Both Wick-Ito-Skorokhod and Stratonovich interpretations of the parabolic
Anderson model (PAM) lead to solutions that are real analytic as functions of
the noise intensity e, and, in the limit e->0, the difference between the two
solutions is of order e^2 and is non-random.Comment: 12 page

Kim, Hyun-Jung

Lototsky, Sergey V.

English

arXiv

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An Asymptotic Comparison of Two Time-Homogeneous PAM Models

Lototsky, Sergey Vladimir

LSU Scholarly Repository (Louisiana State Univ.)

Communications on Stochastic Analysis Volume 12 Number 2 Article 6 10-2018 An Asymptotic Comparison of Two Time-Homogeneous PAM Models Hyun-Jung Kim University of Southern California, Los Angeles, California USA, hyunjungmath@gmail.com Sergey Vladimir Lototsky University of Southern California, Los Angeles, California USA, lototsky@math.usc.edu Follow this and additional works at: https://repository.lsu.edu/cosa  Part of the Analysis Commons, and the Other Mathematics Commons Recommended Citation Kim, Hyun-Jung and Lototsky, Sergey Vladimir (2018) "An Asymptotic Comparison of Two Time-Homogeneous PAM Models," Communications on Stochastic Analysis: Vol. 12: No. 2, Article 6. DOI: 10.31390/cosa.12.2.06 Available at: https://repository.lsu.edu/cosa/vol12/iss2/6 AN ASYMPTOTIC COMPARISON OF TWOTIME-HOMOGENEOUS PAM MODELSHYUN-JUNG KIM* AND SERGEY VLADIMIR LOTOTSKYAbstract. Both Wick-Itô-Skorokhod and Stratonovich interpretations ofthe Parabolic Anderson model (PAM) lead to solutions that are real analyticas functions of the noise intensity ε, and, in the limit ε → 0, the differencebetween the two solutions is of order ε2 and is non-random.1. IntroductionLet W = W (x), x ∈ [0, π] be a standard Brownian motion on a complete prob-ability space (Ω,F ,P). With no loss of generality, we assume that all realizationsof W are in C1/2−((0, π)), that is, Hölder continuous of every order less than 1/2.Consider the equations∂u⋄(t, x; ε)∂t=∂2u⋄(t, x; ε)∂x2+ εu⋄(t, x; ε) ⋄ Ẇ (x), t > 0, 0 < x < π,u⋄(t, 0; ε) = u⋄(t, π; ε) = 0, u⋄(0, x; ε) = φ(x),(1.1)and∂u◦(t, x; ε)∂t=∂2u◦(t, x; ε)∂x2+ εu◦(t, x; ε) ◦ Ẇ (x), t > 0, 0 < x < π,u◦(t, 0; ε) = u◦(t, π; ε) = 0, u◦(0, x; ε) = φ(x).(1.2)Equation (1.1) is the Wick-Itô-Skorokhod formulation of the parabolic Andersonmodel with potential εẆ ; equation (1.2) is the corresponding Stratonovich (orgeometric rough path) formulation. These equations, with ε = 1, are studied in[1] and [2], respectively.The objective of the paper is to show that• The solutions of (1.1) and (1.2) are real-analytic functions of ε: withsuitable functions u(n)⋄ , and u(n)◦ , the equalitiesu⋄(t, x; ε) = u⋄(t, x; 0) +∞∑n=1εnu(n)⋄ (t, x) (1.3)u◦(t, x; ε) = u◦(t, x; 0) +∞∑n=1εnu(n)◦ (t, x) (1.4)Received 2018-8-4; Communicated by the editors.2010 Mathematics Subject Classification. Primary 60H15; Secondary 35R60, 60H40.Key words and phrases. Stratonovich integral, Wick-Itô-Skorokhod integral.* Corresponding author.185           Serials Publications                  www.serialspublications.com Communications on Stochastic Analysis Vol. 12, No. 2 (2018) 185-196186 HYUN-JUNG KIM AND SERGEY VLADIMIR LOTOTSKYhold for all t > 0, x ∈ [0, π], ε > 0, and every realization of W ;• The first two terms in (1.3) and (1.4) are the same so that|u⋄(t, x; ε)− u◦(t, x; ε)| = O(ε2), ε → 0, (1.5)for all t > 0 and x ∈ [0, π], and every realization of W .Equalities (1.3) and (1.4) are in the spirit of [5]. Equality (1.5) is similar to [9,Proposition 4.1]; see also [8].The precise statement of the main result is in Section 2, and the proof is inSections 3, 4, and 5.2. The Main ResultDenote by p = p(t, x, y) the heat semigroup on [0, π] with zero boundary con-ditions:p(t, x, y) =2π∞∑k=1e−k2t sin(kx) sin(ky), t > 0, x, y ∈ [0, π]. (2.1)Let φ = φ(x) be a continuous function on [0, π], and let u = u(t, x) be the solutionof the heat equation∂u(t, x)∂t=∂2u(t, x)∂x2, t > 0, 0 < x < π,u(t, 0) = u(t, π) = 0, u(0, x) = φ(x),(2.2)that is,u(t, x) =∫ π0p(t, x, y)φ(y)dy. (2.3)Next, define the function u = u(t, x) byu(t, x) =∫ t0∫ π0p(t− s, x, y)u(s, y) dW (y) ds. (2.4)That is, u is the mild solution of∂u(t, x)∂t=∂2u(t, x)∂x2+ u(t, x)Ẇ (x), t > 0, 0 < x < π,u(t, 0) = u(t, π) = u(0, x) = 0.(2.5)Because u is non-random, no stochastic integral is required to define u.Proposition 2.1. If φ ∈ C((0, π)), then u is a continuous function of t and x forall t > 0 and x ∈ [0, π].Proof. This follows by the Kolmogorov continuity criterion: u is a Gaussian ran-dom field and direct computations showE(u(t+ τ, x+ h)− u(t, x))2 ≤ C(t)(τ2 + h2)1/4 maxx∈[0,π]|φ(x)|;cf. [1, Sections 6 and 7]. □STATIONARY PAM 187Next, define the functions u(n)⋄ = u(n)⋄ (t, x), n = 0, 1, 2 . . ., t ≥ 0, x ∈ [0, π], byu(0)⋄ (t, x) = u(t, x), and, for n ≥ 1, u(n)⋄ is the mild solution of∂u(n)⋄ (t, x)∂t=∂2u(n)⋄ (t, x)∂x2+ u(n−1)⋄ (t, x) ⋄ Ẇ (x), t > 0, 0 < x < π,u(n)⋄ (t, 0) = u(n)⋄ (t, π) = u(n)⋄ (0, x) = 0.(2.6)In other words,u(n)⋄ (t, x) =∫ t0∫ π0p(t− s, x, y)u(n−1)⋄ (s, y) ⋄ dW (y) ds, n ≥ 1, (2.7)and, in particular, u(1)⋄ = u.Similarly, define the functions u(n)◦ = u(n)◦ (t, x), n = 0, 1, 2 . . ., t ≥ 0, x ∈ [0, π],by u(0)◦ (t, x) = u(t, x), and, for n ≥ 1, u(n)◦ is the mild solution of∂u(n)◦ (t, x)∂t=∂2u(n)◦ (t, x)∂x2+ u(n−1)◦ (t, x) ◦ Ẇ (x), t > 0, 0 < x < π,u(n)◦ (t, 0) = u(n)◦ (t, π) = u(n)◦ (0, x) = 0.(2.8)In other words,u(n)◦ (t, x) =∫ t0∫ π0p(t− s, x, y)u(n−1)◦ (s, y) ◦ dW (y) ds, n ≥ 1, (2.9)and, in particular, u(1)◦ = u.The main result of the paper can now be stated as follows.Theorem 2.2. Let φ ∈ C((0, π)). Then(1) Equality (1.3) holds with u(n)⋄ from (2.7).(2) Equality (1.4) holds with u(n)◦ from (2.9).(3) Equality (1.5) holds andlimε→0ε−2(u◦(t, x; ε)− u⋄(t, x; ε))=∫ π0p(3)(t, x, z)φ(z)dz, (2.10)wherep(3)(t, x, z) =∫ π0∫ t0∫ s0p(t− s, x, y)p(s− r, y, y)p(r, y, z) dr ds dy.The proof is carried out in the following three sections.3. The Wick-Itô-Skorokhod CaseThe objective of this section is the proof of (1.3).The solution of (1.1) is defined as a chaos solution (cf. [6, Theorems 3.10]). It isa continuous in (t, x) function (cf. [1, Sections 6 and 7]) and has a representationas a seriesu⋄(t, x; ε) =∑α∈Juα(t, x; ε)ξα, (3.1)188 HYUN-JUNG KIM AND SERGEY VLADIMIR LOTOTSKYwhereJ ={α = (αk, k ≥ 1) : αk ∈ {0, 1, 2, . . .},∑kαk < ∞},ξα =∏k(Hαk(ξk)√αk!), Hn(x) = (−1)nex2/2 dndxne−x2/2,ξk =∫ π0mk(x) dW (x), mk(x) =√2/π sin(kx),and, with |α| =∑k αk, |(0)| = 0, |ϵ(k)| = 1,u(0)(t, x; ε) = u(t, x),uα(t, x; ε) = ε∑k√αkt∫0π∫0p(t− s, x, y)uα−ϵ(k)(s, y; ε)mk(y) dyds;see [1, Section 3] for details. In particular,∑|α|=nuα(t, x; 1)ξα =∑|α|=n−1∫ t0∫ π0p(t− s, x, y)uα(s, y; 1)ξα ⋄ dW (y) ds. (3.2)Comparing (2.7) with (3.2) shows thatu(n)⋄ (t, x) =∑|α|=nuα(t, x; 1)ξα. (3.3)In other words, (1.3) is equivalent to (3.1).Next,E|u(n)⋄ (t, x)| = E∣∣∣∣∣∣∑|α|=nuα(t, x; 1)ξα∣∣∣∣∣∣ ≤E ∑|α|=nuα(t, x; 1)ξα21/2= ∑|α|=n|uα(t, x; 1)|21/2 ≤ Cn(t)n−n/4 supx∈(0,π)|φ(x)|1/2,where the last inequality follows by [1, Theorem 4.1]. As a result,∑n≥0εnE∣∣u(n)⋄ (t, x)∣∣ < ∞,that is, the series converges absolutely with probability one for all t > 0, x ∈ [0, π],and ε ∈ R.This concludes the proof of (1.3).STATIONARY PAM 1894. The Stratonovich CaseThe objective of this section is to prove (1.4). To simplify the presentation, weuse the following notations:Λ = (−∆)1/2, Hθ = Λ−θ(L2((0, π)), ∥ · ∥θ = ∥Λθ · ∥L2((0,π)), θ ∈ R,p ∗ g(t, s, x) =∫ π0p(t− s, x, y)g(s, y) dy,where ∆ is the Laplace operator on (0, π) with zero boundary conditions and p isthe heat kernel (2.1).By direct computation,∥p ∗ g∥γ(t, s) ≤ CT,θ (t− s)−θ/2∥g∥γ−θ(s), θ > 0, γ ∈ R, t ∈ (s, T ]; (4.1)cf. [3, Lemma 7.3].Consider the equation∂v(t, x)∂t=∂2v(t, x)∂x2+ v(t, x) ◦ Ẇ (x) + f(t, x) ◦ Ẇ (x), t > 0, (4.2)which includes (1.2) as a particular case. By definition, a solution (classical, mild,generalized, etc.) of (4.2) is a suitable limit, as ϵ → 0, of the correspondingsolutions of∂vϵ(t, x)∂t=∂2vϵ(t, x)∂x2+ vϵ(t, x)Vϵ(x) + f(t, x)Vϵ(x), t > 0, (4.3)where Vϵ, ϵ > 0 are smooth functions on [0, π] such thatsupϵ∥∥∥∥∫ Vϵ∥∥∥∥C1/2< ∞, limϵ→0supx∈[0,π]∣∣∣∣∫ x0Vϵ(y)dy −W (x)∣∣∣∣ = 0.By [2, Theorem 3.5],• The generalized solution of (4.2) is the same as the generalized solution ofthe equationvt =(vx +W (x)v +W (x)f)x−W (x)vx −W (x)fx; (4.4)the subscripts t and x, as in fx, denote the corresponding partial deriva-tives;• The mild solution of (4.2) is the solution of the integral equationv(t, x) =∫ t0p ∗((f + v)W)x(t, s, x) ds−∫ t0p ∗((f + v)xW)(t, s, x) ds+∫ π0p(t, x, y)φ(y)dy.(4.5)On the one hand, mild and generalized solutions of (4.2) are the same: just usemk as the test functions. On the other hand, different definitions of the solutionlead to different regularity results.190 HYUN-JUNG KIM AND SERGEY VLADIMIR LOTOTSKYBy standard parabolic regularity, if φ ∈ H0 and f ∈ L2((0, T );Hγ), γ ∈(1/2, 1], then there is a unique generalized solution of (4.4) in the normal triple(H1,H0,H−1) andv ∈ L2((0, T );H1)∩C((0, T );H0)(4.6)for every realization of W ; cf. [4, Theorem 3.4.1]. Note that we cannot claimv ∈ C((0, T );Hγ)even if φ ∈ Hγ . In fact, because W ∈ C1/2− is a point-wise multiplier in Hγ for γ ∈ (−1/2, 1/2) [3, Lemma 5.2], an attempt to find atraditional regularity result for equation (4.4) in a normal triple (Hr+1,Hr,Hr−1)leads to an irreconcilable pair of restrictions on r: to have Wf ∈ L2((0, T );Hr)we need r < 1/2, whereas to have Wfx ∈ L2((0, T );Hr−1) we need r − 1 > −1/2or r > 1/2.Accordingly, to derive a bound on ∥v∥γ(t) for t > 0, we use the mild formulation(4.5).Proposition 4.1. Let γ ∈ (1/2, 1), f ∈ L2((0, T );Hγ), φ ∈ H0, and let v be themild solution of (4.2) with v|t=0 = φ. Then, for every T > 0 and every realizationof W , there exists a number C◦ such that∥v∥γ(t) ≤ C◦(t−γ/2∥φ∥0 +∫ t0(t− s)−γ∥f∥γ(s) ds). (4.7)Proof. Throughout the proof, C denotes a number depending on γ, T , and thenorm of W in the space C1−γ . The value of C can change from one instance toanother. With no loss of generality, we assume that φ and f are smooth functionswith compact support.To begin, let us show that if V is the mild solution of∂V (t, x)∂t=∂2V (t, x)∂x2+ f(t, x) ◦ Ẇ (x), t > 0,V (t, 0) = V (t, π) = 0, V |t=0 = φ, then∥V ∥γ(t) ≤ Ct−γ/2∥φ∥0 + C∫ t0(t− s)−γ∥f∥γ(s) ds. (4.8)Indeed, by (4.5),V (t, x) =∫ t0p ∗ (fW )x(t, s, x) ds−∫ t0p ∗(fxW)(t, s, x) ds+∫ π0p(t, x, y)φ(y)dy.Using (4.1) with θ = γ,∥∥∥∥∫ π0p(t, ·, y)φ(y)dy∥∥∥∥γ≤ Ct−γ/2∥φ∥0.Then∥V ∥γ(t) ≤∫ t0∥p∗(fW )x∥γ(t, s) ds+∫ t0∥p∗(fxW)∥γ(t, s) ds+Ct−γ/2∥φ∥0. (4.9)To estimate the first term on the right hand side of (4.9), we use (4.1) withθ = 2γ. Then∥p ∗ (fW )x∥γ(t, s) ≤ C(t− s)−γ∥(fW )x∥−γ(s) ≤ C(t− s)−γ∥fW∥1−γ(s),STATIONARY PAM 191and, because W ∈ C1/2−((0, π)) is a (point-wise) multiplier in H1−γ ,∥fW∥1−γ(s) ≤ CW ∥f∥1−γ(s);recall that 0 < 1− γ < 1/2. Finally, as 1− γ < γ,∥p ∗ (fW )x∥γ(t, s) ≤ C(t− s)−γ∥f∥γ(s). (4.10)To estimate the second term on the right hand side of (4.9), we use (4.1) withθ = 1. Then∥p ∗(fxW )∥γ(t, s) ≤C√t− s∥fxW∥γ−1(s) ≤C√t− s∥fx∥γ−1(s),that is,∥p ∗(fxW)∥γ(t, s) ≤ C(t− s)−1/2∥f∥γ(s). (4.11)To establish (4.8), we now combine (4.9), (4.10), and (4.11), keeping in mind that(t− s)−1/2 ≤ C(t− s)−γ because γ > 1/2.Next, (4.8) applied to (4.2) implies∥v∥γ(t) ≤ C∫ t0(t− s)−γ∥v∥γ(s) ds+ C∫ t0(t− s)−γ∥f∥γ(s) ds+ Ct−γ/2∥φ∥0,and then a generalization of the Gronwall inequality (e.g. [10, Corollary 2]) leadsto (4.7). □Corollary 4.2. If φ ∈ H0 and γ ∈ (1/2, 1), then, for every T > 0, a > 0, andevery realization of W , there exists a number C̃◦ such that the mild solution of(1.2) satisfiessup|ε|<a∥u◦(t, ·, ε)∥γ ≤ C̃◦ t−γ/2∥φ∥0, t ∈ (0, T ]. (4.12)Next, define the functions u(n),ε◦ = u(n),ε◦ (t, x), n = 0, 1, 2 . . . , t ≥ 0, x ∈ [0, 1],ε ∈ R, by u(0),ε◦ (t, x) = u◦(t, x; ε) and, for n ≥ 1,∂u(n),ε◦ (t, x)∂t=∂2u(n),ε◦ (t, x)∂x2+ εu(n),ε◦ (t, x) ◦ Ẇ (x) + u(n−1),ε◦ (t, x) ◦ Ẇ (x),u(n),ε◦ (t, 0) = u(n),ε◦ (t, π) = 0, u(n),ε◦ (0, x) = 0.(4.13)In particular,u(n),0◦ (t, x) = u(n)◦ (t, x).Note that all equations in (4.13) are of the form (4.2).Proposition 4.3. If φ ∈ H0, then, for every γ ∈ (1/2, 2/3) and every realizationof W ,limε→ε01(ε− ε0)n∥∥∥u◦(t, ·; ε)− n∑k=0(ε− ε0)ku(k),ε0◦ (t, ·)∥∥∥γ= 0, (4.14)n ≥ 0, ε0 ∈ R, t ≥ 0.192 HYUN-JUNG KIM AND SERGEY VLADIMIR LOTOTSKYProof. Throughout the proof, C denotes a number depending on γ, T , ε0, and thenorm of W in the space C1−γ . Definev(n)ε (t, x) =1(ε− ε0)n(u◦(t, x; ε)−n∑k=0(ε− ε0)ku(k),ε0◦ (t, x)). (4.15)By (1.2),n∑k=0(ε− ε0)k∂u(k),ε0◦ (t, x)∂t=n∑k=0(ε− ε0)k∂2u(k),ε0◦ (t, x)∂x2+ ε0n∑k=0(ε− ε0)ku(k),ε0◦ (t, x) ◦ Ẇ (x)+ (ε− ε0)n∑k=1(ε− ε0)k−1u(k−1),ε0◦ (t, x) ◦ Ẇ (x),(4.16)so that∂v(0)ε (t, x)∂t=∂2v(0)ε (t, x)∂x2+ ε0v(0)ε (t, x) ◦ Ẇ (x) + (ε− ε0)u◦(t, x; ε) ◦ Ẇ (x),∂v(n)ε (t, x)∂t=∂2v(n)ε (t, x)∂x2+ ε0v(n)ε (t, x) ◦ Ẇ (x) + v(n−1)ε (t, x) ◦ Ẇ (x), n ≥ 1,(4.17)v(n)ε (0, x) = 0, n ≥ 0, and (4.14) becomeslimε→ε0∥v(n)ε ∥γ(t) = 0, n ≥ 0, t ≥ 0, ε0 ∈ R. (4.18)Note that all equations in (4.17) are of the form (4.2), and (4.18) trivially holdsfor t = 0. Accordingly, combining the second equation in (4.17) with (4.7),∥v(n)ε ∥γ(t) ≤ C∫ t0(t− s)−γ∥v(n−1)ε ∥γ(s) ds,n ≥ 1, and then, for t > 0, (4.18) follows by induction: for n = 0, (4.12) yields∥v(0)ε ∥γ(t) ≤ |ε− ε0|C∫ t0(t− s)−γ∥u◦(s, ·, ε)∥γ ds≤ C|ε− ε0| ∥φ∥0∫ t0(t− s)−γs−γ/2 ds ≤ C|ε− ε0| ∥φ∥0t1−(3/2)γ → 0, ε → ε0;similarly, for n ≥ 1,∥v(n)ε ∥γ(t) ≤ C(n)|ε− ε0| ∥φ∥0,because 1− (3/2)γ > 0. □Proposition 4.4. If φ ∈ H0 and γ ∈ (1/2, 1), thenlimn→∞cn sup|ε|<a∥u(n),ε◦ ∥γ(t) = 0, t ≥ 0, (4.19)for all c > 0, a > 0, and every realization of W .STATIONARY PAM 193Proof. Throughout this proof, C denotes a number depending on γ, T , a, and thenorm of W in the space C1−γ .Combining (4.13) and (4.7),∥u(n),ε◦ ]∥γ(t) ≤ C∫ t0(t− s)r−1∥u(n−1),ε◦ ∥γ(s) ds.By iteration and (4.12), with r = 1− γ > 0,sup|ε|<a∥u(n),ε◦ ∥γ(t) ≤ Cn∥φ∥0×∫ t0∫ sn−10. . .∫ s20(t− sn)r−1(sn − sn−1)r−1 · · · (s2 − s1)r−1s−γ/21 ds1 · · · dsn= Cn∥φ∥0(Γ(r))nΓ(1− (γ/2))Γ(nr + 1) tnr−(γ/2),where Γ is the Gamma functionΓ(y) =∫ ∞0ty−1e−tdt.Then (4.19) follows by the Stirling formula. □Equality (1.4) now follows:• By the Sobolev embedding theorem, every element, or equivalence, classfrom Hγ , γ > 1/2, has a representative that is a continuous function on[0, π];• By Proposition 4.3 and the Taylor formula,u◦(t, x) = u(t, x) +n∑k=1u(k),0◦ (t, x)εk +Rn(t, x);• By Proposition 4.4,limn→∞Rn(t, x) = 0.This concludes the proof of (1.4).5. The Correction TermThe objective of this section is the proof of (2.10).Using (1.3) and (1.4), and remembering that u(1)◦ = u(1)⋄ = u,limε→0ε−2(u◦(t, x; ε)− u⋄(t, x; ε))= u(2)◦ (t, x)− u(2)⋄ (t, x)=∫ t0∫ π0p(t− s, x, y)(u(s, y) ◦ dW (y)− u(s, y) ⋄ dW (y))ds.(5.1)By definition,ξk ⋄ ξn ={ξkξn, k ̸= n,ξ2n − 1, k = n,194 HYUN-JUNG KIM AND SERGEY VLADIMIR LOTOTSKYand thereforeξkξn − ξk ⋄ ξn ={0 k ̸= n,1, k = n,(5.2)Then (5.2) and [7, Theorem 3.1.2] imply that, for a function f = f(x) of the formf(x) =∞∑k=1fk(x)ξk,with fk non-random and satisfying∑k∫ π0|fk(x)| dx < ∞, (5.3)the following equality holds:∫ π0f(x) ◦ dW (x)−∫ π0f(x) ⋄ dW (x) =∞∑k=1(∫ π0fk(x)mk(x) dx). (5.4)Condition (5.3) ensures that the sum on the right-hand side of (5.4) convergesabsolutely.Next, recall that, by (2.4),u(s, y) =∞∑k=1(∫ π0∫ s0p(s− r, y, z)u(r, z)mk(z) dr dz)ξk.For fixed s ∈ [0, T ] and y ∈ [0, π], defineg(z) =∫ s0p(s− r, y, z)u(r, z)dr, gk =∫ π0g(z)mk(z) dz.Thenu =∞∑k=1gkξk,and (5.3) in this case will follow from uniform, in (s, y) convergence of∞∑k=1|gk|,which, by Bernstein’s theorem [11, Theorem VI.3-1], will, in turn, follow from|g(z + h)− g(z)| ≤ Chδ (5.5)with δ ∈ (1/2, 1) and C independent of s, y, z.Recall thatu(r, z) =∞∑k=1φke−k2r mk(z), φk =∫ π0φ(x)mk(x) dx,and, by integral comparison,∞∑k=1kpe−k2 t ≤ C(p)t(1+p)/2, p ≥ 0.STATIONARY PAM 195Also,| sin(k(z + h))− sin(kz)| ≤ kδhδ, δ ∈ (0, 1),and the maximum principle implies |u(r, z)| ≤ C. Thenp(s− r, y, z) ≤ C√s− r, |p(s− r, y, z + h)− p(s− r, y, z)| ≤ Chδ(s− r)(1+δ)/2,|u(r, z + h)− u(r, z)| ≤ Chδr(1+δ)/2,and (5.5) follows because ∫ s0dr(r(s− r))(1+δ)/2 < ∞for δ ∈ (1/2, 1).We now apply (5.4) to (5.1):∫ t0∫ π0p(t− s, x, y)(u(s, y) ◦ dW (y)− u(s, y) ⋄ dW (y))ds=∫ t0∫ π0∞∑n=1(∫ π0(∫ s0p(s− r, y, z)u(r, z) dr)mn(z) dz)mn(y)p(t− s, x, y) dy ds=∫ π0∫ t0∫ s0p(t− s, x, y)p(s− r, y, y)u(r, y) dr ds dy,which, in view of (2.3) and the Fubini theorem, is the same as (2.10).This concludes the proof of (2.10).Acknowledgment. The authors would like to thank the anonymous referee andthe editors for their positive comments and careful reviews.References1. Kim, H.-J. and Lototsky, S. V.: Time-homogeneous parabolic Wick-Anderson model in onespace dimension: regularity of solution, in: Stoch. Partial Differ. Equ. Anal. Comput. 5,(2017), no. 4, 559–591.2. Kim, H.-J. and Lototsky, S. V.: Heat equation with a geometric rough path potential in onespace dimension: Existence and regularity of solution, https://arxiv.org/abs/1712.08196.3. Krylov, N. V.: An analytic approach to SPDEs, Stochastic Partial Differential Equations,Six Perspectives, Mathematical Surveys and Monographs (B. L. Rozovsky and R. Carmona,eds.), AMS, 1999.4. Lions, J.-L. and Magenes, E.: Problémes aux Limites non Homogènes et Applications, vol.I, Dunod, Paris, 1968.5. Lototsky, S. V.: Small perturbation of stochastic parabolic equations: a power series analysis,J. Funct. Anal. 193 (2002), no. 1, 94–115.6. Lototsky, S. V. and Rozovsky, B. L.: Stochastic partial differential equations driven by purelyspatial noise, SIAM J. Math. Anal. 41 (2009), no. 4, 1295–1322.7. Nualart, D.: Malliavin Calculus and Related Topics, 2nd ed., Springer, New York, 2006.8. Wan, X.: A note on stochastic elliptic models, Comput. Method. Appl. M. 199 (2010),no. 45-48, 2987–2995.9. Wan, X.: A discussion on two stochastic elliptic modeling strategies, Commun. Comput.Phys. 11 (2012), 775–796.196 HYUN-JUNG KIM AND SERGEY VLADIMIR LOTOTSKY10. Ye, H., Gao, J., and Ding, Y.: A generalized Gronwall inequality and its application to afractional differential equation, J. Math. Anal. Appl. 328 (2007), no. 2, 1075–1081.11. Zygmund, A.: Trigonometric Series, vol. I, 3rd ed., Cambridge Mathematical Library, Cam-bridge University Press, 2002.Hyun-Jung Kim: Department of Mathematics, USC, Los Angeles, CA 90089, USAE-mail address: hyunjungmath@gmail.comURL: https://hyunjungkim.orgSergey Vladimir Lototsky: Department of Mathematics, USC, Los Angeles, CA90089, USAE-mail address: lototsky@math.usc.eduURL: http://www-bcf.usc.edu/∼lototsky

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An Asymptotic Comparison of Two Time-homogeneous PAM Models

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