VMSTA Modern Stochastics: Theory and Applications 2351-6054 2351-6046 2351-6046 VTeXMokslininkų g. 2A, 08412 Vilnius, Lithuania VMSTA146 10.15559/19-VMSTA146 Research Article Estimates for distribution of suprema of solutions to higher-order partial differential equations with random initial conditions KozachenkoYuriyykoz@ukr.neta OrsingherEnzoenzo.orsingher@uniroma1.itb SakhnoLyudmylalms@univ.kiev.uaa https://orcid.org/0000-0002-0880-3751 VasylykOlgaovasylyk@univ.kiev.uaa Department of Probability Theory, Statistics and Actuarial Mathematics, Taras Shevchenko National University of Kyiv, 64 Volodymyrska str., 01601, Kyiv, Ukraine Department of Statistical Sciences, Sapienza University of Rome, P.le Aldo Moro, 5, 00185, Rome, Italy Corresponding author. 2019 1712201900118 2572019 10102019 16112019 © 2019 The Author(s). Published by VTeX2019 Open access article under the CC BY license.

In the paper we consider higher-order partial differential equations from the class of linear dispersive equations. We investigate solutions to these equations subject to random initial conditions given by harmonizable φ-sub-Gaussian processes. The main results are the bounds for the distributions of the suprema for solutions. We present the examples of processes for which the assumptions of the general result are verified and bounds are written in the explicit form. The main result is also specified for the case of Gaussian initial condition.

Higher-order dispersive equations random initial conditions harmonizable processes sub-Gaussian processes distribution of sumpremum of solution entropy methods 35G10 35R60 60G20 60G60
Introduction

Numerous recent studies are concerned with evolution equations of the form ut+ j=1l2j+1ux2j+1+ukux=0,l,kN, which are dispersive equations of order 2l+1 with a convective term ukux ; equations with coefficients of more general form and different kinds of nonlinearity are also the subject of active research.

The most celebrated equation of this class is the Korteweg–De Vries (KdV) equation ut=3ux3+uux, which describes the evolution of small amplitude long waves in fluids and other media.

The Kawahara equation with dispersive terms of the third and fifth orders ut+uux+α3ux3+β5ux5=0 is used to model various dispersive phenomena such as plasma waves, capilarity-gravity water waves, etc. in situations when the cubic dispersive term is weak or not sufficient. In the most recent research, generalisations of Equations (1.1), (1.2) have been suggested and treated.

In the physical and mathematical literature the existence, uniqueness and analytic properties of solutions to the initial value problem have been intensively investigated for various linear and nonlinear dispersive equations. Boundary value problems for such equations were also considered. We refer, for example, to the comprehensive study undertaken in the book by Tao , among many other books and papers on the topic.

One should note the importance of the study of constant coefficient linear dispersive equations for its own sake and also because this provides prerequisites for the theory of nonlinear dispersive equations, since the latter are often obtained by perturbation of the linear theory (). Developing the theory of linear equations is also essential for describing those evolution phenomena where the linear effects compensate or dominate nonlinear ones. In such situations, one can expect that the nonlinear solutions display almost the same behavior as the linear ones.

In the probabilistic literature significant attention has been paid to the equations of the form ut=kmmuxm,xR,t>0,m2. 0,\hspace{2.5pt}m\ge 2.\]]]>

The investigation of fundamental solutions to the equation (1.3) can be traced back to works by Bernštein and Lévy. Such solutions are sign-varying and, based on them, the so-called pseudoprocesses have been introduced and extensively investigated in the literature. Note, that Equations (1.3) of even and odd order possess solutions of different structure and behaviour (for example, see ). We refer to , where a review of the recent results on this topic and additional literature are presented.

We note that in the probabilistic literature equations of the form (1.3) and their generalizations are often called higher-order heat-type equations.

Equations of the form (1.3) subject to random initial conditions were studied in , namely, the asymptotic behavior was analysed for the rescaled solution to the Airy equation with random initial conditions given by weakly dependent stationary processes.

More general odd-order equations of the form ut= k=1Nak2k+1ux2k+1,N=1,2,, subject to the random initial conditions represented by a strictly φ-sub-Gaussian harmonizable processes were considered in . Rigorous conditions were stated therein for the existence of solutions and some distributional properties of solutions were investigated.

The present paper continues the line of research initiated in the papers . Note that in the mathematical literature the initial value problems for partial differential equations have been studied within the framework of different functional spaces, including the most abstract ones. Here we take into consideration Equation (1.4) in the framework of special Banach spaces of random variables, which constitute a subclass of Orlicz spaces of exponential type, more precisely, we deal with the spaces of strictly φ-sub-Gaussian random processes. These spaces play an important role in extensions of properties of Gaussian and sub-Gaussian processes. Basic results on φ-sub-Gaussian processes and fields can be found, for example, in .

The general methods and techniques developed for φ-sub-Gaussian processes, applied to the problems under consideration in the present paper, permit us to obtain bounds for the distributions of suprema of the solutions to the initial value problem for Equation (1.4). The bounds are presented in a form different than those obtained in the paper , and can be more useful in particular situations. In such a way, the results of the present paper complement and extend the results presented in .

To make the paper self-contained, in Sections 2 we present all important definitions and facts on harmonizable φ-sub-Gaussian processes, which will be used in the derivation of the main results. We also formulate the result on the conditions of existence of solutions to (1.4) with the φ-sub-Gaussian initial condition (see  for its derivation). The main result on the bounds for the distributions of supremum of the solutions is stated in Section 3. We present examples of processes for which the assumptions of the general result are verified and bounds are written explicitly. The main result is also specified for the case of Gaussian initial condition.

Preliminaries

Since in the paper we consider a partial differential equation with random initial condition given by a real-valued harmonizable φ-sub-Gaussian process, in this section we present the necessary definitions and facts concerning such processes.

Harmonizable processes

Harmonizable processes are a natural extension of stationary processes to second-order nonstationary ones. Such class of processes allows us to retain advantages of the Fourier analysis. Harmonizable processes were introduced by Loève . Recent developments on this theory are due to Rao  and Swift  among others.

([<xref ref-type="bibr" rid="j_vmsta146_ref_012">12</xref>]).

The second-order random function X={X(t),tR} , EX(t)=0 , is called harmonizable if there exists a second-order random function y=y(t),tR , Ey(t)=0 such that the covariance Γy(t,s)=Ey(t)y(s) has finite variation and X(t)=Reitudy(u) , where the integral is defined in the mean-square sense.

([<xref ref-type="bibr" rid="j_vmsta146_ref_012">12</xref>] Loève theorem).

The second-order random function X={X(t),tR} , EX(t)=0 , is harmonizable if and only if there exists a covariance function Γy(u,v) with finite variation such that Γx(t,s)=EX(t)X(s)=RRei(tusv)dΓy(u,v).

In the theorem above, the covariance function Γy is of bounded Vitali variation (see ). This fact guarantees that integral in (2.1) is in the Lebesgue sense. The function Γy is also called the spectral function or bi-measure of the process X.

In what follows, an integral of the type Af(t,s)dg(t,s) is understood as a common Lebesgue–Stieltjes integral, that is, the limit of the sum f(t,s)ΔiΔjg(t,s) , and an integral of the type Af(t,s)|dg(t,s)| is is understood as the limit of the sum f(t,s)|ΔiΔjg(t,s)| .

Below we shall focus on real-valued harmonizable processes.

Real-valued second order random function X={X(t),tR} is called harmonizable, if there exists a real-valued second order function y(u) , Ey(u)=0 , uR , such that X(t)=sintudy(u) or X(t)=costudy(u) and the covariance function Γy(t,s)=Ey(t)y(s) has finite variation. The integral is defined in the mean-square sense.

Real-valued second order function X={X(t),tR} , EX(t)=0 , is harmonizable if and only if there exists the covariance function Γy(u,v) with finite variation such that Γx(t,s)=EX(t)X(s)=RRκ(tu)κ(sv)dΓy(u,v), where κ(v)=cosv or κ(v)=sinv .

The theorem above follows from Theorem 2.1.

<italic>φ</italic>-sub-Gaussian random variables and processes

Here we present some basic facts from the theory of φ-sub-Gaussian random variables and processes, as well as some necessary results.

A continuous even convex function φ is called an Orlicz N-function, if φ(0)=0,φ(x)>0 0$]]> as x0 , and the following conditions hold: limx0φ(x)x=0 , limxφ(x)x= . The function φ defined by φ(x)=supyR(xyφ(y)) is called the Young–Fenchel transform (or convex conjugate) of the function φ . We say that φ satisfies the Condition Q, if φ is such an N-function that lim infx0φ(x)x2=c>0 0$]]>, where the case c= is possible .

In what follows we will always deal with N-functions for which condition Q holds.

Examples of N-functions, for which the condition Q is satisfied: φ(x)=|x|αα,1<α2,φ(x)=|x|αα,|x|1,α>2;|x|2α,|x|1,α>2. 2;\hspace{1em}\\ {} \frac{|x{|^{2}}}{\alpha },\hspace{1em}|x|\le 1,\alpha >2.\hspace{1em}\end{array}\right.\end{aligned}\]]]>

([<xref ref-type="bibr" rid="j_vmsta146_ref_004">4</xref>, <xref ref-type="bibr" rid="j_vmsta146_ref_008">8</xref>]).

Let {Ω,L,P} be a standard probability space. The random variable ζ is said to be φ-sub-Gaussian (or belongs to the space Subφ(Ω) ), if Eζ=0 , Eexp{λζ} exists for all λR and there exists a constant a>0 0$]]> such that the following inequality holds for all λR: Eexp{λζ}exp{φ(λa)} . The space Subφ(Ω) is a Banach space with respect to the norm  τφ(ζ)=inf{a>0:Eexp{λζ}exp{φ(aλ)}}, 0:\mathsf{E}\exp \{\lambda \zeta \}\le \exp \{\varphi (a\lambda )\}\},\]]]> which is called the φ-sub-Gaussian standard of the random variable ζ. Centered Gaussian random variables ζN(0,σ2) are φ-sub-Gaussian with φ(x)=x22 and τφ2(ζ)=Eζ2=σ2 . In the case φ(x)=x22 , φ-sub-Gaussian random variables are simply sub-Gaussian. A family Δ of random variables ζSubφ(Ω) is called strictly φ-sub-Gaussian (see ), if there exists a constant CΔ such that for all countable sets I of random variables ζiΔ , iI , the following inequality holds: τφiIλiζiCΔEiIλiζi21/2. The constant CΔ is called the determining constant of the family Δ. The linear closure of a strictly φ-sub-Gaussian family Δ in the space L2(Ω) is the strictly φ-sub-Gaussian with the same determining constant . The random process ζ={ζ(t),tT} is called (strictly) φ-sub-Gaussian if the family of random variables {ζ(t),tT} is (strictly) φ-sub-Gaussian . The following example of strictly φ-sub-Gaussian random process is important for our study. The solutions of partial differential equations considered in the next sections are of the same form as in this example. ([<xref ref-type="bibr" rid="j_vmsta146_ref_005">5</xref>]). Let K be a deterministic kernel and suppose that the process X={X(t),tT} can be represented in the form X(t)=TK(t,s)dξ(s), where ξ(t) , tT , is a strictly φ-sub-Gaussian random process and the integral above is defined in the mean-square sense. Then the process X(t) , tT , is a strictly φ-sub-Gaussian random process with the same determining constant. The notion of admissible function for the space Subφ(Ω) will be used to state the conditions of the existence of solutions of partial differential equations considered in the paper and to write down the bounds for suprema of these solutions. ([<xref ref-type="bibr" rid="j_vmsta146_ref_009">9</xref>, <xref ref-type="bibr" rid="j_vmsta146_ref_010">10</xref>], p. 146). Let Z(u),u0 , be a continuous, increasing function such that Z(u)>0 0$]]> and the function uZ(u) is non-decreasing for u>u0 {u_{0}}$]]>, where u00 is a constant. Then for all u,v0 sinuvZu+u0Zv+u0 Function Z(u),u0 , is called admissible for the space Subφ(Ω) , if for Z(u) conditions of Lemma 2.1 hold and for some ε>0 0$]]> the integral 0εΨlnZ(1)1su0ds converges, where Ψ(v)=vφ(1)(v),v>0 0$]]>. For example, the function Z(u)=uρ , 0<ρ<1 , is an admissible function for the space Subφ(Ω) if φ is defined in (2.2). Characteristic feature of φ-sub-Gaussian random variables is the exponential bounds for their tail probabilities. For φ-sub-Gaussian processes estimates for their suprema are available in different forms, see, for example, the book . To derive our main results, we shall use the following theorem on the distribution of supremum of a φ-sub-Gaussian random process, proved in the paper  (see also ). Let X={X(t),tT} be a φ-sub-Gaussian process and ρX be the pseudometrics generated by X, that is, ρX(t,s)=τφ(X(t)X(s)) , t,sT . Assume that the pseudometric space (T,ρX) is separable, the process X is separable on (T,ρX) and ε0:=suptTτφ(X(t))< . Let r(x),x1 , be a non-negative, monotone increasing function such that the function r(ex),x0 , is convex, and for 0<εε0 Ir(ε):= 0εr(N(v))dv<, where {N(v),v>0} 0\}$]]> is the massiveness of the pseudometric space (T,ρX) , that is, N(v) denotes the smallest number of elements in a v-covering of T, and the covering is formed by closed balls of radius of at most v.

Then for all λ>0 0$]]>, 0<θ<1 and u>0 0$]]> it holds EexpλsuptT|X(t)|Q(λ,θ) and P{suptT|X(t)|u}2A(θ,u), where Q(λ,θ):=expφλε01θr(1)Ir(θε0)θε0,A(θ,u)=exp{φ(u(1θ)ε0)}r(1)Ir(θε0)θε0.

The integrals of the form I(ε):= 0εg(N(v))dv,ε>0, 0,\]]]> with g(v) , v1 , being a nonnegative nondecreasing function, are called entropy integrals. Entropy characteristics of the parametric set T with respect to the pseudometrics ρX(t,s)=τφ(X(t)X(s)) , t,sT , generated by the process X={X(t),tT} , and the rate of growth of metric massiveness N(v) , or metric entropy H(v):=ln(N(v)) , are closely related to the properties of the process X (see  for details).

The integrals (2.8) play an important role in the study of such properties as boundedness and continuity of sample paths of a process, these integrals appear in estimates for modulii of continuity and distribution of supremum.

General results of this kind for φ-sub-Gaussian processes are related to the convergence of the integrals (2.8), where for g(v) one takes Ψ(ln(v)) with Ψ(v)=vφ(1)(v),v>0 0$]]>. Theorem 2.3 is more suitable for the case of “moderate” growth of the metric entropy and can lead to improved inequalities for upper bound for the distribution of supremum of the process, in comparison with more general inequalities involving the integrals based on the above function Ψ (see ). Entropy methods are also used in the modern approximation theory. Theorem 2.3 was applied, for example, in , for developing uniform approximation schemes for φ-sub-Gaussian processes. Solutions of linear odd-order heat-type equations with random initial conditions Let us consider the linear equation k=1Nak2k+1U(t,x)x2k+1=U(t,x)t,t>0,xR, 0,\hspace{0.1667em}x\in \mathbb{R},\]]]> subject to the random initial condition U(0,x)=η(x),xR, and {ak}k=1N are some constants. The next theorem gives the conditions of the existence of the solutions of the equation above with a φ-sub-Gaussian initial condition η(x) (see ). Let η={η(x),xR} be a real harmonizable (see Definition 2.2) and strictly φ-sub-Gaussian random process. Also let Z={Z(u),u0} be a function admissible for the space Subφ(Ω) . Assume that the following integral converges RRλ2N+1μ2N+1Zu0+λ2N+1Zu0+μ2N+1d|Γy(λ,μ)|<. Then U(t,x)=I(t,x,λ)dy(λ) is the classical solution to the problem (2.9)–(2.10), that is, U(t,x) satisfies Equation (2.9) with probability one and U(0,x)=η(x) . Here I(t,x,λ)=κλx+t k=1Nakλ2k+1(1)k, and κ(v)=cosv or κ(v)=sinv for the cases when η(x)=Rcos(xu)dy(u) or η(x)=Rsin(xu)dy(u) respectively. Note that under the condition (2.11) all the integrals RλsIt,x,λdyλ,s=0,1,2,,2N+1, converge uniformly in probability for |x|A and 0tT for all A>0 0$]]>, T>0 0$]]>. This guarantees that the derivatives of orders s=1,2,,2N+1 of solution U(t,x) given by (2.12) exist with probability one. In this sense we can treat U(t,x) as a classical solution. We refer for more details to . Similar result can be stated for the case η(x)=(asinxu+bcosxu)dy(u) , where a and b are some real constants. Let φ(x)=|x|pp , p>1 1$]]>, for sufficiently large x. Then the statement of Theorem 2.4 holds if the following integral converges RRλμ2N+1ln1+λln1+μαd|Γyλ,μ|, where α is a constant such that α>11p 1-\frac{1}{p}$]]> (see ). Main results Some auxiliary estimates Let us consider a separable metric space (T,d) , where T={aitibi,i=1,2} and d(t,s)=maxi=1,2|tisi| , t=(t1,t2) , s=(s1,s2) . Let X={X(t),tT} be a separable φ-sub-Gaussian random process such that ε0=suptTτφ(X(t))< and supd(t,s)h,t,sTτφ(X(t)X(s))σ(h), where {σ(h),0<hmaxi=1,2|biai|} is a monotonically increasing continuous function such that σ(h)0 as h0 , and for 0<εγ0 I˜r(ε):= 0εrb1a12σ(1)(v)+1b2a22σ(1)(v)+1dv<, where γ0=σ(maxi=1,2|biai|) and r(x),x1 , is defined in Theorem 2.3. Then for all 0<θ<1 and u>0 0$]]> it holds P{suptT|X(t)|u}2A˜(θ,u), where A˜(θ,u)=exp{φ(u(1θ)ε0)}r(1)I˜r(min(θε0,γ0))θε0.

To prove this theorem, we apply Theorem 2.3 in the case of the separable metric space (T,d) with T={aitibi,i=1,2} and d(t,s)=maxi=1,2|tisi| , t=(t1,t2) , s=(s1,s2) .

In particular, condition (3.1) means that supd(t,s)h,t,sTτφ(X(t)X(s))=supd(t,s)h,t,sTρX(t,s)σ(h),0<hmaxi=1,2|biai|. From the fact that the process X is separable on (T,ρX) and the function {σ(h),0<hmaxi=1,2|biai|} is a monotonically increasing continuous function, we get that for εγ0 the smallest number of elements in an ε-covering of the pseudometric space (T,ρX) can be estimated as the smallest number of elements in a (σ(1)(ε)) -covering of the metric space (T,d) as follows: N(ε)[(b1a12σ(1)(ε)+1)(b2a22σ(1)(ε)+1)]andN(ε)=1ifε>γ0. {\gamma _{0}}.\]]]>

Hence, from condition (3.2) we get that Ir(ε)= 0εr(N(v))dvI˜r(ε)<,εγ0, that is, conditions of Theorem 2.3 are satified for the process X. Finally, taking into account the estimates above and the properties of the function r we derive the estimate for distribution of supremum of the process X for all 0<θ<1 and u>0 0$]]>: P{suptT|X(t)|u}2exp{φ(u(1θ)ε0)}r(1)Ir(min(θε0,γ0))θε02exp{φ(u(1θ)ε0)}r(1)I˜r(min(θε0,γ0))θε0=2A˜(θ,u). □ As can be seen from Theorem 3.1, it is crucial to guarantee some kind of continuity of X on T in the form (3.1), that is, with respect to the norm τφ induced by the process X itself. Fulfilment of condition (3.1) enables us to write down the upper bound (3.3) for the distribution of supremum of a φ-sub-Gaussian process X={X(t),tT} defined on a separable metric space (T,d) . Below we present as a separate theorem the very useful result giving the conditions for the estimate (3.1) to hold for the field {U(t,x),atb,cxd} representing the solution to (2.9)–(2.10). Let y={y(u),uR} be a strictly φ-sub-Gaussian random process with a determining constant Cy and U(t,x)=I(t,x,λ)dy(λ) , where I(t,x,λ) is given in Theorem 2.4, atb , cxd . Assume that U(t,x) exists and is continuous with probability one (this condition holds if Theorem 2.4 holds). Let Ey(t)y(s)=Γy(s,t) . Assume that {Z(u),u0} is an admissible function for the space Subφ(Ω) . If the integral CZ2=Z(|λ|2+u0)+Z(12| k=1Nakλ2k+1(1)k|+u0)×Z(|μ|2+u0)+Z(12| k=1Nakμ2k+1(1)k|+u0)d|Γy(λ,μ)| converges, then there exists the function σ(h)=2CyCZ(Z(1h+u0))1,0<h<max(ba,dc), such that supt,t1[a,b]:|tt1|hx,x1[c,d]:|xx1|hτφ(U(t,x)U(t1,x1))σ(h). This result was obtained in the paper  as an intermediate statement in the course of the proof of Theorem 5.1. To make the present paper self-contained, we present in Appendix the main steps of the proof. On the distribution of supremum of solution to the problem (<xref rid="j_vmsta146_eq_019">2.9</xref>)–(<xref rid="j_vmsta146_eq_020">2.10</xref>) Now we have all the necessary tools to derive the estimate for the distribution of supremum of the field U(t,x) representing the solution to (2.9)–(2.10). Let y={y(u),uR} be a strictly φ-sub-Gaussian random process with a determining constant Cy and U(t,x)=I(t,x,λ)dy(λ) , where I(t,x,λ) is given in Theorem 2.4, atb , cxd . Assume that for U(t,x) the conditions of Theorem 3.2 hold. Let r(x) , x1 , be a non-negative, monotone increasing function such that the function r(ex),x0 , is convex, and condition (3.2) is satisfied for σ(h) given by (3.5). Then for all 0<θ<1 and u>0 0$]]> the following inequality holds true P{supatbcxd|U(t,x)|>u}2Aˆ(θ,u), u\big\}\le 2\hat{A}(\theta ,u),\]]]> where Aˆ(θ,u)=exp{φ(u(1θ)ε0)}r(1)Iˆr(min(θε0,γ0))θε0, Iˆr(δ)=0δr(ba2(Z(1)(2CZCys)u0)+1)×(dc2(Z(1)(2CZCys)u0)+1)ds,ε0=supatbcxdτφ(U(t,x)),γ0=2CyCZZ(1ϰ+u0),ϰ=max(ba,dc).

The assertion of this theorem follows from Theorems 3.1 and 3.2. Since the conditions of Theorem 3.2 are satisfied, then there exists the function σ(h)=2CyCZ(Z(1h+u0))1,0<h<max(ba,dc) , such that supt,t1[a,b]:|tt1|hx,x1[c,d]:|xx1|hτφ(U(t,x)U(t1,x1))σ(h). In this case, σ(1)(v)=(Z(1)(2CyCZv)u0)1,0<v<2CyCZZ(1ϰ+u0)=γ0, and for ε0 the upper bound (A.1) holds (see Appendix A).

Since the conditions (3.1) and (3.2) of Theorem 3.1 also hold true, the final estimate directly follows.  □

The derivation of our main result is based on Theorem 2.3, and due to this we present the bounds for the distribution of the supremum of the process U(t,x) in the form different than those obtained in the paper . This form of bounds can be more useful in particular situations giving the possibility to calculate the explicit expressions for the bounds.

Now we will specify the statement of Theorem 3.3 for particular choices of the admissible function Z and the function φ.

Consider φ(x)=|x|αα , 1<α2 . Then φ(x)=|x|γγ , where γ2 , and 1α+1γ=1 . Hence, Aˆ(θ,u)=exp{uγ(1θ)γγε0γ}r(1)Iˆr(min(θε0,γ0))θε0.

Now it is necessary to make the following steps:

to check the fulfilment of (3.4) with a particular function Z, admissible for a given φ;

to calculate σ(h) in (3.5);

to choose the function r satisfying (3.2);

to derive an estimate for Aˆ(θ,u) .

So, let us choose the admissible function Z(u)=|u|ρ , 0<ρ1 .

In this case u0=0 , Z(1)(u)=u1ρ , u>0 0$]]>, and CZ2=((|λ|2)ρ+|12 k=1Nakλ2k+1(1)k|ρ) ×((|μ|2)ρ+|12 k=1Nakμ2k+1(1)k|ρ)d|Γy(λ,μ)|122ρ(|λ|ρ+( k=1N|ak||λ|2k+1)ρ)×(|μ|ρ+( k=1N|ak||μ|2k+1)ρ)d|Γy(λ,μ)|. This integral converges if the following integral converges |λμ|(2N+1)ρd|Γy(λ,μ)|<. Note that for the existence of the solution U(t,x) we have to impose the condition (2.11), which for the admissible function Z(u)=|u|ρ takes the form λ(2N+1)(ρ+1)μ(2N+1)(ρ+1)d|Γy(λ,μ)|<, and implies the fulfilment of (3.12). Therefore, CZ2 is well defined. If (3.13) holds true, then we can define σ(h) by means of the formula (3.5), and, for our choice of Z, we have that σ(h)=2CyCZhρ=Chρ,0<ρ1, where we have denoted C=2CyCZ . Therefore, in view of Theorem 3.2, condition (3.6) holds with σ(h) of the form (3.14), that is, we have the Hölder continuity of sample paths of the solution U(t,x) . Let r(v)=vβ1 , 0<β<ρ/2 , then r(1)(v)=(v+1)1/β . For such r(v) and the above choice of Z we have Iˆr(δ)=0δr(ba2(Z(1)(2CZCys)u0)+1)×(dc2(Z(1)(2CZCys)u0)+1)ds0δϰ2(2CZCy)1/ρs1/ρ+12β1ds. Consider δ(0,θε0] and let us choose θ such that ϰ2(2CZCyθε0)1/ρ>1 1$]]>, that is, θ(0,2CZCyε0(ϰ2)ρ) . Then we can write the following estimate: Iˆr(δ)0δ(ϰ(2CZCys)1/ρ)2β1ds=(2CZCy)2β/ρϰ2β12βρ1δ12β/ρδ. Suppose that θε0<γ0 , then r(1)Iˆr(min(θε0,γ0))θε0(2CZCy)2/ρϰ212βρ1/β(θε0)2/ρ, and Aˆ(θ,u)exp{uγ(1θ)γγε0γ}(2CZCy)2/ρϰ212βρ1/β(θε0)2/ρ. If β0 , then (12βρ)1/βe2/ρ , and we obtain Aˆ(θ,u)exp{uγ(1θ)γγε0γ}(2eCZCy)2/ρϰ2(θε0)2/ρ, for θ(0,2CZCyε0(ϰ2)ρ) , and, therefore, for such θ we obtain P{supatbcxd|U(t,x)|>u}2exp{uγ(1θ)γγε0γ}(2eCZCy)2/ρϰ2(θε0)2/ρ. u\big\}\le 2\exp \Big\{-\frac{{u^{\gamma }}{(1-\theta )^{\gamma }}}{\gamma {\varepsilon _{0}^{\gamma }}}\Big\}{(2e{C_{Z}}{C_{y}})^{2/\rho }}{\varkappa ^{2}}{(\theta {\varepsilon _{0}})^{-2/\rho }}.\]]]>

Let y={y(u),uR} be a centered Gaussian random process. Then Cy=1 , φ(x)=x22 , φ(x)=x22 .

Then the bound (3.16) takes place with γ=2 , provided that (3.13) holds.

Note that the constant ρ(0,1] should guarantee the convergence of the integral (3.13), and, therefore, the choice of ρ in (3.16) depends on the integrability properties of Γy .

The choice of the function φ in Example 3.1 is reasoned by the fact that the corresponding class of random process is the natural generalization of Gaussian processes. This example is rather simple and, at the same time, it is very illustrative and instructive. Therefore, the derivations above are worth to be summarized as a separate statement.

Let y={y(u),uR} be a real strictly φ-sub-Gaussian random process with φ(x)=|x|αα , 1<α2 , determining constant Cy and Ey(t)y(s)=Γy(s,t) . Let U(t,x)=I(t,x,λ)dy(λ) , where I(t,x,λ) is given by (2.13), atb , cxd , and ε0=supatbcxdτφ(U(t,x)) . Further, let the constant ρ(0,1] be such that (3.13) holds. Then

(i) U(t,x) exists, is continuous with probability one and for its sample paths the Hölder continuity holds in the form supt,t1[a,b]:|tt1|hx,x1[c,d]:|xx1|hτφ(U(t,x)U(t1,x1))2CZCyhρ, where CZ is defined in (3.10);

(ii) for all 0<θ<1 such that θε0<2CZCy(ϰ/2)ρ with ϰ=max(ba,dc) , γ such that 1α+1γ=1 , and u>0 0$]]> the following inequality holds true P{supatbcxd|U(t,x)|>u}2exp{uγ(1θ)γγε0γ}(2eCZCy)2/ρϰ2(θε0)2/ρ. u\big\}\le 2\exp \Big\{-\frac{{u^{\gamma }}{(1-\theta )^{\gamma }}}{\gamma {\varepsilon _{0}^{\gamma }}}\Big\}{(2e{C_{Z}}{C_{y}})^{2/\rho }}{\varkappa ^{2}}{(\theta {\varepsilon _{0}})^{-2/\rho }}.\]]]> In particular, if the process y={y(u),uR} is Gaussian, then P{supatbcxd|U(t,x)|>u}2exp{u2(1θ)22ε02}(2eCZ)2/ρϰ2(θε)2/ρ u\big\}\le 2\exp \Big\{-\frac{{u^{2}}{(1-\theta )^{2}}}{2{\varepsilon _{0}^{2}}}\Big\}{(2e{C_{Z}})^{2/\rho }}{\varkappa ^{2}}{(\theta \varepsilon )^{-2/\rho }}\]]]> for all 0<θ<1 such that θΓ<2CZ(ϰ/2)ρ and u>0 0$]]>.

Consider φ(x)=exp{|x|}|x|1 , xR . Then φ(x)=(|x|+1)ln(|x|+1)|x|,xR . Hence, Aˆ(θ,u)=expu(1θ)ε0+1lnu(1θ)ε0+1+u(1θ)ε0×r(1)Iˆr(min(θε0,γ0))θε0. Let us take the function Z(u)=lnα(u+1) , u0 , α>1 1$]]>, as an admissible function for the space Subφ(Ω) . In this case, u0=eα1,Z(1)(v)=expv1α1,Z(v+u0)=lnα(v+eα),CZ2=(lnα(|λ|2+eα)+lnα(12| k=1Nakλ2k+1(1)k|+eα))×(lnα(|μ|2+eα)+lnα(12| k=1Nakμ2k+1(1)k|+eα))d|Γy(λ,μ)|. The above integral converges if the following integral converges lnα(|λ|+eα)lnα(|μ|+eα)d|Γy(λ,μ)|. That is, if condition (3.19) holds true, then Theorem 3.3 holds with Iˆr(δ)=0δr(ba2(exp(2CZCys)1α1(eα1))+1)×(dc2(exp(2CZCys)1α1(eα1))+1)ds=0δr(ba2(exp(2CZCys)1αeα)+1)×(dc2(exp(2CZCys)1αeα)+1)ds. Let dc2eα>1 1$]]> and ba2eα>1 1$]]>. That is, we choose some α>max{1,ln(2ba),ln(2dc)} \max \big\{1,\ln \big(\frac{2}{b-a}\big),\ln \big(\frac{2}{d-c}\big)\big\}$]]>. Then Iˆr(δ)0δr(ba)(dc)4exp2(2CZCys)1αds0δrϰ24exp2(2CZCys)1αds. In our case, the function r(v)=lnv , v1 , satisfies the conditions defined in Theorem 2.3 and is convenient for the estimation of Iˆr(δ) . Substituting it in the expression above, we get Iˆr(δ)0δlnϰ24exp2(2CZCys)1αds=δlnϰ24+0δ2(2CZCys)1αds=δlnϰ24+2(2CZCy)1αδ11α11α. Since r(1)(v)=ev,v0 , for such θ(0,1) that θε0<γ0 we obtain r(1)Iˆr(min(θε0,γ0))θε0explnϰ24+2αα12CZCyθε01α and, finally, for all u>0 0\$]]> the following inequality emerges P{supatbcxd|U(t,x)|>u}2Aˆ(θ,u)2expu(1θ)ε0+1lnu(1θ)ε0+1+u(1θ)ε0+lnϰ24+2αα12CZCyθε01α. u\big\}& \displaystyle \le & \displaystyle 2\hat{A}(\theta ,u)\\ {} & \displaystyle \le & \displaystyle 2\exp \left\{-\left(\frac{u(1-\theta )}{{\varepsilon _{0}}}+1\right)\ln \left(\frac{u(1-\theta )}{{\varepsilon _{0}}}+1\right)\right.\\ {} & \displaystyle +& \displaystyle \left.\frac{u(1-\theta )}{{\varepsilon _{0}}}+\ln \left(\frac{{\varkappa ^{2}}}{4}\right)+\frac{2\alpha }{\alpha -1}{\left(\frac{2{C_{Z}}{C_{y}}}{\theta {\varepsilon _{0}}}\right)^{\frac{1}{\alpha }}}\right\}.\end{array}\]]]>

Appendix A. Proof of Theorem <xref rid="j_vmsta146_stat_021">3.2</xref>

The derivation of the bound (3.6) for the process U(t,x) is based on the particular structure of this process and on the use of the property (2.4) of admissible function Z.

Firstly note that the process U(t,x) is separable since U(t,x) is continuous with probability one. The process U(t,x) is strictly φ-sub-Gaussian with the determining constant Cy , and, therefore, we can write: sup|tt1|h|xx1|hτφ(U(t,x)U(t1,x1))Cysup|tt1|h|xx1|hE(U(t,x)U(t1,x1))21/2. We can also estimate ε0=sup(t,x)Dτφ(U(t,x)) , where D={atb , cxd} , as follows: ε0Cysup(t,x)DE|U(t,x)|212Cysup(t,x)D|I(t,x,λ)I(t,x,μ)|d|Γy(λ,μ)|12Cyd|Γy(λ,μ)|12=:Γ.

It is enough to consider one of the representations for initial condition, η(x)=sinxudy(u) or η(x)=cosxudy(u) , since the proofs are similar for both cases.

Let us consider E(U(t,x)U(t1,x1))2 for κ(u)=cos(u) : E(U(t,x)U(t1,x1))2=(I(t,x,λ)I(t1,x1,λ))dy(λ)2|I(t,x,λ)I(t1,x1,λ)||I(t,x,μ)I(t1,x1,μ)|d|Γy(λ,μ)|. We can write |I(t,x,λ)I(t1,x1,λ)|=|cosAcosB| , where A=xλ+t k=1Nakλ2k+1(1)k,B=x1λ+t1 k=1Nakλ2k+1(1)k. Thus |I(t,x,λ)I(t1,x1,λ)|=2|sinA+B2sinBA2|2|sinBA2|=2|sin(C+D)|, where C=λ(x1x)2,D=t1t2 k=1Nakλ2k+1(1)k. Therefore, 2|sin(C+D)|=2|sinCcosD+cosCsinD|2(|sinC|+|sinD|)=2(|sinλ(x1x)2|+|sin(t1t)2 k=1Nakλ2k+1(1)k|). Choose some admissible function {Z(x),x0} for the space Subφ(Ω) . In view of Lemma 2.1, we can write |I(t,x,λ)I(t1,x1,λ)|2Z1(1|xx1|+u0)Z(|λ|2+u0)+2Z1(1|tt1|+u0)Z(12| k=1Nakλ2k+1(1)k|+u0). Thus, we obtain: sup|tt1|h|xx1|hτφ(U(t,x)U(t1,x1))Cysup|tt1|h|xx1|hE(U(t,x)U(t1,x1))21/2Cy2Z(1h+u0) Z(|λ|2+u0)+Z(12| k=1Nakλ2k+1(1)k|+u0)×Z(|μ|2+u0)+Z(12| k=1Nakμ2k+1(1)k|+u0)d|Γy(λ,μ)|1/2=CyCZ2Z(1h+u0). For κ(u)=sinu we have the same inequality. So, in the notations of Theorem 3.1 σ(h)=2CyCZ(Z(1h+u0))1,0<h<max(ba,dc).

Acknowledgement

We are grateful to editors and reviewers for valuable remarks and suggestions, which helped us to improve the paper significantly.

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