Aim and motivation. In this paper, we study sample paths properties of a class of sub-Gaussian type random fields X ( t ), t ∈ 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 { sup t ∈ T | X ( t ) | > u } for bounded T and to evaluate a rate of growth of X over an unbounded domain V by considering upper bounds for P { sup t ∈ V | X ( t ) | f ( t ) > u } 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 consideration 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 φ-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 studying sample paths of a stochastic process X ( t ), t ∈ 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 conditions for the 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 [5], 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 φ-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 φ-sub-Gaussian random processes and fields. Recall that a random variable ξ is sub-Gaussian if its moment generating function is majorized by that of a Gaussian centered random variable η ∼ N ( 0 , σ 2 ): E exp ( λ ξ ) ≤ E exp ( λ η ) = exp ( σ 2 λ 2 / 2 ) . The generalization of this notion to the classes of φ-sub-Gaussian random variables is introduced as follows (see, [5, Ch.2], [9], [19], [25]).

Consider a continuous even convex function φ such that φ ( 0 ) = 0, φ ( x ) > 0 as x ≠ 0 and lim x → 0 φ ( x ) x = 0, lim x → ∞ φ ( x ) x = ∞. Note that such functions are called Orlicz N-functions. Suppose that φ additionally satisfies lim inf x → 0 φ ( x ) x 2 = c > 0, where the case c = ∞ is possible.

Let φ be the function with the above properties and { Ω , F , P } be a standard probability space. The random variable ζ is φ-sub-Gaussian, or belongs to the space Sub φ ( Ω ), if E ζ = 0, E exp { λ ζ } exists for all λ ∈ R and there exists a constant a > 0 such that the following inequality holds for all λ ∈ R: (1) E exp { λ ζ } ≤ exp { φ ( λ a ) } . The random field X ( t ), t ∈ T, is called φ-sub-Gaussian if the random variables { X ( t ) , t ∈ T } are φ-sub-Gaussian.

The space Sub φ ( Ω ) is a Banach space with respect to the norm (see [9, 19]) τ φ ( ζ ) = inf { a > 0 : E exp { λ ζ } ≤ exp { φ ( a λ ) } .

For a φ-sub-Gaussian random variable ζ the following estimate for its tail probability holds: (2) P { | ζ | > u } ≤ 2 exp − φ ∗ u τ φ ( ζ ) , u > 0 , where the function φ ∗ defined by φ ∗ ( x ) = sup y ∈ R ( x y − φ ( y ) ) is called the Young–Fenchel transform (or Legendre transform, or convex conjugate) of the function φ. It was stated in [5] (Corollary 4.1, p. 68) that, moreover, a random variable ζ is a φ-sub-Gaussian if and only if E ζ = 0 and there exist constants C > 0, D > 0 such that P { | ζ | > u } ≤ C exp − φ ∗ u D .

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

The class of φ-sub-Gaussian random variables includes centered compactly supported distributions, reflected Weibull distributions, centered bounded distributions, Gaussian, Poisson distributions. In the case when φ = x 2 2 , the notion of φ-sub-Gaussianity reduces to the classical sub-Gaussianity. The main theory for the spaces of φ-sub-Gaussian random variables and stochastic processes was presented in [5, 9, 19] followed by numerous further studies. Various classes of φ-sub-Gaussian processes and fields were studied, in particular, in [4, 11, 16–18, 22].

The property of φ-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 ∈ T | X ( t ) | > u } of φ-sub-Gaussian stochastic process X were derived in various forms in the monograph [5] basing on entropy methods.

We will base our study on the following theorem (see [5], Theorems 4.1–4.2, pp. 100, 105).

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 (3) are called entropy integrals. Entropy characteristics of the parameter set T with respect to the pseudometric ρ X generated by the process X and the rate of growth of the metric massiveness N ( v ) = N ρ 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 [5]).

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 massiveness N d of T with respect to the metric d and also have a bound for the function ρ X ( t , s ) = τ φ ( X ( t ) − X ( s ) ) in terms of d ( t , s ). Then Theorem 1 implies the following result.

Theorem 2 has been mainly used in the literature with a choice of the metric space ( T , d ) of the form: T = { a i ≤ t i ≤ b i , i = 1 , 2 } and d ( t , s ) = max i = 1 , 2 | t i − s i |, t = ( t 1 , t 2 ), s = ( s 1 , s 2 ) (see, for example, [4, 17, 12] for application to the analysis of solutions to the heat equation and higher order heat-type equations with random initial conditions).

In the paper we study a particular class of φ-sub-Gaussian random fields with φ = | x | α α , α ∈ ( 1 , 2 ], which is a natural generalization of Gaussian and sub-Gaussian random fields. Gaussian and sub-Gaussian cases are involved in our consideration, with the choice α = 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 ] × [ a 2 , b 2 ] or T = [ 0 , + ∞ ) × [ − A , A ] with the so-called anisotropic metric d ( t , s ) = ∑ i = 1 , 2 | t i − s i | H i , H i ∈ ( 0 , 1 ] , i = 1 , 2 . Theorem 2 will serve as the main tool in our study. Note that the above metric is useful for studying anisotropic random fields, which have different geometrical and statistical properties for different directions and also for space-time random fields, where one needs to treat spatial and temporal variables in a different way. This is the case, for example, for random fields arising as solutions to stochastic partial differential equations. We can refer to papers [20], [27] (among others), where the use of such metric was essential for investigating sample paths properties of some models of anisotropic Gaussian random fields.

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], [10], among many others (see references therein). In the present paper we consider the stochastic heat equation: (6) ∂ u ∂ t = ∂ 2 u ∂ x 2 + ∂ 2 W ∂ t ∂ x , ( t , x ) ∈ ( 0 , T ] × R , (7) u ( 0 , x ) = u 0 ( x ) , x ∈ R , where W is a centered Gaussian process which is white in time and is fractional Brownian motion in space with index H ≤ 1 2 . Namely, following [10] we consider the case of W with the covariance functional E W ( ϕ ) W ( ψ ) = ∫ 0 T ∫ R F ϕ ( t , · ) ( y ) F ψ ( t , · ) ( y ) μ ( d y ) d t for any ϕ , ψ ∈ C 0 ∞ ( [ 0 , T ] , R ) (F denotes the Fourier transform with respect to the space variable), where the spectral measure μ is of the form μ ( d y ) = C H | y | 1 − 2 H d y , C H = Γ ( 2 H + 1 ) sin ( π H ) 2 π (we give more details and references in Section 3). The problem (6)–(7) was considered, for example, in [10] and it was stated therein that under some continuity and boundedness conditions on u 0 ( x ) there exists a unique mild solution u ( t , x ), ( t , x ) ∈ ( 0 , T ] × R, satisfying the Hölder condition (8) ‖ u ( t , x ) − u ( s , y ) ‖ L p ≤ c Δ ( ( t , x ) , ( s , y ) ) ρ ∧ H with some constant c = c ( p , T , H ), and where Δ ( ( t , x ) , ( s , y ) ) = | t − s | 1 2 + | x − y | is a parabolic metric, ρ 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 (8) in another form for the case of p = 2, with the constant c given by a closed expression, and then we 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 φ-sub-Gaussian random fields with φ = | x | α α , α ∈ ( 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.

Consider the process X ( t ), t ∈ T, from the class of φ-sub-Gaussian processes with φ = | x | α α , 1 < α ≤ 2. This class is a natural generalization of Gaussian and sub-Gaussian processes, which correspond to α = 2.

For the function φ ( x ) = | x | α α , 1 < α ≤ 2, we have φ ( − 1 ) ( x ) = ( α x ) 1 / α , x > 0, and the Young–Fenchel transform φ ∗ ( x ) = | x | β β , where β > 0 is such that 1 β + 1 α = 1, that is, β = α α − 1 .

The entropy integrals (3) and (5) take the form I φ ( ε ) = ∫ 0 ε ( ln ( N ρ X ( u ) ) ) ) 1 β d u and (9) I ˜ φ ( ε ) = ∫ 0 ε ( ln ( N d ( σ ( − 1 ) ( u ) ) ) ) 1 β d u , and the bounds in Theorems 1 and 2 will be based on these integrals.

As we can see, for such function φ the integrals appear in a quite simple form and can be evaluated for particular metrics d. Note that for more general φ sometimes it is more convenient to use entropy integrals of another form (see, e.g., [11, 16, 18]).

We next consider φ-sub-Gaussian fields X ( t ) , t ∈ T, with the parameter set T = [ a 1 , b 1 ] × [ a 2 , b 2 ] endowed with the anisotropic metric (10) d ( t , s ) = ∑ i = 1 , 2 | t i − s i | H i , H i ∈ ( 0 , 1 ] , i = 1 , 2 .