Indiana Universitymypage.iu.edu/~temam/papers/403_CPT16.pdf · Available online at ScienceDirect J. Differential Equations 260 (2016) 2926–2972 The Stampacchia maximum principle

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J. Differential Equations 260 (2016) 2926–2972

www.elsevier.com/locate/jde

The Stampacchia maximum principle for stochasticpartial differential equations and applications

Mickaël D. Chekroun a, Eunhee Park b,∗, Roger Temam b

a Department of Atmospheric and Oceanic Sciences, University of California, Los Angeles, CA, USAb Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN, USA

Received 29 July 2015

Available online 10 November 2015

Abstract

Stochastic partial differential equations (SPDEs) are considered, linear and nonlinear, for which we estab-lish comparison theorems for the solutions, or positivity results a.e., and a.s., for suitable data. Comparison theorems for SPDEs are available in the literature. The originality of our approach is that it is based on the use of truncations, following the Stampacchia approach to maximum principle. We believe that our method, which does not rely too much on probability considerations, is simpler than the existing approaches and to a certain extent, more directly applicable to concrete situations. Among the applications, boundedness re-sults and positivity results are respectively proved for the solutions of a stochastic Boussinesq temperature equation, and of reaction–diffusion equations perturbed by a non-Lipschitz nonlinear noise. Stabilization results to a Chafee–Infante equation perturbed by a nonlinear noise are also derived.© 2015 Elsevier Inc. All rights reserved.

MSC: 35R60; 60H15; 35B50; 35B51

Keywords: Stochastic partial differential equations; Maximum principle; Stampacchia comparison techniques;Truncation techniques; Nonlinear multiplicative noise

* Corresponding author.E-mail addresses: [email protected] (M.D. Chekroun), [email protected] (E. Park),

[email protected] (R. Temam).

http://dx.doi.org/10.1016/j.jde.2015.10.0220022-0396/© 2015 Elsevier Inc. All rights reserved.

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M.D. Chekroun et al. / J. Differential Equations 260 (2016) 2926–2972 2927

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29272. The stochastic partial differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2929

2.1. The functional framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29292.2. The stochastic framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29302.3. The stochastic parabolic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29332.4. The functions f and F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2934

3. The Itô formula for E ∫ |u−|2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2935

3.1. Preparatory steps in finite-dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29363.2. Passage to the limit as m → ∞ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29373.3. First conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29473.4. Passage to the limit as ε → 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2948

4. Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29504.1. More general elliptic operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29504.2. More general boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29534.3. Another function Fε . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29544.4. Parabolic equations perturbed by a nonlinear noise: the main result . . . . . . . . . . . . . 29554.5. Stochastic parabolic equations with a polynomial drift and nonlinear noise . . . . . . . . 2956

5. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29585.1. Application to the perturbed heat equation by a nonlinear noise . . . . . . . . . . . . . . . . 29585.2. A stochastic Chafee–Infante equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29605.3. Comparison theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29615.4. Removal of the nonlinear drift: comparison and stabilization results . . . . . . . . . . . . . 29645.5. The stochastically perturbed temperature Boussinesq equation . . . . . . . . . . . . . . . . 29655.6. Application to a harvesting model arising in population dynamics . . . . . . . . . . . . . . 29665.7. Non-Lipschitz multiplicative noises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2968

Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2969Appendix A. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2970References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2971

1. Introduction

Comparison theorems appear naturally in the context of stochastic partial differential equa-tions corresponding to parabolic equations of the second order in space and driven by a multi-plicative white noise using generally probabilistic tools. Such results have been derived in e.g. [2,4,13–15,19,31,27,28,32,35,39].

An exhaustive comparison of our results with the existing results available in the literature is out of the scope of this article. One can however point out some aspects that are relevant for appli-cations. In particular, our approach allows for drift terms that may depend on the space variables as in [31], but applies to more general cylindrical multiplicative Hilbert space valued Brownian motions and to more general elliptic operators and boundary conditions than considered in [31]or in [4]. The latter framework is particularly suitable for models arising in population dynamics; see Section 5.6 below.

In [32], the case of cylindrical multiplicative noise was also considered, but our approach al-lows for more general nonlinear noises. In particular, the important case of a non-Lipschitz noise such as arising in the Reggeon-Field Theory (RFT) of directed percolation [30,44] is accessible within our approach; see Section 5.7.

2928 M.D. Chekroun et al. / J. Differential Equations 260 (2016) 2926–2972

At the same time, the case of a nonlinear noise depending on the solution’s gradient such as in [19] is not considered here, although our approach is not limited to stochastic reaction–diffusion equations and includes the case of drift terms depending nonlinearly or linearly on the solution’s gradient; see Section 5.5.

In all the aforementioned applications, our treatment allows for spatially inhomogeneous mul-tiplicative nonlinear noises. The relevance of such noises is well-known in physics, and many experimental or numerical observations of self-organized behavior or phase transitions arising out of such noises have been documented [30,44].

Motivated by such applications, our approach for comparing solutions of SPDEs and proving positivity is different than those of the aforementioned works on the topic, and we believe it is simpler and to a certain extent, more directly applicable to concrete situations. It is essentially based on the use of truncations and the Stampacchia approach to the maximum principle [38]. Typically, considering the solution of an SPDE driven by a white noise

du =N (u)dt + σ(u)dW, (1.1)

where u is a function of x, t , and ω, ω ∈ � the stochastic variable and x ∈ M the spatial variable, we prove e.g., the positivity a.e., a.s. of u, by showing that its negative part u− vanishes where

u− = max(−u,0). (1.2)

In the context of partial differential equations (PDEs), this is usually shown by proving that∫M

∣∣u−(x, t)∣∣2 dx = 0, ∀t ≥ 0, (1.3)

and in the context of SPDEs, we prove it by showing that

F(u(t)

) = E

∫M

∣∣u−(x, t)∣∣2 dx = 0, ∀t ≥ 0. (1.4)

In order to establish a result like (1.4), we need to consider the Itô differential dF(u(t)

)of F for

which we face the double difficulties of F not being C2, and of being a nonlinear functional of infinite-dimensional solutions of Eq. (1.1). We overcome these difficulties by considering smooth C2-approximations Fε of F (see Section 2.4), and by deriving the corresponding expressions associated with finite dimensional Galerkin approximations um of u.

To pass to the limit as m → ∞ in the corresponding stochastic integrals, our key tool relies on [16, Lemma 2.1] recalled hereafter as Lemma 3.2 to make the expository as much self-contained as possible.

We obtain then an Itô formula in Section 3.3 that holds for the aforementioned approximation Fε applied to the solutions of parabolic equations perturbed by nonlinear noises considered in Section 2; see (3.21) in Proposition 3.3.

The passage to the limit as ε → 0 is finally dealt with in Section 3.4, and the desired Itô formula for F defined in (1.4) is finally obtained and extended in Section 4; see formulas (3.24)and (4.18) in Proposition 3.4 and Theorem 4.1, respectively.


This work is at the interface of probability and partial differential equations. Hence we give at times some detailed explanations which may be well-known for some readers but not for others.

The article is organized as follows. In Section 2 we introduce a typical stochastic parabolic equation involving a Laplacian, and describe the deterministic and stochastic backgrounds (Sec-tions 2.1 to 2.3). Then, in Section 2.4 we introduce a C2 approximation fε of f (u) = u−, and the corresponding functionals F(u) = ∫ |u−|2, Fε = ∫ (

fε(u))2. In Section 3 we aim to derive

the Itô formula for E ∫ |u−|2; we start first with the Itô formula for Fε(u), via a Galerkin approx-

imation um in finite dimension (Section 3.1). We then pass to the limit m → ∞ (Section 3.2). The first conclusions with ε > 0 fixed appear in Section 3.3 and the passage to the limit ε → 0 is performed in Section 3.4. Various generalizations are given in Section 4, more general elliptic op-erators, more general boundary conditions, and certain nonlinear equations. The applications to the maximum principle itself appear in Section 5, positivity of solutions, a.e., a.s., boundedness, and comparison of solutions. In particular, stabilization results for the Chafee–Infante equation perturbed by a nonlinear noise (Section 5.4), as well as boundedness and positivity results for the solutions of stochastic Boussinesq temperature equations (Section 5.5), are derived. We obtain also a result of existence of a positive solution for a harvesting model arising in population dy-namics (Section 5.6); and we finally consider non-Lipschitz multiplicative noises (Section 5.7). Appendix A contains some technical results.

2. The stochastic partial differential equations

2.1. The functional framework

Let H = L2(M) and V = H 10 (M) where M is a regular open bounded set in Rd . We define

the scalar products (·, ·) and ((·, ·)) in H and V by

(u, v) =∫M

u(x)v(x)dx,

and

((u, v)) = (∇u,∇v),

respectively.We also define the corresponding norms | · |H and ‖ · ‖ on H and V respectively by

|u|H = (u,u)12 , and ‖u‖ = ((u,u))

12 .

Although we will consider more general elliptic operators and boundary conditions in Sec-tion 4 below, we consider first to fix ideas, the standard operator A : V → V ′ corresponding to −� in L2(M) with Dirichlet boundary condition on ∂M.

It is well-known that there exists a Hilbert basis, {φj }j≥1, of H made of smooth eigenfunc-tions of A, that is φj ∈ H 1

0 (M) ∩ C∞(M) and

Aφj = λjφj , 0 < λ1 ≤ λ2 ≤ · · · ;see e.g. [3, Thm. 9.31].


We consider the finite-dimensional subspace

Hm = span{φ1, . . . , φm}, (2.1)

associated with the orthogonal projection Pm from H onto this space. Given this projection and

an element v =∞∑

j=1

ξjφj in H , we denote by vm, the projection Pmv, i.e.,

vm = vm(x) =m∑

j=1

ξjφj (x), with ξj = (v,φj ). (2.2)

2.2. The stochastic framework

We briefly recall here some aspects of stochastic analysis in Hilbert spaces, used hereafter. For an extended treatment of this topic we refer to [20]. Throughout this article, we will work with a given stochastic basis,

S = (�,F, {Ft }t≥0,P, {Wk}k≥1),

where {Wk} denotes a sequence of mutually independent standard one-dimensional Brownian motions adapted to a complete, right continuous filtration {Ft}t≥0 of σ -algebras1 on a complete probability space (�, F, P).

Let U be an auxiliary separable real Hilbert space endowed with a Hilbert basis {ej }j≥1. We then consider W(t, ·; ω) the U-valued stochastic processes, formally represented for the moment, as the following series

W(t, ·;ω) =∞∑l=1

Wl(t,ω)el(·), (2.3)

for a given realization ω in � and t ≥ 0.To make sense of the representation (2.3) we need to recall some basic facts about Hilbert–

Schmidt operators. In that respect, given any pair (U, X) of separable Hilbert spaces, we denote by

L2(U,X) = {R ∈ L(U,X) :∞∑

k=1

|R · ek|2X < ∞}, (2.4)

the set of Hilbert–Schmidt operators from U to X. We endow this set with the inner product 〈R, S〉L2(U,X) = ∑

k〈Rek, Sek〉X , so that L2(U, X) can be considered itself as a Hilbert space.

1 I.e. F0 contains all P-null sets (complete), and Ft = Ft+ := ⋂s>t Fs (right-continuous).


If U ⊂ U0 is a second auxiliary Hilbert space U0 such that the inclusion

γ : U→ U0

u �→ u, (2.5)

is Hilbert–Schmidt, then the series (2.3) converges in L2(�, U0) for almost every t .A classical example of such an inclusion is given by considering e.g. Au = u − uxx with

periodic boundary conditions on (0, 2π). In that case, by taking U = L2(0, 2π) and the ek’s (resp. λk’s) to be the normalized eigenfunctions (resp. eigenvalues) of A in U, and by choosing U0 to be the space

D(A−r/2) := {v =∑

k

αkek : ‖v‖2−r/2 =∑k≥1

λ−rk α2

k < ∞},

we have then, for 2r > 1, that

‖γ ‖L2(U,U0) =(∑

k

λ−rk

)1/2< ∞,

since λ−rk = O(k−2r ) in such a case. The series (2.3) converges thus here in L2(�, D(A−r/2))

for 2r > 1 since

‖W‖2L2(�,D(A−r/2))

=∑

k

E|Wk(t, ·)|2‖γ ek‖2−r/2

= t ‖γ ‖2L2(U,U0)

< ∞. (2.6)

In the general case, the stochastic process W is referred to as a cylindrical Brownian motion on U adapted to {Ft }t≥0, whose representation (2.3) converges typically on a larger Hilbert space U0 for which the inclusion U ⊂ U0 is Hilbert–Schmidt.2 Physically speaking, such a process is closely related to a space–time white noise3 because of the homogeneous mix of all eigenfunc-tions in (2.3). In many applications such as in climate dynamics, we would like however to allow for state-dependent (“multiplicative”), or state-independent (“additive”) Gaussian white noises or a mixture of both that take into account for spatial inhomogeneity in terms of spatial correlations; see e.g. [43,42]. For instance, in [42], it has been shown on a double-gyre ocean model forced by a stochastic wind-stress, that regime transitions consistent with observations can be produced from a spatially inhomogeneous white noise whereas such transitions do not appear without a spatially inhomogeneous stochastic forcing nor with spatially homogeneous stochastic forcing.

2 For instance U0 given by

U0 :={v =

∑k

αkek :∑k

α2k k−2 < ∞

}

is such a space, when endowed with the norm ‖v‖ := ∑k α2

kk−2.

3 Loosely speaking, the latter is the time derivative of a cylindrical Wiener process.


We refer to [30,44] for other physical contexts in which spatially inhomogeneous multiplicative noises arise naturally.

Motivated by such contexts, we consider stochastic forcing of the form σ(u) dW — that allow for spatially inhomogeneity — but for the purpose of this article regarding the derivation of maximum principles, we will restrict our attention to the state-dependent case in which σ(u)

does explicitly depend on u. Application to an example from geophysics will be discussed in Section 5.5.

We shall assume throughout this work that

σ = σ(u, t) : H × [0, T ] −→ L2(U,H) (2.7)

is B(H ⊗ [0, T ], B(L2(U, H))

)-measurable essentially bounded in time, and continuous in u,

{Ft }t≥0-adapted, and such that the following sublinear growth condition is satisfied

‖σ(u, t)‖L2(U,H) ≤ c0(1 + |u|H ), for a.e. t and for all u ∈ H, (2.8)

where c0 > 0 is independent of t so that σ is uniformly bounded in t . Specific forms of σwill be sometimes considered later on, and additional assumptions on σ will be provided when needed; see Section 5.3. Furthermore, certain additional hypotheses on σ may be necessary to guarantee the existence of solutions for the SPDEs that we consider. However these hypotheses are not recalled in this article since, as we recall below, we do not discuss the issue of existence of solutions of the SPDEs that we consider, but rely for the existence on available references, including [20,16,17,21].

In particular, relying on [16, Lemma 2.1] (recalled below as Lemma 3.2), we refer to [16] for existence results of pathwise or martingale solutions where typically σ is assumed to be locally Lipschitz and to satisfy a version of the growth condition (2.8) for Hilbert–Schmidt norm built out from more regular spaces than H .4

Finally, we recall that the formalism of cylindrical Brownian motion such as recalled above, allows us to make sense to Itô stochastic integrals for which the integrand is an X-valued process G that is predictable, and satisfies

G ∈ L2(�,L2loc([0,∞),L2(U,X))). (2.9)

In that case, taking Gk = G · ek , one may define the Itô stochastic integral

Mt :=t∫

0

GdW =∑

k

t∫0

Gk dWk, (2.10)

as an element of M2X , that is, the space of all X-valued square integrable martingales; see [20]

for a construction.

4 For instance a sublinear growth of σ(u) measured in the L2(U, D(A))-norm can be additionally assumed.


2.3. The stochastic parabolic equations

Let u be a solution of the stochastic evolution equation

du + νAudt = b(t)dt + σ(u, t)dW, (2.11)

such that

u ∈ L2(�;L2(0, T ;V )) ∩L2(�;L∞(0, T ;H)), (2.12)

and that emanates from an initial data u0 in L2(�; H), i.e. u(0) = u0 a.s.We do not specify if u is a pathwise or martingale solution of (2.11). Indeed, except possibly

for technical hypotheses on σ (see e.g. hypotheses (2.10) and (2.11) in [16]), the only difference is that, for martingale solutions the stochastic basis is changed, as provided by the Skorohod theorem, whereas for the pathwise solutions the stochastic basis is the initial one. This difference does not affect the study below. Note also that, by (2.12), u is a weak solution of the SPDEs, in the PDE sense, according to the terminology in [16] and [17].

We will also make use of the decomposition u =∞∑

j=1

ξjφj , with ξj = ξj (t, ω) and φj = φj (x)

are the eigenfunctions of A in D(A) ⊂ H , so that Au becomes ∞∑

j=1

ξjλjφj and b =∞∑

j=1

bjφj

with bj = (b, φj ).

Remark 2.1 (Notation). For σ(u) ∈ L2(0, T ; L2(U, H)) and W =∞∑l=1

Wlel we introduce the

notation σ(u)dW to represent formally

σ(u)dW =∞∑l=1

σ(u) · eldWl

=∞∑

i,l=1

(σ (u) · el, φi)φidWl

=∞∑

i,l=1

σ ilφidWl,

where we have used the decomposition

σ(u) · el =∑

i

σ ilφi, σ il := (σ (u) · el, φi),

which makes sense since, from assumptions, σ(u) · el ∈ H and {φi} is a Hilbert basis of H . Note that this representation can be made rigorous when the first equality is understood in its stochastic integral form such as recalled in (2.10) and the process u possesses the appropriate regularity properties.


2.4. The functions f and F

We introduce a scalar function fε which is C2 and such that

fε(u) = 0 ⇐⇒ u ≥ 0.

We assume that fε has a linear growth at −∞, that it is increasing and satisfies the relations{ |fε(u)| ≤ c1(1 + |u|), |f ′ε(u)| ≤ c1,

|f ′′ε (u)| ≤ c1, |fε(u)f ′′

ε (u)| ≤ c1.(2.13)

It is elementary to see that, e.g., the following function fε satisfies these conditions and we define a function f :

fε(u) =

⎧⎪⎨⎪⎩−u if u < −ε,

− 3

ε4u5 − 8

ε3u4 − 6

ε2u3 if −ε ≤ u < 0,

0 if u ≥ 0,

f ′ε(u) =

⎧⎪⎨⎪⎩−1 if u < −ε,

−15

ε4u4 − 32

ε3u3 − 18

ε2u2 if −ε ≤ u < 0,

0 if u ≥ 0,

(2.14)

f ′′ε (u) =

⎧⎪⎨⎪⎩0 if u < −ε,

−60

ε4u3 − 96

ε3u2 − 36

ε2u if −ε ≤ u < 0,

0 if u ≥ 0,

f (u) = u− ={−u if u ≤ 0,

0 if u ≥ 0.

We now introduce, for u ∈ H , the functional

Fε(u) =∫M

(fε(u(x))

)2dx,

and the functional F :

F(u) =∫M

(f (u(x))

)2dx =

∫M

|u−(x)|2 dx, (2.15)

which we will use subsequently, where we write u− = max{−u, 0}. We note that for u, v, w ∈ H

〈DFε(u), v〉 =∫M

2fε(u)f ′ε(u)v dx,

〈D2Fε(u)v,w〉 =∫

2{fε(u)f ′′

ε (u) + |f ′ε(u)|2}vw dx.

M


We will also need to differentiate the functional Fmε = Fε ◦ Pm; for u, v, w ∈ H we find

〈DFε(Pmu), v〉 =∫M

2fε(Pmu)f ′ε(Pmu)Pmv dx, (2.16)

〈D2Fε(Pmu)v,w〉 =∫M

2{fε(Pmu)f ′′

ε (Pmu) + ∣∣f ′ε(Pmu)

∣∣2}PmvPmw dx

= 2m∑

i,j=1

∫M

�ε(Pmu)(v,φi)(w,φj )φiφj dx, (2.17)

where

�ε(u) = �ε(u(x)) = fε(u(x))f ′′ε (u(x)) + (

f ′ε(u(x))

)2. (2.18)

We note that

∣∣�ε(u)∣∣ ≤ c2, (2.19)

where c2 = c1 +c21 from (2.13). It is obvious that �ε is a non-negative continuous function which

is uniformly bounded in u.

Remark 2.2. The above expressions do not make sense for F because u− and hence F is not C2. This justifies the need of approximating F by a C2 function. However, as we said in the Intro-duction, the final result that we pursue will be the same as if F were C2.

3. The Itô formula for EEE∫ |u−|2

We aim to derive an Itô formula for F(u)(t) = F(u(t)) when u satisfies (2.12) and (2.11). For that purpose we study the Itô formula for Fε(u)(t) = Fε(u(t)) and we will eventually pass to the limit as ε → 0.

We proceed by approximation in finite dimension and for this aim we introduce um = Pmu

which is a solution of {dum + νAum dt =Pmb dt + Pmσ(u)dW(t),

um(0)=um0 ,

(3.1)

where um0 = Pmu0, Pmb = ∑m

j=1 bjφj with bj = (b, φj ), and

Pmσ(u)dW(t) =m∑

i=1

∞∑l=1

σ ilφidWl(t) (3.2)

with σ il = (σ (u) · el, φi).


3.1. Preparatory steps in finite-dimension

We now consider Fε(Pmu)(t) = Fε((Pmu)(t)), that is

Fε(Pmu) =∫M

(fε

((Pmu)(x, t)

))2

dx a.e. t and a.s.

The Itô formula in finite dimension gives:

Fε(um(t)) = Fε(um(0)) +t∫

0

〈DFε(um(s)),Pm[σ(u(s))dW(s)]〉

+t∫

0

〈DFε(um(s)),Pmb(s) + ν�um(s)〉ds

+ 1

2

t∫0

m∑i,j=1

∞∑l=1

〈D2Fε(um(s))i,j , σilσ jlφiφj 〉ds, (3.3)

where

〈DFε(um(t)),Pmσ(u(t))dW(t)〉 =∫M

2fε(um(x, t))f ′ε(um(x, t))Pm[σ(u(x, t))dW(t)]dx,

〈DFε(um(t)),Pmb(t) + ν�um(t)〉 =∫M

2fε(um(x, t))f ′ε(um(x, t))Pmb(x, t)dx

−∫M

2ν�ε(um(x, t))ν∇um(x, t)∇um(x, t)dx,

m∑i,j=1

∞∑l=1

〈D2Fε(um(t)))i,j

, σ ilσ jlφiφj 〉

= 2m∑

i,j=1

∞∑l=1

∫M

�ε(um(x, t))φi(x)φj (x)σ il(u(x, t))σ jl(u(x, t))dx,

where �ε is as in (2.18). We then write

Fε(um(t)) = Imε0(u)(t) + Im

ε1(u)(t) + Imε2(u)(t) + Im

ε3(u)(t) + Imε4(u)(t), (3.4)

where


Imε0(u)(t) = Fε

(um(0)

),

Imε1(u)(t) =

t∫0

∫M

2fε(um(x, s))f ′ε(um(x, s))Pm[σ(u(x, s))dW(s)]dx,

Imε2(u)(t) =

t∫0

∫M

2fε(um(x, s))f ′ε(um(x, s))Pmb(x, s)dx ds,

Imε3(u)(t) = −

t∫0

∫M

2ν�ε(um(x, s))∣∣∇um(x, s)

∣∣2 dx ds,

Imε4(u)(t) =

t∫0

∫M

m∑i,j=1

∞∑l=1

�ε(um(x, s))φi(x)φj (x)σ il(u(x, s))σ jl(u(x, s))dx ds.

It is clear that all these expressions make sense a.s. and for every t ∈ (0, T ), and we now aim to pass to the limit in (3.4) as m → ∞.

3.2. Passage to the limit as m → ∞

We aim to derive the limit of Imεj (u)(t) for each j = 0, . . . , 4, as m → ∞. To pass to the limit

in these expressions we will interplay between strong convergences in H and a.e. convergences in M or in M × (0, T ). We start with Im

ε0(u)(t). Since u0 ∈ L2(�; H), we obtain

Pmu0(ω) −→ u0(ω) in H strongly a.e. ω.

Since ∣∣Pmu0(ω)

∣∣H

≤ |u0(ω)|H for a.e. ω, by the Lebesgue dominated convergence theorem we have ∫

�

∫M

∣∣Pmu0(x,ω) − u0(x,ω)∣∣2 dx dω =

∫�

∣∣Pmu0(ω) − u0(ω)∣∣2H

dω

−→ 0 as m → ∞.

Hence, there exists a subsequence, still denoted by m, such that

Pmu0(x,ω) −→ u0(x,ω) a.e. x, and a.s.

We will use the following lemma for the convergence of the differentials of F and for the limit of Im

ε0(u)(t).

Lemma 3.1. Let g be a real continuously differentiable function. Assume that there exists a constant c such that


g(u) ≤ c(1 + |u|), |g′(u)| ≤ c.

Then, as m → ∞, for u and u0 as above:

{g(Pm(u)

) −→ g(u) in L2((0, T ) ×M

)a.s.,

g(Pm(u0)

) −→ g(u0) in L2(M) a.s.(3.5)

Proof. By the assumption we find for a.e. ω

t∫0

∫M

∣∣g(Pmu(x, s)

) − g(u(x, s)

)∣∣2 dx ds ≤ c2

t∫0

∫M

∣∣Pmu(x, s) − u(x, s)∣∣2 dx ds

≤ c2

t∫0

∣∣Pmu(s) − u(s)∣∣2H

ds.

Since Pmu(t) −→ u(t) in H for a.e. t , ω and ∣∣Pmu(t)

∣∣H

≤ |u(t)|H for a.e. t , ω, the Lebesgue dominated convergence theorem yields that

t∫0

∣∣Pmu(s) − u(s)∣∣2H

ds −→ 0 a.s.

Hence we conclude that g(Pm(u)

) −→ g(u) in L2((0, T ) × M

)a.s. and (3.5)1 follows.

For (3.5)2, we derive similarly that∫M

∣∣g(Pmu0(x)

) − g(u0(x)

)∣∣2dx ≤ c2

∫M

∣∣Pmu0(x) − u0(x)∣∣2

dx

≤ c2∣∣Pmu0 − u0

∣∣2H

.

Since Pmu0 −→ u0 in H strongly as m → ∞ a.s.,

g(Pm(u0)

) −→ g(u0) in L2(M) a.s. �We apply Lemma 3.1 to the case where g(u) = fε(u) to find that, as m → ∞

Imε0(u)(t) =

∫M

(fε

(Pmu0(x)

))2 dx

−→∫M

(fε

(u0(x)

))2 dx = F(u0).


Remark 3.1. Since u ∈L2(�; L∞(0, T ; H)

), we observe that

Pmu(t,ω) −→ u(t,ω) in H strongly for a.e. t and ω.

Indeed ∣∣Pmu(t, ω)

∣∣H

≤ |u(t, ω)|H for a.e. t and ω. By the Lebesgue dominated convergence theorem we have

∫�

T∫0

∫M

∣∣Pmu(x, t,ω) − u(x, t,ω)∣∣2 dx dt dω =

∫�

T∫0

∣∣Pmu(t,ω) − u(t,ω)∣∣2H

dt dω

−→ 0 as m → ∞.

Hence, there exists a subsequence, still denoted by m, such that

Pmu(x, t,ω) −→ u(x, t,ω) for a.e. x, t, and ω. (3.6)

For Imε1(u)(t), we first note that

E

∞∑l=1

∣∣∣∣T∫

0

σ(u(t)) · el dWl(t)

∣∣∣∣2

H

= E

∞∑l=1

∫M

∣∣∣∣T∫

0

σ(u(x, t)) · el dWl(t)

∣∣∣∣2

dx

= E

∞∑l=1

∫M

T∫0

∣∣σ(u(x, t)) · el

∣∣2 dt dx

= E

T∫0

∞∑l=1

∣∣σ(u(t)) · el

∣∣2H

dt

= E

T∫0

∥∥σ(u(t))∥∥2L2(U,H)

dt,

which implies

t∫0

σ(u(x, s))dW(s) =∞∑l=1

t∫0

σ(u(x, s)) · el dWl(s) ∈ L2(�,H).

We have

Imε1(u)(t) =

t∫ ∫�ε(um(x, s))Pm[σ(u(x, s))dW(s)]dx,

0 M


where �ε(u) = 2fε(u)f ′ε(u). By Lemma 3.1 with g(u) = fε(u)f ′

ε(u) we obtain

fε(um)f ′ε(um) −→ fε(u)f ′

ε(u) in L2((0, T ) ×M)

a.s. (3.7)

We now show that

t∫0

Pm[σ(u(s))dW(s)] −→t∫

0

σ(u(s))dW(s) in probability in H.

For that purpose, we introduce the auxiliary Hilbert space U0 ⊃ U via

U0 = {v =∞∑l=1

αkek :∞∑

k=1

α2k

k2< ∞},

and recall the following convergence result from [16] that will allow us to pass to the limit in the stochastic integrals.

Lemma 3.2. (See [16].) Let (�, F, P) be a fixed probability space, X a separable Hilbert space. Consider a sequence of stochastic bases Sm = (�, F, {Fm

t }t≥0, P, Wm), that is a se-quence so that each Wm is a cylindrical Brownian motion (over U) with respect to Fm

t . As-sume that {Gm}m≥1 are a collection of X-valued Fm

t predictable processes such that Gm ∈L2(0, T ; L2(U, X)) a.s. Let S = (�, F, {F}t≥0, P, W) be a stochastic basis. Assume that G ∈L2(0, T ; L2(U, X)), which is Ft predictable. If

Wm −→ W in probability in C([0, T ];U0),

Gm −→ G in probability in L2(0, T ;L2(U,X)),

then

t∫0

Gm dWm −→t∫

0

GdW, in probability in L2(0, T ;X).

We apply this lemma with Sm = S = (�, F, {Ft }t≥0, P, W), X = H , and Wm(t) = W(t)

with W(t) =∞∑l=1

elWl(t) for m = 1, 2, . . . , and for t ∈ [0, T ]. We define

Gm ∈ L2(0, T ;L2(U,H)) and G ∈ L2(0, T ;L2(U,H)),

by setting Gm(t), G(t) ∈ L2(U, H) for t ∈ [0, T ] such that

Gm(t) · el = Pm(σ(u(t)) · el)

=m∑

σ il(u(t))φi,

i=1


G(t) · el = σ(u(t)) · el, for l = 1,2, . . . .

To use Lemma 3.2, it suffices to prove that Gm −→ G in L2(0, T ; L2(U, H)) a.s. and hence in probability. We have

T∫0

‖Gm(t) − G(t)‖2L2(U,H) dt =

T∫0

∞∑l=1

∣∣Gm(t) · el − G(t) · el

∣∣2H

dt

=T∫

0

∞∑l=1

∣∣∣∣ m∑i=1

σ il(u(t))φi −∞∑i=1

σ il(u(t))φi

∣∣∣∣2

H

dt

=T∫

0

∞∑l=1

∣∣∣∣ ∞∑i=m+1

σ il(u(t))φi

∣∣∣∣2

H

dt

=T∫

0

∞∑l=1

∫M

∣∣∣∣ ∞∑i=m+1

σ il(u(x, t))φi(x)

∣∣∣∣2

dx dt

=T∫

0

∞∑l=1

∫M

∣∣σ(u(x, t)) · el − Pm[σ(u(x, t)) · el]∣∣2 dx dt.

By (2.8), it is clear that

T∫0

∫M

∞∑l=1

∣∣σ(u(x, t)) · el

∣∣2 dx dt (3.8)

=T∫

0

∞∑l=1

∣∣σ(u(t)) · el

∣∣2H

dt

=T∫

0

∥∥σ(u(t))∥∥2L2(U,H)

dt

≤ c0

T∫0

(1 + |u(t)|2H

)dt < ∞ a.s., (3.9)

which implies ‖σ(u)‖2L2(U,H)

∈ L1(0, T ) and ∞∑∣∣σ(u) · el

∣∣2 ∈ L1((0, T ) ×M) a.s.
l=1


Since

∞∑l=1

∣∣∣∣σ(u(t)) · el − Pm

[σ(u(t)) · el

]∣∣∣∣2

H

≤ 4∞∑l=1

∣∣σ(u(t)) · el

∣∣2H

= 4‖σ(u(t))‖2L2(U,H),

which belongs to L1(0, T ) from (3.8), and since Pm

[ ∞∑l=1

σ(u(t)) · el

]converges to

∞∑l=1

σ(u(t)) ·el in H for a.e. t and a.s., we then find by the Lebesgue dominated convergence theorem that

t∫0

∫M

∞∑l=1

∣∣σ(u(x, s)) · el − Pm[σ(u(x, s)) · el]∣∣2 dx ds

=t∫

0

∞∑l=1

∣∣∣∣σ(u(s)) · el − Pm

[σ(u(s)) · el

]∣∣∣∣2

H

ds

−→ 0 a.s. as m → ∞. (3.10)

Hence,

T∫0

‖Gm(t) − G(t)‖2L2(U,H) dt −→ 0 as m → ∞ a.s.,

which yields Gm −→ G in probability in L2(0, T ; L2(U, H)). We apply Lemma 3.2 to obtain

∫M

∣∣∣∣T∫

0

∞∑l=1

σ(u(x, t)) · el − Pm[σ(u(x, t)) · el]dWl(t)

∣∣∣∣2

dx −→ 0 in probability.

By (3.7) we also have �ε(um) −→ �ε(u) in L2((0, T ) ×M

)a.s., �ε as in (3.7). Thus we find

that for a.e. t ∈ [0, T ]

Imε1(u)(t) =

t∫0

∫M

�ε(um(x, s))Pm[σ(u(x, s))dW(s)]dx

−→t∫

0

∫M

�ε(u(x, s))σ (u(x, s))dW(s)dx

=t∫

0

∫M

2fε(u(x, s))f ′ε(u(x, s))σ (u(x, s))dW(s)dx in probability. (3.11)

We now find the limit of Im(u)(t). Since b ∈ L2(�; L2(0, T ; H)

), we obtain that, for m → ∞
ε2

{Pmb(t) −→ b(t) in H for a.e. t and ω,∣∣Pmb(t)
∣∣H

≤ |b(t)|H for a.e. t and ω.(3.12)

We can also find a subsequence, still denoted m, such that

Pmb(x, t,ω) −→ b(x, t,ω) a.e. in x, t, and ω.

We then obtain ∫M

∣∣Pmb(x, s) − b(x, s)∣∣2 dx = ∣∣Pmb(s) − b(s)

∣∣2H

≤ 4∣∣b(s)

∣∣2H

a.s.,

which implies by the Lebesgue dominated convergence theorem that

t∫0

∣∣Pmb(s) − b(s)∣∣2H

ds −→ 0 a.s., (3.13)

which means also that Pmb −→ b in L2((0, T ) × M) a.s. From (3.7) and (3.13), we then find that

t∫0

(fε(um(s))f ′

ε(um(s)),Pmb(s))

ds

−→t∫

0

(fε(u(s))f ′

ε(u(s)), b(s))

ds a.s.,

which implies

Imε2(u)(t) =

t∫0

∫M

2fε(um(x, s))f ′ε(um(x, s))Pmb(x, s)dx ds

−→t∫

0

∫M

2fε(u(x, s))f ′ε(u(x, s))b(x, s)dx ds a.s. as m → ∞.

For Imε3(u)(t), we find, integrating by parts and using the Dirichlet boundary condition and

f ′ε(0) = 0:

Imε3(u)(t) = −2ν

t∫ ∫∇(

fε(um(x, s))f ′ε(um(x, s))

) · ∇um(x, s)dx ds

0 M


= −2ν

t∫0

∫M

{∣∣f ′ε(um(x, s))

∣∣2 + fε(um(x, s))f ′′ε (um(x, s))

}|∇um(x, s)|2 dx ds

= −2ν

t∫0

∫M

�ε(um(x, s))∣∣∇um(x, s)

∣∣2 dx ds,

with �ε as in (2.18). Since Pm is also an orthogonal projection in H 10 (M), we also infer that

∇um(t) −→ ∇u(t) in H for a.e. t and a.s. Then∫M

∣∣∇Pmu(x, t) − ∇u(x, t)∣∣2 dx = ∣∣∇Pmu(t) − ∇u(t)

∣∣2H

≤ 4∣∣∇u(t)

∣∣2H

a.e. t and ω,

and by the Lebesgue dominated convergence theorem we obtain

t∫0

∣∣∇Pmu(s) − ∇u(s)∣∣2H

ds −→ 0 a.s. (3.14)

We note that, up to a subsequence,

�ε(um(x, t,ω)) −→ �ε(u(x, t,ω)) for a.e. x, t, and ω. (3.15)

We then infer that

t∫0

∫M

∣∣�ε(um(x, s))∇um(x, s) − �ε(u(x, s))∇u(x, s)∣∣2 dx ds

≤ 4

t∫0

∫M

∣∣�ε(um(x, s)) − �ε(u(x, s))∣∣2∣∣∇u(x, s)

∣∣2 dx ds

+ 4

t∫0

∫M

∣∣�ε(um(x, s))∣∣2∣∣∇um(x, s) − ∇u(x, s)

∣∣2 dx ds

≤ c

t∫0

∫M

∣∣�ε(um(x, s)) − �ε(u(x, s))∣∣|∇u(x, s)|2 dx ds

+ c

t∫ ∣∣∇um(s) − ∇u(s)∣∣2H

ds.

0


By (3.15), we find that ∣∣�ε(um(x, t)) − �ε(u(x, t))

∣∣|∇u(x, t)|2 −→ 0 for a.e. x, t , and ω. Since we have

∣∣�ε(um) − �ε(u)∣∣|∇u|2 ≤ c|∇u|2 ∈ L1

(0, T ; L1(M)

)a.s., we obtain from the

Lebesgue dominated convergence theorem that

t∫0

∫M

∣∣�ε(um(x, s)) − �ε(u(x, s))∣∣|∇u(x, s)|2 dx ds −→ 0 a.s.

This is combined with (3.14) to show that

t∫0

∫M

∣∣�ε(um(x, s))∇um(x, s) − �ε(u(x, s))∇u(x, s)∣∣2

dx ds −→ 0 a.s.,

that is,

�ε(um)∇um −→ �ε(u)∇u in L2((0, T ) ×M)

a.s.

This yields from (3.14) that(�ε(um)∇um,∇um

)L2(

(0,T )×M) −→ (�ε(u)∇u,∇u

)L2(

(0,T )×M).Hence,

Imε3(u)(t) = −2ν

t∫0

∫M

�ε(um(x, s))|∇um(x, s)|2 dx ds

−→ −2ν

t∫0

∫M

�ε(u(x, s))|∇u(x, s)|2 dx ds, a.s. as m → ∞.

For Imε4(u)(t) we remind that

Imε4(u)(t) =

t∫0

∫M

�ε(um(x, s))

m∑i,j=1

∞∑l=1

[σ il(u)σ jl(u)φi(x)φj (x)

]dx ds.

We then have

t∫0

∫M

∣∣∣∣�ε(um(x, s))

m∑i,j=1

∞∑l=1

σ ilσ jlφi(x)φj (x) − �ε(u(x, s))

∞∑i,j=1

∞∑l=1

σ ilσ jlφi(x)φj (x)

∣∣∣∣dx ds

≤t∫ ∫ ∣∣�ε(u(x, s)) − �ε(um(x, s))

∣∣∣∣∣∣ ∞∑l=1

∞∑i,j=1

σ ilσ jlφi(x)φj (x)

∣∣∣∣dx ds

0 M


+t∫

0

∫M

∣∣�ε(um(x, s))∣∣∣∣∣∣ ∞∑

l=1

∞∑i,j=m+1

σ ilφi(x)σ jlφj (x)

∣∣∣∣dx ds

≤t∫

0

∫M

∣∣�ε(u(x, s)) − �ε(um(x, s))∣∣∣∣∣∣ ∞∑

l=1

( ∞∑i=1

σ ilφi(x)

)2∣∣∣∣dx ds

+t∫

0

∫M

∣∣�ε(um(x, s))∣∣∣∣∣∣ ∞∑

l=1

( ∞∑i=m+1

σ ilφi(x)

)2∣∣∣∣dx ds

≤t∫

0

∫M

∣∣�ε(u(x, s)) − �ε(um(x, s))∣∣ ∞∑

l=1

∣∣σ(u(x, s)) · el

∣∣2 dx ds

+ c

t∫0

∫M

∞∑l=1

∣∣∣∣σ(u(x, s)) · el − Pm

[σ(u(x, s)) · el

]∣∣∣∣2

dx ds. (3.16)

For a.e. t ∈ [0, T ] and ω∫M

∣∣�ε(u(x, t)) − �ε(um(x, t))∣∣ ∞∑

l=1

∣∣σ(u(x, t)) · el

∣∣2 dx ≤ c

∞∑l=1

∫M

∣∣σ(u(x, t)) · el

∣∣2 dx

= c‖σ(u(t))‖2L2(U,H).

Let us recall that ‖σ(u)‖2L2(U,H)

∈ L1(0, T ) from (3.8). From (3.15) we also find that

∫M

∣∣�ε(um(x, t)) − �ε(u(x, t))∣∣ ∞∑

l=1

∣∣σ(u(x, t)) · el

∣∣2 dx −→ 0 for a.e. t, ω as m → ∞.

We obtain by the Lebesgue dominated convergence theorem that

t∫0

∫M

∣∣�ε(u(x, s)) − �ε(um(x, s))∣∣ ∞∑

l=1

∣∣σ(u(s)) · el

∣∣2 dx ds −→ 0 a.s.

By (3.10) we infer from (3.16) that

Imε4(u)(t) =

t∫0

∫M

�ε(um(x, s))

m∑i,j=1

∞∑l=1

σ ilσ jlφi(x)φj (x)dx ds

−→t∫

0

∫M

�ε(u(x, s))

∞∑l=1

(σ(u(s)) · el

)2 dx ds a.s.


3.3. First conclusions

We summarize the results of the previous sections in the following proposition.

Proposition 3.3. Let σ = σ(u, t) : H × [0, T ] −→ L2(U, H) be essentially bounded in time, continuous in u, {Ft }t≥0-adapted, and such that (2.8) holds, that is

‖σ(u, t)‖L2(U,H) ≤ c(1 + |u|H ) for all t ≥ 0 and u ∈ H. (3.17)

Let b ∈ L2(�; L2(0, T ; H)). Suppose

u ∈ L2(�;L2(0, T ;V )) ∩L2(�;L∞(0, T ;H)), (3.18)

where u =∞∑

j=1

ξjφj with ξj = ξj (t, ω) and φj = φj (x) and that u satisfies (2.11).

Then for a.e. t ∈ [0, T ],

Fε(u(t)) =∫M

(fε(u(x))

)2dx

= Iε0(u)(t) + Iε1(u)(t) + Iε2(u)(t) + Iε3(u)(t) + Iε4(u)(t), (3.19)

where

Iε0(u)(t) = Fε

(u(0)

),

Iε1(u)(t) =t∫

0

∫M

2fε(u(x, s))f ′ε(u(x, s))[σ(u(x, s))dW(s)]dx,

Iε2(u)(t) =t∫

0

∫M

2fε(u(x, s))f ′ε(u(x, s))b(x, s)dx ds,

Iε3(u)(t) = −t∫

0

∫M

2ν�ε(u(x, s))∣∣∇u(x, s)

∣∣2 dx ds,

Iε4(u)(t) =t∫

0

∫M

∞∑i,j=1

∞∑l=1

�ε(u(x, s))φi(x)φj (x)σ il(u(x, s))σ jl(u(x, s))dx ds, (3.20)

with �ε(u) as in (2.18), all integrals being well defined. Furthermore we find that for a.e. t ∈[0, T ],


EFε(u(t)) = EFε

(u(0)

) +E

t∫0

∫M

�ε(u(x, s))b(x, s)dx ds

− 2νE

t∫0

∫M

�ε(u(x, s))∣∣∇u(x, s)

∣∣2 dx ds

+E

t∫0

∫M

�ε(u(x, s))

[ ∞∑l=1

σ(u(x, s)) · el

]2

dx ds, (3.21)

where �ε(u) = 2fε(u)f ′ε(u).

3.4. Passage to the limit as ε → 0

We will now pass to the limit on each term in (3.21) as ε → 0. For E Fε(u(t)), we find that for 0 < t < T

∣∣Fε(u(t)) − F(u(t))∣∣ =

∣∣∣∣ ∫{−ε<u<0}

(3

ε4u5 + 8

ε3u4 + 6

ε2u3

)2

− u2 dx

∣∣∣∣≤

∫{−ε<u<0}

c ε2 dx

≤∫M

c ε2 dx

−→ 0, a.s. as ε −→ 0,

which implies, see (2.15), that for 0 < t < T , as ε −→ 0

EFε(u(t)) −→ EF(u(t)) = E

∫M

∣∣u−(x, t)∣∣2 dx.

We similarly obtain that, as ε −→ 0:

EFε(u(0)) −→ EF(u(0)) = E

∫M

∣∣u−(x,0)∣∣2

dx.

On the other hand, with �ε as in (3.7),

E

t∫ ∫ ∣∣(�ε(u(x, s)) + 2u−(x, s))b(x, s)

∣∣dx ds

0 M


= E

t∫0

∫M

2χ{−ε<u<0}∣∣(fε(u(x, s))f ′

ε(u(x, s)) + u−(x, s))b(x, s)

∣∣dx ds

≤ E

t∫0

∫M

c ε∣∣b(x, s)

∣∣dx ds.

Since b ∈ L2(�; L2(0, T ; H)), we infer that, for 0 < t < T ,

E

t∫0

∫M

�ε(u(x, s))b(x, s)dx ds −→ −2E

t∫0

∫M

u−(x, s)b(x, s)dx ds as ε −→ 0.

We now consider the next term and write:

E

t∫0

∫M

�ε(u(x, s))∣∣∇u(x, s)

∣∣2 dx ds

= E

t∫0

∫M

χ{u<−ε}∣∣∇u(x, s)

∣∣2 dx ds +E

t∫0

∫M

χ{−ε<u<0}�ε(u(x, s))∣∣∇u(x, s)

∣∣2 dx ds.

Due to (2.19), we infer from the Lebesgue dominated convergence theorem that the last term in the right-hand side converges to 0 as ε −→ 0 and the first term converges to

E

t∫0

∫M

χ{u<0}∣∣∇u(x, s)

∣∣2 dx ds. Hence, by the properties of the truncations, see [38]:

−2νE

t∫0

∫M

�ε(u(x, s))∣∣∇u(x, s)

∣∣2 dx ds −→ −2νE

t∫0

∫M

∣∣∇u−(x, s)∣∣2 dx ds

as ε −→ 0.

We similarly observe that

E

t∫0

∫M

�ε(u(x, s))

[ ∞∑l=1

σ(u(x, s)) · el

]2

dx ds

= E

t∫0

∫M

χ{u<−ε}[ ∞∑

l=1

σ(u(x, s)) · el

]2

dx ds

+E

t∫ ∫χ{−ε<u<0}�ε(u(x, s))

[ ∞∑l=1

σ(u(x, s)) · el

]2

dx ds. (3.22)

0 M


Since σ ∈ L2(0, T ; L2(U, H)), by the Lebesgue dominated convergence theorem, when ε −→ 0, this sum converges to

E

t∫0

∫M

[ ∞∑l=1

σ(−u−(x, s)) · el

]2

dx ds. (3.23)

We obtain from the above argument that an analogue of (3.21) holds for F = F(u).

Proposition 3.4. Let σ = σ(u, t) : H × [0, T ] −→ L2(U, H) be essentially bounded in time, continuous in u, {Ft }t≥0-adapted, and satisfies (2.8). Let u ∈ L2(�; L2(0, T ; V )) ∩L2(�; L∞(0, T ; H)) where u =

∞∑j=1

ξjφj with ξj = ξj (t, ω) and φj = φj (x) that satisfies (2.11)

and let b ∈ L2(�; L2(0, T ; H)). Then for a.e. t ∈ [0, T ],

E

∫M

∣∣u−(x, t)∣∣2 dx = E

∫M

∣∣u−0 (x)

∣∣2 dx − 2E

t∫0

∫M

u−(x, s)b(x, s)dx ds

− 2νE

t∫0

∫M

∣∣∇u−(x, s)∣∣2 dx ds

+E

t∫0

∫M

[ ∞∑l=1

σ(−u−(x, s)) · el

]2

dx ds. (3.24)

4. Generalizations

We now present various natural generalizations of Propositions 3.3 and 3.4 to various other equations and boundary conditions. All conclusions appear in Theorem 4.1 below.

4.1. More general elliptic operators

We first extend the results of the previous section by generalizing equation (2.11), replacing −� with a more general elliptic operator. We consider an elliptic operator

A= − ∂

∂xi

(aij (x)

) ∂

∂xj

, (4.1)

where (aij ) is symmetric. We assume that A is coercive, that is, there exists a constant α > 0such that

aij (x)ξiξj ≥ α|ξ |2 for a.e. x ∈M and for all ξ ∈ Rd . (4.2)

Here and below we use the Einstein convention of summation of repeated indices, i, j = 1, . . . , d .


We also suppose that the aij are uniformly bounded on M so that there exists a constant M > 0 such that

|aij (x)ξiηj | ≤ M|ξ ||η| for a.e. x ∈ M, and for all ξ, η ∈Rd . (4.3)

We denote by A the abstract operator from V = H 10 (M) into V ′ associated with A and the

Dirichlet boundary condition. We restrict ourselves for the moment to the Dirichlet boundary condition, but we show below how the results extend to more general boundary conditions. We also denote by {φj }j∈N the set of eigenvectors of A satisfying

Aφj = λjφj j = 1,2, . . . ,

where 0 < λ1 ≤ λ2 ≤ · · · , and φj ∈ V . We now consider u which satisfies equation (2.11) with this new definition of A. To generalize the results of Proposition 3.3 to this case we proceed exactly as before, but now the φj are the eigenvectors of this new operator A as indicated above, and Pm is the orthogonal projection in H onto Hm = span{φ1, . . . , φm}. Hence um satisfies (3.1)with a different definition for A and Pm.

Remark 4.1. From the coercivity of A we have, for a.e. x, t and a.s.:

m∑i,j=1

aij

∂u

∂xj

∂u−

∂xi

=m∑

i,j=1

aij

(∂u+

∂xj

− ∂u−

∂xj

)∂u−

∂xi

=m∑

i,j=1

−aij

∂u−

∂xj

∂u−

∂xi

≤ −α|∇u−|2≤ 0.

We proceed exactly as before and the only difference for proving the analogue of (3.19) is in the proof that Im

ε3(t) converges to Iε3(t). Since fε(u) = 0 and �ε(u) = 0 if u ≥ 0 we note that

Imε3(t) = −

t∫0

∫M

2fε

(um(x, s)

)f ′

ε

(um(x, s)

)νAum(x, s)dx ds

= −2ν

t∫0

∫M

∂

∂xi

(fε

(um(x, s)

)f ′

ε

(um(x, s)

))aij (x)

∂

∂xj

um(x, s)dx ds

= −2ν

t∫0

∫M

�ε

(um(x, s)

)aij (x)

∂um

∂xi

∂u−m

∂xj

dx ds

= −2ν

t∫ ∫�ε

(um(x, s)

)aij (x)

∂u−m

∂xi

∂u−m

∂xj

dx ds, (4.4)

0 M


where, as in (2.18),

�ε(u) = �ε(u(x)) = fε(u(x))f ′′ε (u(x)) + (

f ′ε(u(x))

)2.

Similarly,

Iε3(t) = −2ν

t∫0

∫M

�ε

(u(x, s)

)aij (x)

∂u−

∂xi

∂u−

∂xj

dx ds.

We know, as observed in (3.15), that up to extracting a subsequence still denoted m,

�ε(um(x, t,ω)) −→ �ε(u(x, t,ω)) for a.e. x, t, and ω. (4.5)

We then obtain

∣∣∣∣t∫

0

∫M

�ε

(um(x, s)

)aij (x)

∂u−m

∂xi

∂u−m

∂xj

− �ε

(u(x, s)

)aij (x)

∂u−

∂xi

∂u−

∂xj

dx ds

∣∣∣∣≤

t∫0

∫M

∣∣∣∣�ε

(um(x, s)

) − �ε

(u(x, s)

)∣∣∣∣∣∣∣∣aij (x)∂u−

∂xi

∂u−

∂xj

∣∣∣∣dx ds

+t∫

0

∫M

|aij (x)|∣∣∣∣�ε

(um(x, s)

)∣∣∣∣∣∣∣∣∂u−m

∂xi

∂u−m

∂xj

− ∂u−

∂xi

∂u−

∂xj

∣∣∣∣dx ds

≤ M

t∫0

∫M

∣∣∣∣�ε

(um(x, s)

) − �ε

(u(x, s)

)∣∣∣∣∣∣∇u−(x, s)∣∣2 dx ds

+ c2M

t∫0

∫M

(∣∣∇u−m(x, s)

∣∣ + ∣∣∇u−(x, s)∣∣)∣∣∇u−

m(x, s) − ∇u−(x, s)∣∣dx ds,

where c2 is as described in (2.19). We infer from the previous result that, as m −→ ∞

t∫0

∫M

∣∣∣∣�ε

(um(x, s)

) − �ε

(u(x, s)

)∣∣∣∣∣∣∇u−(x, s)∣∣2 dx ds −→ 0 a.s., (4.6)

using the Lebesgue dominated convergence theorem and the fact that �ε is bounded by c2. On the other hand,


t∫0

∫M

(∣∣∇u−m(x, s)

∣∣ + ∣∣∇u−(x, s)∣∣)∣∣∇u−

m(x, s) − ∇u−(x, s)∣∣dx ds

≤ (using Schwarz inequality and |∇um|H ≤ |∇u|H )

≤ 2

( t∫0

∣∣∇u(s)∣∣2H

ds

) 12( t∫

0

∣∣∇u−m(s) − ∇u−(s)

∣∣2H

ds

) 12

.

Since Pm is also an orthogonal projection in H 10 (M), as in (3.14), we obtain that

t∫0

∣∣∇u−m − ∇u−∣∣2

Hds −→ 0 a.s. (4.7)

We then find from (4.6) and (4.7) that

Imε3(t) −→ Iε3(t) as m −→ ∞.

In conclusion Proposition 3.3 extends to these operators A which are more general than −�

provided (4.2) and (4.3) hold.

Remark 4.2. To show (4.7) we used the fact that the mappings u −→ u+, u−, |u| are continuous from H 1(M) into itself and from H 1

0 (M) into itself. We also use below the fact that these mappings are continuous from H 1

�1(M) into itself, where

H 1�1

(M) := {u ∈ H 1(M) : u = 0 on �1 with �1 ⊂ ∂M}.

4.2. More general boundary conditions

The results above extend also to more general boundary conditions. Indeed perusing the proof of Proposition 3.3 (in both cases A = −�, and general A), we see that we only use the boundary conditions in the treatment of the term Im

3 , more precisely in the integration by parts based on the Stokes formula in (4.4) and then the boundary term on ∂M vanishes. Any other classical boundary condition for which the boundary term vanishes and which will yield the same results, e.g. Neumann boundary condition:

aij

∂u

∂xj

ni = 0, (4.8)

where n = (n1, . . . , nd) is the outward unit normal vector on ∂M. We might also consider a mixed boundary condition such as⎧⎨⎩

u = 0 on �1,

∂u

∂nA:=aij

∂u

∂xj

ni = 0 on �2,(4.9)

where �1 and �2 are two open components of ∂M such that ∂M = �1 ∪ �2.


Proposition 3.3 extends to all these cases, the space V being now H 1�1

(M) defined in Re-mark 4.2.

4.3. Another function Fε

The previous results are used below to prove that the solutions of (2.11) are non-negative under certain assumptions. One may want also to prove that the solutions are a.e. and a.s. bounded by a number N > 0. For that purpose, instead of Fε and F we will consider

Fε(u) = Fε(N − u), F (u) = F(N − u). (4.10)

We then find that the results similar to those in Proposition 3.3 can be proved exactly in the same way, that is

Fε(u(t)) =∫M

(fε(N − u(x))

)2dx

= Jε0(u)(t) + Jε1(u)(t) + Jε2(u)(t) + Jε3(u)(t) + Jε4(u)(t), (4.11)

where

Jε0(u)(t) = Fε

(u(0)

),

Jε1(u)(t) =t∫

0

∫M

2fε(u(x, s))fε′(u(x, s))[σ(u(x, s))dW(s)]dx,

Jε2(u)(t) =t∫

0

∫M

2fε(u(x, s))fε′(u(x, s))b(x, s)dx ds,

Jε3(u)(t) = −t∫

0

∫M

2ν�ε(u(x, s))aij

∂u

∂xi

∂u

∂xj

dx ds,

Jε4(u)(t) =t∫

0

∫M

∞∑i,j=1

∞∑l=1

�ε(u(x, t))φi(x)φj (x)σ il(u(x, t))σ jl(u(x, t))dx ds, (4.12)

where fε(u) = fε(N − u), and �ε(u) = fε(u)f ′′ε (u) + (

f ′ε(u)

)2. Moreover we see that for a.e. t ∈ [0, T ],

E

∫M

(fε(N − u(x, t))

)2dx = E

∫M

(fε(N − u0(x))

)2dx +E

t∫0

∫M

�ε(u(x, s))b(x, s)dx ds

− 2νE

t∫ ∫�ε(u(x, s))aij

∂u

∂xi

∂u

∂xj

dx ds

0 M


+E

t∫0

∫M

�ε(u(x, s))

[ ∞∑l=1

σ(u(x, s)) · el

]2

dx ds, (4.13)

where

�ε(u) = 2fε(u)f ′ε(u). (4.14)

4.4. Parabolic equations perturbed by a nonlinear noise: the main result

We infer from the previous arguments the main theorem below for stochastically perturbed parabolic equations.

Theorem 4.1. Let σ : [0, T ] × H −→ L2(U, H) be given as in Proposition 3.3 satisfying (2.8)and let b be given in L2(�; L2(0, T ; H)). Suppose that

u ∈ L2(�;L2(0, T ;V )) ∩L2(�;L∞(0, T ;H)), (4.15)

and that u satisfies (2.11), where A corresponds to the differential operator A in (4.1)–(4.3)associated with the boundary conditions (4.9).

Then for a.e. t ∈ [0, T ],

Fε(u(t)) =∫M

(fε(u(x))

)2dx

= Iε0(u)(t) + Iε1(u)(t) + Iε2(u)(t) + Iε3(u)(t) + Iε4(u)(t),

where

Iε0(u)(t) = Fε

(u(0)

),

Iε1(u)(t) =t∫

0

∫M

2fε(u(x, s))f ′ε(u(x, s))[σ(u(x, s))dW(s)]dx,

Iε2(u)(t) =t∫

0

∫M

2fε(u(x, s))f ′ε(u(x, s))b(x, s)dx ds,

Iε3(u)(t) = −t∫

0

∫M


∂u

∂xi

∂u

∂xj

dx ds,

Iε4(u)(t) =t∫

0

∫M

∞∑i,j=1

∞∑l=1

�ε(u(x, s))φi(x)φj (x)σ il(u(x, s))σ jl(u(x, s))dx ds, (4.16)

all integrals being well defined. Furthermore we find that for a.e. t ∈ [0, T ],


EFε(u(t)) = EFε

(u(0)

) +E

t∫0

∫M

2fε(u(x, s))f ′ε(u(x, s))b(x, s)dx ds

−E

t∫0

∫M


∂u

∂xi

∂u

∂xj

dx ds

+E

t∫0

∫M

�ε(u(x, s))

[ ∞∑l=1

σ(u(x, s)) · el

]2

dx ds. (4.17)

In addition if Fε = Fε(u) = Fε(N −u), for N > 0 given, then the relations (4.11), (4.12), (4.13), and (4.14) hold as well. Furthermore for a.e. t ∈ [0, T ]

E

∫M

∣∣u−(x, t)∣∣2 dx = E

∫M

∣∣u−0 (x)

∣∣2 dx − 2E

t∫0

∫M


− 2νE

t∫0

∫M

aij

∂u−

∂xi

∂u−

∂xj

dx ds

+E

t∫0

∫M

[ ∞∑l=1

σ(−u−(x, s)) · el

]2

dx ds. (4.18)

Remark 4.3. We can also prove the analogue of (4.18) for the function F = F (u), which gives for a.e. t ∈ [0, T ]:

E

∫M

∣∣(N − u(x, t))−∣∣2 dx = E

∫M

∣∣(N − u0(x))−∣∣2 dx + 2E

t∫0

∫M

(N − u)−(x, s)b(x, s)dx ds

− 2νE

t∫0

∫M

aij

∂(N − u)−

∂xi

∂(N − u)−

∂xj

dx ds

+E

t∫0

∫M

χ{u≥N}[ ∞∑

l=1

σ(u(x, s)) · el

]2

dx ds. (4.19)

4.5. Stochastic parabolic equations with a polynomial drift and nonlinear noise

We now consider the following stochastic non-linear equations with a polynomial nonlinear-ity,

du +Audt + r(u)dt = b(t)dt + σ(u, t)dW, (4.20)


supplemented with the boundary conditions (4.9), with A given by (4.1)–(4.3), and σ as in The-orem 4.1. Here r(u) denotes a polynomial of odd degree given by

r(u) = a0 + · · · + aquq−1, with aq > 0, q even.

We assume, for 1/p + 1/q = 1 with 1 < p < ∞, that

b ∈ L2(�;L2(0, T ;H)) +Lp

(�;Lp(0, T ;Lp(M))

). (4.21)

One can show the existence of solutions to (4.20), that satisfy

u ∈ L2(�;L2(0, T ;V )) ∩L2(�;L∞(0, T ;H)

) ∩Lq(�;Lq(0, T ;Lq(M))

). (4.22)

The results in Theorem 4.1 can be extended to this case. However the difficulty in the proof is that r(u) and part of b belong to Lp(�; Lp(0, T ; Lp(M))), with p < 2, so that the projection Pm cannot be applied to these terms. We overcome this difficulty by an approximation procedure which relies on the following truncation function

hK(s) =⎧⎨⎩

K if s > K,

s if −K ≤ s ≤ K,

−K if s < −K,

(4.23)

and call uK the solution of

duK +AuKdt + r(hK(uK))dt = hK(b)dt + σdW, (4.24)

with the same boundary and initial conditions as for (4.20). It is clear that r(hK(uK)) and hK(b)

belong to L2(�; L2(0, T ; H)

)and that Theorem 4.1 applies with b replaced by

b = hK(b) − r(hK(uK)).

By using the tools and methods of [21,16,17], one can prove that for any K > 0, there existsuK satisfying (4.15). Furthermore, as K → ∞, uK converges weakly to u in

L2(�;L2(0, T ;V )) ∩L2(�;L∞(0, T ;H)

) ∩Lq(�;Lq(0, T ;Lq(M))

). (4.25)

For instance, in the case r(u) = u3, the choice of q = 4 and p = 43 , allows us to conclude to

the existence of solutions belonging to (4.22), by using the corresponding equation (4.24) with truncated nonlinearity. The results of Theorem 4.1 apply thus to this particular case.


5. Applications

5.1. Application to the perturbed heat equation by a nonlinear noise

We now consider the heat equation in the form

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

du +Audt =b(t)dt + σ(u, t)dW in M,

u=0 on �1,

∂u

∂nA=0 on �2,

u|t=0 =u0,

(5.1)

where, as in (4.9), �1 and �2 are two open parts of ∂M such that ∂M = �1 ∪ �2, and ∂

∂nAwas

defined in (4.9).In general, we assume hereafter, instead of (2.8), that

‖σ(u, t)‖L2(U,H) ≤ c′0|u|H , ∀t ≥ 0, ∀u ∈ H, (5.2)

and at times we will assume that σ is globally Lipschitz in u and uniformly in time, that is

‖σ(u∗, t) − σ(u, t)‖L2(U ,H) ≤ c|u∗ − u|H for a.e. t and for all u, u∗ ∈ H. (5.3)

As before we denote by | · |H the norm in H = L2(M), and then analogously to (3.24), we obtain

E|u−(t)|2H = E|u−0 |2H − 2E

t∫0

∫M


− 2E

t∫0

∫M

aij

∂u−

∂xi

∂u−

∂xj

dx ds

+E

t∫0

∫M

[ ∞∑l=1

(σ(−u−) · el

)(x, s)

]2

dx ds. (5.4)

Note that (5.4) is not restricted to σ satisfying (5.2) and actually holds for σ satisfying (3.17); see Section 5.7 for an application under this latter assumption.

Remark 5.1. It is noteworthy that (5.2) is not purely technical as it excludes the case of additive noise for which it is well-known that the maximum principle is not valid and that actually a solution can jump from one metastable state to another, even for a definite sign initial data; see e.g. [18] for the case b(u) = u − u3.


We now assume that

u0 ≥ 0, b ≥ 0 a.e., a.s. (5.5)

Then in view of Remark 4.2 we obtain, for Y(t) = E|u−(t)|2H :

Y(t) ≤ c

t∫0

Y(s)ds. (5.6)

Thanks to Y(0) = 0, we obtain using Gronwall’s lemma that Y(t) = 0 for all t ≥ 0 and hence

u(x, t) ≥ 0 a.e. x, t, a.s. (5.7)

Next, in view of (4.13), we replace equation (5.1)1 by

du + (Au + λu)dt = b dt + σdW, (5.8)

where λ > 0, everything else being unchanged. The earlier results apply, by replacing b by b =b − λu. We then assume that

u0 ≤ N1, b ≤ N2, a.e. and a.s. (5.9)

Then, in view of (4.13), and assuming (5.2), we find that

E|(N − u)−(t)|2H ≤ E|(N − u0)−|2H − 2λE

t∫0

∫M

u(x, s)(N − u(x, s))− dx ds

+ 2E

t∫0

∫M

b(x, s)(N − u(x, s))− dx ds

− 2E

t∫0

∫M

aij

∂(N − u(x, s))−

∂xi

∂(N − u(x, s))−

∂xj

dx ds

+E

t∫0

∫M

χ{u≥N}[ ∞∑

l=1

σ(u(x, s)) · el

]2

dx ds. (5.10)

Now we assume that σ satisfies (2.8) and (5.3) and (see Remark 5.2) that

σ(N, t) = 0, (5.11)

for some N satisfying


N ≥ max(N1,

N2

λ

), (5.12)

with N1 and N2 as in (5.9). Then the last integral in (5.10) is bounded by

cE

t∫0

∫M

|(N − u)−|2 dx ds

and we obtain for Z(t) = E|(N − u)−(t)|2H that

Z(t) ≤ c

t∫0

Z(s)ds,

from which it follows that Z(t) = 0 for all t ≥ 0. We infer from this that

u ≤ N a.e. x, t, a.s. (5.13)

We could similarly prove a lower bound for u, u ≥ −N for N ≥ 0 a.e. and a.s.

Remark 5.2. In [1], L. Arnold introduces, for finite-dimensional SDEs, the concept of a multi-plicative noise with respect to a function g(t) when g(t) solves both the unperturbed equation and the equation with the noise added. This definition can be readily extended to SPDEs and such a situation occurs for the SPDE considered in Section 5.5. Here we could extend this concept by introducing the concept of (sub or) supermultiplicative noise with respect to a function g when g is a (sub or) supersolution of both the unperturbed equation and the equation with the noise added. It is clear in view of (5.12) that g(t) ≡ N is a supersolution of (5.1) when σ = 0. And we just proved that it is a supersolution of (5.1) when σ(N, t) = 0.

5.2. A stochastic Chafee–Infante equation

Let us briefly extend the positivity result to one of the non-linear equations discussed in Sec-tion 4.5, namely ⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

du + (Au + u3 − λu)dt =b(t)dt + σ(u, t)dW, in M,

u=0 on �1,

∂u

∂nA=0 on �2,

u|t=0 =u0.

(5.14)

Here the assumptions on A, b, σ are the same as in Section 4.5, and λ is a real-valued param-eter. Eq. (5.14) can be viewed as a stochastic version of the famous Chafee–Infante equation [8]in the case A = −� and b ≡ 0, known also as the stochastic Ginzburg–Landau equation in the physics literature; see e.g. [44]. If furthermore σ(u) = u, this model defines what is known as the multiplicative noise universality class [10]. If u remains positive (a.e. and a.s.) as long as u0


is, then by means of the Hopf–Cole transformation, u = exp(h), the new field follows a Kardar–Parisi–Zhang (KPZ) equation [34] with a potential V (h) = λh − 2−1e2h [26,46]:

dh = (λ − e2h + �h + (∇h)2)dt + dWt. (5.15)

We refer to [10] for references and a detailed historical account on the KPZ equation. We show below how our approach allows us to ensure in a straightforward fashion that u is positive when-ever u0 is.

Assuming (5.2) and (5.5), we find indeed

E|u−(t)|2H ≤ 2E

t∫0

∫M

u3(x, s)u−(x, s)dx ds + c2E

t∫0

∫M

|u−(x, s)|2 dx ds,

where c = |λ| + c′0, c′

0 as in (5.2). We observe that u3u− ≤ 0 a.e. x, t and a.s. We then obtain again (5.6) and conclude that Y(t) = 0 for all t ≥ 0, which yields

u ≥ 0 a.e. x, t, a.s. (5.16)

5.3. Comparison theorems

As a corollary to the previous results we aim to derive more general comparison theorems of the form u ≤ v.

For that purpose, let us start with SPDEs whose drift term is affine and forced by a stochastic state-dependent nonlinear noise as considered in Theorem 4.1. Let us consider a solution u to the following problem: {

du + Audt =b dt + σ(u, t)dW,

u(0)=u0,(5.17)

We assume in what follows that σ is such that a local solution u of (5.17) is well-defined in the appropriate functional setting.5 We refer to [25] for existence results of such solutions where σis assumed to be locally Lipschitz and to satisfy a local version of the growth condition (2.8). A case of nonlinear noise that is even non-Lipschitz is considered in Section 5.7.

Now let u∗ be the solution of (5.17), emanating from u∗0 �= u0, and corresponding to b∗ �= b.

Our goal is to show, under the assumptions of Theorem 4.1, that if u0 ≤ u∗0 and b ≤ b∗

a.e. and a.s., then u ≤ u∗ a.e. and a.s. First, observe that v = u∗ − u satisfies{dv + Av dt = (b∗ − b)dt + (σ (u∗, t) − σ(u, t))dW,

v(0)=u∗0 − u0,

(5.18)

and that v(0) ≥ 0 and b∗ − b ≥ 0 a.e., a.s. The problem (5.18) is thus analogous to (5.17) in which the stochastic term σ(u, t)dW has been replaced by (σ (u∗, t) − σ(u, t))dW .

5 Pathwise or martingale solution as explained above.


As explained below, it is not difficult to show that the analogues of Propositions 3.3 and 3.4, hold in that case, with u, b, and u0 replaced respectively by v, b∗ − b, and u∗

0 − u0, on one hand, and σ(u) replaced by σ(u∗) − σ(u) = σ(v + u) − σ(u), on the other.

The modification of Proposition 3.3 consists of noting that Iε1 in (3.19), is replaced by Iε1given by

Iε1 = Iε1(v)(t) =t∫

0

∫M

2fε(v(x, s))f ′ε(v(x, s))[σ(u∗(x, s)) − σ(u(x, s))]dW(s)dx, (5.19)

for t ∈ [0, T ].Then the analogue of (3.21) holds and when we want to pass to the limit as ε → 0, we ought

to consider the term similar to (3.22) which reads here as follows:

E

t∫0

∫M

�ε

(v(x, s)

)[ ∞∑l=1

(σ((v + u)(x, s)) − σ(u(x, s))

) · el

]2

dx ds

= E

t∫0

∫M

χ{v<−ε}[ ∞∑

l=1

(σ((v + u)(x, s)) − σ(u(x, s))

) · el

]2

dx ds

+E

t∫0

∫M

χ{−ε<v<0}�ε

(v(x, s)

)[ ∞∑l=1

(σ((v + u)(x, s)) − σ(u(x, s))

) · el

]2

dx ds.

In the passage to the limit as ε → 0, the second term tends to zero, and the first term converges to

E

t∫0

∫M

χ{v<0}[ ∞∑

l=1

(σ((v + u)(x, s)) − σ(u(x, s))

) · el

]2

dx ds,

which is the same as

E

t∫0

∫M

[ ∞∑l=1

(σ((u − v−)(x, s)) − σ(u(x, s))

) · el

]2

dx ds. (5.20)

We now supplement the growth condition (2.8) with the Lipschitz condition (5.3). We, however, do not need here to assume (5.2). We obtain, in the end, instead of (5.4), for a.e. t ∈ [0, T ],

E|v−(t)|2H ≤ cE

t∫|v−(s)|2H ds.

0


With v−(0) = 0 we conclude again that E|v−(t)|2H = 0 for a.e. t ∈ [0, T ], that is, we have

u∗(x, t) ≥ u(x, t) for a.e. x ∈ M, and t ∈ [0, T ], and a.s. (5.21)

Corollary 5.1. Let u and u∗ be the solutions of (5.17) emanating respectively from u0 and u∗0,

and driven respectively by b and b∗. Under the assumptions of Theorem 4.1, if we assume fur-thermore (5.3) and that

u0 ≤ u∗0 and b ≤ b∗ a.e., a.s., (5.22)

we then have

(u∗ − u)− = 0 a.e. x, t, and a.s.,

that is

u ≤ u∗ a.e. x, t, and a.s.

Remark 5.3. The conclusion of Corollary 5.1 holds for the case where b = f (u), b∗ = g(u), with f (u) ≤ g(u), assuming that equation (5.17) has solutions with such reaction terms. We obtain that if u0 ≤ u∗

0 a.e. and a.s., then u ≤ u∗ a.e. and a.s.

We can also consider an equation like (4.20) with an odd nonlinear drift term. For (4.20), all what we need to observe is that

(r(u∗) − r(u)

)(u∗ − u) ≥ −λ|u∗ − u|2

for some λ > 0 such that r(u) + λu is monotone, that is e.g.

λ

2≥ min

u∈R( − r(u)

).

Then the analogue of the inequality (5.6) is replaced by

Y(t) ≤ (c + λ)

t∫0

Y(s)ds,

and the conclusion follows similarly, i.e.

Y(t) = E∣∣(u∗ − u)−(t)

∣∣H

= 0,

which yields to

u∗ ≥ u a.e. x, t, a.s. (5.23)


5.4. Removal of the nonlinear drift: comparison and stabilization results

We go back to the stochastic Chafee–Infante equation of Section 5.2 that we consider for simplicity in the following one-dimensional setting (compare also with (4.20)):⎧⎪⎨⎪⎩

du = (uxx + λu − u3)dt + σ(u)dW, 0 < x < l, t > 0,

u(0, t) = u(l, t) = 0, t > 0,

u(x,0) = u0(x), 0 < x < l,

(5.24)

where λ ∈R and φx = ∂φ/∂x, as above. We denote by u(x, t, ω; u0) the solution of (5.24) (which can be shown to exist). The results above show that

If u0 ≥ 0 a.e. x ∈ (0, l) and a.s., then u ≥ 0 a.e. x ∈ (0,1), t > 0, and a.s. (5.25)

We consider now the following stochastic equation in which the nonlinear drift term in (5.24)has been dropped, ⎧⎨⎩

du = (uxx + λu)dt + σ(u)dW, 0 < x < l, t > 0,

u(0, t) = u(l, t) = 0, t > 0,

u(x,0) = u0(x), 0 < x < l,

(5.26)

and whose solution is denoted by uL = uL(x, t, ω; u0).We assume hereafter that σ satisfies the Lipschitz hypothesis (5.3), and call c = k this Lips-

chitz constant. Observing that −u3 ≤ 0 whenever u ≥ 0, one can, with the comparison result of Section 5.3 show that

uL(x, t,ω;−|u0|) ≤ u(x, t,ω;u0) ≤ uL(x, t,ω; |u0|), (5.27)

for a.e. x ∈ (0, l) and t > 0, and a.s., where |u0| just represents the absolute value of u0, |u0|(x) = |u0(x)|. For instance to prove the right inequality of (5.27), it is sufficient to note that the order-preserving property u(x, t, ω; u0) ≤ u(x, t, ω; |u0|) holds (a.e. and a.s.) since u0 ≤ |u0|, and that u(x, t, ω; |u0|) ≤ uL(x, t, ω; |u0|) by the comparison result of Section 5.3 since u ≥ 0(and thus −u3 ≤ 0) is ensured here following Section 5.2.

Let λ1 be the first eigenvalue of −∂xx on (0, l), that is,

|u|L2(0,l) ≤ λ−11 |ux |L2(0,l), ∀u ∈ H 1

0 (0, l).

Our stochastic framework being compatible with that of [9], we can check that Condi-tion (e3.2) of [9] is satisfied with α = 2, and λ therein replaced by 2λ + k2, allowing us thus to apply [9, Theorem 2.2]. As a consequence, if

(k2 + 2λ) < 2λ21, (5.28)

then there exist constants C > 0, and γ > 0 such that

E|uL(t)|2 2 ≤ Ce−γ t , (5.29)
L (0,l)


and hence, in view of (5.27), we infer that

E|u(t)|2L2(0,l)≤ Ce−γ t . (5.30)

A result like (5.30) has an interesting consequence in term of stabilization. For that purpose, let us recall that when σ ≡ 0 the semigroup associated with Eq. (5.24) possesses at least two locally stable non-zero equilibria when λ > λ1 while the solution u ≡ 0 is unstable; see e.g. [8,29].

Therefore (5.30) shows that any nonlinear noise for which (5.28) is satisfied while λ > λ1, has a stabilizing effect since E|u(t)|2L2(0,l)

−→ 0 as t → ∞. In particular, such a stabilization

effect takes place when l < π (ensuring λ1 > 1), λ = λ1(1 + ε) and ε > 0 and k2 are sufficiently small, i.e. above but near the criticality and for a nonlinear noise with a small Lipschitz constant. For stabilization results of the Chafee–Infante equation perturbed by a sum of multiplicative Stratonovich linear terms, we refer to [6].

5.5. The stochastically perturbed temperature Boussinesq equation

We consider the domain M = (0, L1) × (0, L2) where the direction Oy is the vertical ascend-ing direction. The Boussinesq equation in M consists in finding the velocity u = (u1, u2), the pressure p, and the temperature θ satisfying

⎧⎪⎪⎪⎨⎪⎪⎪⎩∂u∂t

+ (u · ∇)u − ν�u + ∇p = e2(θ − θ2),

∂θ

∂t+ (u · ∇)θ − κ�θ = 0,

∇ · u = 0,

(5.31)

together with suitable initial and boundary conditions; in particular θ2 is the given temperature at the top, y = L2, and θ1, with θ1 ≥ θ2, is the given temperature at the bottom, y = 0. Here κ and νare positive constants and e2 is the unit vector on Oy. See e.g. [45,23] for the functional setting of these equations. We also assume, for instance, periodicity in the x-direction. The stochastic version of this equation is studied in [16]. Although this problem is not mentioned explicitly, the general framework considered in [16] applies to the 2D Boussinesq equation: The existence of global martingale or pathwise solutions is established in [16], which are also strong in the PDE sense. We want to establish that if the initial temperature θ0 satisfies

θ2 ≤ θ0 ≤ θ1

a.e. and a.s., then, for all times

θ2 ≤ θ ≤ θ1 a.e. and a.s. (5.32)

Remark 5.4. The result (5.32), interesting by itself, is also essential for proving the existence of a global stochastic attractor, an issue that we do not address here; see related issues in e.g. [23,45,5] in the deterministic context and in e.g. [7] and [24] in the stochastic context.


In order to prove (5.32) we only consider the stochastic heat equation, assuming that u is known. Hence

dθ + ((u · ∇)θ − ν�θ

)dt = σ(θ)dW, (5.33)

where σ satisfies (5.2) and (5.3), with H = L2(M). Then, changing notations, we apply (5.4)to (θ − θ2)

−. We observe that the boundary terms in (5.4) disappear because (θ − θ2)− = 0 at

y = 0 and y = L2, as θ1 ≥ θ2 and by periodicity in the x-direction. Assuming furthermore that the noise is multiplicative with respect to the constant functions θ1 and θ2, that is (see [1] and Remark 5.2),

σ(θ2, t) = σ(θ1, t) = 0,

we arrive at

E|(θ − θ2)−(t)|2H ≤ E|(θ0 − θ2)

−|2H + 2E

t∫0

∫M

[(u · ∇)θ

](θ − θ2)

− dx ds

+ cE

t∫0

∫M

|(θ − θ2)−|2 dx ds. (5.34)

The first term in the right-hand side of (5.34) vanishes, as well as the second one which is equal to

E

t∫0

∫M

(u · ∇)|(θ − θ2)−|2 dx ds,

and we see by integration by parts that this term vanishes because ∇ · u = 0 and because of the boundary conditions in the x- and y-directions. We note that this integration by part is legitimate for the strong solutions and can be proven by approximation for the weak solutions. We arrive at an equation identical to (5.6), with now Y(t) = E|(θ − θ2)

−(t)|2 and we conclude similarly that Y(t) = 0 for all t ≥ 0, and hence

θ ≥ θ2 a.e., and a.s.

The proof of θ ≤ θ1 is similar, considering now (θ1 − θ)−.

5.6. Application to a harvesting model arising in population dynamics

We now consider an application to the following stochastic version of harvesting models aris-ing in population dynamics [12,40,41]:

du − ∂(

aij

∂u)

dt + u(ν(x)u − μ(x))dt + δh(x)ρε(u)dt = σ(u) ◦ dW(t). (5.35)
∂xi ∂xj


This equation differs from the classical Fisher equation [22] (also known as the Kolmogorov–Petrovsky–Piskunov equation [33]), by its spatially-dependent coefficients, μ(x), ν(x) and h(x), and its state-dependent noise term, σ(u) ◦ dWt , understood here in the sense of Stratonovich.

The domain M is assumed to be a sufficiently smooth bounded domain of RN (N ≥ 1) so that the Neumann boundary condition holds:

∂u

∂n= 0, on ∂M, (5.36)

where n denotes the outward unit normal to ∂M.Following [12,40], we assume that μ, ν, h with 0 ≤ h ≤ 1 are deterministic functions that

belong to L∞(M), with ν ≥ 0. We also assume that δ is a positive constant and that ρε is a C1

function such that

ρε ∈ C1(R), ρ′ε ≥ 0, ρε(s) = 0, ∀s ≤ 0, and ρε(s) = 1 ∀s ≥ ε, (5.37)

so that ρε is a C1-approximation of the Heaviside function.The last term in the drift part of (5.35)–(5.36), δh(x)ρε(u), corresponds to a quasi-constant-

yield harvesting term. With such a harvesting function, the yield is constant in time whenever u ≥ ε, when σ ≡ 0. In [40], conditions for which solutions to (5.35)–(5.36) remain larger than ε, have been identified when σ ≡ 0.

The state-dependent noise term, σ(u) ◦ dWt , accounts typically for growth fluctuations and intraspecific competition effects which are not modeled properly by the nonlinear term alone, N (u, x) := u(μ(x) − ν(x)u). We refer to [12,40,41] for more details about the biological inter-pretation of models such as (5.35), when σ ≡ 0 and to [11] for further analysis regarding the biological implications when σ �≡ 0.

We focus hereafter on the existence of a positive solution to (5.35)–(5.36) when u0 ≥ 0 a.e., a.s., which will be an important ingredient in the analysis of (5.35) in [11]. In particular, such a result ensures the relevance of such stochastic harvesting models, since u representing a density of population, σ has to be chosen so that u remains positive when u0 is. We show hereafter that this is the case when σ satisfies (5.2).

For that purpose, we first rewrite the SPDE (5.35), in the Itô sense, namely

du − ∂

∂xi

(aij

∂u

∂xj

)dt + u(ν(x)u − μ(x))dt + δh(x)ρε(u)dt = 1

2σ ′(u)σ (u)dt + σ(u)dW(t).

Here aij is as in (4.2) and (4.3), and σ ′(u) denotes the differential of σ with respect to u.To show the existence of a positive solution to (5.35)–(5.36) when u0 ≥ 0 a.e., a.s., we con-

sider first the following substitutive equation

du − ∂

∂xi

(aij

∂u

∂xj

)dt + u(ν(x)|u| − μ(x))dt + δh(x)ρε(u)dt

= 1

2σ ′(u)σ (u)dt + σ(u)dW(t). (5.38)

For this modified equation, the nonlinear part has now good sign properties as u → ±∞, and the existence of solutions to (5.38) supplemented by the boundary condition (5.36) can be


ensured by standard techniques as for (5.14); see also Section 4.5. We then find with some minor modifications that

E|u−(t)|2H ≤ E|u−0 |2H −E

t∫0

∫M

δh(x)ρε(u)u−(x, s)dx ds

+ (|μ|L∞ + c′)Et∫

0

∫M

|u−(x, s)|2 dx ds. (5.39)

With δ, h, ρε ≥ 0 and assuming that u0 ≥ 0 a.e., a.s., we arrive at

Y(t) ≤ (|μ|L∞ + c′)t∫

0

Y(s)ds,

where Y(t) = E|u−(t)|2H again. Hence, the Gronwall lemma implies that Y(0) = 0 which means

u ≥ 0 a.e. x, t, a.s.

From this we deduce that |u| = u a.e. x, t , a.s., so that equation (5.38) is the same as equa-tion (5.35) when u0 ≥ 0, and in the end we have found a positive solution u of (5.35)–(5.36)such that u lives in

L2(�;L∞(0, T ;L2(M))) ∩L2(�;L2(0, T ;H 1(M))

)∩L3(�;L3(0, T ;L3(M))). (5.40)

5.7. Non-Lipschitz multiplicative noises

We consider here the case when σ(u) = √u for u ≥ 0. We could also similarly consider

σ(u) = √u(1 − u) for 0 ≤ u ≤ 1; both cases are known as Wright–Fisher models arising in

population dynamics; see e.g. [36,37].Interestingly, the case b(u) = λu − u2 (compare to (5.35)) and σ(u) = √

u is not limited to population dynamics and corresponds actually to the Reggeon-Field Theory (RFT) model of directed percolation; see [30,44] and references therein. The RFT is known to describe phase transitions in a vast class of systems with absorbing states.

Motivated by such problems, we consider below the existence of positive solutions for σ(u) =√u+. The existence of a solution can be obtained from [16] whose general framework applies;

see Appendix A for the details. Since σ satisfies (3.17), we can apply (5.4) and, assuming u0 ≥ 0, b ≥ 0, a.e., a.s., we find that

E|u−(t)|2H ≤ 0.

Indeed for σ(u) = √u+, the term σ(u−) appearing in (5.4) vanishes. Hence u ≥ 0 a.e., and a.s.,

that is u+ = u, and with [16] we have proven the existence of a positive solution for

du + (Au − b)dt = √udt, (5.41)


with the same boundary and initial conditions as in (5.1). When A = −�, [48] proves the ex-istence and uniqueness of a solution when σ(u) = √|u|. Combining these results we have the existence and uniqueness of a positive solution to (5.41). We note that we do not describe explic-itly here the classes of solutions.

Remark 5.5. We note that the procedure above for proving the positivity of u does not apply directly to the solutions obtained in [47,48] using σ(u) = √|u|. Indeed in this context we would obtain instead of (5.6), (

y′(t))2 ≤ c y(t), (5.42)

with y(t) = E

t∫0

|u−(s)|H ds, (y′(t)

)2 = (E|u−(t)|H

)2 ≤ E|u−(t)|2H , and equation (5.42) has a

non-zero solution (y(t) = c t2/4) although y(0) = y′(0) = 0. Hence (5.42) does not guarantee uniqueness in this context and we instead used σ(u) = √

u+.

Remark 5.6. We can similarly consider⎧⎪⎨⎪⎩du − div(k∇u)dt + g(u)dt = σ(u)dW(t),

∂u

∂n= 0 on ∂M,

u(x,0) = u0(x), x ∈M,

(5.43)

where g(u) is e.g. equal to u(u − 1) and σ(u) = u(u − 1). The existence of solutions to (5.43)follows from many sources again, and we want to show that 0 ≤ u ≤ 1 a.e. and a.s. if 0 ≤ u0 ≤ 1a.e. and a.s. For this purpose we replace g by

g(u) = u+(u − 1)+.

The existence of solutions is proven similarly for (5.43) with g replaced by g. Applying now (5.4)with b = −g, we see that g(u)u− = 0 a.e. so that the term involving the nonlinearity g disappears and we arrive again at u− = 0. Similarly, by considering

E∣∣(1 − u)−(t)

∣∣2H

,

the term g(u)(1 − u)− vanishes and we obtain (1 − u)− = 0, showing in the end that we have a solution of (5.43) such that

0 ≤ u ≤ 1, a.e. and a.s.

Acknowledgments

E.P. and R.T. were supported by National Science Foundation grants DMS-1206438 and DMS-1510249, and by the research fund of Indiana University. M.C. was supported by grant N00014-12-1-0911 from the Multi-University Research Initiative of the Office of Naval Re-search, and by the National Science Foundation grant OCE-1243175.


The authors want to warmly thank Chuntian Wang for many helpful discussions at the start of this project.

Appendix A

We mentioned in Section 5.7 that the existence result in [16] does not require σ to be Lips-chitz, and that a continuity hypothesis would suffice. We note however that the Lipschitz property of σ is needed for the proof of uniqueness in [16].

Instead of a general result, we discuss below how the proof of existence in [16] would apply when σ(u) = √

u+. A perusal of the proof in [16] shows that we only need to prove that, if, for k → ∞,

uk −→ u in L2(�;L2(0, T ;L2(M))), (A.1)

then

σ(uk) =√

u+k −→ σ(u) = √

u+ in L2(�;L2(0, T ;L2(M))). (A.2)

Note first that, by extraction of a subsequence, (A.1) implies that

uk −→ u a.e. x and t, and a.s. (A.3)

Now in order to derive (A.2) we write

E

T∫0

∫M

|σ(uk) − σ(u)|2 dx dt

= E

T∫0

∫M

∣∣∣∣√u+k − √

u+∣∣∣∣2

dx dt

= E

T∫0

∫{u+

k <1}

∣∣∣∣√u+k − √

u+∣∣∣∣2

dx dt +E

T∫0

∫{u+

k ≥1}

∣∣∣∣√u+k − √

u+∣∣∣∣2

dx dt.

For the first integral, we observe that

χ{u+k <1}

∣∣∣∣√u+k − √

u+∣∣∣∣2

≤ ∣∣1 + √u∣∣2

,

and since this function is integrable, (A.3) and the Lebesgue dominated convergence theorem imply that the first integral converges to 0 as k → ∞.

The integrand in the second integral is bounded by∣∣u+k − u+∣∣2∣∣√u+k + √

u+∣∣2≤

∣∣u+k − u+∣∣2∣∣1 + √

u+∣∣2≤ ∣∣u+

k − u+∣∣2,

so that by (A.1), the second integral also converges to 0 as k → ∞.


Remark A.1. The reasoning above can be extended to other classes of functions σ : R −→ R

which are continuous and sublinear (hypothesis (2.8)), and satisfy certain Hölder properties.

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