominate
over gluons in the NLO cross sections for $J/\psi$ at $y\sim 0$. This is
because the LO and NLO gluon amplitudes tend to cancel, which opens room for a
significant quark contribution. Because of this the $J/\psi$ cross sections
reflect the gluon nPDFs in a nontrivial way, and certainly they do not scale
with the gluon nPDFs squared like in LO [2]. Whether this remains so also at
NNLO is yet to be seen. For $\Upsilon$, the gluon dominance is again
recovered, as is seen in the right panel of Fig. 2.

Figure 3: As Fig. 1 left panel, but with PDF uncertainties at the “optimal”
scale. From [3], with the new CMS data [12] added.

The propagation of the nPDF uncertainties to the NLO $J/\psi$ cross section is
shown in Fig. 3. The EPPS21 and nNNPDF3.0 error bands include both the nPDF
and the PDF uncertainties, while nCTEQ15WZSIH gives only the nuclear ones. The
NLO calculation at the optimal scale is consistent with the data within the
promisingly moderate EPPS21 error band. The data can be expected to have
constraining power on the global fits, provided that the calculation can be
brought into a better theoretical control. With the nNNPDF3.0 central set, the
NLO $y$-distributions are narrower than those with EPPS21, increasing the
tension with the forward/backward data, but interestingly with the
nCTEQ15WZSIH central set we do catch also these data. In nCTEQ15WZSIH, the
$s$-quark distribution is strongly enhanced, which (with the quark dominance)
enhances the cross sections. Thus, the studied $J/\psi$ process may a bit
surprisingly turn out to be a probe of the poorly known $s$-quark
distributions. Again, whether this feature persists also at NNLO, remains to
be seen.

Figure 4: Upper left: Scale dependence of the same-energy ratios of the
coherent photoproduction cross sections of $J/\psi$ in O+O and Pb+Pb collisons
at the LHC, obtained with EPPS21 nPDFs. Lower left: The same but at the
optimal scales of EPPS21, nNNPDF3.0 and nCTEQ15WZSIH, along with their
uncertainties. Right: Different-energy ratios with their scale- and PDF
uncertainties at $y=0$. Figures based on [3].

In Fig. 4 , we demonstrate that the significant scale uncertainty can be tamed
by forming ratios of $J/\psi$ cross sections in O+O and Pb+Pb UPCs [3]. In the
same-energy ratios plotted in the upper left panel, the factor of 55 scale
uncertainty at $y=0$ of Fig. 1 shrinks down to a factor of 1.7. The lower left
panel shows how these ratios still remain promisingly sensitive to the nPDF
uncertainties. Also in the different-energy ratios at $y=0$, shown in the
right panel of Fig. 4, the scale uncertainties are at most of the same
magnitude as the PDF uncertainties.

Next, we repeat the NLO calculation for coherent $\Upsilon$ photoproduction in
Pb+Pb UPCs at the LHC [4]. We now also model the GPDs via the Shuvaev
transform [13]. From the $\gamma$+p baseline case shown in the left panel of
Fig. 5, we see that HERA $\gamma$+p, and LHC p+p/Pb data favor smaller scales,
thus we vary the scale $\mu$ between $M_{\Upsilon}/4$ and $M_{\Upsilon}$. As a
larger scale process, the scale dependence for $\Upsilon$ becomes clearly
weaker than for $J/\psi$. And, as we saw in Fig. 2, gluons now dominate, thus
coherent $\Upsilon$ photoproduction in Pb+Pb UPCs probes gluons more directly
(at NLO) than $J/\psi$. The left panel of Fig. 5 also shows that the effects
due to the GPD modeling here remain quite small. We can also see that the NLO
cross sections fall consistently below the HERA/LHC data, indicating again the
need for, e.g., NNLO and NRQCD corrections. To make use of the baseline data,
we adopt a data-driven method for obtaining the photon-nucleus cross sections,
writing

$\sigma^{\gamma A\rightarrow\Upsilon A}(W)=\left[{\sigma^{\gamma
A\rightarrow\Upsilon A}(W)}/{\sigma^{\gamma p\rightarrow\Upsilon
p}(W)}\right]_{\text{pQCD}}\sigma^{\gamma p\rightarrow\Upsilon
p}_{\text{fit}}(W),$ (6)

where the NLO calculation (the pQCD ratio) gives the nuclear effects and the
HERA/LHC data fit the normalization. Folding these with the photon fluxes then
gives our data-driven predictions shown in the right panel of Fig. 5. Due to
the pQCD ratio, the scale uncertainties become now smaller than those of the
nuclear GPDs/PDFs, and the GPD effects are seen to become negligible.

Figure 5: Left: Coherent photoproduction cross section of $\Upsilon$ in
$\gamma$+p collisions, obtained with CT18ANLO-based nGPDs, CT18ANLO nPDFs [14]
and their uncertainties, compared with HERA and LHC data (for data refs., see
[4]). Right: Data-driven NLO prediction for coherent photoproduction of
$\Upsilon$ in Pb+Pb UPCs at the LHC, for EPPS21-based nGPDs and nPDFs at three
scales. The error bands in both panels are for $\mu=m_{b}$. Figures from [4].

##  4 Conclusions and outlook

We presented the first NLO study of coherent photoproduction of $J/\psi$ and
$\Upsilon$ in $A$+$A$ UPCs at the LHC using collinear factorization. For
$J/\psi$, a significant scale uncertainty but more moderate nPDF-originating
uncertainties were found. Within the nPDF uncertainties, the NLO calculation
at a specific optimal scale agrees with all the available LHC data. The PDF
uncertainties exceed the data error bars, suggesting PDF-constraining power
for the data. For $\Upsilon$, the scale uncertainties are considerably
smaller. In both cases, however, there clearly is room for further
corrections, such as those arising from NNLO pQCD and NRQCD. The GPD modeling
effects via the Shuvaev transform were found to be small in the NLO cross
sections. Quarks were found to dominate the NLO cross sections for $J/\psi$,
suggesting that $J/\psi$ could be a probe of the elusive $s$-quark
distributions. For $\Upsilon$, the gluons were the dominant component,
suggesting $\Upsilon$ to be a more direct gluon probe than $J/\psi$. Finally,
we made a HERA/LHC-data-driven prediction for the $y$-differential coherent
photoproduction cross section of $\Upsilon$, to be tested in the future LHC
measurements.111 We acknowledge the financial support from the MagnusEhrnrooth
foundation (T.L.), the Academy of Finland Projects No. 308301 (H.P.) and No.
330448 (K.J.E.). This research was funded as a part of the Center of
Excellence in Quark Matter of the Academy of Finland (Projects No. 346325 and
No. 346326). This research is part of the European Research Council Project
No. ERC-2018-ADG-835105 YoctoLHC.

## References

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Investigation of sample paths properties of sub-Gaussian type random fields,
with application to stochasic heat equations

Olha Hopkalo
Department of Economic Cybernetics, Faculty of Economics,
Taras Shevchenko National University of Kyiv, Kyiv, Ukraine,
E-mail<EMAIL_ADDRESS>
Lyudmyla Sakhno
Department of Probability Theory, Statistics and Actuarial Mathematics,
Faculty of Mechanics and Mathematics,
Taras Shevchenko National University of Kyiv, Kyiv, Ukraine,
E-mail<EMAIL_ADDRESS>
###### Abstract

The paper presents bounds for the distributions of suprema for a particular
class of sub-Gaussian type random fields defined over spaces with anisotropic
metrics. The results are applied to random fields related to stochastic heat
equations with fractional noise: bounds for the tail distributions of suprema
and estimates for the rate of growth are provided for such fields.

Key words: sub-Gaussian type random fields, distribution of supremum, rate of
growth, stochastic heat equation, fractional noise

##  1 Introduction and preliminaries

Aim and motivation. In this paper, we study sample paths properties of a class
of sub-Gaussian type random fields $X(t)$, $t\in T$, focusing on the case
where a parameter set $T$ is endowed with an anisotropic metric and imposing
some kind of Hölder continuity condition on the field $X$. Our aim is to
establish upper bounds for the distribution of the supremum
$P\big{\\{}\sup_{t\in T}|X(t)|>u\big{\\}}$ for bounded $T$ and to evaluate a
rate of growth of $X$ over an unbounded domain $V$ by considering upper bounds
for $P\big{\\{}\sup_{t\in V}\frac{|X(t)|}{f(t)}>u\big{\\}}$ for a properly
chosen continuous function $f$. The study is motivated by applications to
random fields related to stochastic heat equations. Extensive recent
investigations of such equations resulted, in particular, in establishing the
Hölder continuity of solutions in various settings. It is quite natural and
appealing to make further steps towards considerarion of different functionals
of solutions. We evaluate the distribution of suprema of solutions and their
asymptotic rate of growth.

Approach and tools. We present bounds for distributions of suprema assuming
$X$ to belong to a particular class of $\varphi$-sub-Gaussian random fields
(to be defined below), which provides a generalization of Gaussian and sub-
Gaussian fields. To derive the results we apply entropy methods. Recall that
the entropy approach in studing sample paths of a stochastic process $X(t)$,
$t\in T$, requires to evaluate entropy characteristics of the set $T$ with
respect to a particular metrics generated by the process $X$. The origins of
this approach are due to Dudley, who stated coditions for boundedness of
Gaussian processes in the form of convergence of metric entropy integrals
(which we call now Dudley entropy integrals). We address for corresponding
references, e.g., to [1] and [4], where in the latter one the entropy approach
was extended to different classes of processes, more general than Gaussian
ones.

Some facts from the general theory of $\varphi$-sub-Gaussian random variables
and fields. Note that it is important for applications to go beyond the
Gaussianity assumption in considered models and possible extensions are
provided by sub-Gaussian and $\varphi$-sub-Gaussian random processes and
fields. Recall that a random variable $\xi$ is sub-Gaussian if its moment
generating function is majorized by that of a Gaussian centered random
variable $\eta\sim N(0,\sigma^{2})$:

$\textsf{E}\exp(\lambda\xi)\leq\textsf{E}\exp(\lambda\eta)=\exp(\sigma^{2}\lambda^{2}/2).$

The generalization of this notion to the classes of $\varphi$-sub-Gaussian
random variables is introduced as follows (see, [4, Ch.2], [7], [16], [20]).

Consider a continuous even convex function $\varphi$ such that $\varphi(0)=0$,
$\varphi(x)>0$ as $x\neq 0$ and $\lim\limits_{x\to 0}\frac{\varphi(x)}{x}=0$,
$\lim\limits_{x\rightarrow\infty}\frac{\varphi(x)}{x}=\infty.$ Note that such
functions are called Orlicz N-functions. Suppose that $\varphi$ satisfies
additionally: $\lim\inf\limits_{x\to 0}\frac{\varphi(x)}{x^{2}}=c>0,$ where
the case $c=\infty$ is possible.

Let $\varphi$ be the function with the above properties and
$\\{\Omega,\cal{F},\mathsf{P}\\}$ be a standard probability space. The random
variable $\zeta$ is $\varphi$-sub-Gaussian, or belongs to the space
$\mathrm{Sub}_{\varphi}(\Omega)$, if $\textsf{E}\zeta=0,$
$\textsf{E}\exp\\{\lambda\zeta\\}$ exists for all $\lambda\in\mathbb{R}$ and
there exists a constant $a>0$ such that the following inequality holds for all
$\lambda\in\mathbb{R}$

$\textsf{E}\exp\\{\lambda\zeta\\}\leq\exp\\{\varphi(\lambda a)\\}.$ (1.1)

The random field $X(t)$, $t\in T$, is called $\varphi$-sub-Gaussian if the
random variables $\\{X(t),t\in T\\}$ are $\varphi$-sub-Gaussian.

The space $\mathrm{Sub}_{\varphi}(\Omega)$ is a Banach space with respect to
the norm (see [7, 16]):

$\tau_{\varphi}(\zeta)=\inf\\{a>0:\textsf{E}\exp\\{\lambda\zeta\\}\leq\exp\\{\varphi(a\lambda)\\}.$

For $\varphi$-sub-Gaussian random variable $\zeta$ the following estimate for
its tail probability holds:

$P\\{|\zeta|>u\\}\leq
2\exp\left\\{-\varphi^{*}\left(\frac{u}{\tau_{\varphi}(\zeta)}\right)\right\\},\,\,u>0,$
(1.2)

where the function $\varphi^{*}$ defined by
$\varphi^{*}(x)=\sup_{\begin{subarray}{c}y\in\mathbb{R}\end{subarray}}(xy-\varphi(y))$
is called the Young-Fenchel transform (or Legendre transform, or convex
conjugate) of the function $\varphi$. It was stated in [4] (Corollary 4.1, p.
68) that, moreover, a random variable $\zeta$ is a $\varphi$-sub-Gaussian if
and only if $\textsf{E}\zeta=0$ and there exist constants $C>0$, $D>0$ such
that

$P\\{|\zeta|>u\\}\leq
C\exp\left\\{-\varphi^{*}\left(\frac{u}{D}\right)\right\\}.$ (1.3)

This second characterization of $\varphi$-sub-Gaussian random variable by the
tail behavior of its distribution is important for practical applications.

The class of $\varphi$-sub-Gaussian random variables includes centered
compactly supported distributions, reflected Weibull distributions, centered
bounded distributions, Gaussian, Poisson distributions. In the case when
$\varphi=\frac{x^{2}}{2}$, the notion of $\varphi$-sub-Gaussianity reduces to
the classical sub-Gaussianity. The main theory for the spaces of
$\varphi$-sub-Gaussian random variables and stochastic processes was presented
in [4, 7, 16] followed by numerous further studies. Various classes of
$\varphi$-sub-Gaussian processes and fields were studied, in particular, in
[3, 9, 13, 14, 15, 17].

The property of $\varphi$-sub-Gaussianity allows to evaluate different
functionals of the stochastic processes, in particular, the behavior of their
suprema.

Estimates for distribution of supremum $P\\{\sup_{t\in T}|X(t)|>u\\}$ of
$\varphi$-sub-Gaussian stochastic process $X$ were derived in various forms in
the monograph [4] basing on entropy methods.

We will base our study on the following theorem (see [4], Theorems 4.1–4.2,
pp. 100, 105).

######  Theorem 1.1 ([4]).

Let $X(t)$, $t\in T,$ be a $\varphi$-sub-Gaussian process and $\rho_{X}$ be
the pseudometrics generated by $X$, that is,
$\rho_{X}(t,s)=\tau_{\varphi}(X(t)-X(s)),$ $t,s\in T.$ Suppose further that

  1. (i)

the pseudometric space $(T,\rho_{X})$ is separable, the process $X$ is
separable on $(T,\rho_{X})$;

  2. (ii)

$\varepsilon_{0}:=\sup\limits_{t\in T}\tau_{\varphi}(X(t))<\infty$;

  3. (iii)

$I_{\varphi}(\varepsilon_{0}):=\int\limits_{0}^{\varepsilon_{0}}\Psi(\ln(N(v)))\,dv<\infty,$
(1.4)

where $\Psi(v)=\frac{v}{\varphi^{(-1)}(v)}$ and $N(v)=N_{\rho_{X}}(v)$, $v>0$,
is the metric massiveness of the pseudometric space $(T,\rho_{X})$, that is,
the smallest number of elements in a $v$-covering of $T$ by closed balls,
w.r.t. the metric $\rho_{X}$, of a radius at most $v$.

Then for all $\lambda>0$ and $0<\theta<1$

$\textsf{E}\exp\\{\lambda\sup_{{t}\in T}|X(t)|\\}\leq 2Q(\lambda,\theta),$
(1.5)

where

$Q(\lambda,\theta)=\exp\Big{\\{}\varphi\Big{(}\frac{\lambda\varepsilon_{0}}{1-\theta}\Big{)}+\frac{2\lambda}{\theta(1-\theta)}I_{\varphi}(\theta\varepsilon_{0})\Big{\\}},$
(1.6)

and for $0<\theta<1$,
$u>\frac{2I_{\varphi}(\theta\varepsilon_{0})}{\theta(1-\theta)}$

$P\big{\\{}\sup_{{t}\in T}|X({t})|>u\big{\\}}\leq 2A(u,\theta),$ (1.7)

where

$A(u,\theta)=\exp\Big{\\{}-\varphi^{*}\Big{(}\frac{1}{\varepsilon_{0}}\Big{(}u(1-\theta)-\frac{2}{\theta}I_{\varphi}(\theta\varepsilon_{0})\Big{)}\Big{)}\Big{\\}}.$
(1.8)

In the above theorem and in what follows we denote by $f^{(-1)}$ the inverse
function for a function $f$.

The integrals of the form (1.4) are called entropy integrals. Entropy
characteristics of the parameter set $T$ with respect to the pseudometric
$\rho_{X}$ generated by the process $X$ and the rate of growth of the metric
massiveness $N(v)=N_{\rho_{X}}(v)$, $v>0$, or metric entropy $H(v):=\ln(N(v)$
are important for the study of sample paths properties of the underlying
process $X$ (see [4]).

Consider now a metric space $(T,d)$, with an arbitrary metric $d$ and suppose
that this metric space is separable. Suppose that we can evaluate the metric
massivness $N_{d}$ of $T$ with respect to the metric $d$ and also have a bound
for the function $\rho_{X}(t,s)=\tau_{\varphi}(X(t)-X(s))$ in terms of
$d(t,s)$. Then Theorem 1.1 implies the following result.

######  Theorem 1.2.

Let $X(t)$, $t\in T,$ be a $\varphi$-sub-Gaussian process and $T$ be supplied
with a metric $d$. Assume that

  1. (i)

the metric space $(T,d)$ is separable, the process $X$ is separable on
$(T,d)$;

  2. (ii)

$\varepsilon_{0}:=\sup\limits_{t\in T}\tau_{\varphi}(X(t))<\infty$;

  3. (iii)

there exists a monotonically increasing continuous function $\sigma(h)$,
$0<h\leq\sup_{t,s\in T}d(s,t)$ such that $\sigma(h)\rightarrow 0$ as
$h\rightarrow 0$ and

$\sup_{\begin{subarray}{c}d({t},{s})\leq h,\\\ {t},{s}\in T\\\
\end{subarray}}\tau_{\varphi}(X({t})-X({s}))\leq\sigma(h),$ (1.9)

and for $0<\varepsilon\leq\gamma_{0}$

$\widetilde{I}_{\varphi}(\varepsilon):=\int\limits_{0}^{\varepsilon}\Psi(\ln(N_{d}(\sigma^{(-1)}(v)))\,dv<\infty,$
(1.10)

where $\Psi(v)=\frac{v}{\varphi^{(-1)}(v)}$, $N_{d}(v)$, $v>0$, is the metric
massiveness of the metric space $(T,d)$, $\gamma_{0}=\sigma(\sup_{t,s\in
T}d(t,s))$.

Then the statement of Theorem 1.1 holds for $\lambda>0$ and $0<\theta<1$ such
that $\theta\varepsilon_{0}\leq\gamma_{0}$ with $Q(\lambda,\theta)$ and
$A(u,\theta)$ changed for the $\widetilde{Q}(\lambda,\theta)$ and
$\widetilde{A}(u,\theta)$ which correspond to the integral
$\widetilde{I}_{\varphi}$:

$\textsf{E}\exp\\{\lambda\sup_{{t}\in T}|X(t)|\\}\leq
2\widetilde{Q}(\lambda,\theta)$ (1.11)

and for
$u>\frac{2\widetilde{I}_{\varphi}(\theta\varepsilon_{0})}{\theta(1-\theta)}$

$P\big{\\{}\sup_{{t}\in T}|X({t})|>u\big{\\}}\leq 2\widetilde{A}(u,\theta),$
(1.12)

where

$\widetilde{Q}(\lambda,\theta)=\exp\Big{\\{}\varphi\Big{(}\frac{\lambda\varepsilon_{0}}{1-\theta}\Big{)}+\frac{2\lambda}{\theta(1-\theta)}\widetilde{I}_{\varphi}(\theta\varepsilon_{0})\Big{\\}},$
(1.13)
$\widetilde{A}(u,\theta)=\exp\Big{\\{}-\varphi^{*}\Big{(}\frac{1}{\varepsilon_{0}}\Big{(}u(1-\theta)-\frac{2}{\theta}\widetilde{I}_{\varphi}(\theta\varepsilon_{0})\Big{)}\Big{)}\Big{\\}}.$
(1.14)

###### Proof.

Theorem 1.2 follows immediately from Theorem 1.1. We have from (2.5) that
$\sup_{\begin{subarray}{c}d({t},{s})\leq h,\\\ {t},{s}\in T\\\
\end{subarray}}\rho_{X}(t,s)\leq\sigma(h)$, therefore, the smallest number of
elements in an $\varepsilon$-covering of $(T,\rho_{X})$ can be bounded by the
smallest number of elements in a $\sigma^{(-1)}(\varepsilon)$-covering of
$(T,d)$: $N_{\rho_{X}}(\varepsilon)\leq N_{d}(\sigma^{(-1)}(\varepsilon))$.
This implies the estimate
$I_{\varphi}(\varepsilon)\leq\widetilde{I}_{\varphi}(\varepsilon)$, as
$\varepsilon\leq\gamma_{0}$, and the statement of the theorem follows. ∎

Theorem 1.2 has been mainly used in the literature with a choice of the metric
space $(T,d)$ of the form: $T=\\{a_{i}\leq t_{i}\leq b_{i},\,i=1,2\\}$ and
$d({t},{s})=\max\limits_{i=1,2}|t_{i}-s_{i}|,$ ${t}=(t_{1},t_{2}),$
${s}=(s_{1},s_{2}).$ (see, for example, [3, 14, 10] for application to the
analysis of solutions to heat equation and higher order heat-type equations
with random initial conditions).

In the paper we study a particular class of $\varphi$-sub-Gaussian random
fields with $\varphi=\frac{|x|^{\alpha}}{\alpha}$, $\alpha\in(1,2]$, which as
a natural generalization of Gaussian and sub-Gaussian random fileds. Gaussian
and sub-Gaussian cases are involved in our consideration, with the choice
$\alpha=2$. We study the sample paths of such fields for the case of the
parameter set $T$ of the form $T=[a_{1},b_{1}]\times[a_{2},b_{2}]$ or
$T=[0,+\infty)\times[-A,A]$ with the so-called anisotropic metric
$d(t,s)=\sum_{i=1,2}|t_{i}-s_{i}|^{H_{i}}$, $H_{i}\in(0,1]$, $i=1,2$. Theorem
1.2 will serve as the main tool in our study.

Stochastic heat equations with fractional noises. Stochastic heat equations
have been studied in various settings: with a time-space white noise, with
generalizations of noise in space and/or in time, and also by considering
differential operators more general than the Laplacian. The case of fractional
noises was considered, e.g., in the recent papers [2], [8], among many others
(see references therein). In the present paper we consider the stochastic heat
equation:

$\displaystyle\frac{\partial u}{\partial t}=\frac{\partial^{2}u}{\partial
x^{2}}+\frac{\partial^{2}W}{\partial t\partial
x},~{}~{}(t,x)\in(0,T]\times\mathbb{R},$ (1.15) $\displaystyle
u(0,x)=u_{0}(x),x\in\mathbb{R},$ (1.16)

where $W$ is a centered Gaussian process which is white in time and is
fractional Brownian motion in space with index $H\leq\frac{1}{2}$. The problem
(1.15)–(1.16) was considered, for example, in [8] and it was stated that under
some continuity and boundedness conditions on $u_{0}(x)$ there exists a unique
mild solution $u(t,x)$, $(t,x)\in(0,T]\times\mathbb{R}$, satisfying the Hölder
condition

$\|u(t,x)-u(s,y)\|_{L^{p}}\leq c\Delta((t,x),(s,y))^{\rho\wedge H}$ (1.17)

with some constant $c=c(p,T,H)$ and where
$\Delta((t,x),(s,y))=|t-s|^{\frac{1}{2}}+|x-y|$ is a parabolic metric, $\rho$
is an index in the Hölder condition imposed on $u_{0}(x)$. For our
consideration, the bounds for the increments $u(t,x)-u(s,y)$ in $L_{2}$ norm
will be important. In view of this, we restate the bound (1.17) in another
form for the case of $p=2$, with the constant $c$ given by a closed
expression, and then use this bound to derive the results on the distribution
of suprema and on the rate of growth for random fields representing the
solution.

Contents. The paper is organized as follows. In Section 2 we study
$\varphi$-sub-Gaussian random fields with
$\varphi=\frac{|x|^{\alpha}}{\alpha}$, $\alpha\in(1,2]$. In Section 2.1 we
present the estimates for the tail distribution of suprema on the bounded
domain and in Section 2.2 we state the results on the rate of growth of random
fields over unbounded domains. Section 3 presents applications of the results
of sections 2.1 and 2.2 to random fields related to stochastic heat equations
with fractional noise.

##  2 $Sub_{\varphi}(\Omega)$ processes with
$\varphi=\frac{|x|^{\alpha}}{\alpha}$, $\alpha\in(1,2]$, defined on spaces
with anisotropic metrics

Consider the process $X(t)$, $t\in T$, from the class of $\varphi$-sub-
Gaussian processes with $\varphi=\frac{|x|^{\alpha}}{\alpha}$, $1<\alpha\leq
2$. This class is a natural generalisation of Gaussian and sub-Gaussian
processes, which correspond to $\alpha=2$.

For the function $\varphi(x)=\frac{|x|^{\alpha}}{\alpha}$, $1<\alpha\leq 2$,
we have $\varphi^{(-1)}(x)=(\alpha x)^{1/\alpha}$, $x>0$, and the Young-
Fenchel transform $\varphi^{*}(x)=\frac{|x|^{\beta}}{\beta}$, where $\beta>0$
is such that $\frac{1}{\beta}+\frac{1}{\alpha}=1$, that is,
$\beta=\frac{\alpha}{\alpha-1}$.

The entropy integrals (1.4) and (1.10) become of the form

$I_{\varphi}(\varepsilon)=\int\limits_{0}^{\varepsilon}\Big{(}\ln(N_{\rho_{X}}(u)))\Big{)}^{\frac{1}{\beta}}\,du$

and

$\widetilde{I}_{\varphi}(\varepsilon)=\int\limits_{0}^{\varepsilon}\Big{(}\ln(N_{d}(\sigma^{(-1)}(u)))\Big{)}^{\frac{1}{\beta}}\,du,$
(2.1)

and the bounds in Theorems 1.1 and 1.2 will be based on these integrals.

As we can see, for such function $\varphi$ the integrals appear in a quite
simple form and can be evaluated for particular metrics $d$. Note that for
more general $\varphi$ sometimes it is more convenient to use entropy
integrals of another form (see, e.g., [9, 13, 15]).

We next consider $\varphi$-sub-Gaussian fields $X(t),t\in T$, with the
parameter set $T=[a_{1},b_{1}]\times[a_{2},b_{2}]$ endowed with the
anisotropic metric

$d(t,s)=\sum_{i=1,2}|t_{i}-s_{i}|^{H_{i}},H_{i}\in(0,1],i=1,2.$ (2.2)

###  2.1 Estimates for the distribution of suprema

######  Theorem 2.1.

Let $X(t),t\in T$, be a $\varphi$-sub-Gaussian field with
$\varphi(x)=\frac{|x|^{\alpha}}{\alpha}$, $\alpha\in(1,2]$,
$T=[a_{1},b_{1}]\times[a_{2},b_{2}]$ with the metric $d(t,s)$ defined by
(2.2), $\beta=\frac{\alpha}{\alpha-1}$. Suppose that the field $X$ satisfies
conditions (i)-(iii) of Theorem 1.2 with $\sigma(h)=ch^{\gamma}$, $c>0$,
$\gamma\in(0,1]$ and $\gamma\beta\neq 1$.

Then for all $\lambda>0$ and $\theta\in(0,1)$

$\textsf{E}\exp\left\\{\lambda\sup\limits_{t\in T}|X(t)|\right\\}\leq
2\exp\left\\{\frac{1}{\alpha}\Big{(}\frac{\lambda\varepsilon_{0}}{1-\theta}\Big{)}^{\alpha}+\frac{2\lambda}{\theta(1-\theta)}\Big{(}\theta\varepsilon_{0}\Big{)}^{1-\frac{1}{\gamma\beta}}c_{1}\right\\}$
(2.3)

and for all $\theta\in(0,1),\theta\varepsilon_{0}<\gamma_{0}$ and
$u>\frac{2}{\theta(1-\theta)}\big{(}\theta\varepsilon_{0}\big{)}^{1-\frac{1}{\gamma\beta}}c_{1}$,
we have

$\mathsf{P}\\{\sup\limits_{t\in T}|X(t)|>u\\}\leq
2\exp\Big{\\{}-\frac{1}{\beta}\Big{(}\frac{u(1-\theta)}{\varepsilon_{0}}-\frac{2}{\theta}\big{(}\theta\varepsilon_{0}\big{)}^{1-\frac{1}{\gamma\beta}}c_{1}\Big{)}^{\beta}\Big{\\}},$
(2.4)

where
$c_{1}=\frac{2^{\frac{1}{\beta}}c^{\frac{1}{\gamma\beta}}}{1-\frac{1}{\gamma\beta}}\sum\limits_{i=1,2}\frac{1}{H_{i}}\big{(}\frac{T_{i}}{2}\big{)}^{\frac{H_{i}}{\beta}}$,
with $T_{i}=b_{i}-a_{i}$, $i=1,2$.

###### Proof.

We apply Theorem 1.2. We need to estimate the entropy integral
$\widetilde{I}_{\varphi}(\varepsilon)$ given by (1.10) for the particular
$\varphi,\sigma,d$ under consideration. For the metric $d$ given by (2.2) we
can write the following estimate for the metric massivenes

$N_{d}(\varepsilon)\leq\prod_{i=1,2}\Big{(}\frac{T_{i}}{2(\frac{\varepsilon}{2})^{\frac{1}{H_{i}}}}+1\Big{)}=\prod_{i=1,2}\Big{(}\frac{2^{\frac{1}{H_{i}}}T_{i}}{2\varepsilon^{\frac{1}{H_{i}}}}+1\Big{)}.$

This estimate can be deduced from the observation that a rectangle
$[-(\frac{\varepsilon}{2})^{\frac{1}{H_{1}}},(\frac{\varepsilon}{2})^{\frac{1}{H_{1}}}]\times[-(\frac{\varepsilon}{2})^{\frac{1}{H_{2}}},(\frac{\varepsilon}{2})^{\frac{1}{H_{2}}}]$
is contained in the ball in metric $d$ with center $(0,0)$ and radius
$\varepsilon$, which is given as
$B(\varepsilon)=\\{(x_{1},x_{2}):|x_{1}|^{H_{1}}+|x_{2}|^{H_{2}}\leq\varepsilon\\}$.

Now, for the given functions $\varphi$ and $\sigma$ (note that
$\sigma^{(-1)}(u)=(\frac{u}{c})^{\frac{1}{\gamma}}$), we consider the entropy
integral (2.1):

$\displaystyle\widetilde{I}_{\varphi}(\varepsilon)=\int\limits_{0}^{\varepsilon}\Big{(}\ln(N_{d}(\sigma^{(-1)}(u)))\Big{)}^{\frac{1}{\beta}}\,du$
$\displaystyle\leq\int\limits_{0}^{\varepsilon}\Big{(}\ln\prod_{i=1,2}\Big{(}\frac{2^{\frac{1}{H_{i}}}T_{i}}{2(\sigma^{(-1)}(u))^{\frac{1}{H_{i}}}}+1\Big{)}\Big{)}^{\frac{1}{\beta}}\,du$
$\displaystyle=\int\limits_{0}^{\varepsilon}\Big{(}\ln\prod_{i=1,2}\Big{(}\frac{T_{i}2^{\frac{1}{H_{i}}}c^{\frac{1}{\gamma
H_{i}}}}{2u^{\frac{1}{\gamma H_{i}}}}+1\Big{)}\Big{)}^{\frac{1}{\beta}}\,du.$

For any $0<\varkappa\leq 1$,
$\ln(1+x)=\frac{1}{\varkappa}\ln(1+x)^{\varkappa}\leq\frac{x^{\varkappa}}{\varkappa}$,
we apply this inequality for each term in the product in the above formula
choosing $\varkappa=H_{i},i=1,2$:

$\displaystyle\widetilde{I}_{\varphi}(\varepsilon)$
$\displaystyle\leq\int\limits_{0}^{\varepsilon}\sum_{i=1,2}\frac{1}{H_{i}}\Big{(}\frac{T_{i}2^{\frac{1}{H_{i}}}c^{\frac{1}{\gamma
H_{i}}}}{2u^{\frac{1}{\gamma
H_{i}}}}\Big{)}^{\frac{H_{i}}{\beta}}\,du=\sum_{i=1,2}\int\limits_{0}^{\varepsilon}\frac{1}{H_{i}}\Big{(}\frac{T_{i}}{2}\Big{)}^{\frac{H_{i}}{\beta}}2^{\frac{1}{\beta}}c^{\frac{1}{\gamma\beta}}\,\frac{du}{u^{\frac{1}{\gamma\beta}}}$
$\displaystyle=\frac{\varepsilon^{1-\frac{1}{\gamma\beta}}}{1-\frac{1}{\gamma\beta}}2^{\frac{1}{\beta}}c^{\frac{1}{\gamma\beta}}\sum_{i=1,2}\frac{1}{H_{i}}\Big{(}\frac{T_{i}}{2}\Big{)}^{\frac{H_{i}}{\beta}}=\varepsilon^{1-\frac{1}{\gamma\beta}}c_{1}.$

∎

###  2.2 Estimates for the rate of growth

Consider now the field $X(t_{1},t_{2})$, $(t_{1},t_{2})\in V$, defined over
the unbounded domain $V=[0,+\infty)\times[-A,A]$.

Let $f(t)>0$, $t\geq 0$, be a continuous strictly increasing function and
$f(t)\to\infty$ as $t\to\infty$.

Introduce the sequence $b_{0}=0$, $b_{k+1}>b_{k}$, $b_{k}\to\infty$,
$k\to\infty$.

We will use the following notations:

$l_{k}=b_{k+1}-b_{k}$, $V_{k}=[b_{k},b_{k+1}]\times[-A,A]$, $k=0,1,\dots$,
$f_{k}=f(b_{k})$,

$\varepsilon_{k}=\sup_{(t_{1},t_{2})\in V_{k}}\tau_{\varphi}(X(t_{1},t_{2}))$,
and suppose that $0<\varepsilon_{k}<\infty$;

$\gamma_{k}=c_{k}\max_{(t_{1},t_{2}),(s_{1},s_{2})\in
V_{k}}(d((t_{1},t_{2}),(s_{1},s_{2})))^{\gamma}$, where $c_{k}$ are from (2.5)
below, $\tilde{\theta}=\inf_{k}\frac{\gamma_{k}}{\varepsilon_{k}}$,

$\beta=\frac{\alpha}{\alpha-1}$.

######  Theorem 2.2.

Let $X(t_{1},t_{2})$, $(t_{1},t_{2})\in V$, be a $\varphi$-sub-Gaussian
separable field with $\varphi=\frac{|x|^{\alpha}}{\alpha}$, $\alpha\in(1,2]$.
Supose further that

  1. (i)

$\sup_{\begin{subarray}{c}d((t_{1},t_{2}),(s_{1},s_{2}))\leq h,\\\
(t_{1},t_{2}),(s_{1},s_{2})\in
V_{k}\end{subarray}}\tau_{\varphi}(X(t_{1},t_{2})-X(s_{1},s_{2}))\leq
c_{k}h^{\gamma},c_{k}>0,0<\gamma\leq 1;$ (2.5)

  2. (ii)

$C=\sum_{k=0}^{\infty}\frac{\varepsilon_{k}}{f_{k}}<\infty$.

  3. (iii)

$S=\sum_{k=0}^{\infty}\frac{\varepsilon^{1-\frac{1}{\gamma\beta}}_{k}c_{1}(k)}{f_{k}}<\infty$,
where
$c_{1}(k)=\Big{(}\frac{1}{H_{1}}(l_{k}/2)^{\frac{H_{1}}{\beta}}+\frac{1}{H_{2}}A^{\frac{H_{2}}{\beta}}\Big{)}\frac{2^{\frac{1}{\beta}}c_{k}^{\frac{1}{\gamma\beta}}}{1-\frac{1}{\gamma\beta}}$.

Then

  1. (i)

for any $\theta\in(0,\min(1,\widetilde{\theta}))$ and any $\lambda>0$

$\textsf{E}\exp\left\\{\lambda\sup\limits_{(t_{1},t_{2})\in
V}\frac{|X(t_{1},t_{2})|}{f(t_{1})}\right\\}\leq
2\exp\left\\{\frac{\lambda^{\alpha}}{\alpha}\left(\frac{C}{1-\theta}\right)^{\alpha}+\frac{2\lambda}{(1-\theta)\theta^{\frac{1}{\gamma\beta}}}S\right\\};$
(2.6)

  2. (ii)

for any $\theta\in(0,\min(1,\widetilde{\theta}))$ and
$u>\frac{2S}{(1-\theta)\theta^{\frac{1}{\gamma\beta}}}$

$\mathsf{P}\left\\{\sup\limits_{(t_{1},t_{2})\in
V}\frac{|X(t_{1},t_{2})|}{f(t_{1})}>u\right\\}\leq
2\exp\left\\{-\frac{1}{\beta
C^{\beta}}\left(u(1-\theta)-2S\theta^{-\frac{1}{\gamma\beta}}\right)^{\beta}\right\\}.$
(2.7)

###### Proof.

We use the scheme of the proof which is similar that in [6] (Theorem 2.4), [9]
(Theorem 4). Using (2.3) we can write the estimate:

$\displaystyle I(\lambda)$
$\displaystyle=\textsf{E}\exp\left\\{\lambda\sup\limits_{(t_{1},t_{2})\in
V}\frac{|X(t_{1},t_{2})|}{f(t_{1})}\right\\}\leq\textsf{E}\exp\left\\{\lambda\sum_{k=0}^{\infty}\sup\limits_{(t_{1},t_{2})\in
V_{k}}\frac{|X(t_{1},t_{2})|}{f_{k}}\right\\}\leq$
$\displaystyle\leq\prod_{k=0}^{\infty}\Big{(}\textsf{E}\exp\left\\{\lambda\frac{r_{k}}{f_{k}}\sup\limits_{(t_{1},t_{2})\in
V_{k}}|X(t_{1},t_{2})|\right\\}\Big{)}^{\frac{1}{r_{k}}}\leq
2\prod_{k=0}^{\infty}Q_{k}(\lambda,\theta)^{\frac{1}{r_{k}}},$

where

$Q_{k}(\lambda,\theta)=\exp\left\\{\frac{1}{\alpha}\Big{(}\frac{\lambda\varepsilon_{k}r_{k}}{(1-\theta)f_{k}}\Big{)}^{\alpha}+2\lambda\frac{r_{k}}{f_{k}}\frac{1}{(1-\theta)\theta^{\frac{1}{\gamma\beta}}}\varepsilon_{k}^{1-\frac{1}{\gamma\beta}}c_{1}(k)\right\\},$
$c_{1}(k)=\Big{(}\frac{1}{H_{1}}(l_{k}/2)^{\frac{H_{1}}{\beta}}+\frac{1}{H_{2}}A^{\frac{H_{2}}{\beta}}\Big{)}\frac{2^{\frac{1}{\beta}}c_{k}^{\frac{1}{\gamma\beta}}}{1-\frac{1}{\gamma\beta}},$

and let here $r_{k}$, $k\geq 0$ are such that
$\sum_{k=0}^{\infty}\frac{1}{r_{k}}=1$. This implies that

$I(\lambda)\leq
2\exp\left\\{\frac{1}{\alpha}\Big{(}\frac{\lambda}{1-\theta}\Big{)}^{\alpha}\sum_{k=0}^{\infty}\Big{(}\frac{\varepsilon_{k}r_{k}}{f_{k}}\Big{)}^{\alpha}r_{k}^{-1}+\frac{2\lambda}{(1-\theta)\theta^{\frac{1}{\gamma\beta}}}\sum_{k=0}^{\infty}\frac{\varepsilon_{k}^{1-\frac{1}{\gamma\beta}}c_{1}(k)}{f_{k}}\right\\}.$

Choose $r_{k}=\frac{f_{k}}{\varepsilon_{k}}C$, where
$C=\sum_{k=0}^{\infty}\frac{\varepsilon_{k}}{f_{k}}$, then

$I(\lambda)\leq
2\exp\left\\{\frac{1}{\alpha}\Big{(}\frac{\lambda}{1-\theta}\Big{)}^{\alpha}C^{\alpha}+\frac{2\lambda}{(1-\theta)\theta^{\frac{1}{\gamma\beta}}}\sum_{k=0}^{\infty}\frac{\varepsilon_{k}^{1-\frac{1}{\gamma\beta}}c_{1}(k)}{f_{k}}\right\\}.$

Therefore,

$\textsf{E}\exp\left\\{\lambda\sup\limits_{(t_{1},t_{2})\in
V}\frac{|X(t_{1},t_{2})|}{f(t_{1})}\right\\}\leq
2\exp\left\\{\frac{\lambda^{\alpha}}{\alpha}\Big{(}\frac{C}{1-\theta}\Big{)}^{\alpha}+\frac{2\lambda}{(1-\theta)\theta^{\frac{1}{\gamma\beta}}}S\right\\},$

and for all $\lambda>0$, $u>0$, $0<\theta<\widetilde{\theta}$

$\displaystyle P\left\\{\sup\limits_{(t_{1},t_{2})\in
V}\frac{|X(t_{1},t_{2})|}{f(t_{1})}>u\right\\}$
$\displaystyle\leq\exp\\{-\lambda
u\\}\textsf{E}\exp\left\\{\lambda\sup\limits_{(t_{1},t_{2})\in
V}\frac{|X(t_{1},t_{2})|}{f(t_{1})}\right\\}\leq$ $\displaystyle\leq
2\exp\left\\{\frac{\lambda^{\alpha}}{\alpha}\Big{(}\frac{C}{1-\theta}\Big{)}^{\alpha}+\frac{2\lambda
S}{(1-\theta)\theta^{\frac{1}{\gamma\beta}}}-\lambda u\right\\}.$

We minimise the right-hand side with respect to $\lambda$ and obtain for
$u>\frac{2S}{(1-\theta)\theta^{\frac{1}{\gamma\beta}}}$

$\displaystyle P\left\\{\sup\limits_{(t_{1},t_{2})\in
V}\frac{|X(t_{1},t_{2})|}{f(t_{1})}>u\right\\}$ $\displaystyle\leq
2\exp\left\\{-\Big{(}u-\frac{2S}{(1-\theta)\theta^{\frac{1}{\gamma\beta}}}\Big{)}^{\frac{\alpha}{\alpha-1}}\Big{(}\frac{C}{1-\theta}\Big{)}^{-\frac{\alpha}{\alpha-1}}\frac{\alpha-1}{\alpha}\right\\}=$
$\displaystyle=2\exp\left\\{-\frac{\alpha-1}{\alpha}C^{-\frac{\alpha}{\alpha-1}}\left(u(1-\theta)-2S\theta^{-\frac{1}{\gamma\beta}}\right)^{\frac{\alpha}{\alpha-1}}\right\\}.$

∎

######  Theorem 2.3.

Let $X(t_{1},t_{2})$, $(t_{1},t_{2})\in V$, be a $\varphi$-sub-Gaussian
separable field with $\varphi=\frac{|x|^{\alpha}}{\alpha}$, $\alpha\in(1,2]$.
Supose further that

  1. (i)

$\sup_{\begin{subarray}{c}d((t_{1},t_{2}),(s_{1},s_{2}))\leq h,\\\
(t_{1},t_{2}),(s_{1},s_{2})\in
V_{k}\end{subarray}}\tau_{\varphi}(X(t_{1},t_{2})-X(s_{1},s_{2}))\leq
c_{k}h^{\gamma},c_{k}>0,0<\gamma\leq 1;$

  2. (ii)

there exists $\delta>0$ and a constant $c(\delta)$ such that for
$(t_{1},t_{2})\in V$

$\tau_{\varphi}(X(t_{1},t_{2}))\leq c(\delta)t_{1}^{\delta};$

  3. (iii)

$\widetilde{C}=c(\delta)\sum_{k=0}^{\infty}\frac{b_{k+1}^{\delta}}{f_{k}}<\infty$,

  4. (iv)

$\widetilde{S}=(c(\delta))^{1-\frac{1}{\gamma\beta}}\sum_{k=0}^{\infty}\frac{b_{k+1}^{1-\frac{1}{\gamma\beta}}c_{1}(k)}{f_{k}}<\infty$,
where
$c_{1}(k)=\Big{(}\frac{1}{H_{1}}(l_{k}/2)^{\frac{H_{1}}{\beta}}+\frac{1}{H_{2}}A^{\frac{H_{2}}{\beta}}\Big{)}\frac{2^{\frac{1}{\beta}}c_{k}^{\frac{1}{\gamma\beta}}}{1-\frac{1}{\gamma\beta}}$.

Then

  1. (i)

for any $\theta\in(0,\min(1,\widetilde{\theta}))$ and any $\lambda>0$

$\textsf{E}\exp\left\\{\lambda\sup\limits_{(t_{1},t_{2})\in
V}\frac{|X(t_{1},t_{2})|}{f(t_{1})}\right\\}\leq
2\exp\left\\{\frac{\lambda^{\alpha}}{\alpha}\left(\frac{\widetilde{C}}{1-\theta}\right)^{\alpha}+\frac{2\lambda}{(1-\theta)\theta}\widetilde{S}\right\\};$
(2.8)

  2. (ii)

for any $\theta\in(0,\min(1,\widetilde{\theta}))$ and
$u>\frac{2(c(\delta))^{1-\frac{1}{\gamma\beta}}\widetilde{S}}{(1-\theta)\theta^{\frac{1}{\gamma\beta}}}$

$\mathsf{P}\left\\{\sup\limits_{(t_{1},t_{2})\in
V}\frac{|X(t_{1},t_{2})|}{f(t_{1})}>u\right\\}\leq
2\exp\left\\{-\frac{1}{\beta\widetilde{C}^{\beta}}\left(u(1-\theta)-2\widetilde{S}\theta^{\frac{1}{\gamma\beta}}\right)^{\beta}\right\\}.$
(2.9)

The proof is analogous to that of Theorem 2.2.

######  Remark 2.1.

The bound (2.7) can be rewritten in another form. We can choose
$\theta=u^{-\frac{\gamma\beta}{\gamma\beta+1}}$ and then under conditions of
Theorem 2.1 for $u>(1+2S)^{\frac{\gamma\beta}{\gamma\beta+1}}$ the following
bound holds

$\mathsf{P}\left\\{\sup\limits_{(t_{1},t_{2})\in
V}\frac{|X(t_{1},t_{2})|}{f(t_{1})}>u\right\\}\leq
2\exp\left\\{-\frac{1}{\beta
C^{\beta}}\left(u-u^{\frac{1}{\gamma\beta+1}}(1+2S)\right)^{\beta}\right\\}.$

Correspondingly, under conditions of Theorem 2.3 for
$u>(1+2\widetilde{S})^{\frac{\gamma\beta}{\gamma\beta+1}}$ we can write the
bound

$\mathsf{P}\left\\{\sup\limits_{(t_{1},t_{2})\in
V}\frac{|X(t_{1},t_{2})|}{f(t_{1})}>u\right\\}\leq
2\exp\left\\{-\frac{1}{\beta\widetilde{C}^{\beta}}\left(u-u^{\frac{1}{\gamma\beta+1}}(1+2\widetilde{S})\right)^{\beta}\right\\}.$

######  Theorem 2.4.

Suppose that for the field $X(t_{1},t_{2})$, $(t_{1},t_{2})\in V$, and
function $f$ conditions of Theorem 2.3 are satisfied. Then there exists a
random variable $\xi$ such that with probability one

$|X(t_{1},t_{2})|\leq f(t_{1})\xi,$ (2.10)

and $\xi$ satisfies the following assumption:

$\mathsf{P}\left\\{\xi>u\right\\}\leq
2\exp\left\\{-\frac{1}{\beta\widetilde{C}^{\beta}}\left(u-u^{\frac{1}{\gamma\beta+1}}(1+2\widetilde{S})\right)^{\beta}\right\\}$
(2.11)

for $u>(1+2\widetilde{S})^{\frac{\gamma\beta}{\gamma\beta+1}}$.

###### Proof.

The theorem is a corollary of Theorem 2.3. We denote
$\xi=\sup\limits_{(t_{1},t_{2})\in V}\frac{|X(t_{1},t_{2})|}{f(t_{1})}$, then
$\xi$ satisfies (2.11) and for all $\omega$ we have (2.10). ∎

##  3 Application to stochastic heat equation with fractional noise

Consider the stochastic heat equation:

$\displaystyle\frac{\partial u}{\partial t}=\frac{\partial^{2}u}{\partial
x^{2}}+\frac{\partial^{2}W}{\partial t\partial
x},~{}~{}(t,x)\in(0,T]\times\mathbb{R},$ (3.1) $\displaystyle
u(0,x)=u_{0}(x),x\in\mathbb{R},$ (3.2)

with the following assumption about the noise:

$\bf{A.1.}$ $W$ is a centered Gaussian process which is white in time and is
fractional Brownian motion in space with index $H\leq\frac{1}{2}$.

Note that there exists an extensive literature on SPDEs driven by Gaussian
noises and involving various differential operators.

As for the stochastic heat equation, it has been studied in various settings:
as that one driven by time-space white noise, further with generalization of
noise in space or in time, and also by considering differential operators more
general than the Laplacian.

Since we are going to apply the results of the previous section, we are
interested in the results for solution to the problem (3.1)-(3.2), which allow
to write down the bounds for $\textsf{E}|u(t,x)-u(s,y)|^{2}$.

We will base our study on the results from the recent papers [2, 8] on
reqularity properties of solution to stochastic heat equation.

We consider the problem (3.1)-(3.2) with assumption A.1. on the noise and
imposing the following assumption on the initial condition $u_{0}$.

$\bf{A.2.}$ The process $u_{0}(x),x\in\mathbb{R}$, is continuous, possesses
uniformly bounded $p$-th moments for $p\geq 2$ and is stochastically Hölder
continuous with $\rho\in(0,1]$, i.e., for all $p\geq 1$

$\|u_{0}(x)-u_{0}(y)\|_{L^{p}}\leq
L_{0}(p)|x-y|^{\rho},~{}~{}x,y\in\mathbb{R},$

where $L_{0}(p)$ is a constant and $\|.\|_{L^{p}}$ denotes the norm in
$L^{p}(\Omega,\mathbb{R})$:
$\|u\|_{L^{p}}=\left(\textsf{E}|u|^{p}\right)^{\frac{1}{p}}$, $p\geq 1$.

From [8] (see Theorem 1.1) it follows that under assumptions A.1. and A.2.
equation (3.1) has a unique mild solution
$u(t,x),(t,x)\in(0,T]\times\mathbb{R}$, satisfying
$\sup\limits_{(t,x)\in(0,T]\times\mathbb{R}}\textsf{E}|u(t,x)|^{p}<\infty$ and
the Hölder condition

$\|u(t,x)-u(s,y)\|_{L^{p}}\leq c\Delta((t,x),(s,y))^{\rho\wedge H}$

with some constant $c=c(p,T,H)$ and where
$\Delta((t,x),(s,y))=|t-s|^{\frac{1}{2}}+|x-y|$ is the parabolic metric.

The mild solution is defined as the random field

$u(t,x)=\int_{\mathbb{R}}G_{t}(x-y)u_{0}(y)\,dy+\int^{t}_{0}\int_{\mathbb{R}}G_{t-\theta}(x-\eta)W(d\theta,d\eta)=\omega(t,x)+V(t,x),$
(3.3)

where $G_{t}(x)$ is the Green’s function (fundamental solution) of the
equation $(\frac{\partial}{\partial t}-\frac{\partial^{2}}{\partial
x^{2}})u=0$, that is,

$G_{t}(x)=\frac{1}{\sqrt{4\pi t}}\exp\left(-\frac{|x|^{2}}{4t}\right).$

We refer for the rigorous definitions of the integrals in (3.3), for example,
to [8], [2].

In the next theorem we state the result on Hölder continuity of the fields
$\omega(t,x)$ and $V(t,x)$ in the form, where explicit expressions for the
constants are given.

######  Theorem 3.1.

Let assumptions A.1. and A.2. hold. Then the following bounds hold:

$\|\omega(t,x)\|_{L^{2}}\leq\sup\limits_{x\in\mathbb{R}}\|u_{0}(x)\|_{L^{2}}\leq
c_{0};$ (3.4) $\|V(t,x)\|_{L^{2}}\leq A(H)t^{\frac{H}{2}};$ (3.5)
$\|\omega(t,x)-\omega(s,y)\|_{L^{2}}\leq
c_{\omega}\left(|t-s|^{\frac{\rho}{2}}+|x-y|^{\rho}\right);$ (3.6)
$\|V(t,x)-V(s,y)\|_{L^{2}}\leq
c_{V}\left(|t-s|^{\frac{H}{2}}+|x-y|^{H}\right),$ (3.7)

where the constants $c_{\omega},c_{V},A(H)$ are given by formulas (3.8),
(3.9), (3.10) below.

###### Proof.

For the proof we use the reasoning similar to that applied for more general
case in [8] (Theorem 1.1) (see also [2]). Therefore, we present only the main
steps, our interest is in keeping the values of constants involved on each
step through all the chain of bounds. We have

$\displaystyle\textsf{E}|\omega(t,x)-\omega(s,x)|^{2}$
$\displaystyle=\textsf{E}\left|\int_{\mathbb{R}}\left(G_{t}(x-y)-G_{s}(x-y)\right)u_{0}(y)\,dy\right|^{2}$
$\displaystyle=\textsf{E}\left|\int_{\mathbb{R}}G_{t-s}(y)\left(\int_{\mathbb{R}}G_{s}(x-z)(u_{0}(z-y)-u_{0}(z))\,dz\right)\,dy\right|^{2}$
$\displaystyle\leq\int_{\mathbb{R}}G_{t-s}(y)\int_{\mathbb{R}}G_{s}(x-z)\textsf{E}\left|u_{0}(z-y)-u_{0}(z)\right|^{2}\,dzdy$
$\displaystyle\leq
L\int_{\mathbb{R}}G_{t-s}(y)|y|^{2\rho}\,dy=LC_{1}|t-s|^{\rho},$

since

$\displaystyle\int_{\mathbb{R}}G_{h}(y)|y|^{2\rho}\,dy$
$\displaystyle=\int_{\mathbb{R}}\frac{1}{\sqrt{4\pi
h}}\exp\left(-\frac{y^{2}}{4h}\right)|y|^{2\rho}\,dy=\frac{1}{\sqrt{4\pi
h}}\,2\int_{0}^{\infty}\exp\left(-\frac{y^{2}}{4h}\right)y^{2\rho}\,dy$
$\displaystyle=\frac{4^{\rho}}{\sqrt{\pi}}\Gamma(\rho+\frac{1}{2})h^{\rho}=C_{1}h^{\rho},$

where $C_{1}=\frac{4^{\rho}}{\sqrt{\pi}}\Gamma(\rho+\frac{1}{2})$,
$L=L_{0}(2)$, and we have used above the property
$\int_{\mathbb{R}}G_{t}(y)\,dy=1$ and assumption A.2. We now consider the more
general increment:

$\displaystyle\textsf{E}|\omega(t,x)-\omega(s,y)|^{2}$ $\displaystyle\leq
2\left(\textsf{E}|\omega(t,x)-\omega(s,x)|^{2}+\textsf{E}|\omega(s,x)-\omega(s,y)|^{2}\right)$
$\displaystyle\leq
2LC_{1}|t-s|^{\rho}+2\textsf{E}\left|\int_{\mathbb{R}}\left(G_{s}(x-z)-G_{s}(y-z)\right)u_{0}(z)\,dz\right|^{2}$
$\displaystyle=2LC_{1}|t-s|^{\rho}+2\textsf{E}\left|\int_{\mathbb{R}}G_{s}(z)\left(u_{0}(x-z)-u_{0}(y-z)\right)\,dz\right|^{2}$
$\displaystyle\leq 2LC_{1}|t-s|^{\rho}+2L^{2}|x-y|^{2\rho}\leq
c\left(|t-s|^{\rho}+|x-y|^{2\rho}\right),$

where $c=2L\max(C_{1},L)$, and using again assumption A.2. Therefore,

$\displaystyle\left(\textsf{E}|\omega(t,x)-\omega(s,y)|^{2}\right)^{\frac{1}{2}}\leq
c_{\omega}\left(|t-s|^{\frac{\rho}{2}}+|x-y|^{\rho}\right),~{}c_{\omega}=\sqrt{c}.$
(3.8)

Considering $V(t,x)$, we follow again the similar lines as those in [8]
keeping the track of constants. We can write:

$\displaystyle\textsf{E}\left|V(t,x)-V(s,y)\right|^{2}=$
$\displaystyle\textsf{E}\left|\int_{0}^{t}\int_{\mathbb{R}}G_{t-\theta}(x-\eta)W(d\theta,d\eta)-\int_{0}^{s}\int_{\mathbb{R}}G_{s-\theta}(y-\eta)W(d\theta,d\eta)\right|^{2}$
$\displaystyle=$
$\displaystyle\textsf{E}\left|\int_{0}^{t}\int_{\mathbb{R}}G_{t-\theta}(x-\eta)W(d\theta,d\eta)\right.$
$\displaystyle+\int_{0}^{s}\int_{\mathbb{R}}\left(G_{t-\theta}(x-\eta)-G_{s-\theta}(x-\eta)\right)W(d\theta,d\eta)$
$\displaystyle+\left.\int_{0}^{s}\int_{\mathbb{R}}\left(G_{s-\theta}(x-\eta)-G_{s-\theta}(y-\eta)\right)W(d\theta,d\eta)\right|^{2}$
$\displaystyle\leq$ $\displaystyle
3C_{H}\left(\int_{0}^{t-s}\int_{\mathbb{R}}\left|FG_{\theta}(\xi)\right|^{2}|\xi|^{1-2H}\,d\xi
d\theta\right.$
$\displaystyle+\int_{0}^{s}\int_{\mathbb{R}}\left|FG_{t-s+\theta}(\xi)-FG_{\theta}(\xi)\right|^{2}|\xi|^{1-2H}\,d\xi
d\theta$
$\displaystyle+\left.\int_{0}^{s}\int_{\mathbb{R}}\left(1-\cos(\xi(x-y))\right)|FG_{\theta}(\xi)|^{2}|\xi|^{1-2H}\,d\xi
d\theta\right)$ $\displaystyle=$ $\displaystyle 3C_{H}(I_{1}+I_{2}+I_{3}),$

here above by $F$ attached to a function we mean the Fourier transform, we use
that $FG_{t}(\xi)=\exp(-t\xi^{2})$ and also the isometry relation (see, e.g.,
[8, Theorem 2.1], [2, Theorem 2.7]):

$\textsf{E}\left|\int_{0}^{t}\int_{\mathbb{R}}\varphi(s,y)W(ds,dy)\right|^{2}=\int_{0}^{t}\int_{\mathbb{R}}\left|F\varphi(s,\cdot)(y)\right|^{2}\mu(dy)ds,$

where $\mu$ is the spectral measure involved in the representation of the
covariance of $W$ and

$\mu(dy)=C_{H}|y|^{1-2H}dy,\,\,\,C_{H}=\frac{\Gamma(2H+1)\sin(\pi H)}{2\pi}$

(see for details [8], [2]). Evaluate now the integrals $I_{1},I_{2},I_{3}$.

$I_{1}=\int_{0}^{t}\int_{\mathbb{R}}\exp(-2s\xi^{2})|\xi|^{1-2H}\,d\xi
ds=2^{-(H+1)}\frac{\Gamma(H+1)}{H}t^{H}=c_{1,H}t^{H},$

where $c_{1,H}=2^{-(H+1)}\frac{\Gamma(H+1)}{H}$;

$\displaystyle I_{2}$
$\displaystyle=\int_{0}^{s}\int_{\mathbb{R}}\left|FG_{t-s+\theta}(\xi)-FG_{\theta}(\xi)\right|^{2}|\xi|^{1-2H}\,d\xi
d\theta$
$\displaystyle=\int_{0}^{s}\int_{\mathbb{R}}\exp(-2\theta\xi^{2})\left(1-\exp(-(t-s)\xi^{2})\right)^{2}\times|\xi|^{1-2H}\,d\xi
d\theta$
$\displaystyle=\int_{\mathbb{R}}\frac{1-\exp(-2s\xi^{2})}{2|\xi|^{2}}\left(1-\exp(-(t-s)\xi^{2})\right)^{2}|\xi|^{1-2H}\,d\xi$
$\displaystyle\leq\frac{1}{2}\int_{\mathbb{R}}\frac{\left(1-\exp(-(t-s)\xi^{2})\right)^{2}}{|\xi|^{2-1+2H}}\,d\xi$
$\displaystyle=(t-s)^{H}\frac{1}{2}\int_{\mathbb{R}}\frac{\left(1-\exp(-u^{2})\right)^{2}}{u^{1+2H}}\,du=c_{2,H}(t-s)^{H},$

where
$c_{2,H}=\frac{1}{2}\int_{\mathbb{R}}\frac{\left(1-\exp(-u^{2})\right)^{2}}{u^{1+2H}}\,du$;

$\displaystyle I_{3}$
$\displaystyle=\int_{0}^{s}\int_{\mathbb{R}}\left(1-\cos(\xi(x-y))\right)\exp(-2\theta\xi^{2})|\xi|^{1-2H}\,d\xi
d\theta$
$\displaystyle=\int_{\mathbb{R}}\frac{1-\exp(-2s\xi^{2})}{2|\xi|^{2}}\left(1-\cos(\xi(x-y))\right)|\xi|^{1-2H}\,d\xi$
$\displaystyle\leq\int_{\mathbb{R}}\left(1-\cos(\xi(x-y))\right)\frac{1}{2}|\xi|^{-1-2H}\,d\xi=c_{3,H}(x-y)^{2H},$

where

for $H<\frac{1}{2}$: $c_{3,H}=\int_{0}^{\infty}\left(1-\cos
x\right)x^{-1-2H}\,dx=(2H)^{-1}\Gamma(1-2H)\cos(H\pi)$,

for $H=\frac{1}{2}$: $c_{3,H}=\frac{\pi}{2}$.

(We used here Lemma D.1. from [2]).

Therefore,

$\textsf{E}\left|V(t,x)-V(s,y)\right|^{2}\leq
3C_{H}((c_{1,H}+c_{2,H})|t-s|^{H}+c_{3,H}|x-y|^{2H})$

and

$\left(\textsf{E}\left|V(t,x)-V(s,y)\right|^{2}\right)^{\frac{1}{2}}\leq
c_{V}\left(|t-s|^{\frac{H}{2}}+|x-y|^{H}\right),$

where

$\displaystyle c_{V}=\sqrt{3C_{H}\max((c_{1,H}+c_{2,H}),c_{3,H})}.$ (3.9)

We also have the bound

$\left(\textsf{E}\left|V(t,x)\right|^{2}\right)^{\frac{1}{2}}\leq
A(H)t^{\frac{H}{2}},$

where

$\displaystyle A(H)=\sqrt{C_{H}c_{1,H}}.$ (3.10)

∎

For the next results we will use an additional assumption on the initial
condition $u_{0}$ and will need the following definition([12]):

A family $\Delta$ of $\varphi$-sub-Gaussian random variables is called
strictly $\varphi$-sub-Gaussian if there exists a constant $C_{\Delta}$
(called determining constant) such that for all countable sets $I$ of random
variables ${\zeta_{i}}\in\Delta$, $i\in I$, the inequality holds:
$\tau_{\varphi}\big{(}\sum_{i\in I}\lambda_{i}\zeta_{i}\big{)}\leq
C_{\Delta}\big{(}\textsf{E}\left(\sum_{i\in
I}\lambda_{i}\zeta_{i}\right)^{2}\big{)}^{1/2}.$ Random field $\zeta(t)$,
$t\in T$, is called strictly $\varphi$-sub-Gaussian if the family of random
variables $\\{\zeta(t),t\in T\\}$ is strictly $\varphi$-sub-Gaussian.

Let $K$ be a deterministic kernel and $\eta(x)=\int_{T}K(x,y)d\xi(y)$, $x\in
X,$ where $\xi(y)$, $y\in Y$, is a strictly $\varphi$-sub-Gaussian field and
the integral is defined in the mean-square sense. Then $\eta(x)$, $x\in X$, is
strictly $\varphi$-sub-Gaussian field with the same determining constant (see
[12]).

######  Theorem 3.2.

Let assumptions A.1 and A.2 hold and assume that $u_{0}(x)$ is strictly
$\varphi$-sub-Gaussian random field with
$\varphi(x)=\frac{|x|^{\alpha}}{\alpha}$, $\alpha\in(1,2]$, and let
$c_{\varphi}$ be the determining constant. Then for fields $\omega(t,x)$ and
$V(t,x)$, $(t,x)\in D=[a_{1},b_{1}]\times[a_{2},b_{2}]$, the following
estimates hold:

  1. (i)

$\mathsf{P}\left\\{\sup\limits_{(t,x)\in D}|\omega(t,x)|>u\right\\}\leq
2\exp\left\\{-\frac{1}{\beta}\left(\frac{u(1-\theta)}{c_{0}c_{\varphi}}+2(\theta
c_{0}c_{\varphi})^{-\frac{1}{\beta}}\widetilde{c}_{1}\right)^{\beta}\right\\}$
(3.11)

for $u>\frac{2}{(1-\theta)\theta}(\theta
c_{0}c_{\varphi})^{1-\frac{1}{\beta}}\widetilde{c}_{1}$, where
$\widetilde{c}_{1}=\frac{2^{\frac{1}{\beta}}(c_{\omega}c_{\varphi})^{\frac{1}{\beta}}}{1-\frac{1}{\beta}}\left(\frac{2}{\rho}\big{(}\frac{T_{1}}{2}\big{)}^{\frac{\rho}{2\beta}}+\frac{1}{\rho}\big{(}\frac{T_{2}}{2}\big{)}^{\frac{\rho}{\beta}}\right)$,
$T_{i}=b_{i}-a_{i}$, $i=1,2$, $\beta=\frac{\alpha}{\alpha-1}$.

  2. (ii)

$\mathsf{P}\left\\{\sup\limits_{(t,x)\in D}|V(t,x)|>u\right\\}\leq
2\exp\left\\{-\frac{1}{2}\left(\frac{u(1-\theta)}{\varepsilon_{V}}+2(\theta\varepsilon_{V})^{-\frac{1}{2}}\widetilde{\widetilde{c}}_{1}\right)^{2}\right\\}$

for
$u>\frac{2}{(1-\theta)\theta}(\theta\varepsilon_{V})^{\frac{1}{2}}\widetilde{\widetilde{c}}_{1}$,
where $\varepsilon_{V}=A(H)b_{1}^{H/2}$,
$\widetilde{\widetilde{c}}_{1}=2\sqrt{2c_{V}}\left(\frac{2}{H}\big{(}\frac{T_{1}}{2}\big{)}^{\frac{H}{4}}+\frac{1}{H}\big{(}\frac{T_{2}}{2}\big{)}^{\frac{H}{2}}\right)$,
$T_{i}=b_{i}-a_{i}$, $i=1,2$.

###### Proof.

Statement $(i)$ follows from Theorem 2.1 and the estimates (3.4),(3.6) and we
use that $\tau_{\varphi}(\omega(t,x))\leq c_{\varphi}\|\omega(t,x)\|_{L^{2}}$.
Statement $(ii)$ follows from Theorem 2.1 with $\alpha=2$ (since the field
$V(t,x)$ is Gaussian) and the estimate (3.7). ∎

The next theorem presents the power upper bound for the asymptotic growth of
the trajectories of the field $V(t,x)$.

######  Theorem 3.3.

Let assumptions A.1 and A.2 hold. Then for any $p>1$ there exist a random
variable $\xi(p)$ such that for any $(t,x)\in V$, with probability one

$|V(t,x)|\leq((t^{\frac{H}{2}}(\log t)^{p})\vee 1)\xi(p),$

where $\xi(p)$ satisfies assumption (2.11) with $\gamma=1,\beta=2$ and some
constants $\widetilde{C}$ and $\widetilde{S}$.

###### Proof.

We apply Theorem 2.4 and Theorem 2.3. Condition $(i)$ holds since $V(t,x)$ is
continuous, condition $(ii)$ holds in view of (3.5).

Consider conditions $(iii)$ and $(iv)$. Let $f(t)=(t^{\frac{H}{2}}|\log
t|^{p})\vee 1$ for $t>0$ and some $p>1$. Let us choose $b_{k}=e^{k},k\geq 0$.

Then $f_{k}=b_{k}^{\frac{H}{2}}(\log b_{k})^{p}=e^{k\frac{H}{2}}(\log
e^{k})^{p}=e^{k\frac{H}{2}}k^{p}$ and we have

$\widetilde{C}=A(H)\sum_{k=0}^{\infty}\frac{b_{k+1}^{\frac{H}{2}}}{f_{k}}=A(H)\left(e^{\frac{H}{2}}+\sum_{k=1}^{\infty}\frac{e^{\frac{H}{2}}}{k^{p}}\right)=A(H)e^{\frac{H}{2}}\left(1+\sum_{k=1}^{\infty}\frac{1}{k^{p}}\right)<\infty.$

Consider

$\widetilde{S}=(A(H))^{\frac{1}{2}}\sum_{k=0}^{\infty}\frac{b_{k+1}^{\frac{H}{4}}c_{1}(k)}{f_{k}}=(A(H))^{\frac{1}{2}}\sum_{k=0}^{\infty}\frac{e^{(k+1)\frac{H}{4}}c_{1}(k)}{e^{k\frac{H}{2}}k^{p}},$

where
$c_{1}(k)=\frac{2}{H}e^{k\frac{H}{4}}(\frac{e-1}{2})^{\frac{H}{4}}+A^{\frac{H}{2}},$
and we can see that $\widetilde{S}<\infty$.

Therefore, conditions of Theorem 2.4 hold true, and applying this theorem with
$f(t)=(t^{\frac{H}{2}}|\log t|^{p})\vee 1$ we obtain the statement of Theorem
3.3.

∎

We consider now an assumption on the initial condition which can be used
instead of assumption A.2.

$\bf{A.3.}$ The process $u_{0}(x),x\in\mathbb{R}$ is a real, measurable, mean-
square continuous stationary stochastic process.

Let $B(x),x\in\mathbb{R},$ is a covariance function of the process
$u_{0}(x),x\in\mathbb{R}$, with the representation

$B(x)=\int_{\mathbb{R}}\cos(\lambda x)dF(\lambda),$ (3.12)

where $F(\lambda)$ is a spectral measure, and for the process itself we can
write the spectral representation

$u_{0}(x)=\int_{\mathbb{R}}e^{i\lambda x}Z(d\lambda),$ (3.13)

The stochastic integral is considered as $L_{2}(\Omega)$ integral. Orthogonal
random measure $Z$ is such that $\textsf{E}|Z(d\lambda)|^{2}=F(d\lambda)$.

Then the field $\omega$ can be writen in the form

$\omega(t,x)=\int_{\mathbb{R}}\exp\Big{\\{}i\lambda x-\mu
t\lambda^{2}\Big{\\}}Z(d\lambda)$ (3.14)

and its covariance function has the following representation

$Cov\Big{(}\omega(t,x),\omega(s,y)\Big{)}=\int_{\mathbb{R}}\exp\Big{\\{}i\lambda(x-y)-\mu\lambda^{2}(t-s)\Big{\\}}F(d\lambda)$
(3.15)

(see, [9]).

######  Theorem 3.4.

Let assumption A.3 holds. Then

$\|\omega(t,x)\|_{L^{2}}\leq\Big{(}\int_{\mathbb{R}}F(d\lambda)\Big{)}^{\frac{1}{2}},$
(3.16)

and if for some $\varepsilon\in(0,\frac{1}{2}]$

$c^{2}(\varepsilon):=\int_{\mathbb{R}}\lambda^{2\varepsilon}F(d\lambda)<\infty,$
(3.17)

then the following estimate holds:

$\|\omega(t,x)-\omega(s,y)\|_{L^{2}}\leq
c(\varepsilon)\big{(}4^{1-\varepsilon}|x-y|^{2\varepsilon}+|t-s|^{\varepsilon}\big{)}^{1/2}.$
(3.18)

###### Proof.

We have

$\textsf{E}\big{(}\omega(t,x)-\omega(s,y)\big{)}^{2}=\int_{\mathbb{R}}|b(\lambda)|^{2}F(d\lambda),$

where

$b(\lambda)=\exp\\{i\lambda x\\}\exp\\{-\mu\lambda^{2}t\\}-\exp\\{i\lambda
y\\}\exp\\{-\mu\lambda^{2}s\\},$

and we can estimate

$\displaystyle|b(\lambda)|^{2}$
$\displaystyle\leq\Big{(}1-\exp\Big{\\{}-\mu\lambda^{2}|t-s|\Big{\\}}\Big{)}^{2}+4\sin^{2}\Big{(}\frac{1}{2}\lambda(x-y)\Big{)}$
$\displaystyle\leq\Big{(}\min(\lambda^{2}|t-s|,1)\Big{)}^{2}+4\min\Big{(}\frac{1}{2}|\lambda||x-y|,1\Big{)}^{2}$
$\displaystyle\leq\Big{(}\lambda^{2}|t-s|\Big{)}^{2\varepsilon_{1}}+4\Big{(}\frac{1}{2}|\lambda||x-y|\Big{)}^{2\varepsilon_{2}}$

for any $\varepsilon_{1},\varepsilon_{2}\in(0,1]$. Let us choose
$\varepsilon:=\varepsilon_{1}=\varepsilon_{2}/2$, $\varepsilon\in(0,1/2]$, and
suppose $\int_{\mathbb{R}}\lambda^{2\varepsilon}F(d\lambda)<\infty$. Then we
can write the bound:

$\displaystyle\int_{\mathbb{R}}|b(\lambda)|^{2}F(d\lambda)\leq\Big{(}\int_{\mathbb{R}}\lambda^{2\varepsilon}F(d\lambda)\Big{)}\big{(}4^{1-\varepsilon}|x-y|^{2\varepsilon}+|t-s|^{\varepsilon}\big{)},$

which implies (3.18).The estimate (3.16) follows from (3.15). ∎

In view of Theorem 3.4, under assumption A.3 and assuming $u_{0}$ to be
strictly $\varphi$-sub-Gaussian, we can write the estimate for the tail
distribution of supremum of $\omega(t,x)$ which is analogous to (3.11), where
the constants $c_{0}$ and $c_{\omega}$ will come now from (3.16), (3.17).

In the example below we present the process which can be used as initial
condition, for which (3.17) is satisfied.

######  Example 3.1.

Let the initial condition $u_{0}(x)$, $x\in\mathbb{R}$, be a Gaussian
stationary process with the spectral density

$f(\lambda)=\frac{\sigma^{2}}{(1+\lambda^{2})^{2\alpha}},\
\lambda\in\mathbb{R}.$ (3.19)

The corresponding covariance function is of the form:

$B_{\eta}(x)=\frac{\sigma^{2}}{\sqrt{\pi}\Gamma(2\alpha)}\Big{(}\frac{|x|}{2}\Big{)}^{2\alpha-1/2}K_{2\alpha-1/2}(|x|),\
x\in\mathbb{R},$ (3.20)

where $K_{\nu}$ is the modified Bessel function of the second kind, in
particular, $K_{1/2}(x)=\sqrt{\frac{\pi}{2x}}e^{-x}$. Covariances (3.20)
constitute the so-called Matérn class, a parameter $\nu=2\alpha-1/2>0$
controls the level of smoothness of the stochastic process.

Note that the Gaussian stochastic process with the above covariance and
spectral density can be obtained as solution to the following fractional
partial differential equation:

$\Big{(}1-\frac{d^{2}}{dx^{2}}\Big{)}^{\alpha}\eta(x)=w(x),\ x\in\mathbb{R},$

with $w$ being a white noise: $\textsf{E}w(x)=0$ and
$\textsf{E}w(x)w(y)=\sigma^{2}\delta(x-y)$ (see, e.g. [5, Thm. 3.1]).

The Matérn model is popular in spatial statistics and modeling random fields
(with corresponding adjustment of (3.20) for $n$-dimensional case). The
relation between the spatial Matérn covariance model and stochastic partial
differential equation $(\mu-\Delta)^{\alpha}\eta(x)=w(x),\
x\in\mathbb{R}^{n},$ was established by Whittle in 1963 and since then has
been widely used in various applied and theoretical contexts.

For the stationary initial condition $u_{0}$ with spectral density (3.19) the
condition (3.17) holds and we are able to calculate the constant
$c(\varepsilon)$ defined in (3.17). We have

$\int_{\mathbb{R}}\frac{\lambda^{2\varepsilon}}{(1+\lambda^{2})^{2\alpha}}=\int_{0}^{\infty}\frac{t^{\varepsilon+1/2-1}}{(1+t)^{\varepsilon+1/2+2\alpha-\varepsilon-1/2}}\,dt={\cal{B}}(\varepsilon+1/2,2\alpha-\varepsilon-1/2),$

where ${\cal B}$ is Beta-function, $\varepsilon\in(0,1/2]$,
$2\alpha-\varepsilon-1/2>0$ and we used the formula
$\int_{0}^{\infty}\frac{t^{\mu-1}}{(1+t)^{\mu+\nu}}\,dt={\cal{B}}(\mu,\nu)$.

Therefore, in this case we obtain
$c^{2}(\varepsilon)=\sigma^{2}{\cal{B}}(\varepsilon+1/2,2\alpha-\varepsilon-1/2)$.
In particular, having in (3.19) $\alpha>1$ and choosing $\varepsilon=1/2$ we
get $c(1/2)=\sigma^{2}\frac{1}{\alpha-1}$.

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