Stochastic evolution equations with Wick-polynomial nonlinearities

We study nonlinear parabolic stochastic partial differential equations with Wick-power and Wick-polynomial type nonlinearities set in the framework of white noise analysis. These equations include the stochastic Fujita equation, the stochastic Fisher-KPP equation and the stochastic FitzHugh-Nagumo equation among many others. By implementing the theory of $C_0-$semigroups and evolution systems into the chaos expansion theory in infinite dimensional spaces, we prove existence and uniqueness of solutions for this class of SPDEs. In particular, we also treat the linear nonautonomous case and provide several applications featured as stochastic reaction-diffusion equations that arise in biology, medicine and physics.


Introduction
We study stochastic nonlinear evolution equations of the form u t (t, ω) = A u(t, ω) + n k=0 a k u ♦k (t, ω) + f (t, ω), t ∈ (0, T ] (1) where u(t, ω) is an X−valued generalized stochastic process; X is a certain Banach algebra and A corresponds to a densely defined infinitesimal generator of a C 0 −semigroup. The nonlinear part is given in terms of Wick-powers u ♦n = u ♦n−1 ♦u = u♦ . . . ♦u, n ∈ N, where ♦ denotes the Wick product. The Wick product is involved due to the fact that we allow random terms to be present both in the initial condition u 0 and the driving force f . This leads to singular solutions that do not allow to use ordinary multiplication, but require a renormalization of the multiplication, which is done by introducing the Wick product into the equation. The Wick product is known to represent the highest order stochastic approximation of the ordinary product [16]. In our previous paper [14] we treated the case of linear stochastic parabolic equations with Wick-multiplicative noise which includes the case n = 1. The present paper is an extension of [14] to nonlinear equations, where the nonlinearity is generated by a Wickpolynomial function leading to stochastic versions of Fujita-type equations u t = Au + u ♦n + f , FitzHugh-Nagumo equations u t = Au + u ♦2 − u ♦3 + f , Fisher-KPP equations u t = Au + u − u ♦2 + f and Chaffee-Infante equations u t = Au + u ♦3 − u + f . These equations have found ample applications in ecology, medicine, engineering and physics. For example, the FitzHugh-Nagumo equation is used to study electrical activity of neurons in neurophysiology by modeling the conduction of electric impulses down a nerve axon. The Fisher-Kolmogorov-Petrovsky-Piskunov equation provides a model for the spread of an epidemic in a population or for the distribution of an advantageous gene within a population. Other applications in medicine involve the modeling of cellular reactions to the introduction of toxins, and the process of epidermal wound healing. In plasma physics it has been used to study neutron flux in nuclear reactors, while in ecology it models flame propagation of fire outbreaks. Thus, the study of their stochastic versions, when some of the input factors is disturbed by an external noise factor and hence it becomes randomized, is of immense importance. For instance, a stochastic version of the FitzHugh-Nagumo equation has been studied in [1] and [3], while the stochastic Fisher-KPP equation has been studied in [10] and [19].
We implement the Wiener-Itô chaos expansion method combined with the operator semigroup theory in order to prove the existence and the uniqueness of a solution for (1). Using the chaos expansion method any SPDE can be transformed into a lower triangular infinite system of PDEs (also known as the propagator system) that can be solved recursively. Solving this system, one obtains the coefficients of the solution to (1). In order to solve the propagator system, we exploit the intrinsic relationship between the Wick product and the Catalan numbers that was discovered in [11] where the authors considered the stochastic Burgers equation. We build upon these ideas in order to solve a general class of stochastic nonlinear equations (1).
The plan of exposition is as follows: In the introductory section we recall upon basic notions of C 0 −semigroups, evolution systems and white noise theory including chaos expansions of generalized stochastic processes. In Section 2, which represents the main part of the paper, we prove existence and uniqueness of the solution to (1) for the case when a 0 = a 1 = · · · = a n−1 = 0 and a n = 1. This normalization is made for technical simplicity to illustrate the method of solving and to put out in details all building blocks of the formulae involved. In Section 3 we treat the general case of (1) and provide some concrete examples.

Evolution systems
We fix the notation and recall some known facts about evolution systems (see [20,Chapter 5]). Let X be a Banach space. Let {A(t)} t∈[s,T ] be a family of linear operators in X such that A(t) : D(A(t)) ⊂ X → X, t ∈ [s, T ] and let f be an X−valued function f : [s, T ] → X. Consider the initial value problem for all t ∈ (s, T ] and u satisfies (2), then u is a classical solution of (2). A two parameter family of bounded linear operators S(t, s), 0 ≤ s ≤ t ≤ T on X is called an evolution system if the following two conditions are satisfied: Clearly, if S(t, s) is an evolution system associated with the homogeneous evolution problem (2), i.e. if f ≡ 0, then a classical solution of (2) is given by for every finite sequence 0 ≤ s ≤ t 1 ≤ t 2 ≤ · · · ≤ t k ≤ T, k = 1, 2, . . Let {A(t)} t∈[s,T ] be a stable family of infinitesimal generators of C 0 −semigroups on X such that the domain D(A(t)) = D is independent of t and for every x ∈ D, A(t)x is continuously differentiable in X. If f ∈ C 1 ([s, T ], X) then for every x ∈ D the evolution problem (2) has a unique classical solution u given by From the proof of [20,Theorem 5.3,p. 147] one can obtain Since t → A(t) is continuous in B(D, X) and (t, s) → S(t, s) is strongly continuous for all 0 ≤ s ≤ t ≤ T, we have additionally that the solution u to (2) exhibits the regularity property u ∈ C 1 ([s, T ], X) and d dt u(t)| t=s = A(s)x + f (s). Recall that the evolution system S(t, s) satisfies: Remark 1.1. Considering infinitezimal generators depending on t, we follow the standard approach of Yosida (cf. [24], [12]). We refer to [18] for a method based on an equivalent operator extension problem (see also references in [18]). The chaos expansion approach, which is the essence of our paper, requires the existence results for the propagator system i.e. for the coordinate-wise deterministic Cauchy problems. For this purpose we demonstrate the applications of the hyperbolic Cauchy problem given in [20].

Generalized stochastic processes
Denote by (Ω, F , µ) the Gaussian white noise probability space (S ′ (R), B, µ), where Ω = S ′ (R) denotes the space of tempered distributions, B the Borel sigma-algebra generated by the weak topology on S ′ (R) and µ the Gaussian white noise measure corresponding to the characteristic function given by the Bochner-Minlos theorem. We recall the notions related to L 2 (Ω, µ) (see [9]). The set of multi-indices I is (N N 0 ) c , i.e. the set of sequences of non-negative integers which have only finitely many nonzero components. Especially, we denote by 0 = (0, 0, 0, . . .) the zero multi-index with all entries equal to zero, the length of a multi-index is |α| = ∞ i=1 α i for α = (α 1 , α 2 , . . .) ∈ I and α! = ∞ i=1 α i !. We will use the convention that α − β is defined if α n − β n ≥ 0 for all n ∈ N, i.e., if α − β ≥ 0.
The Wiener-Itô theorem (sometimes also referred to as the Cameron-Martin theorem) states that one can define an orthogonal basis {H α } α∈I of L 2 (Ω, µ), where H α are constructed by means of Hermite orthogonal polynomials h n and Hermite functions ξ n , Then, every F ∈ L 2 (Ω, µ) can be represented via the so called chaos expansion Denote by ε k = (0, 0, . . . , 1, 0, 0, . . .), k ∈ N the multi-index with the entry 1 at the kth place. Denote by H 1 the subspace of L 2 (Ω, µ), spanned by the polynomials H ε k (·), k ∈ N. The subspace H 1 contains Gaussian stochastic processes, e.g. Brownian motion is given by the chaos expansion B(t, ω) = ∞ k=1 t 0 ξ k (s)ds H ε k (ω). Denote by H m the mth order chaos space, i.e. the closure in L 2 (Ω, µ) of the linear subspace spanned by the orthogonal polynomials H α (·) with |α| = m, m ∈ N 0 . Then the Wiener-Itô chaos expansion states that L 2 (Ω, µ) = ∞ m=0 H m , where H 0 is the set of constants in L 2 (Ω, µ).
Changing the topology on L 2 (Ω, µ) to a weaker one, T. Hida [8] defined spaces of generalized random variables containing the white noise as a weak derivative of the Brownian motion. We refer to [8], [9] for white noise analysis.
Their topological duals, the stochastic distribution spaces, are given by formal sums: The space of test random variables is (S) ρ = p≥0 (S) ρ,p , ρ ≥ 0 endowed with the projective topology. Its dual, the space of generalized random variables is (S) −ρ = p≥0 (S) −ρ,−p , ρ ≥ 0 endowed with the inductive topology.
For ρ = 0 we obtain the space of Hida stochastic distributions (S) −0 and for ρ = 1 the Kondratiev space of generalized random variables (S) −1 . It holds that where ֒→ denotes dense inclusions. Usually the values of ρ are restricted to ρ ∈ [0, 1] in order to establish the S−transform (see [8], [9]) when solving SPDEs, but in our case values ρ > 1 may be considered as well.
The time-derivative of the Brownian motion B(t, ω) = ∞ k=1 t 0 ξ k (s)ds H ε k (ω) exists in a generalized sense and belongs to the Kondratiev space (S) −1,−p for p ≥ 5 12 . We refer it as the white noise and its formal expansion is given by W (t, ω) = ∞ k=1 ξ k (t)H ε k (ω). We extended in [21] the definition of stochastic processes to processes with the chaos expansion form U(t, ω) = α∈I u α (t)H α (ω), where the coefficients u α are elements of some Banach space of functions X. We say that U is an X-valued generalized stochastic process, i.e.
We have proved in [22] that the differentiation of a stochastic process can be carried out componentwise in the chaos expansion, i.e. due to the fact that (S) −ρ is a nuclear space it holds that C k ([0, T ], (S) −ρ ) = C k [0, T ]⊗(S) −ρ wherê ⊗ denotes the completion of the tensor product which is the same for the ε−completion and π−completion. In the sequel, we will use the notation ⊗ instead of⊗. Hence C k [0, T ]⊗ (S) −ρ,−p and C k [0, T ] ⊗ (S) ρ,p denote subspaces of the corresponding completions. We keep the same notation when C k [0, T ] is replaced by another Banach space. This means that a stochastic process U(t, ω) is k times continuously differentiable if and only if all of its coefficients u α (t), α ∈ I are in C k [0, T ].
The same holds for Banach space valued stochastic processes i.e. elements of In addition, if X is a Banach algebra, then the Wick product of the stochastic processes F = α∈I f α H α and G = β∈I g β H β ∈ X ⊗ (S) −ρ,−p is given by and F ♦G ∈ X ⊗ (S) −ρ,−(p+k) for all k > 1 (see [9]). The nth Wick power is defined by Throughout the paper we will assume that X is a Banach algebra.

be a coordinatewise operator that corresponds to a family of deterministic operators
(see [14,Section 2]). We are looking for a solution of (3) as an X-valued stochastic process The chaos expansion representation of the Wick-square is given by Then, for arbitrary n ∈ N, it can be shown that the nth Wick-power is given by where t ∈ [0, T ], ω ∈ Ω. The functions r α,n (t), t ∈ [0, T ], α ∈ I, n > 1 contain only the coordinate functions u β , β < α. Moreover, we recall that the Wick power u ♦n of a stochastic process u ∈ X ⊗ (S) −1,−p is an element of X ⊗ (S) −1,−q , for q > p + n − 1, see [9].
We rewrite all processes that figure in (3) in their corresponding Wiener-Itô chaos expansion form and obtain Due to the orthogonality of the base H α this reduces to the system of infinitely many deterministic Cauchy problems: with t ∈ (0, T ] and ω ∈ Ω. Let B α,n (t) = A α + n u n−1 0 (t) Id and g α, for all α > 0. Then, the system (7) can be written in the form Note that the inhomogeneous part g α,n in (8) does not contain any of the functions u β , β < α for |α| = 1, while for |α| > 1 it involves also u β , β < α. Hence, we distinguish these two cases.
Each solution u to (3) can be represented in the form (4) and hence its coefficients u 0 and u α for α > 0 must satisfy (6) and (8) respectively. Vice versa, if the coefficients u 0 and u α for α > 0 solve (6) and (8) respectively, and if the series in (4) represented by these coefficients exists in X ⊗ (S) −1 , then it defines a solution to (3). (6) and for every α ∈ I \ {0}, the coefficient u α is a classical solution to (8).
We assume that the following hold: with a common domain D α = D, α ∈ I, dense in X. We assume that there exist constants m ≥ 1 and w ∈ R such that The action of A is given by (A2) The initial value u 0 = α∈I u 0 α H α ∈ D, i.e. u 0 α ∈ D for every α ∈ I and there exists p ≥ 0 such that Remark 2.1. Particularly, if A 0 = ∆ is the Laplace operator and f 0 ≡ 0, then (6) belongs to the class of Fujita equations studied by Fujita, Chen and Watanabe [6,7]. The authors proved that for a nonnegative initial condition (10) has a unique classical solution on some [0, T 1 ). Moreover, if p > 1 + 2 N then there exist a positive bounded solution. The Fujita equation (10) apart from an interest per se also acts as a scaling limit of more general superlinear equations whose nonlinearities exhibit a polynomial growth rate. Originally, it has been developed to describe molecular concentration of a solution subjected to centrifugation and sedimentation.
Remark 2.2. In general, equations of the form (6), i.e. the deterministic equation for α = 0 can be solved by the Fixed Point Theorem [25]. Thus, in order to check if condition (A4-n) holds, one has to apply fixed point methods or other established methods for deterministic PDEs. The solution to (6) will usually blow-up in finite time. Especially the description of blow-up in the Sobolev supercritical regime poses a challenge that has been tackled in several papers (e.g. [7], [15] for the Fujita equation). We stress that our equation (3) and hence also (6) is given on a finite time interval, which is assumed to provide a solution on the entire interval (we restrict our considerations form the very start to the interval where no blow-up appears). Now we focus on solving (8) for α > 0. Lemma 2.3. Let the assumptions (A1)-(A4-n) be fulfilled. Then for every α > 0 the evolution system (8) has a unique classical solution u α ∈ C 1 ([0, T ], X).
Proof. First, for every α > 0, we consider the family of operators B α, The perturbation n u n−1 0 (t)Id : X → X, t ∈ [0, T ] is a family of uniformly bounded linear operators such that Thus, for every α > 0, the family {A α + n u n−1 0 (t)Id} t∈[0,T ] is a stable family of infinitesimal generators with stability constants m and w + nM n−1 n m. By assumption (A4-n) the function u 0 ∈ C 1 ([0, T ], X) so we obtain continuous differentiability of (A α + n u n−1 0 (t)Id)x, t ∈ [0, T ] for every x ∈ D and for every α > 0. Additionally, the domain of the operators n u n−1 0 (t)Id is the entire space X which implies that all of the operators B α,n (t), t ∈ [0, T ] have a common domain D(B α,n (t)) = D(A α ) = D not depending on t. Notice here that assumption (A1) additionally provides the same domain D of the family {B α,n (t)} t∈[0,T ] for all α > 0.
Now we proceed with four technical lemmas that will be used in the sequel.
Proof. This is a direct consequence of [11,Proposition 2.3]. More precisely, in [11] authors proved that |α|! ≤ q α α! if a sequence q = (q k ) k∈N satisfies < 1, the sequence (2N) 2 = ((2k) 2 ) k∈N satisfies a required property. Proof. Let c > 0 and choose s ≥ 0 such that c ≤ 2 s . Then, for q > s + 1, In the next lemma, for the sake of completeness, we give some useful properties of the well known Catalan numbers, see for example [23].
Lemma 2.6. A sequence {c n } n∈N defined by the recurrence relation is called the sequence of Catalan numbers. The closed formula for c n is a multiple of the binomial coefficient, i.e. the solution of the Catalan recurrence (15) is The Catalan numbers satisfy the growth estimate c n ≤ 4 n , n ≥ 0.

Proof of the main theorem
The statement of the main theorem is as follows. Proof. The proof of Theorem 2.8 will be given by induction with respect to n ∈ N in Theorems 2.9 and 2.10. We will prove in the first one that the statement of the main theorem holds for n = 2. Since it is technically pretty challenging to write down the proof of the inductive step for arbitrary n ∈ N, in Theorem 2.10 the proof is given for n = 3 by reducing the problem to the case n = 2. In the same way one can reduce the problem for arbitrary n ∈ N to the case n − 1.
u(0, ω) = u 0 (ω), The chaos expansion representation of the Wick-square is given by (5). Applying the Wiener-Itô chaos expansion to the nonlinear stochastic equation (17) one obtain which reduces to the system of infinitely many deterministic Cauchy problems: with t ∈ (0, T ] and ω ∈ Ω.
Recall that for all α > 0, so the system (19) can be written in the form Proof. According to Lemma 2.3 for every α > 0 the evolution equation (20) has an unique classical solution u α ∈ C 1 ([0, T ], X). Thus, the generalized stochastic process u(t, ω) = α∈I u α (t)H α (ω), t ∈ [0, T ], ω ∈ Ω has coefficients that are all classical solutions to the corresponding deterministic equation (20), hence in order to show that u is an almost classical solution to (17) one has to prove that u ∈ C([0, T ], X) ⊗ (S) −1 .
Let R α , α > 0, be defined as follows: It is a direct consequence of the definition of the numbers R α , α > 0, and it can be shown by induction with respect to the length of the multi-index α > 0 that (see [11,Section 5]) Lemma 2.7 shows that the numbers R α , α > 0 satisfy Further on, by (27), where c n = 1 n+1 2n n , n ≥ 0 denotes the nth Catalan number (more information on Catalan numbers is provided in Lemma 2.6). Using Lemma 2.4, (28), (29) and (16) we obtain that for α ∈ I * , |α| > 1 the estimatioñ holds. Finally, from the definition ofL α , α > 0 we obtain Notice that the upper estimate also holds for |α| > 1, Taking that s > 0 is such that 2 s ≥ 16c 2 , according to Lemma 2.5, we obtain for q > 2p + s + 5.
In the sequel we prove the existence of the almost classical solution of the Cauchy problem for t ∈ [0, T ], ω ∈ Ω. Applying the Wiener-Itô chaos expansion method to the nonlinear stochastic equation (30) reduces to the system of infinitely many deterministic Cauchy problems: with t ∈ (0, T ] and ω ∈ Ω. Let B α,3 (t) = A α + 3u 2 0 (t) Id and for all α > 0, then, the system (32) can be written in the form Proof. According to Lemma 2.3 for every α > 0 the evolution equation (34) has an unique classical solution u α ∈ C 1 ([0, T ], X) given in the form (14). Thus, the generalized stochastic process u(t, ω), represented in the chaos expansion form (4), has coefficients that are all classical solutions to the corresponding deterministic equation (34). Hence, in order to show that u is an almost classical solution to (30), one has to prove that u ∈ C([0, T ], X)⊗(S) −1 .

It is clear that for
From (12) we obtain that By (21), (22), (12) and (35) we obtain which leads to the estimate where m 3 = m + m w 3 . For |α| = 2 we have two different forms of the multiindex. First, for α = 2ε k , k ∈ N from (33) we obtain the form of the inhomogeneous part g 2ε k , Then, together with (14) we obtain Thus, where a 1 = m + m w 3 (3M 3 m 2 3 e 2w 3 T K + 1). In the second case, for α = ε k + ε j , k = j, k, j ∈ N from (33) we obtain the form g ε k +ε j ,3 (t) = 6u 0 (t) u ε k (t) u ε j (t) + f ε k +ε j (t) of the inhomogeneous part of (14). By applying (36) and (22) it can be estimated as . Then, (14) combined with the previous estimate lead to Then, we obtained where a 2 = m + m w 3 (6M 3 m 2 3 e 2w 3 T K + 1). Finaly, from (37) and (38) we obtain the estimate for all |α| = 2 L α ≤ a 2 e w 3 T K (2N) pα . For |α| > 2 we deal with general form of the inhomogeneous part of (34) The solution to (34) is of the form We underline that in the previous inductive steps, we obtained the estimates of L α−θ = sup t∈[0,T ] u α−θ (t) for all 0 < θ < α. Then, where m 3 = m + m w 3 . In order to estimate L α for |α| > 2 we consider two possibilities: (a) Define R α for |α| ≥ 1 in the following inductive way then, using Lemma 2.7, we obtain the estimate
(b) We assume, in the second case, that there exists α ∈ I, |α| ≥ 2 such that Consider the most complicated case. Then, we would have that the inequality (41) is fulfilled for all α ∈ I. Then, (39) reduces to where we used inequality L β > 0<γ<β L β−γ L γ for β < α. Further, we have At this point, we can repeat the proof of Theorem 2.9. Particularly, using the notation m ′ 3 = (3M 3 + 1) m 3 and K ′ = K 3M 3 +1 , the following inequality corresponds to the inequality (26), since K ′ < 1, and the proof continues in the same manner as the one from Theorem 2.9, i.e. the proof of solvability of the equation (17) with the Wick-square nonlinearity.
Remark 2.11. Note here that if the almost classical solution u to (3) satisfies u ∈ D = DomA then u is a classical solution to (3).
We assume the following: is a coordinatewise operator depending on t that corresponds to a family of deterministic operators A α (t) : D(A α ) ⊂ X → X, α ∈ I. For every α ∈ I the operator family {A α (t)} t∈[0,T ] is a stable family of infinitesimal generators of C 0 −semigroups on X with stability constants m > 1 and w ∈ R not depending on α, therefore the corresponding evolution systems S α (t, s) satisfy The domain D(A α (t)) = D is independent of t ∈ [0, T ] and α ∈ I. For every x ∈ D the function A α (t)x, t ∈ [0, T ] is continuously differentiable in X for each α ∈ I.

Extensions and applications
Our results can be extended to a far more general case of stochastic evolution equation of the form with a Wick-polynomial type of nonlinearity p ♦ n (u) = n k=0 a k u ♦k = a 0 + a 1 u + a 2 u ♦2 + a 3 u ♦3 + . . . a n u ♦n , where a n = 0 and a k , 0 ≤ k ≤ n are either constants or deterministic functions. Equation (44) generalizes equation (3) and it can be solved by the very same method presented in the paper, provided that one stipulates that the corresponding deterministic version of (44) has a solution and modifies assumption (A4 − n) correspondingly. Hence, we replace (A4 − n) with the following assumption: (A4-pol-n) The Cauchy problem a k u k = a 0 + a 1 u + a 2 u 2 + a 3 u 3 + . . . a n u n , is a classical polynomial of degree n corresponding to the Wick-polynomial (45).
We extend Theorem 2.8, and for the sake of technical simplicity, present only a procedure for solving (44) for n = 3, but note that the general case may be done mutatis mutandis.
First we note that from the form of the process (4) and from the form of its Wick-powers (5), as well as from (31) we obtain the expansion of the Wick-polynomial nonlinearity When summing up the corresponding coefficients, the expression (47) transforms to where p ′ 3 , p ′′ 3 and p ′′′ 3 denote the first, the second and the third derivative of the polynomial (46), respectively. Thus, by applying the Wiener-Itô chaos expansion method to the nonlinear stochastic problem (44) we obtain the system of infinitely many deterministic Cauchy problems:

Examples
We present two classes of stochasic reaction-diffusion equations that belong to the class of problems (44).

Stochastic generalized Fisher-KPP equation
The deterministic nonlinear equation of the form (52) with F (u) = au(1 − u) is called the Fisher equation (also known as the Kolmogorov-Petrovsky-Piskunov equation). Such equations occur in phase transition problems arising in biology, ecology, plasma physics [4,13] etc. Particularly, such an equation provides a deterministic model for the density of a population living in an environment with a limited carrying capacity. It also describes the wave progression of an epidemic outbreak or the spread of an advantageous gene within a population. Other applications in medicine involve the modeling of cellular reactions to the introduction of toxins, voltage propagation through a nerve axon, and the process of epidermal wound healing [2]. In other research areas it has been also used to study flame propagation of fire outbreaks, and neutron flux in nuclear reactors.
Stochastic models that include random effects due to some external (enviromental) noise were studied in the framework of white noise analysis [10], where the authors proved the existence of the traveling wave solution. In the same setting, the stochastic KPP equation, i.e. heat equations with semilinear potential and perturbation by a multiplicative noise were considered in [19]. Under suitable assumptions, by applying the Itô calculus, existence of a unique strong traveling wave solution was proven, and an implicit Feyman-Kac-like formula for the solution was presented. Here we consider a generalized Wick-version of the stochastic Fisher-KPP equation u t (t, ω) = A u(t, ω) + u(t, ω) − u ♦2 (t, ω) + f (t, ω), t ∈ (0, T ] u(0, ω) = u 0 (ω), ω ∈ Ω, which can be solved by applying Theorem 3.1.

Conclusion
In this paper we have presented a methodology for solving stochastic evolution equations involving nonlinearities of Wick-polynomial type. However, the applications and extensions of the theory do not stop here. In place of the nonlinearity u ♦2 , one might consider u♦u x and with appropriate modifications solve the stochastic Burgers-type equation u t = u xx + u♦u x +f or the stochastic KdV equation u t = u xxx +u♦u x +f , coalesced into the form u t = Au+u♦u x +f . One can also replace the nonlinearity u ♦n by u♦|u| n−1 , where the modulus of a complex-valued stochastic process is understood as |u| = α∈I |u α |H α , and find explicit solutions to the stochastic nonlinear Schrödinger equation (i )u t = ∆u + u♦|u| n−1 + f .