ON THE POSITIVITY OF SOLUTIONS OF SYSTEMS OF STOCHASTIC PDES

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1 ON THE POSITIVITY OF SOLUTIONS OF SYSTEMS OF STOCHASTIC PDES JACKY CRESSON 1,2, MESSOUD EFENDIEV 3, AND STEFANIE SONNER 3,4 On the occasion of the 75 th birthday of Prof. Dr. Dr.h.c. Wofgang L. Wendand (1) Laboratoire de Mathématiques Appiquées de Pau, Université de Pau et des Pays de Adour, avenue de Université, BP 1155, 6413 Pau Cedex, France (2) Institut de Mécanique Céeste et de Cacu des Éphémérides, Observatoire de Paris, 77 avenue Denfert-Rochereau, 7514 Paris, France (3) Hemhotz Zentrum München, Institut für Biomathematik und Biometrie, Ingostädter Landstrasse 1, D Neuherberg, Germany (4) BCAM Basque Center for Appied Mathematics, Mazzaredo 14, 489 Bibao, Spain Abstract. We study the positivity of soutions of a cass of semi-inear paraboic systems of stochastic partia differentia equations by considering random approximations. For the famiy of random approximations we derive expicit necessary and sufficient conditions such that the soutions preserve positivity. These conditions impy the positivity of the soutions of the stochastic system for both Itô s and Stratonovich s interpretation of stochastic differentia equations. 1. Introduction We study the positivity of soutions of systems of semi-inear paraboic equations under stochastic perturbations. The systems are of the form du (x, t) = ( m A i(x, D)u i (x, t) + f (x, t, u(x, t)) ) (1) dt + q j gj(x, t, u(x, t))dw j t, = 1,..., m, where x O, O R n is a bounded domain, and t >. The function u = (u 1,..., u m ) is a vector-vaued, A i are inear eiptic operators of second order, and the noninearity f = (f 1,..., f m ) takes vaues in R m. Moreover, we assume that {W j t, t } j N is a famiy of independent standard scaar Wiener processes on the canonica Wiener space (Ω, F, P), and dw j t denotes the corresponding Itô differentia. The non-negative parameters q j are normaization factors and the functions gj are rea-vaued, = 1,..., m, j N. The boundary conditions are given by the operators (B 1,..., B m ), B (x, D)u (x, t) = x O, t >, = 1,..., m, 1991 Mathematics Subject Cassification. 6H15, 35K6, 47H7, 37L3, 65M6, 35K57, 92C15, 35B5, 92B99. Key words and phrases. Necessary and sufficient conditions, Positive cone, Semi-inear paraboic systems, Stochastic PDEs. 1

2 2 JACKY CRESSON 1,2, MESSOUD EFENDIEV 3, AND STEFANIE SONNER 3,4 and the soution satisfies the initia conditions u (, x) = u (x) x Ō, = 1,..., m. We interpret the stochastic system in the sense of Itô, denote by (A, f, g) stochastic systems of the form (1), and the corresponding unperturbed deterministic system by (A, f, ). Our aim is to derive expicit conditions on the coefficient functions of the operators A i and the functions f and g to ensure that the soutions of System (1) preserve positivity. The expicit characterization is important as it aows to verify mathematica modes arising in various appications, where the soutions describe positive quantities (see [5]). In this case, that is, if soutions emanating from non-negative initia data remain non-negative as ong as they exist, we say that the system satisfies the positivity property. To study the positivity of soutions of the stochastic system we construct a famiy of random PDEs such that its soutions converge in expectation to the soution of the stochastic system. We are in particuar interested in characterizing the cass of stochastic perturbations g such that the famiy of random approximations satisfies the positivity property, which then impies the positivity of soutions of the stochastic system. Moreover, we prove that the positivity is preserved for both, Itôs and Stratonovich s interpretation of stochastic differentia equations (see [7]). Appications that fa into the cass of stochastic modes (1) are for instance predator-prey systems under stochastic perturbations. We give an exampe that was discussed in [2] (Section 5), further appications can be found in [3] (Section 6). The deterministic mode is formuated as reaction-diffusion system for the predator u and the prey v in a bounded spatia domain O R 3, ( ) t u = t v where the matrix D(u, v) is of the form ( ( β1 v ) D(u, v) = u ( ) ( u u + D(u, v), v v) ( cβ 2 v ) ) [ ( u γ β2 v )], and the soutions satisfy homogeneous Neumann boundary conditions. The constants c and γ are positive, and the functions β 1, β 2 : R + R + non-negative. It can be verified by our deterministic positivity criterion (see Theorem 3 in Section 2.2) that the system preserves positivity. The mode incudes a certain uncertainty since it is impossibe to determine the exact mode parameters γ, β 1 and β 2 (see [2]). One possibiity to take this into account is to add noise, which eads to the foowing stochastic mode ( ) ( ) t u u = + [ D(u, v) + dw t v v t Id ] ( ) u (2), v where Id denotes the identity matrix. Our main resut (Theorem 5 in Section 3.2) impies that the positivity of soutions is aso preserved by the stochastic system (2), and this is vaid u

3 ON THE POSITIVITY OF SOLUTIONS OF SYSTEMS OF STOCHASTIC PDES 3 independent of the choice of Itô s or Stratonovich s interpretation of stochastic differentia equations. The outine of the paper is as foows: In the introductory sections we present our probem, expain our strategy to study the positivity property of stochastic systems and formuate our main resut. We specify the cass of deterministic systems in Section 2 and formuate necessary and sufficient conditions for the positivity of soutions of semi-inear paraboic PDEs. The positivity property of stochastic systems is anaysed in Section 3. Our proof is based on the deterministic resut and essentiay uses an approximation theorem for stochastic systems. This random approximation theorem was obtained by Chueshov-Vuiermot in [3] and is recaed in Subsection 3.1. Finay, in Subsection 3.2 we formuate and prove our positivity criterion for stochastic systems Deterministic case. Random equations can be interpreted pathwise and aow to appy deterministic methods. The random approximations ead to a famiy of non-autonomous paraboic systems. For autonomous deterministic systems of quasi-inear and semi-inear PDEs necessary and sufficient conditions for the positivity of soutions were obtained in [5]. Generaizing the resuts for non-autonomous deterministic systems of the form (A, f, ) we deduce the foowing criterion (see Theorem 3 in Section 2). Theorem 1. Under appropriate conditions on the coefficient functions of the operators A and B the deterministic system (A, f, ) satisfies the positivity property if and ony if the differentia operators are diagona, and the components of the interaction function satisfy f (x, t, u 1,..., u 1,, u +1,..., u m ) for x O, t, u k, k, = 1,..., m. This theorem yieds expicit conditions on the coefficients of the differentia operator A and the interaction function f, which are easy to verify, and aows to vaidate mathematica modes (see [8], Section 4.3, p.76, and [5]). As a consequence, it suffices to consider stochastic systems with diagona differentia operators ( ) (3) du (x, t) = A (x, D)u (x, t) + f (x, t, u(x, t)) dt + q j gj(x, t, u(x, t))dw j t, = 1,..., m, where x O and t >. In the seque we denote by (f, g) the system of SPDEs (3) and the corresponding unperturbed deterministic system by (f, ) Stochastic case. To anayse the stochastic probem (f, g) with diagona differentia operator A we use a Wong-Zakaï type approximation theorem, which was obtained by Chueshov- Vuiermot in [3], and yieds a famiy of random approximations (f ɛ,ω, ) for the stochastic system. The soutions of the random approximations do not converge to the soution of the

4 4 JACKY CRESSON 1,2, MESSOUD EFENDIEV 3, AND STEFANIE SONNER 3,4 origina system, but to the soution of a modified stochastic system. Hence, we first construct in Section 3.2 an auxiiary stochastic system (F, g) such that the soutions of the corresponding random approximations (F ɛ,ω, ) converge to the soution of our origina probem (f, g). We appy the deterministic resut to derive expicit necessary and sufficient conditions for the positivity property of the random systems (F ɛ,ω, ). Since these conditions are preserved by the random approximations and are invariant under the transformation reating the origina and the modified system, our main resut yieds an expicit characterization of the stochastic perturbations g and interaction functions f to ensure that the stochastic system (f, g) satisfies the positivity property. Furthermore, the conditions for the positivity property of the random PDEs are invariant under the transformation reating the equations obtained through Itô s and Stratonovich s interpretation. Our main resut, stated in the foowing theorem, is therefore independent of the choice of interpretation: Theorem 2. Let (f, g) be a system of stochastic PDEs of the form (3), which is interpreted in the sense of Itô or Stratonovich. Then, the associated famiy of random approximations (F ɛ,ω, ) satisfies the positivity property if and ony if the interaction term satisfies f (x, t, u 1,..., u 1,, u +1,..., u m ) x O, t, for u k, and the stochastic perturbation g fufis gj(x, t, u 1,..., u 1,, u +1,..., u m ) = x O, t, for u k, for a j N and k, = 1,..., m. positivity property. In this case, the stochastic system (f, g) satisfies the Remark 1. Up to the author s knowedge for systems of stochastic PDEs ony sufficient conditions for the positivity of soutions are known. Initiay, we were hoping to obtain a stronger resut. Namey, that the stochastic system (f, g) satisfies the positivity property if and ony if the stochastic perturbation g and the interaction function f satisfy the conditions in Theorem 2. P. Koteenez proved this equivaence in [6] for scaar paraboic equations. His proof is not based on random approximations. Whie the sufficiency of the conditions is formuated in our theorem, showing the necessity is more invoved since we cannot deduce the non-negativity of soutions of the random approximations from the non-negativity of the soutions of the stochastic system. Hence, we cannot directy appy the necessary conditions known in the deterministic case. 2. The Deterministic Case - Necessary and Sufficient Conditions for Positivity

5 ON THE POSITIVITY OF SOLUTIONS OF SYSTEMS OF STOCHASTIC PDES Semi-inear Systems of Paraboic PDEs. Necessary and sufficient conditions for autonomous systems of semi-inear and quasi-inear reaction-diffusion-convection-equations were studied in [5]. We cannot directy appy these resuts in our case since the Wong- Zakaï approximations ead to a famiy of paraboic systems with time-dependent interaction functions. In this section we formuate a generaization of one of the theorems in [5] aowing for non-autonomous interaction functions and arbitrary inear eiptic differentia operators of second order. For its proof we refer to [4], which uses the same methods and ideas as appied in the mentioned artice. To be more precise, we consider the foowing cass of systems of semi-inear paraboic equations m (4) t u (x, t) = A i(x, D)u i (x, t) + f (x, t, u(x, t)) x O, t >, where u = (u 1,..., u m ) is a vector-vaued function, and O R n, n N, is a bounded domain with smooth boundary O. Assumptions on the operator A The inear second order differentia operators A i (x, D) are defined as n n A i(x, D) = a i kj (x) x k xj + a i k (x) x k for x O, i, = 1,..., m. k, k=1 Compared to the setting in [3] we omit the zero-order terms in the operator A as for our probem it seems more natura to absorb these terms in the interaction function f. We assume that the coefficient functions satisfy a i kj = ai jk, and the operators are uniformy eiptic, n a i kj (x)ζ kζ j µ ζ 2 for a x O, ζ R n, i, = 1,..., m. k, Moreover, a coefficient functions of the operator A are continuousy differentiabe and bounded in the domain O. Assumptions on the boundary operators B The boundary vaues of the soution are determined by the operators n B (x, D) = b (x) + δ b k (x) x k = 1,..., m, k=1 where δ {, 1}. The functions b k, b are smooth on the boundary O and satisfy b. Moreover, we assume b 1 for δ =, and b = (b 1,..., b m) is an outward pointing, nowhere tangent vector-fied on the boundary O.

6 6 JACKY CRESSON 1,2, MESSOUD EFENDIEV 3, AND STEFANIE SONNER 3,4 Assumptions on the non-inear interaction term f For the interaction function we assume that the partia derivatives u f exist and are continuous, = 1,..., m. Moreover, we assume that for x O and t > the functions f = f (x, t, u) and u f = u f (x, t, u) are bounded for bounded vaues of u A Positivity Criterion. To formuate our criterion for the positivity of soutions we define the positive cone in L 2 (O; R m ). Definition 1. The positive cone is the set of a componentwise non-negative functions in L 2 (O; R m ), K + := {u L 2 (O; R m ) u i a.e. in O, i = 1,..., m}. Furthermore, we say that System (4) satisfies the positivity property if every soution u(, ; u ) : O [, T ] R m originating from non-negative initia data u K + remains non-negative (as ong as it exists); that is, u(, t; u ) K + for t [, T ]. Thereby, [, T ] denotes the maxima existence interva of the soution. Our concern is not to study the existence of soutions but their quaitative behaviour. Hence, in the seque we assume that for any initia data u K + there exists a unique soution, and for t > the soution satisfies L -estimates. The foowing theorem provides a criterion for the positivity property of System (4) and generaizes the previous resut for semi-inear systems in [5]. Theorem 3. Let the operators A and B be defined as in the beginning of this section and the above conditions on the coefficient functions of the operators and interaction functions be satisfied. Moreover, we assume the initia data u K + is smooth and fufis the compatibiity conditions. Then, System (4) satisfies the positivity property if and ony if the matrices ( ) a i kj 1 i, m and ( a i ) k are diagona for a 1 j, k n, and the components of the 1 i, m reaction term satisfy (5) f i (x, t, u 1,..., }{{},..., u m ), for x O, t >, u k, i, k = 1... m. i This theorem can be proved by extending the method appied in [5], for a detaied proof we refer to the forthcoming artice [4]. Consequenty, it suffices to consider stochastic perturbations of systems of semi-inear PDEs of the form (3). A Wong-Zakaï approximation theorem for such systems was obtained in [3].

7 ON THE POSITIVITY OF SOLUTIONS OF SYSTEMS OF STOCHASTIC PDES 7 3. The Stochastic Case - Necessary and Sufficient Conditions for the Positivity of Random Approximations We use a Wong-Zakaï-type approximation theorem and associate to a given stochastic system (f, g) a suitabe famiy of random approximations. The resut of the previous section yieds necessary and sufficient conditions for the positivity property of the famiy of random PDEs. Since the soutions of the random approximations converge in expectation to the soution of the origina probem, theses conditions ensure the positivity property of the stochastic system Wong-Zakaï Approximation and Random Systems of PDEs. E. Wong and M. Zakaï ([1],[11]) studied the reation between ordinary and stochastic differentia equations and introduced a smooth approximation of the Brownian motion to approximate stochastic integras by ordinary integras. Doing so, they obtain an approximation of the stochastic differentia equation by a famiy of random differentia equations. However, when the smoothing parameter tends to zero the random soutions do not converge to the soution of the origina stochastic differentia equation, but a modified one. The appearing correction term is caed Wong-Zakaï correction term. The Wong-Zakaï approximation theorem has been generaized in various directions. In this section, we briefy reca the main resut by Chueshov-Vuiermot in [3] about a Wong-Zakaï-type approximation theorem for a cass of stochastic systems of semi-inear paraboic PDEs, which is, in particuar, appicabe for the systems we consider. Assumptions on the stochastic perturbations We assume {W j t, t } j N is a famiy of mutuay independent standard scaar Wiener processes on the canonica Wiener space (Ω, F, P), and dw j t denotes the corresponding Itô differentia. The non-negative parameters q j are normaization factors. Moreover, the functions gj : O [, T ] R R are smooth and assumed to be bounded for bounded vaues of the soution, where j N, = 1,..., m Smooth Predictabe Approximation of the Wiener Process. A genera notion of a smooth predictabe approximation of the Wiener process is defined by Chueshov and Vuiermot in [3] (Definition 4.1, p.144). In the foowing, we wi take their main exampe as a definition (Proposition 4.2, p.1441). Let {W t, t } be a standard scaar Wiener process on the probabiity space (Ω, F, P) with fitration {F t, t }. The smooth predictabe approximation of {W t, t } is the famiy of random processes {W ɛ (t), t } ɛ> defined by W ɛ (t) = φ ɛ (t τ)w τ dτ,

8 8 JACKY CRESSON 1,2, MESSOUD EFENDIEV 3, AND STEFANIE SONNER 3,4 where φ ɛ (t) = ɛ 1 φ(t/ɛ), and φ(t) is a function with the properties φ C 1 (R), suppφ [, 1], 1 φ(t)dt = 1. We wi need the foowing resut ([3], p.1442), which states that the derivative of the smooth predictabe approximation W ɛ, denoted by W ɛ, can be written as a stochastic integra of the form (6) As a consequence, W ɛ (t) = t t ɛ φ ɛ (t τ)dw τ, t ɛ. W ɛ is Gaussian, which wi be fundamenta in our proof Predictabe Smoothing of Itô s Probem and Random Systems. Using the previousy defined famiy of smooth predictabe approximations {W j ɛ (t), t } ɛ>,j N of the Wiener processes {W j t, t } j N the predictabe smoothing of Itô s probem (3) is the famiy of random equations (7) du (x, t) = ( A (x, t, D)u (x, t) + f (x, t, u(x, t)) ) dt + ( where = 1,..., m. Using our notation, we are ed to the foowing definition: q j g j (x, t, u(x, t)) W j ɛ (t) ) dt, Definition 2. The smooth random approximation of the stochastic system (f, g) of PDEs with respect to the smooth predictabe approximation {W ɛ (t), t } ɛ> is the famiy of random PDEs (f ɛ,ω, ), where f ɛ,ω(x, t, u(x, t)) = f (x, t, u(x, t)) + q j g j(x, t, u(x, t)) W j ɛ (t) ɛ > A Wong-Zakaï Approximation Theorem. Foowing Chueshov and Vuiermot ([3], p.1436) we consider mid soutions of the stochastic system of PDEs (f, g): In the foowing definition the famiy {U(t), t } denotes the inear semigroup generated by the operator A = (A 1,..., A m ) in L 2 (O; R m ) with domain Here, B indicates the boundary operator and W 2,2 B (O; Rm ) := {u W 2,2 (O; R m ) : Bu = }. W k,2 (O) := {u L 2 (O) : D α u L 2 (O) for a α k}. Definition 3. A random function u(x, t, ω) = (u 1 (x, t, ω),..., u m (x, t, ω)) is caed a mid soution of the stochastic probem (f, g) in the space V = W 1,2 B (O; Rm ) on the interva [, T ], if u(t) = u(x, t, ω) C(, T ; L 2 (Ω O)) is a predictabe process such that T E u(t) 2 V dt <,

9 ON THE POSITIVITY OF SOLUTIONS OF SYSTEMS OF STOCHASTIC PDES 9 and satisfies the integra equation (8) u(t) = U(t)u + t U(t τ)f(τ, u(τ))dτ + where we assume that a integras in (8) exist. For further detais we refer to [3] and [1]. t q j U(t τ)g j (τ, u(τ))dw j (τ, ω), Definition 4. Let (f, g) be a stochastic system of PDEs and u be its mid soution. We say that the mid soutions u ɛ of a famiy of random PDEs (F ɛ,ω, ) converge to the mid soution of the stochastic system (f, g) if T im ɛ E u(t) u ɛ (t) 2 W 1,2 (O;R m ) dt =. Thereby, the function u ɛ is a mid soution of the famiy of random PDEs (F ɛ,ω, ) if it satisfies the integra equation u ɛ (t) = U(t)u + t U(t τ)f ɛ,ω (τ, u ɛ (τ))dτ. The main resut of Chueshov and Vuiermot in [3] is the foowing (Theorem 4.3, p.1443): Theorem 4. Assume that the stated assumptions on the operators A and B and the functions f and g are satisfied. Moreover, et q j <, the initia data u CB 2 (O; Rm ) be F -measurabe and E u r C 2 (O) < for some r > 8. We assume the associated system of random PDEs (f ɛ,ω, ) has a mid soution u ɛ beonging to the cass C(, T ; L r (Ω, X α,p )) for a α < 1 and p > 1, and for this soution there exists a constant C independent of ɛ such that sup t [,T ] E u ɛ r L p (O) C for a p > 1. Then, the mid soutions u ɛ converge to a soution u cor of the corrected stochastic system of PDEs (f cor, g) when ɛ tends to zero, where fcor = f + 1 m qj 2 g i gj j 2 u i for = 1,... m. The spaces X α,,p in Theorem 4 denote the fractiona power spaces associated to the operator A. For further detais we refer to [3] A Positivity Criterion for Systems of Stochastic PDEs. Our aim is to study the positivity property of the stochastic system (f, g) of the form (3). Hence, in the seque we assume that a unique soution of the stochastic initia vaue probem exists, and the soutions of the random approximations converge to the soution of the modified stochastic system (cf. Theorem 4). Sufficient conditions for the existence and uniqueness of soutions under even

10 1 JACKY CRESSON 1,2, MESSOUD EFENDIEV 3, AND STEFANIE SONNER 3,4 more genera assumptions can be found in the artice [3] (Section 3). Since the soutions of the random approximations do not converge to the soution of the origina system we construct an auxiiary stochastic system as foows: Let (F, g) be a stochastic system of the form (3). The corresponding famiy of random approximations (F ɛ,ω, ), ɛ >, ω Ω is expicit, depends on the definition of the smooth approximation W ɛ of the Wiener process {W (t), t }, and is given by Fɛ,ω = F + q j gjẇ ɛ j = 1,..., m. Theorem 4 states that the soutions of the random systems converge in expectation to the soution of the modified stochastic system (F cor, g), where Fcor = F + 1 m qj 2 g i gj j 2 u i = 1,..., m. To anayse a given stochastic system (f, g) of the form (3) we therefore construct an auxiiary system (F, g) (see Equation (9) beow) such that the soutions of the associated system of random PDEs (F ɛ,ω, ) converge to the soutions of our origina system (f, g). We then use the deterministic positivity criterion to derive necessary and sufficient conditions for the positivity property of the famiy of random approximations (F ɛ,ω, ). Finay, we show that this property is preserved by the transformation reating the origina system and the modified system and by passing to the imit when ɛ goes to zero. Let (f, g) be a given system of SPDEs. If we define the auxiiary stochastic system (F, g) by ( ) (9) F = f 1 qj 2 gj 1 gj 2 u g gm j j u m = 1,..., m, then, the soutions of the famiy of random PDEs (F ɛ,ω, ) converge to the soution of the origina stochastic system (f, g). Theorem 3 states that the deterministic system (f, ) satisfies the positivity property if and ony if the the interaction term satisfies Condition (5) in Theorem 3. foowing definition. Definition 5. We say that the function f : O R + R m R m, f(x, t, u) = (f 1 (x, t, u),..., f m (x, t, u)), satisfies the positivity condition if it satisfies Property (5). This motivates the

11 ON THE POSITIVITY OF SOLUTIONS OF SYSTEMS OF STOCHASTIC PDES 11 The next emma wi be essentia for the proof of our main resut. Lemma 1. Let (f, g) be a given stochastic system of PDEs. We assume that the functions g j are twice continuousy differentiabe with respect to u k and satisfy (1) gj(x, t, u 1,..., }{{},..., u m ) = x O, t >, u k, for a j N and k, = 1,..., m. Then, the foowing statements are equivaent: (a) The function f satisfies the positivity condition. (b) The modified function F satisfies the positivity condition. (c) The associated random functions F ɛ,ω satisfy the positivity condition for a ɛ > and ω Ω. Proof. The proof is a simpe computation. Let j N and 1 m. Since gj is continuousy differentiabe with respect to u and satisfies gj (x, t, u1,..., u 1,, u +1,..., u m ) =, we can represent it in the form gj (x, t, u) = u G j (x, t, u) with a continuousy differentiabe function G j. We obtain for the sum appearing in the Wong-Zakaï correction term m gj i gj u i = m which eads to an associated function F of the form F = f 1 m qj 2 g i gj j 2 u i = f 1 2 gj i (u G j ) u i = gju i G j u i + g (u G j ) j u, i q 2 j i g i ju G j u i + g j (u G j ) u. Due to Assumption (1) we note that the modified function F satisfies the positivity condition if and ony if f satisfies the positivity condition since a correction terms vanish if u =. Finay, the associated system of random PDEs (F ɛ,ω, ) is given by Fɛ,ω = F + q j gj Wɛ j. Condition (1) therefore impies Fɛ,ω(x, t, u 1,..., }{{},..., u m ) = F (x, t, u 1,..., }{{},..., u m ) = f (x, t, u 1,..., }{{},..., u m ), which concudes the proof of the emma. Appying Lemma 1 we derive necessary and sufficient conditions for the positivity property of the random approximations. Theorem 5. Let (f, g) be a system of stochastic PDEs and (F ɛ,ω, ) be the associated famiy of random approximations. We assume that the functions gj are twice continuousy differentiabe

12 12 JACKY CRESSON 1,2, MESSOUD EFENDIEV 3, AND STEFANIE SONNER 3,4 with respect to u k, for a j N and k, = 1,..., m. Then, the famiy of random approximations (F ɛ,ω, ) satisfies the positivity property for a ω and (sufficienty sma) ɛ > if and ony if f satisfies the positivity condition and the stochastic perturbation g fufis Property (1). In this case, the stochastic system (f, g) satisfies the positivity property. Proof. Sufficiency: By assumption, the function f satisfies the positivity condition. Since the stochastic perturbation fufis Property (1), Lemma 1 impies the positivity condition for random functions F ɛ,ω, ω Ω, ɛ >. We now appy the deterministic positivity criterion (Theorem 3) to concude the non-negativity of the soutions of the random approximations. Finay, the Wong-Zakaï approximation theorem states that the soutions of the random approximations (F ɛ,ω, ) converge in expectation to the soution of the stochastic system (f, g), which impies that the stochastic system (f, g) satisfies the positivity property. Necessity: We assume the famiy of random PDEs (F ɛ,ω, ) satisfies the positivity property. By Theorem 3 this is equivaent to the positivity condition for the random functions F ɛ,ω; that is, (11) F ɛ,ω(x, t, ũ) = F (x, t, ũ) + q j g j(x, t, ũ) W j ɛ (t) x O, t >, where ũ := (u 1,..., }{{},..., u m ), u k, k, = 1,..., m. The derivative of the smooth approximation W ɛ (t) of the Wiener process can be represented as the stochastic integra (6) and takes arbitrary vaues. Assuming that the function g j (x, t, u1,..., u 1,, u +1,..., u m ) is not identicay zero, then for sufficienty sma ɛ > we aways find an ω Ω such that the inequaity (11) is vioated. This proves the necessity of the condition on the stochastic perturbation. If Property (1) hods, the positivity condition for the famiy of random approximations is equivaent to the positivity condition for the function f by Lemma 1. The same resut is vaid if we appy Stratonovich s interpretation of stochastic differentia equations. In other words, the positivity property of soutions of the stochastic system is independent of the choice of interpretation, which was stated in Theorem 2 in the introduction. Coroary 1. Let (f, g) be a system of stochastic (Itô) PDEs. We assume the hypothesis of Theorem 5 are satisfied and the famiy of random approximations (F ɛ,ω, ) satisfies the positivity property. Then, the stochastic system (f, g) Strat obtained when we use Stratonovich s interpretation of the stochastic differentia equations satisfies the positivity property. Proof. The Wong-Zakaï correction term coincides with the transformation reating Ito s and Stratonovich s interpretation of the stochastic system (see [9], Section 6.1). That is, the soutions of the random approximations (f ɛ,ω, ) converge to the soution of the given stochastic

13 ON THE POSITIVITY OF SOLUTIONS OF SYSTEMS OF STOCHASTIC PDES 13 system, when interpreted in the sense of Stratonovich. Hence, the coroary is an immediate consequence of Theorem 5 and Lemma 1. The intuitive interpretation of the condition on the stochastic perturbation is the foowing: In the critica case, when one component of the soution approaches zero, the stochastic perturbation needs to vanish. Otherwise, the positivity of the soution cannot be guaranteed. Acknowedgements. We thank the anonymous referees for vauabe comments and remarks that greaty improved the presentation of our resuts. The third author woud ike to thank for the financia funding by the ERC Advanced Grant FP NUMERIWAVES starting from June 212. References [1] Amann H., Invariant Sets and Existence Theorems for Semiinear Paraboic and Eiptic Systems, Journa of Mathematica Anaysis and Appications, No. 65, pp , [2] Assing S., Comparison of Systems of Stochastic Partia Differentia Equations, Stochastic Processes and Their Appications 82, pp , [3] Chueshov I.D., Vuiermot P.A., Non-Random Invariant Sets for some Systems of Paraboic Stochastic Partia Differentia Equations, Stochastic Anaysis and Appications Vo. 22, No. 6, pp , 24. [4] Cresson J., Efendiev M.A., Sonner S., Necessary and Sufficient Conditions for the Positivity of Soutions of Stochastic PDEs, in preparation. [5] Efendiev M., Sonner S., On Verifying Mathematica Modes with Diffusion, Transport and Interaction, Current Advances in Noninear Anaysis and Reated Topics, GAKUTO Internationa Series Mathematica Sciences and Appications, Vo. 32, pp , 21. [6] Koteenez P., Comparison Methods for a Cass of Function Vaued Stochastic Partia Differentia Equations, Probabiity theory and reated fieds, Vo. 93, 1-19, [7] Oksenda B., Stochastic Differentia Equations: An Introduction with Appications, 6th Edition, Springer- Verag, Berin-Heideberg, 23. [8] Pave N.H., Differentia Equations, Fow Invariance and Appications, Res. Notes Math., Vo. 113, Pitman, Boston, [9] Twardowska K., Wong-Zakai Approximations of Stochastic Differentia Equations, Acta Appicandae Mathematicae 43, , [1] Wong E., Zakaï M., On the Reationship between Ordinary and Stochastic Differentia Equations, Internat. J. Engin. Sci. 3, , [11] Wong E., Zakaï M., Riemann-Stiejes Approximations of Stochastic Integras, Z. Wahrscheinichkeitstheorie verw. Geb. 12, 87-97, 1969.