Properties of the Parabolic Anderson Model and the Anderson Polymer Model

Abstract

In this article we examine some properties of the solutions of the parabolic Anderson model. In particular we discuss intermittency of the field of solutions of this random partial differential equation, when it occurs and what the field looks like when intermittency doesn't hold. We also explore the behavior of a polymer model created by a Gibbs measure based on solutions to the parabolic Anderson equation.

1. Introduction

It has been twenty years since the publication of the seminal work “Parabolic Anderson Problem and Intermittency” by Carmona and Molchanov. Their memoir has inspired an enormous amount of research on the subject in the intervening years. In this paper we hope to give an account of what is now known about the behavior of solutions of the parabolic Anderson equation as well as the behavior of typical paths under the Anderson polymer measure. Perhaps the initial and still most compelling reason for studying the parabolic Anderson model is physical. In the three-dimensional case it provides a model for the growth of magnetic fields in young stars. In addition, it also has an interpretation as a population growth model. Further, since the work of [1] it has also provided a model for a polymer in a random media. Furthermore, it is a part of the theory of stochastic partial differential equations. Besides its interest as what can be described as a canonical object, in the sense that Brownian motion is a canonical object, its interest also derives from its relations to many other important models. In particular, it is a close relative of other models, for example, the stepping stone model [2], catalytic branching [3], super random walk, the Burger's equation [4], and the KPZ equation [4]. The original motivation of Anderson concerned the question of whether there were bound states for electrons in crystals with impurities. The crystal structure is taken to be 𝐙^𝑑 and the impurities are modeled by means of an 𝑖𝑖𝑑 random field {𝜉_𝑥∶𝑥∈𝐙^𝑑}. The phenomena of localization can be expressed as the existence of eigenfunctions of the Hamiltonian 𝐻=Δ+𝜉.

The equation satisfied by a magnetic field generated by a turbulent flow leads naturally to an analogous parabolic equation with a time varying random field {𝑣(𝑡,𝑥)∶𝑡∈[0,∞),𝑥∈𝐑³} as opposed to the time stationary field {𝜉_𝑥∶𝑥∈𝐙^𝑑} which arises in the original localization question. The difference is that the random medium changes rapidly in the case of the magnetic field generated by turbulent flow whereas the impurities in the localization can be considered to be unchanging in time. In the latter case, the fluctuations in the medium are slow compared to the phenomena of interest, capture of electrons.

The magnetic field in a star is generated by the turbulent flow of electrical charges. Turbulent flows are modeled by means of a randomly evolving velocity field, see, for example, [5] or [6] for a canonical example. Let {𝑣(𝑡,𝑥)∶𝑡∈[0,∞),𝑥∈𝐑³} denote such a velocity field on 𝐑³. Typically, this field is incompressible, Markov in time and Gaussian together with other spectral properties. The magnetic field generated by charges carried by the Lagrangian flow of {𝑣(𝑡,𝑥)∶𝑡∈[0,∞),𝑥∈𝐑³},𝐇=(𝐻₁,𝐻₂,𝐻₃) is incompressible and evolves according to an equation of the form 𝜕𝐇_𝑖+𝑣𝜕𝑡^𝑗∇_𝑗𝐇_𝑖=𝜅Δ𝐇_𝑖+𝐇_𝑗∇_𝑗𝑣_𝑖,𝐇(0,𝑥)=𝐇₀(𝑥), (1) where in this case Δ is the standard Laplacian on 𝐑³. This equation has been studied in [7–11], to name but a few references. We mention now some of the results of [9], on a tractable version of the model which one gets when 𝐑^𝑑 is replaced by 𝐙^𝑑 and in the environment one takes the time correlation in the velocity field to 0. In this version of the model, the field {𝑣(𝑡,𝑥)∶𝑡∈[0,∞),𝑥∈𝐑³} is replaced by a 3×3 matrix Wiener process, {𝑤(𝑡,𝑥)∶𝑡≥0,𝑥∈𝐙³} on some probability space (Ω,ℱ_𝑡,𝑡≥0,𝑄) satisfying 𝑑𝑤_𝑖𝑗(𝑡,𝑥)𝑑𝑤_𝑘𝑙𝑠,𝑦=𝛿₀(𝑡−𝑠)𝛿_𝑖𝑘𝛿_𝑗𝑙𝑑𝑡. (2) Then (1) satisfied by 𝐇 takes the form 𝑑𝐇_𝑖(𝑡,𝑥)=𝜅Δ𝐇_𝑖()+𝑡,𝑥_3𝑗=1𝐇_𝑗(𝑡,𝑥)𝜕𝑤_𝑖𝑗()𝐇()𝑡,𝑥,𝑖=1,2,3,0,𝑥=𝐇₀(𝑥), (3) where Δ, the discrete Laplacian, is given by Δ𝑓(𝑥)=∑_{𝑦∶‖𝑥−𝑦‖=1}(𝑓(𝑦)−𝑓(𝑥)) and 𝜕𝑤_𝑖𝑗 is the Stratonovich differential of 𝑤_𝑖𝑗. Converting to the Itô differential instead leads to the following equivalent form: 𝑑𝐇_𝑖(𝑡,𝑥)=𝜅Δ𝐇_𝑖(𝑡,𝑥)+³_𝑗=1𝐇_𝑗(𝑡,𝑥)𝑑𝑤_𝑖𝑗()+1𝑡,𝑥2𝐇_𝑖()𝐇()𝑡,𝑥𝑑𝑡,𝑖=1,2,3,0,𝑥=𝐇₀(𝑥). (4)

The average (over the media) of the magnetic field, 𝐌₁(𝑡,𝑥)≡𝐸[𝐇(𝑡,𝑥)], (5) is easily seen to satisfy 𝜕𝐌_1𝑖()𝜕𝑡𝑡,𝑥=𝜅Δ𝐌_1𝑖()+1𝑡,𝑥2𝐌_1𝑖(𝑡,𝑥),𝑖=1,2,3. (6) Since the spectrum of Δ is [−4𝑑,0]=[−12,0], for 𝑑=3, this equation implies that first moment Lyapunov exponent satisfies 𝛾(1)≡lim_𝑡→∞1𝑡log𝐌₁(=1𝑡,𝑥)2. (7) In [9], by simply using Itô's formula, an equation was derived for the average (over the media) magnetic field energy 𝐌₂𝑡,𝑥,𝑦≡𝐸_𝑄𝐇(𝑡,𝑥)𝐇𝑡,𝑦. (8) This satisfies 𝜕𝐌₂Δ𝜕𝑡𝑡,𝑥,𝑦=𝜅_𝑥+Δ_𝑦𝐌₂𝑡,𝑥,𝑦+𝐌₂𝑡,𝑥,𝑦+3𝛿_𝑥𝑦𝐌₂𝑡,𝑥,𝑦. (9) Under an assumption of homogeneity on 𝐇₀, one has 𝐌₂(0,𝑥,𝑦)=𝐦₂(0,𝑥−𝑦) and so for all time 𝐌₂(𝑡,𝑥,𝑦)=𝐦₂(𝑡,𝑥−𝑦), where 𝜕𝐦₂()𝜕𝑡𝑡,𝑥=2𝜅Δ_𝑥𝐦₂(𝑡,𝑥)+𝐦₂(𝑡,𝑥)+3𝛿₀(𝑥)𝐦₂(𝑡,𝑥). (10) Writing 𝐧₂(𝑡,𝑥)=𝑒^−𝑡𝐦₂(𝑡,𝑥), this becomes 𝜕𝐧₂()𝜕𝑡𝑡,𝑥=2𝜅Δ_𝑥𝐧₂(𝑡,𝑥)+3𝛿₀(𝑥)𝐧₂(𝑡,𝑥). (11) In the physically relevant dimension 𝑑=3, taking the Fourier transform of the eigenvalue equation 2𝜅Δ_𝑥𝜓(𝑥)+3𝛿₀(𝑥)𝜓(𝑥)=𝜆𝜓(𝑥) reveals that the operator 2𝜅Δ_𝑥+3𝛿₀(𝑥) possesses a positive eigenvalue 𝜆₀(𝜅) if and only if 2𝜅3<𝜅_𝑐𝑟≡∫_𝐓3𝑑𝜙4∑³_𝑖=11−cos𝜙₁, (12) where 𝐓³ is the three-dimensional torus. This implies that the solution 𝐦₂(𝑡,𝑥) satisfies 𝛾(2)≡lim_𝑡→∞1𝑡log𝐦₂(𝑡,𝑥)=1,if2𝜅3≥𝜅_𝑐𝑟,𝛾(2)≡lim_𝑡→∞1𝑡log𝐦₂(𝑡,𝑥)=1+𝜆₀(𝜅),if2𝜅3≤𝜅_𝑐𝑟. (13) This implies 𝛾(2)=2𝛾(1),𝜅>𝜅_𝑐𝑟, (14) whereas 𝛾(2)=𝜆₀(𝜅)+2𝛾(1)>2𝛾(1),𝜅≤𝜅_𝑐𝑟. (15) This last inequality is the definition of full intermittency, and the second moment grows strictly faster than twice the first moment. As a consequence, the field 𝐇(𝑡,𝑥) has widely separated large peaks. This explains the well-known phenomena of sun spots. They are widely separated sites of high magnetic field energy. The main question of interest in astrophysics is whether the magnitude of 𝐇 grows exponentially, a.s., in other words, is the a.s. Lyapunov exponent positive? This is the question of whether lim_𝑡→∞1𝑡log‖𝐇(𝑡,𝑥)‖=𝜆(𝜅)>0? (16) A further question regards the physically relevant asymptotic behavior of the a.s. Lyapunov exponent 𝜆(𝜅) as 𝜅→0. Since 𝜅 is the inverse Reynolds number, which in this situation is very small. Another interesting question is whether 𝑒^−𝑡/2𝐇(𝑡,𝑥) has a limiting distribution. These questions have an affirmative answer in the scalar case and remain difficult open problems in the multidimensional model just discussed.

2. Parabolic Anderson Model

The commonly studied parabolic Anderson equation is a scalar version of the magnetic field equation (1) and most progress has occurred in 𝐙^𝑑 and we will treat this case first. The velocity field {𝑣(𝑡,𝑥)∶𝑡≥0,𝑥∈𝐑^𝑑} may be replaced by a random environment that is either a stationary in time random field {𝜉_𝑥∶𝑥∈𝐙^𝑑} or an evolving random field {𝜁_𝑥(𝑡)∶𝑥∈𝐙^𝑑,𝑡≥0}. Typically, the variables 𝜉_𝑥 and 𝜁_𝑥,𝑥∈𝐙^𝑑, in both cases are assumed to be 𝑖𝑖𝑑. The stationary in time field can be thought of as a model where the phenomena of interest evolves much more rapidly than the evolution of the environment. The nonstationary case models a phenomena whose evolution is on a time scale comparable to the time scale of the evolution of the random environment. In this paper we will only discuss the nonstationary case. The enormous literature has been developed in this time stationary case. A partial random sample of works on this topic includes [12–17]. We will only discuss the nonstationary case. When considering the discrete model, that is, 𝐙^𝑑, we take the operator Δ to be the discrete Laplacian as mentioned above, whereas it is the ordinary Laplacian when the model is set in 𝐑^𝑑. The parameter 𝜅>0 in the model is called the diffusivity. The parabolic Anderson model is defined as a parabolic partial differential equation. A canonical version of this model is with the random environment provided by a white noise potential. In the 𝐙^𝑑 case, one takes 𝒲={𝑊_𝑥∶𝑥∈𝐙^𝑑} to be 𝑖𝑖𝑑 standard, one-dimensional Brownian motions defined on some probability space (Ω,ℱ_𝑡,𝑄), where ℱ_𝑡=𝜎{𝑊_𝑥(𝑠)∶𝑥∈𝐙^𝑑,0≤𝑠≤𝑡}. This field is assumed to be 𝛿-correlated in time and space, that is, 𝐸[𝑊_𝑥(𝑡)𝑊_𝑦(𝑠)]=𝑠∧𝑡𝛿₀(𝑥−𝑦) where 𝐸 denotes expectation with respect to 𝑄. The differential form of the parabolic Anderson equation is then 𝑑𝑢(𝑡,𝑥)=𝜅Δ𝑢(𝑡,𝑥)𝑑𝑡+𝑢(𝑡,𝑥)𝜕𝑊_𝑥(𝑡),𝑢(0,𝑥)=𝑢₀(𝑥),𝑥∈𝐙^𝑑,𝑡≥0. (17) The notation 𝜕𝑊_𝑥(𝑠) indicates the Stratonovich differential, (see [18] for a description of Stratonovich differentials) and this is preferred over the Itô differential because of the simplicity of the Feyneman-Kac representation which will appear shortly. The equivalent integral formulation is 𝑢(𝑡,𝑥)=𝑢₀(𝑥)+𝜅^𝑡₀()+Δ𝑢𝑠,𝑥𝑑𝑠_𝑡0𝑢(𝑠,𝑥)𝜕𝑊_𝑥(𝑠),𝑥∈𝐙^𝑑,𝑡≥0. (18) The differential formulation can be expressed converting to the Itô differential by 𝑑𝑢(𝑡,𝑥)=𝜅Δ𝑢(𝑡,𝑥)𝑑𝑠+𝑢(𝑡,𝑥)𝑑𝑊_𝑥(𝑠)+12𝑢()𝑢()𝑡,𝑥𝑑𝑡,0,𝑥=𝑢₀(𝑥),𝑥∈𝐙^𝑑,𝑡≥0. (19) At times it will be convenient to discuss the Itô solution defined by 𝑑𝑢_𝐼(𝑡,𝑥)=𝜅Δ𝑢_𝐼(𝑡,𝑥)𝑑𝑠+𝑢_𝐼(𝑡,𝑥)𝑑𝑊_𝑥(𝑠),𝑢_𝐼(0,𝑥)=𝑢₀(𝑥),𝑥∈𝐙^𝑑,𝑡≥0. (20) The relation between the two is that 𝑢_𝐼(𝑡,𝑥)=𝑒^−𝑡/2𝑢(𝑡,𝑥). Typically, it is assumed that 𝑢₀≥0 and that it is bounded from above, which guarantees existence and uniqueness of the solution. The positivity of 𝑢₀ assures that 𝑢(𝑡,𝑥)≥0 for all (𝑡,𝑥)∈[0,∞)×𝐙^𝑑. Fundamental information on this equation, including existence and uniqueness results, and its applications are contained in [19]. The field of solutions 𝒰(𝑡)≡{𝑢(𝑡,𝑥)∶𝑥∈𝐙^𝑑} exhibits interesting behavior as revealed by the growth properties of its moments which will be discussed at length below. At the risk of stating what is obvious, we stress that the random variables in the field {𝑢(𝑡,𝑥)∶𝑥∈𝐙^𝑑} are dependent. The correlation structure of this field is examined in detail in Theorem 2 later in this paper.

The solution of (20) has a Feynman-Kac representation as an average over a path space. This is done by means of a family of measures {𝑃^𝜅_𝑥∶𝑥∈𝐙^𝑑}, on the set 𝒟([0,∞),𝐙^𝑑).𝒟([0,∞),𝐙^𝑑) is the space of right-continuous paths possessing left limits from [0,∞)→𝐙^𝑑 which have a finite number of jumps on any compact subset of [0,∞). Define 𝒟_𝑇=𝒟([0,𝑇),𝐙^𝑑) analogously. We let {𝑋_𝑡∶𝑡≥0} be the canonical process on 𝒟([0,∞),𝐙^𝑑), that is, 𝑋_𝑡(𝜔)=𝜔(𝑡) for 𝜔∈𝒟([0,∞),𝐙^𝑑). Then the measures {𝑃^𝜅_𝑥∶𝑥∈𝐙^𝑑} are taken to be the ones that make {𝑋_𝑡,𝑡≥0} the pure jump Markov process on 𝐙^𝑑 with generator 𝜅Δ. This process satisfies at 𝑃^𝜅_𝑥(𝑋₀=𝑥)=1 and 𝑋 waits at 𝑥 for a random amount of time 𝜏=inf{𝑡∶𝑋_𝑡≠𝑥} which is exponentially distributed with parameter 2𝑑𝜅,𝑃^𝜅_𝑥(𝜏≥𝑡)=𝑒^{−2𝑑𝜅𝑡}, and then selects, using the uniform measure on its neighbors, one of the 2𝑑 neighbors of 𝑥 and jumps there and then proceeds afresh as if starting from time 0 at the new position.

The solution to (20) can then be expressed as 𝑢(𝑡,𝑥)=𝐸^𝜅_𝑥𝑒^{∫𝑡0𝑑𝑊𝑋(𝑡−𝑠)(𝑠)}𝑢₀𝑋_𝑡. (21) Unless explicitly mentioned otherwise, we will take 𝑢₀(𝑥)≡1 so that 𝑢(𝑡,𝑥)=𝐸^𝜅_𝑥𝑒^{∫𝑡0𝑑𝑊𝑋(𝑡−𝑠)(𝑠)}. (22)

3. Relations to Other Processes and Equations

The parabolic Anderson equation is closely related to equations for many other models. Probably the best known connection to another equation is provided by the Hopf-Cole transformation. If we take a solution 𝑢(𝑡,𝑥) to a parabolic Anderson equation 𝜕𝑢()()()𝑢()𝜕𝑡𝑡,𝑥=𝜅Δ𝑢𝑡,𝑥+𝐹𝑡,𝑥𝑡,𝑥,𝑥∈𝐑^𝑑, (23) with a spatially smooth random force term 𝐹(𝑡,𝑥) and set 𝑓(𝑡,𝑥)=∇𝐹(𝑡,𝑥) and 𝑣(𝑡,𝑥)=−2𝜅∇log𝑢(𝑡,𝑥), then 𝑣(𝑡,𝑥) satisfies Burger's equation 𝜕𝑣()+⟨𝑣()⟩𝑣()()().𝜕𝑡𝑡,𝑥𝑡,𝑥,∇𝑡,𝑥=𝜅Δ𝑣𝑡,𝑥+𝑓𝑡,𝑥 (24) Similarly, setting ℎ=log𝑢, where 𝑢 solves (23), yields a solution of the KPZ equation, see, for example, [4], 𝜕ℎ()()+‖(𝜕𝑡𝑡,𝑥=𝜅Δℎ𝑡,𝑥∇ℎ𝑡,𝑥)‖²+𝐹(𝑡,𝑥). (25)

Equation (20) may be cast as a particular case of a pair of equations driven by two 𝑖𝑖𝑑 jointly Gaussian, standard Brownian fields, {𝑊^𝑖_𝑡(𝑥)∶𝑥∈𝐙^𝑑},𝑖=1,2 which also relates it to the mutually catalytic branching and stepping stone models. Assume the correlations between the fields is controlled by a parameter ℎ∈[−1,1] by 𝐸𝑊¹_𝑡(𝑥)𝑊²_𝑠𝑦=ℎ𝑡∧𝑠𝛿₀𝑥−𝑦. (26)

An interesting observation of Etheridge and Fleischmann, [20], neatly ties the symbiotic branching and stepping stone models together with the parabolic Anderson model by taking different values for the parameter ℎ. Consider the following pair of equations: 𝑑𝑢(𝑡,𝑥)=𝜅Δ𝑢(𝑡,𝑥)𝑑𝑡+𝑢(𝑡,𝑥)𝑣(𝑡,𝑥)𝑑𝑊¹_𝑥(𝑡),𝑑𝑣(𝑡,𝑥)=𝜅Δ𝑣(𝑡,𝑥)𝑑𝑡+𝑢(𝑡,𝑥)𝑣(𝑡,𝑥)𝑑𝑊²_𝑥(𝑡),𝑢(0,𝑥)=𝑢₀(𝑥),𝑣(0,𝑥)=𝑣₀(𝑥). (27) When ℎ=0 in (26), this is known as the mutually catalytic branching model for two interacting populations and has been studied in [3, 20–25] to name just a small random sample.

If ℎ=−1 in (26) and 𝑢₀(𝑥)+𝑣₀(𝑥)≡1, then 𝑢(𝑡,𝑥)+𝑣(𝑡,𝑥)≡1,𝑡≥0 and 𝑢(𝑡,𝑥) solves 𝑑𝑢(𝑡,𝑥)=𝜅Δ𝑢(𝑡,𝑥)𝑑𝑡+𝑢(𝑡,𝑥)(1−𝑢(𝑡,𝑥))𝑑𝑊¹_𝑥(𝑡). (28) This is known as the stepping stone model from population genetics and has been the subject of the works [2, 16, 26] among others.

Finally, when ℎ=1 in (26) and 𝑢₀≡𝑣₀ one gets 𝑢(𝑡,𝑥)≡𝑣(𝑡,𝑥) and 𝑢(𝑡,𝑥) is the solution of (20) with 𝑊_𝑥(𝑡)=𝑊¹_𝑥(𝑡).

Other versions of noise have been considered as well and this can lead to substantial technical difficulties. For example, the recent works [27–31] replace the field 𝒲 by catalysts which are from interacting particle systems such as the voter model or exclusion processes. Among the problems unresolved here is the existence of the a.s. Lyapunov exponent. The subadditive argument which worked so well for the environment 𝒲 fails for these models.

4. Moment Lyapunov Exponents

One of the most interesting properties of the solutions of (20) is that of intermittency. Intermittency is defined in terms of the moment Lyapunov exponents. These are the limits for 𝑝>0 of 𝛾_𝑝(𝜅)=lim_𝑡→∞1𝑡log𝐸^𝜅_𝑥𝑢^𝑝(𝑡,𝑥). (29) The existence of these limits was proved in [19]. For 𝑝=1 one uses (116) to compute 𝛾₁(𝜅). This is easily done with the observation that for any fixed path {𝑋(𝑡−𝑠)∶0≤𝑠≤𝑡}, the random variable 𝐴_𝑡(𝑋)=^𝑡₀𝑑𝑊_{𝑋(𝑡−𝑠)}(𝑠) (30) is Gaussian with mean 0 and variance 𝑡. Thus, by Fubini's theorem, 𝐸[𝑢(𝑡,𝑥)]=𝑒^𝑡/2 (31) so that 𝛾₁(𝜅)=1/2. An interesting quantity, the overlap, arises when computing 𝛾₂(𝜅). Here we will use 𝐸_𝑥^𝜅,⊗2 to denote the product measure of 𝐸^𝜅_𝑥 with itself on 𝒟([0,∞),𝐙^𝑑)×𝒟([0,∞),𝐙^𝑑). Then we use the observation that for two fixed paths, {𝑋(𝑡−𝑠)∶0≤𝑠≤𝑡} and {𝑌(𝑡−𝑠)∶0≤𝑠≤𝑡}, the variables 𝐴_𝑡(𝑋) and 𝐴_𝑡(𝑌) are jointly Gaussian with 𝐸𝐴_𝑡(𝑋)𝐴_𝑡(𝑌)=^𝑡₀𝛿₀𝑋_𝑠−𝑌_𝑠𝑑𝑠. (32) Thus, 𝐴_𝑡(𝑋)+𝐴_𝑡(𝑌) is Gaussian with mean 0 and variance var𝐴_𝑡(𝑋)+𝐴_𝑡(𝑌)=2𝑡+^𝑡₀𝛿₀𝑋_𝑠−𝑌_𝑠𝑑𝑠 (33) by (116), 𝐸𝑢²(𝑡,𝑥)=𝐸𝐸_𝑥^𝜅,⊗2𝑒^{𝐴𝑡(𝑋)+𝐴𝑡(𝑌)}=𝐸_𝑥^𝜅,⊗2𝐸𝑒^{𝐴𝑡(𝑋)+𝐴𝑡(𝑌)}=𝑒^{𝑡+∫𝑡0𝛿0(𝑋𝑠−𝑌𝑠)𝑑𝑠}. (34) This implies 𝛾₂(𝜅)=2𝛾₁(𝜅)+lim_𝑡→∞1𝑡log𝐸_𝑥^𝜅,⊗2𝑒^{∫𝑡0𝛿0(𝑋𝑠−𝑌𝑠)𝑑𝑠}. (35)

For general 𝑝≥1, Hölder's inequality implies 𝛾_𝑝(𝜅)𝑝≤𝛾_𝑝+1(𝜅)𝑝+1,∀𝑝≥1, (36) and 𝑝→𝛾_𝑝(𝜅) is convex. Full intermittency is then the property that 𝛾_𝑝(𝜅) is strictly convex on [1,∞), that is, 𝛾_𝑝(𝜅)𝑝<𝛾_𝑝+1(𝜅)𝑝+1,∀𝑝≥1. (37) It is by now classical and was proven in [19], that in dimensions 𝑑=1,2, full intermittency holds for all 𝜅>0 but in dimensions 𝑑≥3, full intermittency only holds for 0<𝜅<𝜅₂(𝑑) where 𝜅₂(𝑑) is a dimension-dependent constant. This was done in [19] by noting that in dimensions 𝑑=1 and 2 the operator 2𝜅Δ+𝛿₀ has a positive eigenvalue for any 𝜅>0. By contrast, in dimensions 𝑑≥3 there is a positive constant 𝜅₂(𝑑) such that 2𝜅Δ+𝛿₀ has a positive eigenvalue only for 0<𝜅<𝜅₂(𝑑). When there is a positive eigenvalue, 𝜆₀(𝜅), one can conclude that lim_𝑡→∞1𝑡log𝐸_𝑥^𝜅,⊗2𝑒^{∫𝑡0𝛿0(𝑋𝑠−𝑌𝑠)𝑑𝑠}=𝜆₀(𝜅). (38) When the spectrum of 2𝜅Δ+𝛿₀ has no eigenvalue, that is, when in dimension 𝑑≥3 with 𝜅>𝜅₂(𝑑), one has lim_𝑡→∞1𝑡log𝐸_𝑥^𝜅,⊗2𝑒^{∫𝑡0𝛿0(𝑋𝑠−𝑌𝑠)𝑑𝑠}=0. (39) Using this with (120) we have the following consequence due to Carmona and Molchanov [19].

Theorem 1. Full intermittency holds in dimensions 1 and 2 for any 𝜅>0. For each 𝑑≥3, there is a positive 𝜅₂(𝑑) such that for 0<𝜅<𝜅₂(𝑑), full intermittency holds.

Below we will give a probabilistic proof of this result which identifies the value 𝜅₂(𝑑). In addition, in [19], it was shown that there is a sequence 𝜅_𝑝(𝑑) which was proved to be strictly increasing in [32], such that 𝛾_𝑝(𝜅)𝑝=𝛾₁(𝜅),𝜅≥𝜅_𝑝(𝑑),𝛾_𝑝(𝜅)𝑝>𝛾_𝑝−1(𝜅)𝑝−1,𝜅<𝜅_𝑝(𝑑). (40) This has been refined, extended, and improved in the recent work of Greven and den Hollander, [32].

The physical phenomena of intermittency is the property that a random field possesses widely separated high peaks. The most well-known field exhibiting this property is the magnetic field energy in a star. In our sun, this exhibits itself as sun spots where most of the magnetic field energy is concentrated, thereby lowering the temperature and causing the darkening which appears as a spot. Intermittency properties of the field {𝑢(𝑡,𝑥)∶𝑥∈𝐙^𝑑} were established in [19] and this will be discussed below.

5. Itô Solution and Probabilistic Proof of Intermittency

The Itô solution, 𝑢_𝐼(𝑡,𝑥), is the solution when the Itô differential is used in (20) instead of the Stratonovich differential. Recalling this solution satisfies 𝑢_𝐼(𝑡,𝑥)=𝑢₀(𝑥)+𝜅^𝑡₀Δ𝑢_𝐼()+𝑠,𝑥𝑑𝑠_𝑡0𝑢_𝐼(𝑠,𝑥)𝑑𝑊_𝑥(𝑠),𝑥∈𝐙^𝑑,𝑡≥0. (41) The relation between 𝑢 and 𝑢_𝐼 is simply 𝑢_𝐼(𝑡,𝑥)=𝑒^−𝑡/2𝑢(𝑡,𝑥). (42) Then defining 𝛾^𝐼₂(𝜅)=lim_𝑡→∞(1/𝑡)log𝐸[𝑢²_𝐼(𝑡,𝑥)], it follows that 𝛾₂(𝜅)−2𝛾₁(𝜅)=𝛾^𝐼₂(𝜅), (43) and so full intermittency holds if and only if 𝛾^𝐼₂(𝜅)>0.

We show that in dimensions greater than 2, that the Itô solution 𝑢_𝐼(𝑡,𝑥) has bounded second moment for 𝜅>𝜅₂(𝑑) and that the second moment grows exponentially for 𝜅≤𝜅₂(𝑑). This was shown in [19] using spectral considerations outlined in the previous section, so here we will give a probabilistic argument which was alluded to in [19] but does not seem to have appeared anywhere. First, recall that the jump rate of 𝑋 is 2𝑑𝜅. Thus, if 𝑌 is an independent copy of 𝑋, the jump rate of 𝑋−𝑌 is 4𝑑𝜅, that is, ({(𝑋_𝑠−𝑌_𝑠)∶𝑠≥0},𝑃^𝜅_𝑥)^ℒ=({𝑋_𝑠∶𝑠≥0},𝑃_𝑥^2𝜅). Recalling that 𝐸𝑒^{𝐴𝑡(𝑋)+𝐴𝑡(𝑌)}=𝑒^{𝑡+∫𝑡0𝛿0(𝑋𝑠−𝑌𝑠)𝑑𝑠}, (44) we have 𝐸𝑢²_𝐼(𝑡,𝑥)=𝑒^−𝑡𝐸𝐸_𝑥^𝜅,⊗2𝑒^{𝐴𝑡(𝑋)+𝐴𝑡(𝑌)}=𝑒^−𝑡𝐸_𝑥^𝜅,⊗2𝐸_𝑄𝑒^{𝐴𝑡(𝑋)+𝐴𝑡(𝑌)}=𝐸_𝑥^𝜅,⊗2𝑒^{∫𝑡0𝛿0(𝑋𝑠−𝑌𝑠)𝑑𝑠}=𝐸_𝑥^2𝜅𝑒^{∫𝑡0𝛿0(𝑋𝑠)𝑑𝑠}. (45)

Now we introduce some helpful notation. We let 𝑇₁=𝜏₁=inf{𝑡>0∶𝑋_𝑡≠𝑋₀},𝜌₀=0 and for 𝑗≥1, 𝑇_𝑗=inf𝑡>𝜌_𝑗−1∶𝑋_𝑡≠0,𝜌_𝑗=inf𝑡>𝑇_𝑗∶𝑋_𝑡=0. (46) If the underlying chain is recurrent, then all of these stopping times are finite. If the chain is transient, then only a finite number of these times is finite. Set, for 𝑗≥1, 𝜏_𝑗=𝑇_𝑗−𝜌_𝑗−1,𝑈_𝑗=𝜌_𝑗−𝑇_𝑗. (47) The distribution of 𝜏_𝑗 with respect to 𝑃₀^2𝜅 is exponential with parameter 4𝑑𝜅. For all 𝑗≥1, the random variable 𝜏_𝑗 is the duration of the 𝑗th sojourn at 0 and 𝑈_𝑗 is the duration of the 𝑗th excursion from 0. In the recurrent case, the random variables 𝜏₁,𝜏₂,… and 𝑈₁,𝑈₂,… are all finite and independent. They are also independent in the transient case “up to the time they all become infinite.” For a given 𝑡, if 𝑇_𝑛≤𝑡≤𝑇_𝑛+1, then writing 𝑅_𝑡=∫^𝑡₀𝛿₀(𝑋_𝑠)𝑑𝑠, 𝜏₁+⋯+𝜏_𝑛≤𝑅_𝑡≤𝜏₁+⋯+𝜏_𝑛+1. (48) The skeletal random walk of 𝑋 can now be defined using these stopping times. This is the Markov chain on 𝐙^𝑑 which keeps track of the sites visited by 𝑋, namely, 𝑌₀=𝑋₀,𝑌_𝑛=𝑋_𝑇𝑛+,𝑛≥1. Observe that if 𝑅=∑_𝑛≥0𝛿₀(𝑌_𝑛), then 𝑅 is a geometric random variable with parameter 𝑟=𝑃₀^2𝜅(𝑅=1), the probability of no return to the origin which is independent of 𝜅. For 𝑑≥3, and it is well known that 𝑟>0, see, for example, [2]. Since 𝑅 and the random variables 𝜏_𝑛 are independent, an easy argument shows that the total time spent at the origin is thus ^∞₀𝛿₀𝑋_𝑠𝑑𝑠=^𝑅_𝑛=1𝜏_𝑛 (49) and is an exponentially distributed random variable with parameter 4𝑑𝜅𝑟 with respect to the measure 𝑃₀^2𝜅. If 𝜅>1/4𝑑𝑟, one has lim_𝑡→∞𝐸𝑢²_𝐼(𝑡,0)=𝐸₀^2𝜅𝑒^{∑𝑅𝑛=1𝜏𝑛}=4𝑑𝜅𝑟4𝑑𝜅𝑟−1<∞. (50) We remark that by stationarity of the field 𝒲 and the choice of 𝑢₀≡1, it follows that 𝐸[𝑢²_𝐼(𝑡,𝑥)] is independent of 𝑥. When 𝜅≤1/4𝑑𝑟, one has 𝐸[𝑢²_𝐼(𝑡,0)]=∞. An important consequence of this little computation is the lack of intermittency for 𝜅>𝜅_2,𝑐𝑟≡1/4𝑑𝑟. Indeed, for 𝜅>𝜅_2,𝑐𝑟=1/4𝑑𝑟, lim_𝑡→∞1𝑡log𝐸𝑢(𝑡,𝑥)²=lim_𝑡→∞1𝑡log𝑒^𝑡𝐸𝑢_𝐼(𝑡,𝑥)²=1. (51) Consequently, since 𝛾₁(𝜅)=1, for 𝜅>𝜅_2,𝑐𝑟=1/4𝑑𝑟, 𝛾₂(𝜅)=2𝛾₁(𝜅), (52) and there is no intermittency for large 𝜅 in dimensions 𝑑≥3.

For 𝑑=1 or 2, however, we proceed as follows to show that there is intermittency for all 𝜅>0. It will be useful to make a change of variable and to set up some more notation. First we observe that ({𝑋_{𝑠/4𝑑𝜅}∶𝑠≥0},𝑃_𝑥^2𝜅)^ℒ=({𝑋_𝑠∶𝑠≥0},𝑃_𝑥^1/2𝑑). Thus, 𝐸_𝑥^2𝜅𝑒^{𝐴𝑡(𝑋)}=𝐸_𝑥^1/2𝑑𝑒^{(1/4𝑑𝜅)𝐴𝑡(𝑋)}. (53) We use Theorem 2 and it then suffices to show lim_𝑡→∞(1/𝑡)log𝐸₀^1/2𝑑[𝑒^{(1/4𝑑𝜅)𝐴𝑡(𝑋)}]>0. For simplicity, we denote 1𝛽=.4𝑑𝜅 (54)

Notice that the random variables 𝜏_𝑛 are exponentially distributed with parameter 1 with respect to 𝑃₀^1/2𝑑.

Recall for 𝑑=1 or 2 that 𝑟=0. Choose 𝐾 sufficiently large so that 𝑃₀^1/2𝑑(𝑈₁≤𝐾)≥𝑒^−𝛽/2. Then, 𝑃₀^1/2𝑑𝑈₁+⋯+𝑈_𝑛≤𝐾𝑛≥𝑃₀^1/2𝑑max𝑈₁,…,𝑈_𝑛≤𝐾≥𝑒^{−𝛽𝑛/2}. (55) The central limit theorem implies that 𝑃₀^1/2𝑑(𝜏₁+⋯+𝜏_𝑛≥𝑛)≥1/4 for 𝑛 sufficiently large. Observe that 𝜏₁+⋯+𝜏_𝑛≥𝑛,𝑈₁+⋯+𝑈_𝑛≤𝐾𝑛⊂𝑅_𝑛(𝐾+1)≥𝑛. (56) Hence, 𝐸₀^1/2𝑑𝑒^{𝛽𝑅𝑛(𝐾+1)}≥𝐸₀^1/2𝑑𝑒^{𝛽𝑅𝑛(𝐾+1)}∶𝜏₁+⋯+𝜏_𝑛≥𝑛,𝑈₁+⋯+𝑈_𝑛≤𝐾𝑛≥𝑒^𝛽𝑛𝑃₀^1/2𝑑𝜏₁+⋯+𝜏_𝑛≥𝑛,𝑈₁+⋯+𝑈_𝑛≥1≤𝐾𝑛4𝑒^𝑛/2=14𝑒^{(𝛽/2(𝐾+1))(𝐾+1)𝑛}. (57) Consequently, 𝛾_𝐼(2)=lim_𝑛→∞1(𝐾+1)𝑛log𝐸₀^1/2𝑑𝑒^{𝛽𝑅𝑛(𝐾+1)}≥𝛽2(𝐾+1)>0. (58) As a result, intermittency 𝛾₂(𝜅)>2𝛾₁(𝜅) holds for all 𝜅>0 in dimensions 𝑑=1 and 2.

We return now to 𝑑≥3 so that 𝑟>0 and show that intermittency holds for 𝜅<1/4𝑑𝑟. Again, it suffices to show lim_𝑡→∞(1/𝑡)log𝐸₀^1/2𝑑[𝑒^{𝛽𝐴𝑡(𝑋)}]>0. Recall that the 𝜏_𝑛 are exponential distributed with parameter 1 with respect to 𝑃₀^1/2𝑑. Standard large deviation estimates (see, e.g., [33, Theorem 9.3]) show that for fixed 𝛿>0, 𝑃₀^1/2𝑑𝜏₁+⋯+𝜏_𝑛≥(1+𝛿)𝑛≈𝑒^{𝑛(log(1+𝛿)−𝛿)}. (59) Also, 𝑃₀^1/2𝑑max𝑈₁,…,𝑈_𝑛<∞=𝑒^{𝑛log(1−𝑟)}. (60) As in the previous paragraph, for every 𝜖>0 we can find a 𝐾 such that 𝑃₀^1/2𝑑𝑈₁+⋯+𝑈_𝑛≤𝐾𝑛≥𝑃₀^1/2𝑑max𝑈₁,…,𝑈_𝑛≤𝐾=𝑒^{𝑛(log(1−𝑟)−𝜖)}. (61) Choosing 1+𝛿=(1−𝑟)⁻¹, we get 𝑑𝑃𝜏₁+⋯+𝜏_𝑛≥𝑛𝑈1−𝑟,max₁,…,𝑈_𝑛≤𝐾≈𝑒^{−(𝑛/(1−𝑟))(𝑟+𝜖(1−𝑟))}. (62) Recalling (56), it follows that for all 𝑛 sufficiently large, 𝑃𝑅_𝑛(𝐾+1)≥(1/(1−𝑟))𝑛≥𝑒^{−(𝑛/(1−𝑟))(𝑟+𝜖(1−𝑟))}. (63) If 𝛽>𝑟 and we choose 𝜖 with 𝜖<(𝛽−𝑟)/(4(1−𝑟)), with the corresponding choice of 𝐾 as in (61), then for all sufficiently large 𝑚, 𝐸𝑒^{𝛽𝑅𝑛(𝐾+1)}≥𝑒^{(𝛽/(1−𝑟))𝑛}𝑒^{−(𝑛/(1−𝑟))(𝑟+𝜖(1−𝑟))}≥𝑒^{((𝛽−𝑟−𝜖(1−𝑟))/((1−𝑟)(𝐾+1)))𝑛(𝐾+1)}. (64) Therefore, lim_𝑡→∞(1/𝑡)ln𝐸₀^1/2𝑑[𝑒^{𝛽𝐴𝑡(𝑋)}]≥(𝛽−𝑟−𝜖(1−𝑟))/((1−𝑟)(𝐾+1)), for every 𝜖>0 and intermittency holds in dimension 𝑑=1 or 2 for all 𝜅>0.

6. Covariance Structure and Association

Now we will examine the covariance structure of the field 𝒰(𝑡) in the intermittent regime, that is, when 𝜅<𝜅₂(𝑑) so that 2𝛾₁(𝜅)<𝛾₂(𝜅). Recall that this range for 𝜅 corresponds exactly to the range of 𝜅 for which the operator 2𝜅Δ+𝛿₀ has a positive eigenvalue. We will also establish that this field is associated. The mixed second moments 𝐸[𝑢(𝑡,𝑥)𝑢(𝑡,𝑦)] are significant in understanding the structure of the field 𝒰(𝑡). They are given by 𝐸𝑢(𝑡,𝑥)𝑢𝑡,𝑦=𝑒^𝑡𝐸^𝜅_𝑥⊗𝐸^𝜅_𝑦𝑒^{∫𝑡0𝛿0(𝑋𝑠−𝑌𝑠)𝑑𝑠}=𝑒^𝑡𝐸^2𝜅_𝑥−𝑦𝑒^{∫𝑡0𝛿0(𝑋𝑠)𝑑𝑠}. (65) The asymptotics of the function 𝐸[𝑢(𝑡,𝑥)𝑢(𝑡,𝑦)] can be evaluated as follows. Define 𝑍^𝜅_𝛽,𝑡(𝑥)=𝐸^𝜅_𝑥𝑒^{𝛽∫𝑡0𝛿0(𝑋𝑠)𝑑𝑠}. (66) Then observe that 𝑒^−𝑡𝐸𝑢(𝑡,𝑥)𝑢𝑡,𝑦=𝑍^2𝜅_1,𝑡𝑥−𝑦. (67) But there is a scaling relation 𝑍^2𝜅_1,𝑡𝑥−𝑦=𝐸^2𝜅_𝑥−𝑦𝑒^{∫𝑡0𝛿0(𝑋𝑠)𝑑𝑠}=𝐸^2𝜅_𝑥−𝑦𝑒^{∫02𝜅𝑡𝛿0(𝑋𝑠/2𝜅)𝑑(𝑠/2𝜅)}=𝐸¹_𝑥−𝑦𝑒^{(1/2𝜅)∫02𝜅𝑡𝛿0(𝑋𝑠)𝑑𝑠}=𝑍¹_{1/2𝜅,2𝜅𝑡}𝑥−𝑦 (68) since 𝑌_⋅=𝑋_(2𝜅)⁻¹_⋅ is a rate 2𝑑 simple symmetric random walk on 𝐙^𝐝 with respect to 𝑃_𝑥^2𝜅. This shows that 𝑍^2𝜅_1,𝑡(𝑥)=𝑍¹_{1/2𝜅,2𝜅𝑡}(𝑥). (69) The function 𝑍_𝛽,𝑡(𝑥)≡𝑍¹_𝛽,𝑡(𝑥) arises as the partition function of a homopolymer, [34], and by the Feyman-Kac formula, 𝑍_𝛽,𝑡(𝑥) solves 𝜕𝑍𝜕𝑡^𝛽,𝑡(𝑥)=Δ𝑍_𝛽,𝑡(𝑥)+𝛽𝛿₀(𝑥)𝑍_𝛽,𝑡(𝑥),𝑍_𝛽,0(𝑥)≡1. (70) The spectrum of the operator on the right hand side, 𝐻_𝛽=Δ+𝛽𝛿₀, (71) satisfies spectrum𝐻_𝛽=[−4𝑑,0]∪𝜆₀𝛽,for𝛽>0,𝑑=1,2, (72) and for 𝑑≥3, there is a dimension-dependent 𝛽_𝑐(𝑑) such that spectrum𝐻_𝛽=[−4𝑑,0],for0<𝛽<𝛽_𝑐(𝑑)𝐻,𝑑≥3,spectrum_𝛽=[−4𝑑,0]∪𝜆₀𝛽,for𝛽_𝑐(𝑑)<𝛽,𝑑≥3. (73) In the above, 𝜆₀(𝛽)>0 is a simple eigenvalue for 𝐻_𝛽 and the portion [−4𝑑,0] is purely a.c. spectrum. In fact, one now sees that 𝛽_𝑐(𝑑)=1/2𝜅₂(𝑑) from the section on intermittency. We denote by 𝜓_𝛽 the eigenfunction corresponding to 𝜆₀(𝛽) and note that it is given by, see [34], 𝜓_𝛽(𝑥)=1(2𝜋)^𝑑𝐓𝑑𝑒^{𝑖⟨𝜙,𝑥⟩}𝜆₀𝛽+Φ𝜙𝑑𝜙, (74) where Φ𝜙=2^𝑑_𝑗=11−cos𝜙_𝑗 (75) is the symbol (Fourier transform) of Δ and 𝐓^𝑑 is the 𝑑-dimensional torus. The representation (74) can be used to establish the exponential decay of 𝜓₀, and there is a positive constant 𝑐=𝑐(𝛽,𝑑) such that 𝜓_𝛽(𝑥)≤𝑐𝑒^{−𝑐|𝑥|},𝑥∈𝐙^𝑑. (76) By the spectral theorem, letting 𝐸_𝜆 be the resolution of the identity for the operator 𝐻_1/2𝜅, one has 𝑍_1/2𝜅,𝑡(𝑥)=𝑒^{𝜆0(1/2𝜅)𝑡}𝜓_1/2𝜅(𝑥)𝜓_1/2𝜅𝐿1+⁰_−4𝑑𝑒^𝜆𝑡𝑑𝐸_𝜆1_𝑥,1. (77) Note that 𝑍_1/2𝜅,𝑡(𝑥)=𝐸_𝑥^1/2𝜅[𝑒^{∫𝑡0𝛿0(𝑋(𝑠))𝑑𝑠}]≥1. The following result was used in [19] to prove a central limit theorem for sum of the form ∑_{𝑥∈Λ𝐿}𝑢(𝑡,𝑥) which will be described later in this paper.

Theorem 2. For 𝑑=1,2 and any 𝜅>0 or for 𝑑≥3 and 𝜅<𝜅₂(𝑑)=1/2𝛽_𝑐(𝑑), 𝑍_1/2𝜅,𝑡(𝑥)𝜆−1=exp₀1𝑡×𝜓2𝜅_1/2𝜅(𝑥)𝜓_{1/2𝜅𝐿1(ℝ𝑑)}+𝑞(𝑥)𝑂exp(−𝜖𝑡),𝑡→∞, (78) where_{𝑥∈𝐙𝑑}𝑞(𝑥)<∞. (79) In addition, when there is a positive eigenvalue for 𝐻_1/2𝜅, this eigenfunction satisfies 𝜓_1/2𝜅(𝑥)≤𝑐𝑒^{−𝑐|𝑥|},𝑥∈𝐙^𝑑, (80) where 𝑐=𝑐(1/2𝜅,𝑑)>0 depends on 𝑑 and 𝜅.

This implies exponential decay in the spatial variable for the covariance of solutions of (20). Recalling the equivalence in law stated in (69) it follows from (78) that 𝑍^2𝜅_1,𝑡(𝑥)∼𝑒^{𝜆0(1/2𝜅)2𝜅𝑡}𝜓_1/2𝜅(𝑥)𝜓_1/2𝜅𝐿1+𝑞(𝑥)𝑂𝑒^{−𝜖2𝜅𝑡}. (81)

Corollary 3. For 𝑑=1,2 and any 𝜅>0 or for 𝑑≥3 and 0<𝜅<𝜅₂(𝑑), Cov𝑢(𝑡,𝑥),𝑢𝑡,𝑦=𝑒^𝑡𝑍^2𝜅_1,2𝜅𝑡𝑥−𝑦−1∼𝑒^𝑡𝑒^{𝜆0(1/2𝜅)2𝜅𝑡}𝜓_1/2𝜅𝑥−𝑦𝜓_1/2𝜅𝐿1+𝑞𝑥−𝑦𝑂𝑒^{−𝜖2𝜅𝑡},𝐸𝑢²(𝑡,𝑥)=𝑒^𝑡𝑍^2𝜅_1,𝑡(0)−1∼𝑒^𝑡𝑒^{𝜆0(1/2𝜅)2𝜅𝑡}𝜓_1/2𝜅(0)𝜓_1/2𝜅𝐿1+𝑞(0)𝑂𝑒^{−𝜖2𝜅𝑡}. (82)(83) Consequently, 𝛾₂(𝜅)=2𝛾₁(𝜅)+𝜆₀12𝜅2𝜅=1+𝜆₀12𝜅2𝜅. (84)

Note that (84) gives a quantitative expression for the intermittency condition 𝛾₂(𝜅)>2𝛾₁(𝜅). Since 𝛾₁(𝜅)=1/2, we see that 𝛾₂(𝜅)−2𝛾₁(𝜅)=𝜆₀12𝜅2𝜅. (85) We note that 𝜆₀(1/2𝜅)→0 as 𝜅↗𝜅₂(𝑑). Its rate of decay depends on the dimension.

An important property of the field 𝒰(𝑡) is that the random variables in this field are associated. A collection of random variables {𝑋_𝑘,𝑘∈𝑆}, where 𝑆 is a countable set, is said to be associated if for any 𝑑 and coordinate-wise increasing functions 𝑓,𝑔∶𝐑^𝑑→𝐑, and any finite subcollection 𝑋_𝑘1,𝑋_𝑘2,…,𝑋_𝑘𝑑, it holds that Cov𝑓𝑋_𝑘1,𝑋_𝑘2,…,𝑋_𝑘𝑑,𝑔𝑋_𝑘1,𝑋_𝑘2,…,𝑋_𝑘𝑑≥0. (86) This notion was introduced in [35] and is of course related to the FKG inequality. One important aspect of this property was developed in [36] where Newman established a central limit theorem for the collection {𝑋_𝑘,𝑘∈𝐙^𝑑} under the assumptions that the {𝑋_𝑘,𝑘∈𝐙^𝑑} are associated, stationary and satisfy finite susceptibility Cov𝑋_{𝑘∈𝐙𝑑0},𝑋_𝑘<∞. (87) Note that the bound provided by (82) implies that the field 𝒰(𝑡) has finite susceptibility in the intermittent regime. A classical application of the Newman's central limit theorem is to take the 𝑋_𝑘=𝜎(𝑘),𝜎∈{0,1}^𝐙𝑑, the spins of a ferromagnetic stochastic Ising model and derive a central limit theorem for sums, ∑_𝑘∈Λ𝜎(𝑘), over growing boxes Λ⊂𝐙^𝑑, with respect to a Gibbs state. The spins are correlated, but they possess the property of being associated and stationary with respect to the Gibbs state.

The solutions of the parabolic Anderson equation (20) are associated. The following result was established in [14]. The proof uses a result of Pitt, [37], which states that a necessary and sufficient condition for the associativity of a Gaussian vector is the point-wise nonnegativity of its correlation function. Since associativity is preserved by convergence in distribution, the result below is proved using a simple approximation procedure.

Theorem 4. Let {𝑊_𝑥∶𝑥∈𝐑^𝑑} be a field of 𝑖𝑖𝑑, standard, one-dimensional Brownian motions on some probability space (Ω,𝑄). Then 𝒰(𝑡)={𝑢(𝑡,𝑥)∶𝑥∈𝐙^𝑑}, and the field of solutions of (20) is associated.

7. Almost Sure Lyapunov Exponents

In the previous section we examined the behavior of the moments of 𝑢(𝑡,𝑥). We now turn our attention to the a.s. behavior of the solution of (20); that is, we consider the a.s. Lyapunov exponent defined by 𝜆(𝜅)=lim_𝑡→∞1𝑡log𝑢(𝑡,𝑥). (88) The existence of this limit was first established by Carmona and Molchanov in [19] in the case when either 𝑢₀(𝑥)=𝛿₀(𝑥) or, more generally, when 𝑢₀ has compact support. The technique of proof used Liggett's subadditive ergodic theorem, [38]. The sub-additivity, when 𝑢₀=𝛿₀, is an easy consequence of the Feynman-Kac representation (116) 𝑢(𝑡+𝑟,0)=𝐸^𝜅₀𝑒^{∫0𝑡+𝑟𝑑𝑊𝑋(𝑡+𝑢−𝑠)(𝑠)}𝛿₀𝑋_𝑡+𝑢≥𝐸^𝜅₀𝑒^{∫0𝑡+𝑟𝑑𝑊𝑋(𝑡+𝑟−𝑠)(𝑠)}𝛿₀𝑋_𝑡+𝑟𝛿₀𝑋_𝑡=𝐸^𝜅₀𝑒^{∫𝑡0𝑑𝑊𝑋(𝑡−𝑠)(𝑠)}𝛿₀𝑋_𝑡×𝐸^𝜅₀𝑒^{∫𝑟0𝑑𝑊𝑋(𝑟−𝑠)(𝑡+𝑠)}𝛿₀𝑋_𝑟ℒ=𝑢(𝑡,0)𝑢(𝑟,0). (89) The Markov property is used in going from line 2 to line 3 and this technique broke down at this step in the case of 𝑢₀≡1. The latter case was established in [39] using a block argument from percolation theory. This type of block argument originated in [40]. Using the fact that time increments of the field 𝒲 are independent over disjoint space-time blocks in [0,∞)×𝐙^𝑑, the proof established an oriented percolation scheme and applied a recurrence result from [41] for such schemes.

In view of the application to stellar magnetic fields, a significant aspect of 𝜆(𝜅) is its asymptotic behavior as 𝜅↘0. The exact asymptotics were established independently in and in [42].

The asymptotics are derived from the Feynman-Kac representation (116) through analysis of the Gaussian field {𝐻_𝑇(𝜁)∶𝜁∈𝒟_𝑇} where 𝐻_𝑇(𝜁)=^𝑇₀𝑑𝑊_𝜁(𝑠)(𝑠),𝜁∈𝒟_𝑇. (90) This field places a natural metric 𝜌 on 𝒟_𝑇 by means of 𝜌²𝜁,𝜂=𝐸𝐻_𝑇(𝜁)−𝐻_𝑇𝜂²=2𝑇−^𝑇₀𝛿₀𝜁(𝑠)−𝜂(𝑠)𝑑𝑠. (91)

The index set 𝒟_𝑇 of the field {𝐻_𝑇(𝜁)∶𝜁∈𝒟_𝑇} is too large from the point of view of the metric entropy, see [43] for an explanation of the metric entropy, induced by the metric 𝜌. Thus we restrict the index set by specifying the number of jumps of its elements 𝜁 in the interval. So, using 𝑁(𝛾,𝑡) to denote the number of jumps of 𝜁 in [0,𝑡] we can define the space of paths 𝒟_𝑛,𝑚=𝜁∈𝒟_𝑛∶𝜁∶[0,𝑛]→𝐙^𝑑,𝑁𝛾,𝑛=𝑚. (92)

The superadditive functional 𝐴_𝑛,𝑚=sup_{𝜁∈𝒟𝑛,𝑚}𝐻_𝑛(𝜁) (93) is the supremum of a Gaussian field 𝐻_𝑛(𝜁)∶𝜁∈𝒟_𝑛,𝑚 (94) indexed by the set 𝒟_𝑛,𝑚. This set has a suitably bounded entropy, which, by a theorem of Fernique and Talagrand, implies 𝐸[𝐴_𝑛,𝑛]≤𝐶𝑛. This bound allows, by means of Liggett's subadditive ergodic theorem, the conclusion that there is a positive constant 𝛼 such that lim_𝑛→∞𝐴_𝑛,𝑛𝑛=𝛼,a.s. (95)

An interesting and presumably difficult problem is to determine the proper scale 𝐴_𝑛,𝑛−𝛼𝑛 in order to obtain a limit law. It is conjectured that (𝐴_𝑛,𝑛−𝛼𝑛)/𝑛^2/3 should have a nontrivial limit law, possibly related to the Airy distribution. This conjecture comes from related results arising in random matrix theory such as [44, 45]. Similarly, one would expect nontrivial limit laws for (log𝑢(𝑡,𝑥)−𝜆(𝜅)𝑡)/𝑡^2/3.

The asymptotics established in [39, 42] for 𝜆(𝜅) is that lim_𝜅↘0𝜆(𝜅)()=𝛼log1/𝜅4². (96)

These asymptotics are arrived at by decomposing the Feynman-Kac representation, (116), 𝑢(𝑛,𝑥)^ℒ=^∞_𝑗=0𝐸^𝜅_𝑥𝑒^{∫𝑛0𝐻𝑋(𝑛)}∣𝑁(𝑋,𝑛)=𝑗×𝑃^𝜅_𝑥𝑁(𝑋,𝑛)=𝑗, (97) where ^ℒ= denotes equality in law (distribution.) Note that the only change has been to have the time direction be the same in both 𝑊 and 𝑋. The intuition now is that the conditional expectation in the 𝑗th term should be nearly 𝑒^{𝐴𝑛,𝑗}. One quickly realizes that sum over 𝑗∉[𝑎𝑛,𝑏𝑛] for suitable choices of 𝑎 and 𝑏 is not significant, then only terms of the from 𝑒^{𝐴𝑛,𝑗𝑛} matter. But, by Brownian scaling, 𝐴^ℒ_{𝑛,𝑗𝑛}=(1/𝑗)𝐴_{𝑗𝑛,𝑗𝑛}. Using this in (97) and simple large deviations results for the Poisson distributed 𝑁(𝑋,𝑛) leads to an upper bound for the asymptotics of 𝜆(𝜅). The lower bound comes from looking at a particular path 𝑋 that dominates the Feynamn-Kac expectation and using similar estimates.

Thus, for small 𝜅,𝜆(𝜅)≪𝛾₁(𝜅)≡1/2 which says the a.s. behavior is much smaller than the first moment behavior. This just reflects the fact that the expectation of 𝑢(𝑡,𝑥) is dominated by large values of 𝑢(𝑡,𝑥) which occur with small, but not too small, probability. This is related to the intermittency and will be examined in the section on sums over boxes below.

We would like to point out that 𝜅→𝜆(𝜅) is an increasing function of 𝜅. Also, since 𝐸[𝑢(𝑡,𝑥)]=𝑒^𝑡/2 one has 𝜆(𝜅)≤1/2 for all 𝜅>0. Moreover, it was pointed out in [19] that 𝜆(𝜅)=𝛾₁(𝜅) for 𝜅≥𝜅₂(𝑑). The argument given there goes as follows. First define 𝜆_𝐼(𝜅)=lim_𝑡→∞1𝑡log𝑢_𝐼(𝑡,𝑥). (98) From the fact that 𝑢(𝑡,𝑥)=𝑒^𝑡/2𝑢_𝐼(𝑡,𝑥), it follows that 𝜆_𝐼(𝜅)=−1/2+𝜆(𝜅). Note for 𝜅≥𝜅₂(𝑑) that 𝐸[𝑢²_𝐼(𝑡,𝑥)]≤𝐶(𝑑)<∞ and 𝐸[𝑢_𝐼(𝑡,𝑥)]=1. Thus, for every 𝜖>0, there is a 𝛿>0 such that 𝑃(𝑢_𝐼(𝑡,𝑥)>𝛿)>𝜖. This implies 𝜆_𝐼(𝜅)≥0. But obviously, 𝜆_𝐼(𝜅)≤0 so we conclude that for 𝜅≥𝜅₂(𝑑) one has 𝜆_𝐼(𝜅)=0 and, consequently, 𝜆(𝜅)=1/2.

We end this section with a discussion of the relation between the a.s. Lyapunov exponent and the moment Lyapunov exponents from [39].

Theorem 5. We have the following: 𝜆(𝜅)=𝛾^′₀(𝜅), (99) where (𝑑/𝑑𝑝)𝛾_𝑝(𝜅)=𝛾^′_𝑝(𝜅).

We give a brief sketch of the proof. In [46], it was shown that 𝜆(𝜅)=𝛾^′₀₋(𝜅). In [39] the large deviation estimate for every 𝜖>0 there is a 𝑐(𝜖)>0 such that 𝑄𝑢≥𝑒^{(𝜆(𝜅)+𝜖)𝑡}≤𝑒^{−𝑐(𝜖)𝑡} (100) was established by means of a block argument. Thus, for 𝑝>0, 𝐸𝑢^𝑝(𝑡,𝑥)=𝐸𝑢^𝑝(𝑡,𝑥)∶𝑢(𝑡,𝑥)≤𝑒^{𝑐(𝜖)𝑡}+𝐸𝑢^𝑝(𝑡,𝑥)∶𝑢(𝑡,𝑥)>𝑒^{𝑐(𝜖)𝑡}≤𝑒^{𝑝(𝜆(𝜅)+𝜖)𝑡}+𝐸[𝑢^2𝑝(𝑡,𝑥)]^1/2𝑒^{−𝑐(𝜖)𝑡/2}. (101) Thus, for every 𝜖>0, 𝛾^′₀₊(𝜅)≤lim_𝑝↘0𝜆(𝜅)1+𝜖,𝛾2𝑝^2𝑝(𝜅)−𝜖=𝜆(𝜅)+𝜖 (102) which gives 𝛾^′₀(𝜅)=𝜆(𝜅).

8. Solution of PAM as Interacting Diffusion

An interesting point of view regarding the solution of (20) was proposed in [47, 48] which grew out of work on the stepping stone model in [26]. In [48], Shiga and Shimizu, the authors view the entire field 𝒰_𝐼(𝑡)={𝑢_𝐼(𝑡,𝑥)∶𝑥∈𝐙^𝑑} as a Markov process in a subset of a particular 𝐿¹-space. In [47, 48] and the related works [32, 49], a more general underlying Markov process is used than the one with generator 𝜅Δ. However, for simplicity we will confine our discussion to the case where the operator appearing in (20) is 𝜅Δ. Take any summable collection of positive numbers {𝛾_𝑥∶𝑥∈𝐙^𝑑} that also satisfies for some positive 𝐶, _{𝑦∶|𝑦−𝑥|=1}𝛾_𝑦≤𝐶𝛾_𝑥. (103) Then set for 𝑝>0, ℒ^𝑝𝛾=𝜉∈[0,∞)^𝐙𝑑∶_{𝜉∈𝐙𝑑}𝜉^𝑝_𝑥𝛾_𝑥<∞⊂𝐿^𝑝𝛾. (104) The space ℒ¹(𝛾) is endowed with the product topology. We recall the following theorem of [48].

Theorem 6. Given 𝒰_𝐼(0)∈ℒ¹(𝛾), the SDE (20) has a unique strong solution with (𝒰_𝐼(𝑡))_𝑡≥0⊂ℒ¹(𝛾) and strongly continuous paths in ℒ²(𝛾) a.s. The process (𝒰_𝐼(𝑡))_𝑡≥0 is a Markov process on ℒ¹(𝛾) with semigroup 𝑆_𝑡 which satisfies 𝑆_𝑡𝑓−𝑓=^𝑡₀𝑆_𝑢𝐿𝑓𝑑𝑢 (105) for 𝑓 depending smoothly on only finitely many coordinates and where 𝐿𝐹(𝜉)=𝜅_{𝑥∈𝐙𝑑𝑦∈𝐙𝑑∶𝑦−𝑥=1}𝜅𝜉_𝑦−𝜉_𝑥𝜕𝑓𝜕𝜉_𝑥+12_{𝑥∈𝐙𝑑}𝜉²_𝑥𝜕²𝑓𝜕𝜉²_𝑥. (106) If 𝒰_𝐼(0)∈ℒ²(𝛾) then (𝒰_𝐼(𝑡))_𝑡≥0⊂ℒ²(𝛾) and is a Feller diffusion.

One interesting aspect of this theorem is the door it opens into applying the techniques of diffusion processes to the process 𝒰_𝐼. So one can ask what are the invariant measures for the semigroup 𝑆_𝑡? Since the components of 𝒰_𝐼 are interacting processes, it also brings the point of view of interacting particle systems onto the scene. In the latter field, characterization of all the 𝑆_𝑡-invariant, shift invariant, and ergodic measures is a common theme. The reader is referred to [50] or [51] for examples and many references. One example of such a result is the following due to Shiga from [47]. To describe his results we introduce some notation and relevant objects. Denote by 𝒫(ℒ²(𝛾)) the probability measures on (ℒ²(𝛾),ℬ(ℒ²(𝛾))) where ℬ(ℒ²(𝛾)) is the Borel field on ℬ(ℒ²(𝛾)). Denote by 𝑆^∗_𝑡 the action of the dual of 𝑆_𝑡 on the space 𝒫(ℒ²(𝛾)), that is, with ⟨𝜇,𝑓⟩=∫_ℒ2(𝛾)𝑓(𝑥)𝜇(𝑑𝑥), 𝑆^∗_𝑡𝜇,𝑓=𝜇,𝑓,𝑓∈𝐶_𝑏ℒ²𝛾. (107) Then we can define the invariant measures for the process 𝒰_𝐼 by ℐ=𝜇∈𝒫ℒ²𝛾∶𝑆^∗_𝑡𝜇=𝜇,𝑡≥0. (108) We can also consider the class of probability measures that are invariant with respect to the group of shifts 𝜏_𝑥,𝑥∈𝐙^𝑑, 𝒯=𝜇∈𝒫ℒ²𝛾∶𝜏_𝑥𝜇=𝜇,𝑥∈𝐙^𝑑. (109) Let 𝒯₁=𝜇∈𝒯∶𝜇,𝜉_𝑥<∞,𝑥∈𝐙^𝑑. (110) Consider the initial configuration 𝒰_𝐼(0)=Θ which means all coordinates take on the value 𝜃∈[0,∞). View this as starting the process 𝒰_𝐼 with initial distribution 𝛿_Θ. An important result of Shiga, with additions made by [32, 49], explains the asymptotic behavior of 𝒰_𝐼 in the nonintermittent regime, 𝜅>𝜅₂(𝑑).

Theorem 7. Assume 𝑑≥3 and 𝜅>𝜅₂(𝑑). Then (i) for each 𝜃∈𝐑,lim_𝑡→∞𝑆^∗_𝑡𝛿_Θ=𝜈_𝜃 exists and ⟨𝜈_𝜃,𝜉_𝑥⟩=𝜃 for 𝑥∈𝐙^𝑑;(ii)  the set of extreme points of 𝒮∩𝒯₁ is exactly {𝜈_𝜃∶𝜃∈𝐑}. If 𝜇∈𝒯₁ is ergodic with respect to the group of translations {𝜏_𝑥∶𝑥∈𝐙^𝑑} and ⟨𝜇,𝜉_𝑥⟩=𝜃, then lim_𝑡→∞𝑆^∗_𝑡𝜇=𝜈_𝜃; (111)(iii) the measure 𝜈_𝜃 is associated.

We observe that the association of 𝜈_𝜃 follows from the association of the field 𝒰_𝐼(𝑡) since this property is preserved by limits in distribution. As a complement, Shiga in [47] also gave the behavior of the field 𝒰_𝐼 when 𝑑=1,2 so that the process is in the intermittent regime. More recently, this was extended to 𝑑≥3 by Cox and Greven, [49], and then Greven and den Hollander, [32], into the intermittent region, 0<𝜅<𝜅₂(𝑑). In this case, it turns out that the process 𝒰_𝐼(𝑡) dies out in the following sense that its law tends to the point mass on the configuration of all 0's. By this we mean the element 0∈[0,∞)^𝐙𝑑 all of whose components are 0. We denote the point mass on 0 by