Stochastic comparisons for stochastic heat equation

We establish the stochastic comparison principles, including moment comparison principle as a special case, for solutions to the following nonlinear stochastic heat equation on $\mathbb{R}^d$ \[ \left(\frac{\partial }{\partial t} -\frac{1}{2}\Delta \right) u(t,x) = \rho(u(t,x)) \:\dot{M}(t,x), \] where $\dot{M}$ is a spatially homogeneous Gaussian noise that is white in time and colored in space, and $\rho$ is a Lipschitz continuous function that vanishes at zero. These results are obtained for rough initial data and under Dalang's condition, namely, $\int_{\mathbb{R}^d}(1+|\xi|^2)^{-1}\hat{f}(\text{d} \xi)<\infty$, where $\hat{f}$ is the spectral measure of the noise. We establish the comparison principles by comparing either the diffusion coefficient $\rho$ or the correlation function of the noise $f$. As corollaries, we obtain Slepian's inequality for SPDEs and SDEs.


Introduction
In this paper, we study the stochastic comparison principle (see Definition 1.4) including moment comparison principle for the solutions to the following stochastic heat equation (SHE)    ∂ ∂t − 1 2 ∆ u(t, x) = ρ(u(t, x))Ṁ (t, x), x ∈ R d , t > 0, u(0, ·) = µ(·). (1.1) In this equation, ρ is assumed to be a globally Lipschitz continuous function with The linear case, i.e., ρ(u) = λu, is called the parabolic Anderson model (PAM) [3]. The noisė M is a Gaussian noise that is white in time and homogeneously colored in space. Informally, where δ 0 is the Dirac delta measure with unit mass at zero and f is a nontrivial "correlation function/measure" i.e., a nonnegative and nonnegative definite function/measure that is not identically zero 1 . The Fourier transform of f , which is again a nonnegative and nonnegative definite measure and is usually called the spectral measure, is denoted byf The SPDE (1.1) is understood in its integral form, i.e., the mild solution, We plan to work under weakest possible conditions on (1.1), which include rough initial data and Dalang's condition on f . Let us explain these two conditions in more details. We first note that by the Jordan decomposition, any signed Borel measure µ can be decomposed as µ = µ + − µ − where µ ± are two non-negative Borel measures with disjoint support. Denote |µ| := µ + + µ − . The rough initial data refers to any signed Borel measure µ such that R d e −a|x| 2 |µ|(dx) < +∞ , for all a > 0 , (1.5) where |x| = x 2 1 + · · · + x 2 d denotes the Euclidean norm. It is easy to see that the condition (1.5) is equivalent to the condition that the solution to the homogeneous equation -J 0 (t, x) defined in (2.1) below -exists for all t > 0 and x ∈ R d . Existence and uniqueness of a random field solution for rough initial conditions are recently established in [6] (see also [5] and [17]) under Dalang's condition [11], i.e., Υ(β) := (2π) −d R df (dξ) β + |ξ| 2 < +∞ for some and hence for all β > 0; (1.6) Dalang's condition (1.6) is the weakest condition for the correlation function f in order to have a random field solution (in the sense of Definition 2.1). Throughout this paper, we will assume that µ is a nonnegative measure.
Instead of presenting our results in full details, which will be done in Section 1.1, let us first take a look of several examples. Under Dalang's condition and for rough initial data, for either one of the above two scenarios (S-1) or (S-2), we have the following comparison results: (E-1) (Moment comparison principle) For m arbitrary space-time points (t ℓ , x ℓ ) ∈ (0, ∞) × R d (not necessarily to be distinct) and m integers k ℓ ∈ N, ℓ = 1, · · · , m, it holds that (1.7) (E-2) For any (t, x) ∈ (0, ∞) × R d , c > 0 and any integer n ≥ 1, it holds that (1. 8) In particular, by choosing c = J 0 (t, x) (see (2.1) below), (1.8) tells us that all central moments of even orders can be compared. When n = 1, this is a comparison result for the variances.
(E-4) Statement in (1.9) is true with g ℓ (z) being either of the following two functions: with a ℓ , b ℓ , d ℓ ≥ 1 and c ℓ ≥ e.
(E-5) For any m ≥ 1, t m > · · · > t 1 > 0, k 1 , · · · k m ∈ N \ {0}, α 1 , · · · , α m ∈ [2, ∞), and x ℓ j ∈ R d with ℓ = 1, · · · , m and j = 1, · · · , k ℓ such that x ℓ k 1 , · · · , x ℓ k ℓ are distinct points for each ℓ, (1.10) Note that (1.8) is not a special case of (1.7) when n ≥ 2. The oscillatory nature caused by the negative one makes (1.8) non-trivial. One more example, that is slightly different from the above ones, is the following Slepian's inequality for SPDEs: (E-6) (Slepian's inequality for SPDEs) Under the scenario (S-2), if f 1 and f 2 are equal to each other near the origin (see the precise meaning in Corollary 1.7 below), then for all a > 0, t > 0, and x 1 , · · · , x N ∈ R d , For the parabolic Anderson model (PAM), it is well known that the moments enjoy the Feynman-Kac representation, based on which one can obtain very sharp estimates for the moments. The literature is vast and we refer the interested readers to Xia Chen's papers [8,9] and references therein. One may also check the work by Borodin and Corwin [2] where the p-th moment is represented by some multiple contour integrals. Using the sharp estimates of the moments for PAM, intermittent phenomena (i.e., the solution develops tall peaks on small islands of many different scales), have been studied extensively, e.g., see [3,14] for the definition and analysis of intermittency in terms of moments and also [21,22,23] for the study on intermittency based on the macroscopic multi-fractal analysis. However, whenever ρ is nonlinear or whenever the functionals go beyond the moments functionals, much fewer tools are available. The stochastic comparison results of the above kinds, including moment comparison principle, play a fundamental role in this setting.
When the noise is additive, i.e., ρ(u) = constant, the moment comparison principle -Case (E-1) -under the second scenario (S-2) comes from Isserlis' theorem [19] since the solution is a Gaussian random field whose distribution is determined by the spatial correlation function f . On the other hand, to the best of our knowledge, the comparison principle including the moment comparison principle under the second scenario is new for (1.1) with the condition (1.2). As for the first scenario, the moment comparisons principle -Case (E-1) -has been studied recently. In [20], Joseph, Khoshnevisan and Mueller proved one-time comparison of (1.7) for the one-dimensional case, i.e., d = 1, with space-time white noise f = δ 0 , and t 1 = · · · = t m , which was later generalized by Foondun, Joseph and Li in [13] to the multiple-time comparison of the form (1.7) in the higher dimensional case d ≥ 1 with the Riesz kernel It is easy to see that the Riesz kernel with the range of β specified above satisfies Dalang's condition (1.6). In both [20] and [13], the initial conditions are assumed to be the Lebesgue measure µ(dx) = dx. We will generalize these results to cover rough initial data and all possible correlation functions under Dalang's condition (1.6). Moreover, we will cover many other functionals other than moment functionals in Case (E-1). The approximation results in Sections 3 and 4.2 below are interesting by themselves, where we use different approximation procedures which produce strong solutions in this paper rather than mild solutions as in [20,13]. We believe strong solutions are more straightforward and easier to handle when showing approximations.

Statement of the main results
In order to state our main results, we first need to introduce some notation. We first note that under our assumptions, namely, ρ(0) = 0 and the initial data µ being nonnegative, the solutions to (1.1) are nonnegative (see [5,7] and also Theorem 5.5 below). Hence, all function spaces in Definition 1.1 have their domains in R m + for some m ≥ 1.
R + be the set of nonnegative functions on R m + having continuous second order partial derivatives and all second order partial derivatives are nonnegative. Let C 2,v b R m + ; R + be the set of functions in C 2,v R m + ; R + such that all partial derivatives of orders 0, 1 and 2 are bounded. Let C 2,v p R m + ; R + be the set of functions in C 2,v R m + ; R + such that the gradient has at most some polynomial growth, namely, f ∈ C 2,v p (R m + ; R + ), then there exists some constant C > 0 and k ∈ N such that Definition 1.2. Let K be the spatial index set, which could be either R d or Z d or a finite set {0, · · · , d}. Let F K [C 2,v ] denote the set of finite-dimensional nonnegative functions of twice continuously differentiable functions, namely, |K| m=1 x ℓ ∈K: ℓ=1,··· ,m, where |K| is the cardinality of the index set K, which is equal to d for K = {1, · · · , d} and ∞ when there is countably or uncountable many elements in K. In the same way, one can define Let F M and F L denote the set of moment and Laplace functions, i.e., When there is no ambiguity from the context, we often omit the superscript K for these function spaces. Remark 1.3. In [10], F[C 2,v ] and any one in (1.15) are function cones because we will see latter that they are preserved under the certain semigroup operations (and/or multiplication). In contrast, F M and F L are not cones in that sense. It is clear that these sets of functions satisfy the following inclusion relations: (1.18) x) ∈ R + × K}, i = 1, 2, be two random fields, where K is the spatial index set as in Definition 1.2. For some set of functions F, such as those defined in Definition 1.2, and for some n ≥ 1, we say that u 1 and u 2 satisfy the n-time stochastic comparison principle over F with u 1 dominating u 2 if for any 0 < t 1 < · · · < t n < ∞, and F 1 , . . . , F n ∈ F, it holds that Now we are ready to state our main results: Theorem 1.5 (Comparison with respect to diffusion coefficients). Suppose that the correlation function f satisfies Dalang's condition (1.6). Let µ be a nonnegative measure that satisfies (1.5). Let u 1 (t, x) and u 2 (t, x) be two solutions of (1.1), both starting from µ, but with diffusion coefficients ρ 1 and ρ 2 , respectively. If then for any integer n ≥ 1, u 1 and u 2 satisfy the n-time (resp. 1-time) stochastic comparison principle over either We now state the stochastic comparison theorem with respect to two comparable correlation functions f 1 and f 2 . Theorem 1.6 (Comparison with respect to correlations of noises). LetṀ (1) andṀ (2) be two noises with correlation functions f 1 and f 2 , respectively, that satisfy Dalang's condition (1.6). Let µ be a nonnegative measure that satisfies (1.5). Let u 1 (t, x) and u 2 (t, x) be two solutions of (1.1), both starting from µ, with the same diffusion coefficient ρ, driven byṀ (1) andṀ (2) , respectively. If then for any integer n ≥ 1, u 1 and u 2 satisfy the n-time (resp. 1-time) stochastic comparison principle over either We would like to point out that for multiple-time comparison results, working on F M alone won't be sufficient since F M is not a function cone, i.e., it is not preserved under the underlying semigroup and multiplication (see Step 3 of the proof of Theorem 1.15 in Section 4.3 below). One needs to go through the function cone F[C 2,v + ] or F[C 2,v − ] as in [10]. On the other had, as an application of the 1-time comparison principle, we can obtain Slepian's inequality for SPDEs. Let C 2 b (R d ; R + ) denote the set of C 2 functions with bounded partial derivatives of orders 0, 1 and 2.
we have that, for any numbers a i > 0, x i ∈ R d for i = 1, . . . , N , and t ≥ 0, (1.20) In particular, for any a > R, x i ∈ R d for i = 1, . . . , N , and t ≥ 0, the inequality (1.11) it true.
Here, one example for the case (i) in Corollary 1.7 is that d = 1, Interacting diffusions. The proof of the above comparison theorems 1.5 and 1.6 rely on similar comparison results for the following linearly interacting diffusions, which are of interest by themselves. Let K denote a non-empty set with at most countably infinite elements (e.g. where κ > 0 is a fixed constant and we make the following assumptions over this equation: (ii) ρ : R + → R + is a globally Lipschitz function with ρ(0) = 0.
(iii) {M i (t); t ≥ 0} i∈K is a system of correlated Brownian motions with the following covariance structure: where γ : K → R + is a non-negative, symmetric and non-negative definite function.
(iv) u 0 : K → R + is a non-negative function in ℓ 2 (K) such that u 0 (i) > 0 for some i ∈ K. Remark 1.9. Regarding condition (1.22), when the state space K has finite cardinality, it is trivially satisfied. When the underlying random walk is symmetric, i.e., p i,j = p j,i , then this condition is satisfied with Λ = 1.
We say that U = {U (t, i); t ≥ 0, i ∈ K} is a strong solution to (1.21) with the initial data u 0 (·) if it satisfies that for all i ∈ K and t > 0, (1.24) The existence and uniqueness of a strong solution to (1.21) when the driving Brownian motions are independent is well-known (see, e.g., [26]). Since we only need the case when the initial data is in ℓ 2 (K) -(iv) of Assumption 1.8, we won't need the weighted ℓ 2 (K) space as was used in [26]. In [13], p i,j depends only on j − i and it is shown that there is a unique mild solution to (1.21) in L ∞ ([0, T ] × K; L k (Ω)) for any T > 0 and k ≥ 2. The next theorem, on the other hand, we provide a proof of existence and uniqueness of a strong solution in a slightly better space (see (1.25)) and for more general transition probabilities p i,j . As one can see later, a strong solution is easier to handle than a mild solution when showing approximations. In particular, U (t, ·) ∈ ℓ k (K) a.s. for any t ≥ 0 and k ≥ 2. Moreover, for any T > 0 and k ≥ 2, where the constant C > 0 depends only on κ, Lip ρ , γ(0) and Λ.
Note that the discrete nature of the spatial variable enables us to bring the supremum over the spatial variable inside the expectation; see (4.10). This is in general not true when the spatial variable lives in R d . For this interacting diffusions (1.21), we have the following two similar stochastic comparison results: Theorem 1.11 (Comparison with respect to diffusion coefficients). Let U 1 and U 2 be two solutions to (1.21), both starting from u 0 , but with diffusion coefficients ρ 1 and ρ 2 , respectively. Then the condition implies that for any integer n ≥ 1, U 1 and U 2 satisfy the n-time (resp. 1-time) stochastic comparison principle over either   i (t); t ≥ 0} i∈K , respectively. Let γ i be the covariance function for M (i) . Then the condition γ 1 (k) ≥ γ 2 (k), for all k ∈ K implies that for any integer n ≥ 1, U 1 and U 2 satisfy the n-time (resp. 1-time) stochastic comparison principle over either Corollary 1.13 (Slepian's inequality for interacting diffusions). Under the assumptions in Theorem 1.12 and, in addition, we have that, for any numbers a k ∈ R, i k ∈ K for k = 1, . . . , N , and t ≥ 0, In particular, we have that, for any a ∈ R, i k ∈ K for k = 1, . . . , N , and t ≥ 0, (1.28) At the very core of the chain of arguments is the following comparison results for the finite dimensional SDE with C 2 c (R + ) diffusion coefficient. Here C 2 c (R + ) = C 2 c (R + ; R) refers to the functions defined on R + , having compact support and continuous second derivative. Assumption 1.14. In the SDE (1.21), we assume that (i) ρ ∈ C 2 c (R + ) and ρ(u 0 (i)) = 0 for some i ∈ K; and (ii) the cardinality of the index set K is finite. Theorem 1.15. Under Assumption 1.14, the statement in Theorem 1.11 is true with F[C 2,v p,+ ], Theorem 1. 16. Under Assumption 1.14, the statement in Theorems 1.12 is true with Both Theorem 1.11 and Theorem 1.15 are essentially covered by Cox, Fleischmann and Greven [10]. The main difference is that Theorem 1.11 covers a much richer family of functions and another difference is that we have correlated, instead of independent, Brownian motions; See Remark 4.5 below for more details.

Outline of the paper
This paper is organized as follows: After some definitions, notation and preliminaries in Section 2, we provide the approximation procedure which shows that SHE (1.1) with rough initial data and noise whose spatial correlation only satisfies Dalang's condition (1.6) can be approximated by systems of infinite dimensional SDEs (i.e., interacting diffusions on the d-dimensional lattice) in Section 3. Combining the approximation procedures and the comparison theorems for infinite dimensional SDEs (Theorems 1.11 and 1.12, and Corollary 1.13) proves the main theorems 1.5 and 1.6 and also Slepian's inequality for SPDEs -Corollary 1.7 in Section 3.4. It remains to establish Theorems 1.11 and 1.12, and Corollary 1.13, which is done in Section 4. We first prove the existence and uniqueness result -Theorem 1.10 -in Section 4.1. Then we will prove Theorems 1.11 and 1.12 by first showing that a system of infinite dimensional SDEs can be approximated by systems of finite dimensional SDEs with a nice ρ in Section 4.2 and then obtaining the comparison theorems for finite dimensional SDEs following the procedure of Cox, Fleischmann and Greven [10] in Section 4.3. With these preparations, we proceed to prove Theorems 1.11 and 1.12 and Corollary 1.13 in Section 4.4. Finally, in Section 5, we give several examples to cover those in (E-1) -(E-5) above and one application of our approximation results to give another straightforward proof for the weak sample path comparison principle.

Some definitions, notation and preliminaries
Throughout this paper, ||·|| p denotes the L p (Ω)-norm, N := {0, 1, 2, · · · }, Lip ρ refers to the Lipschitz constant for ρ, D i := ∂ ∂x i , and R + : Recall that a spatially homogeneous Gaussian noise that is white in time is an L 2 (Ω)-valued mean zero Gaussian process on a complete probability space (Ω, F, P) Let B b (R d ) be the collection of Borel measurable sets with finite Lebesgue measure. As in Dalang-Walsh theory [11,29], one can extend F to a σ-finite L 2 (Ω)-valued martingale measure Let (F t , t ≥ 0) be the natural filtration generated by M · (·) and augmented by all P-null sets N in F, i.e., Then for any adapted, jointly measurable (with respect to and I(t, x) be the stochastic integral in the mild form (1.3). Hence, the mild form (1.3) can be written as u(t, x) = J 0 (t, x) + I(t, x).
(2) u is jointly measurable with respect to B (0, ∞) × R d × F; Existence and uniqueness of a random field solution for bounded initial data is covered by classical Dalang-Walsh theory [11,29]. For rough initial data, this is established in [4,5,6,17].
A key tool for dealing the rough initial data is the following moment formula. We need first introduce some notation. Denote By the Fourier transform, this function can be written in the following form Define h 0 (t) := 1 and for n ≥ 1, Theorem 2.2 (Moment bounds, Theorem 1.7 of [5]). Under Dalang's condition (1.6), if the initial data µ is a signed measure that satisfies (1.5), then the solution u to (1.1) for any given t > 0 and x ∈ R d is in L p (Ω), p ≥ 2, and then when p ≥ 2 is large enough, there exists some constant C > 0 such that Note that H(t; γ) in (2.6) has genuine exponential growth as proved in the following lemma: 3 Approximation procedure and the proof of Theorems 1.5 and 1.6 The following approximation procedure is interesting by itself, based on which our comparison results are direct consequences (see Step 4). Basically, we show that stochastic heat equations on R d with rough initial condition and driven by Gaussian noise which is white in time and correlated in space can be approximated by systems of interacting diffusions on the d-dimensional lattice. There are several steps in order to achieve this goal.

Step 1 (Regularization of the initial data and noise)
We will need the following approximation results, which were proved in Theorem 1.9 of [5] for L 2 (Ω) case. The generalization to the L p (Ω), p ≥ 2, is straightforward thanks to the moment formula (2.5). (1) Suppose that the initial measure µ satisfies (1.5). If u and u ǫ are the solutions to (1.1) starting from µ and ((µ ψ ǫ ) * G(ǫ, ·))(x), respectively, where (2) Let φ be any continuous, nonnegative and nonnegative definite function with the same initial conditionũ ǫ (0, ·) = µ as u, where Moreover, one can always find such φ so that for all i, j = 1, · · · , d. Proof.
(2) From the direct computation, the spatial covariance function for M ǫ is given by f ǫ,ǫ (x) := (φ ǫ * φ ǫ * f )(x) and f ǫ,ǫ is nonnegative and nonnegative definite and it satisfies strengthened Dalang's condition (2.7) for each ǫ > 0 (see Step 3 of Section 7 of [5]). In addition, the L 2 (Ω) convergence has been established in Theorem 1.7 of [5]. As in the proof of part (1), we need only show that for all t > 0 and p ≥ 2, where we have used the fact that g ∈ L ∞ (R d ) and γ p = 32p Lip 2 ρ , and H ǫ (t; γ p ) is defined in (2.6) with the function k(·) replaced by k ǫ (·). Because where the upper bound is uniform in both ǫ and x.
It is easy to see that φ is a continuous, nonnegative and nonnegative definite function on R d with compact support such that It is clear that θ(·) is an even and C 2 function on R with θ ′ (·) being a continuous odd function and θ ′′ (·) a continuous even function. More precisely, Because f is nonnegative definite, we see that One can also find some constant C > 0 large enough such that for all i, j ∈ {1, · · · , d}, where the right-hand side is a continuous, nonnegative, and nonnegative definite function.
Hence, for any i, j ∈ {1, · · · , d}, where the third inequality is due to the fact that the integrand is a nonnegative definite function and in the last inequality we use the fact that the integrand is a continuous function. This completes the proof of Lemma 3.1.
We also point out that the initial data ((µψ ǫ ) * G(ǫ, ·))(x) in the above theorem has Gaussian tails so that it is in L p (R d ) for any p ∈ [1, ∞]. This will be used in Step 4 of Section 3.4.
Lemma 3.2. Suppose that µ is a (possibly signed) Borel measure that satisfies (1.5). For any δ > ǫ > 0, there exists some constant C = C(ǫ, δ, µ) > 0 such that Proof. Fix δ > ǫ > 0 and denote It is clear that Ψ is a nonnegative and smooth function. Notice that Hence, Ψ(x) is also an integrable function on R d . These facts together imply that Ψ(x) has to be bounded. This proves the lemma.

Step 2 (Mollification of the Laplacian operator)
In this section, we regularize the Laplacian operator by using Yosida's approximation. Thanks to Step 1, we may assume that the initial data µ First, let us view the G(t, x) as an operator, denoted by G(t), as follows: Let I be the identity operator: If (x) := (δ * f )(x) = f (x). For any ǫ ∈ (0, 1), set where the operator R ǫ (t) has a density, denoted by R ǫ (t, x), which is equal to Because f ∈ C 2 (R d ; R + ), the stochastic integral with respect to M (ds, dy) is equivalent to the stochastic integral with respect to M y (ds)dy, where {M x (t), t ≥ 0, x ∈ R d } are Brownian motions starting from zero indexed by x ∈ R d with the following correlation structure . Consider the following stochastic differential equation Since ρ is Lipschitz continuous and ∆ ǫ is a bounded operator, (3.15) has a unique strong solution We will need the following lemma regarding the spatial regularity of u ǫ (t, x). Lemma 3.3. Let u ǫ be a solution to (3.15). If the initial data u 0 ∈ S(R d ), and if the correlation function f in (3.14) is in ∈ C 2 (R d ; R + ) with f ′ (0) ≡ 0 and f ′′ (·) being bounded, then for any ǫ > 0, T > 0, and p ≥ 2, there is a constant C = C(T, p, ǫ, µ, Lip ρ ) > 0 such that for all t ∈ [0, T ] and x, y ∈ R d with |x − y| ≤ K.
Proof. Fix p ≥ 2, T > 0, and ǫ > 0. Let C be a generic constant that may depend on these constants, namely, T , p, ǫ, and Lip ρ . For any t ∈ [0, T ] and x, y ∈ R d , we have that It is clear that The boundedness and regularity of the initial data implies that Hence, we have that Notice that Lemma 3.1 of [5] with α = 1 implies that Therefore, where we note that the constants C depend on ǫ.
As for I 3 , we see that As for I 4 , by (3.14) and the Burkholder-Davis-Gundy inequality, we see that As for I 5 , by the Burkholder-Davis-Gundy inequality, (3.14), and (3.18), we see that Combining these five terms, we see that Finally, an application of Gronwall's lemma proves Lemma 3.3. where u(t, x) is the solution to the same equation (3.15) but with ∆ ǫ replaced by the standard Laplacian operator ∆.
Proof. The proof of this lemma is very similar to the proof of Lemma 3.1. First of all, (7.9) in Step 2 of Section 7 in [5] shows that for any t > 0, Note that in Step 2 of Section 7 of [5], one mollifies the Laplacian, initial data and the noise at the same time. Here, we do that in separate steps. The arguments in this slightly simplified case can be carried out line-by-line. We leave the details for the interested readers. Then, the boundedness of initial data implies that which basically implies (3.20)

Step 3 (Discretization in space)
For δ ∈ (0, 1) and where [x] is the function that rounds half away from zero, e.g., .
For ǫ and δ ∈ (0, 1), and i, j ∈ Z d , let where G d (t, x) is the heat kernel on R d (see (1.4)) and when there is no confusion from the context, we will simply write it as G(t, x). Now we consider the following infinite dimensional SDE: It has a strong solution We first note that (3.22) is a discretization of (3.16): If we replace x in (3.16) by [x] δ and set i = [x/δ], we see that the first and third terms on the r.h.s. of (3.16) becomes the first and third terms on the r.h.s. of (3.22), respectively. The r.h.s. of (3.17) becomes which is equal to the second term on the r.h.s. of (3.22) (then one may put a superscript δ in u ǫ to denote the step size of this discretization). Note that {M ǫ iδ (t), t ≥ 0} i∈Z is a sequence of correlated Brownian motions starting from zero with see (3.14).
The main result of this step is the following lemma: Lemma 3.5. Suppose that the initial data µ is bounded, i.e., µ(dx) = g(x)dx with g ∈ L ∞ (R d ).
Let u ǫ (t, x) be the strong solution (3.16) to (3.15) and for any ǫ, δ ∈ (0, 1), let u δ ǫ (t, [x] δ ) be the system of stochastic differential equations given in (3.21). Then for any t > 0, x ∈ R d and p ≥ 2, it holds that Proof. Fix arbitrary p ≥ 2, t > 0 and x ∈ R d . Notice that For I ǫ,δ 1 , Lemma 3.3 shows that By Lemma 3.3 again (see also (3.24)), For A ǫ,δ 2 and A ǫ,δ 3 , by Minkowski's inequality, we see that For A ǫ,δ 4 , by the same argument as (3.19), we see that Combining these terms we see that ds.
An application of Gronwall's lemma implies that

3.4
Step 4 (Proof of Theorems 1.5, 1.6 and Corollary 1.7) We now combine Steps 1-3 above to construct solutions to the infinite dimensional SDEs (3.21) which converge in L p (Ω) to the unique solution u(t, x) to SHE (1.1) with rough initial data and driven by Gaussian noise whose spatial spectral measure only satisfies Dalang's condition (1.6). Let us fix arbitrary t > 0, x ∈ R d and p ≥ 2.
Step 2 now implies that there exists a strong solution u ǫ 1 ,ǫ ′ 1 ,ǫ 2 (t, x) to (3.15) such that Step 3 shows that there exists a solution u δ Now it is easy to check that (3.21) is of the form (1.21), i.e., Assumption 1.8 is satisfied. In particular, part (iv) of Assumption 1.8 is satisfied thanks to Lemma 3.2. Thus, an application of Theorems 1.11 and 1.12 completes the proof of Theorems 1.5 and 1.6. Note that the two cases, namely, the multiple-time comparison over F[C 2,v p,+ ] or F[C 2,v b,− ] and the single-time comparison over F[C 2,v p ], are treated separately in the proofs of Theorems 1.11 and 1.12 below.
It remains to prove Corollary 1.7. Under condition (i), there exists some ǫ 0 > 0 such that , which, together with the fact that f 1 − f 2 is a nonnegative measure, imply that for all ǫ ∈ (0, ǫ 0 ], f ǫ,ǫ is defined in Lemma 3.1. Thus, the result is a consequence of the approximation procedure and Corollary 1.13. Under condition (ii), since f ℓ ∈ C 2 b (R d ; R + ), ℓ = 1, 2, f ℓ have to satisfy properties in (3.9). Hence, in Step 1, we do not need to mollify the noise, or equivalently, we could set ǫ 1 = 0. We keep the approximations. Then one can apply Corollary 1.13 to conclude this case. This proves Corollary 1.13.

Stochastic comparison principles for interacting diffusions
In this section, we will study the interacting diffusion equations (1.21) and prove Theorems 1.10, 1.11, 1.12, 1.15, and Corollary 1.13.

Existence and uniqueness (Proof of Theorem 1.10)
Proof of Theorem 1.10. To show the existence of a solution, we use the standard Picard iteration. For n = 0, set U (0) (t, i) := u 0 (i) and for any n ≥ 1, define recursively (4.1) Choose and fix an arbitrary integer k ≥ 2. Without loss of generality, we may assume k is an even integer. We first show that all U (n) (t, ·)'s are in ℓ k (K) almost surely for any t ≥ 0. For any random field Z(t, i), define Note that N 1/k β,k (Z) is a norm on the random field. Then by the Minkowski inequality, We will compute the three N 1/k β,k (·) norms in the right-hand side of (4.2). It is clear that As for N β,k (I n ), because k is even, we have that By the inequality k i=1 a i ≤ (a k 1 + · · · + a k k )/k applied to the product over ℓ ′ , we see that where we have used the assumption (1.22) and the fact that j∈K p i,j = 1. Therefore, Now let us consider N β,k (R n ). By the Burkholder-Davis-Gundy inequality, we have that where c k is some universal constant and we have used the fact that k is an even integer. By the same arguments as above, Thus, which implies that Putting (4.3), (4.4) and (4.5) back to (4.2) shows that It is clear that β → C k (β) is a strictly decreasing function for all β ≥ 0. Therefore, by choosing β * to be the unique positive solution to the equation C k (β) = 1/2, we have that Therefore, we have that This implies that U (n) (t, ·) for all n ≥ 1 is well-defined and in L ∞ [0, T ]; L k (Ω; ℓ k (K)) .
Since ρ is globally Lipschitz with Lipschitz constant Lip ρ , following the same process as above, we can have n∈N is a Cauchy sequence in the Banach space with the norm N 1/k β * ,k (·). As a consequence, In addition, using the convergence from U (n) to U in L ∞ [0, T ]; L 2 (Ω; ℓ 2 (K)) , it is easy to see that U satisfies (1.24). The proof of uniqueness follows from a standard argument. We will not repeat here. This completes the proof of Theorem 1.10. Proof. For simplicity, we will only prove the symmetric case, i.e., p i,j = p j,i , in which case, Λ ≡ 1. We first remark that the constant c k in the Burkholder-Davis-Gundy inequality can be chosen to be c k = 2 k k k/2 (see, e.g., [4]). In order to solve the equation C k (β) = 1/2, set x = β −k/2 . Then equivalently, x solves (6κk) k x 2 + 18γ(0)k 2 Lip 2 ρ k/2 x − 3/2 = 0.
By finding the positive solution and then taking the power of −2/k, we see that where we have applied twice the subadditivity property of the function R + ∋ x → x α for α ∈ (0, 1]. Therefore, one can find a constant C depending on κ, γ(0) and Lip ρ such that β * ≤ Ck 2 for all k ≥ 2.

Several approximations
In this subsection, we will reduce the SDE (1.21) to the case in Theorem 1.15. We will need to approximate the solution to (1.21) in the following two cases: The first case is to approximate (1.21) by that with a C 2 c (R + ) diffusion coefficient. This is covered in two steps through Propositions 4.2 and 4.3 below. The second case is to approximate (1.21) by a finite dimensional SDE and this is covered by Proposition 4.4 below.
Since ρ is a globally Lipschitz continuous function with the Lipschitz constant Lip ρ and ρ(0) = 0, it is easy to see that ρ N is also globally Lipschitz such that (4.13) The existence and uniqueness of a strong solution in the space (1.25) follows from Theorem 1.10.
Proof. Let T ≥ t ≥ 0 and fix k ≥ 2. Without loss of generality, we may assume that k is an By Itô's formula By the following Young's inequality for product for all a, b ≥ 0 and k ≥ 2, (4. 15) we see that where we have used the assumption (1.22) and the fact that j∈K p i,j = 1. By (4.12), By Young's inequality (4.15) with k/2, we see that Hence, . Therefore, where the two constants can be chosen as follows: By setting W N (t) := sup s∈[0,t] E ||V N (s, ·)|| k ℓ k (K) , we see that E ρ(U (s ′ , ·)) − ρ N (U (s ′ , ·)) k ℓ k (K) ds. (4.16) Because |ρ N (x)| ≤ Lip ρ |x| for all x ∈ R, the moment bound (1.26) implies that On the other hand, the above inequality shows that for all t ∈ [0, T ]. Hence, the second term on the right-hand side of (4.16) converges to zero as N → ∞ by the dominated convergence theorem. Moreover, as a function of t, this term is in L 1 ([0, T ]). Therefore, an application of Gronwall's lemma completes the proof.
Thanks to Proposition 4.2, we can now assume that ρ is a function with compact support.
Then, it is easy to see that ρ ǫ ∈ C ∞ c and ρ ǫ is globally Lipschitz with the same Lipschitz constant Lip ρ as for ρ. Consider  , respectively, with the same initial data u 0 (·) ∈ ℓ 2 (K) and with ρ being a continuous function with compact support. Then, for any T > 0 and k ≥ 2, it holds that Proof. Since ρ is a continuous function with compact support, we have that ||ρ ǫ || L ∞ (R) ≤ ||ρ|| L ∞ (R) < ∞. On the other hand, since both ρ and ρ ǫ are continuous functions with compact support, ρ ǫ converges to ρ uniformly on any compact set, i.e., Hence, the bounded convergence theorem implies that as ǫ → 0 + . Therefore, one can follow the same arguments as those in proposition 4.2 to complete the proof.
Case 2. It remains to show the approximation by a finite-dimensional SDE when K has countably infinite many elements. Let K i be subsets of K with finite cardinalities such that K 1 ⊂ K 2 ⊂ · · · ↑ K. Consider the following finite system of interacting diffusions: The existence and uniqueness of a strong solution to (4.19) is a standard result. Indeed, one may also follow the proof of Theorem 1.10 to show the existence of a unique strong solution.  21) and (4.19), respectively, with the same diffusion coefficient ρ which is assumed to be globally Lipschitz continuous. Then, for any T > 0 and k ≥ 2, it holds that as m → +∞.
Proof. Let T ≥ t ≥ 0 and fix k ≥ 2. Without loss of generality, we assume that k is an even dU (t, i) = the r.h.s. of the first equation in (1.21), otherwise.
By Itô's formula, we see that, for any i ∈ K m , Theorem 1.10 says that U (t, ·) ∈ ℓ 2 (K) ⊆ ℓ k (K) a.s. for all t ≥ 0, which implies that j∈K\Km U (t, j) k → 0 as m → ∞ a.s. Therefore, thanks to (1.26), the monotone convergence theorem implies that (4.21) In addition, since u 0 (·) ∈ ℓ 2 (K) ⊆ ℓ k (K), we can get As for ||V m (t, ·)|| k ℓ k (Km) , by Young's inequality (4.15), we have that k i∈Km j∈Km , where we have used the assumption (1.22) and the fact that j∈K p i,j = 1. Similarly, by (4.15), where, by Hölder inequality and the fact that j∈K\Km p i,j ≤ 1, Now combine things together and use the fact that ρ is globally Lipschitz to see that Thanks to (4.21) and (4.22), an application of Gronwall's lemma to proves the proposition.

Comparison theorems for finite interacting diffusions
In this subsection, we will prove Theorems 1.15 and 1.16. Before the proof, we first make a remark to comment the difference of our results with those in Cox, Fleischmann and Greven [10].   [10]; see (1.18) for relations of these function spaces. In particular, the multiple-time comparison result in Theorem 1.11 works well for the moment functions F M . However, in order to apply the same comparison results in [10] to the moment functions, one needs to restrict the moment functions to bounded subinterval I ⊂ R + , i.e., one needs to replace R K + in the definition (1.16) by I K ; see Example 6 of [10]. We can make this extension thanks to our stronger approximation results in Section 4. − ], our results are slightly more general, even thought this improvement is not essential because each component of the the diffusion process will live in the compact support of ρ. The major difference here (and also for infinite dimensional SDE case) is that in [10], only the case of independent Brownian motions was studied. So we need to change the infinitesimal generator from to (4.26) below. This change won't bring any new difficulties. The original proof in [10] works line by line.
Although Theorem 1.15 can be proved in the same way as those in Sections 2.1-2.4 of [10] with only minor changes as is explained in part (b) of the above remark, considering that Theorem 1.16 is new, we will streamline the proof of both results altogether. This will also serve as an alternative presentation of the proofs in [10].
Proof of Theorems 1.15 and 1.16. Let the index set K be {1, · · · , d}. Under both Assumptions 1.8 and 1.14, we have a finite dimensional SDE with ρ ∈ C 2 c (R + ). Hence, it is well-known that there exists a unique strong solution U (t, ·) ∈ R d . For ℓ ∈ {1, 2}, let U ℓ be the unique strong solution either corresponding to ρ ℓ in case of Theorem 1.11 or to γ ℓ in case of Theorem 1.12. In the following, we will slightly abuse the notation for the expectation. We may put subscript to denote the initial data and where there is no subscript, the initial data is u 0 (·). Now we need to prove the following two statements: 1. For any integer m ≥ 1, 0 < t 1 < · · · < t m < ∞, and , then for any t ≥ 0, (4.25) Step 1. We start by proving (4.25) for F ∈ F {1,··· ,d} [C 2,v ], which will cover both (4.25) and (4.24) when all t ℓ are the same. For ℓ ∈ {1, 2}, let G ℓ be the infinitesimal generator for u ℓ (t, ·) ∈ R d , that is,  be the corresponding semigroup, namely, Then, (4.25) is equivalent to showing By the integration by parts formula, preserves positivity, i.e., T (ℓ) t g ≥ 0 whenever g ≥ 0. It is also known (see, e.g., Theorem 5.6.1 in [15] that under our assumption on ρ ℓ or ρ, and T (ℓ) t F is continuous in t. Our assumptions on ρ's and γ's assure that for all i, j ∈ {1, · · · , d} and all Hence, we only need to show that D i D j T (2) t F (z) ≥ 0 for all 1 ≤ i, j ≤ d, t > 0 and z ∈ R d + . For simplicity, we define U (t, ·) := U (2) (t, ·), ρ := ρ 2 , G := G (2) and T t := T (2) t , and show that (4.30) Step 2. In this step, we will use Trotter's product formula (see, e.g., Corollary 1.6.7 of [12]) to prove (4.30). Let T (κ,ρ) t denote this semigroup of the d-dimensional diffusion process in (1.21) with drift parameter κ and diffusion coefficient ρ. Trotter's product formula suggests to study the limit of the semigroup T (κ,0) We first study the semigroup T Although U (t, i) and U (t, j) are not independent, they interact only through the random environment M i (t) when i = j. Hence, in (4.31) each component U (t, i) of (U (t, 1), . . . , U (t, d)) has its own equation. Following Cox et al [10], for i, j ∈ {1, · · · , d}, and h 1 , h 2 > 0, denote where e i is the ith unit vector in R d and z ∈ R d + . To avoid triviality, we assume that z ∈ supp(ρ) d . When i = j, (u 0 , u 1 , u 2 , u 12 ) forms a rectangle in the (i, j)-th directions; when i = j, it forms nondecreasing sequence in the i-th direction: u 0 ≤ u 1 ∧ u 2 ≤ u 1 ∨ u 2 ≤ u 12 (here, the inequality u ≥ v for u, v ∈ R d means that each component u i ≤ v i for all 1 ≤ i ≤ d). Let U 0 , U 1 , U 2 , U 12 be the solutions to (4.31) with the initial condition u 0 , u 1 , u 2 , u 12 , respectively, when i = j and with u 0 , u 1 ∧u 2 , u 1 ∨u 2 , u 12 , respectively, when i = j. By the classical comparison principle for the one-dimensional SDEs (see e.g., either [18, Theorem VI.1.1] or [28, Theorem IX.3.7]), we have that with probability one, for all t ≥ 0, (in case of i = j) we see that Notice that the expectation of the left-hand side of (4.33) is finite because ρ has compact support and max i=1,··· ,d Hence, we can take expectation on both sides of (4.33) to see that which, in view of (4.32), is nothing but (4.30) for T (0,ρ) t . Therefore, we have proved (4.30) for the case of F ∈ F[C 2,v ] and no drift (κ = 0). In other words, T Next, we study the semigroup T (κ,0) t , i.e., the case when κ > 0 but ρ ≡ 0 in (1.21). In this case, the system is deterministic: If we view U (t, ·) and the initial data u 0 (·) as column vectors in R d and set A = (p i,j −δ i,j ) 1≤i,j≤d , then we have that U (t, ·) = exp(κAt)u 0 (·). Hence, for F ∈ F[C 2,v ] and z ∈ R d + (viewed as a column vector), T and hence, with F k,m (z) = D k D m F (z), which proves (4.30) for this case. Therefore, T (κ,0) t also preserves the function cone F[C 2,v ]. Now we can apply Trotter's product formula with C 2 (R d + ) as the core to see that where U (t, ·) is the unique solution to (1.21) and U ǫ (t, ·) with ǫ fixed solves a finite-dimensional SDE with C 2 c (R + )-diffusion coefficient.
Case I. We first consider the one-time comparison results over , one can find m ∈ N \ {0} and distinct i 1 , · · · , i m ∈ K such that, by the mean-value theorem, we have that where c ∈ [0, 1]. For any β ≥ 1, by Cauchy-Schwartz inequality, we see that By the growth condition (1.13), there are some constants C > 0 and k ∈ N such that Hence, Thanks to (4.37), sup On the other hand, Hence, F (U ǫ (t, ·)) converges to F (U (t, ·)) in L β (Ω) for all β ≥ 1. The comparison results will be carried thought the limit.
We now prove Corollary 1.13.
Proof of Corollary 1.13. We first consider the case under Assumption 1.14 (i.e., finite dimensional SDEs and K := {1, . . . , d}). Let where φ ǫ,k (z k ) ∈ C 2 (R) are non-increasing and non-negative functions such that φ ǫ,k (z k ) converges to 1 (−∞,a k ] (z k ) as ǫ goes to 0 for each z k ∈ R. It is easy to see that F ǫ is uniformly bounded by some constant, in C 2 (R d ) and D i D j F ǫ ≥ 0 for i = j. On the other hand, the assumption that γ 1 (0) = γ 2 (0) enables us to get where G (i) is the infinitesimal generator of U (i) as in the proof of Theorem 1.15. Hence, following the proof of Theorem 1.15, we get EF ǫ (U 1 (t, 1), . . . , U 1 (t, d)) ≥ EF ǫ (U 2 (t, 1), . . . , U 2 (t, d)) .
Therefore, thanks to the bounded convergence theorem, as ǫ goes to 0, we get

Some examples and one application
In all examples below, we always work either under the settings of Theorem 1.5 or under those of Theorem 1.6, and use u ℓ (t, x), ℓ = 1, 2, to denote corresponding solutions to (1.1).
Finally, let us give one application of approximation results proved in this paper. Here we can give a straightforward proof of the weak sample path comparison principle, which was proved in [7,24] (for one dimensional case) and in [5] for (d-dimensional case).
Theorem 5.5 (Weak sample path comparison principle). Assume that f satisfies Dalang's condition (1.6) and the diffusion coefficient ρ is globally Lipschitz continuous, which is not necessary to vanish at zero. Let u 1 and u 2 be two solutions to (1.1) with the initial measures µ 1 and µ 2 that satisfy (1.5), respectively. If µ 1 ≤ µ 2 , then P (u 1 (t, x) ≤ u 2 (t, x)) = 1 , for all t ≥ 0 and x ∈ R d . (

5.2)
Sketch of the proof. Set v = u 1 − u 2 . Then v satisfies a SHE similar to (1.1) withρ that satisfies ρ(0) = 0. It suffices to show that v(t, x) ≥ 0 a.s. for all (t, x) fixed. As is shown in Section 3.4, one can find v δ On the other hand, v δ ǫ 1 ,ǫ ′