# Some Nonlinear Stochastic Cauchy Problems with Generalized Stochastic Processes

Authors: Victor Dévoué
Journal: International Journal of Analysis
 (2016)

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## Abstract

We study some nonlinear stochastic Cauchy problems in the framework of the(C,E,P)-algebras. We adapt the definitions to this framework. By means of suitable regularizations, we define associated generalized problems. We use our previous results about the wave equation in canonical form to obtain generalized solutions. We compare the generalized solutions with the classical ones when they exist.


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## Body

## 1. Introduction

A possibility in studying stochastic differential equations is to make use of the theory of Colombeau-type generalized functions spaces to overcome the multiplication problem in distribution space. Here, to study some nonlinear stochastic Cauchy problems, we choose to reformulate them correctly in the framework of the(C,E,P)-algebras of Marti [1–3] in order to show that, following the example of the theory of Colombeau [4, 5], these algebras may serve as a tool for treating singular processes in stochastic analysis. Until now, similar studies were made only in Colombeau-type algebras.A(C,E,P)-algebra, where C is overgenerated by a finite set B, is always isomorphic to a Colombeau-type algebra but the asymptotic scale of this last algebra, which can be obtained from the generators of B, is not explicit and difficult to work with. So, by using the overgeneration, we are able to work with asymptotics which are explicit. Therefore, we consider this framework more convenient.We adapt the definitions to the framework of the(C,E,P)-algebras. We interpret generalized stochastic processes on R2 as measurable maps with values in a (C,E,P)-algebra. Using our previous results about our study on the wave equation in canonical form [6, 7], we can prove that some nonlinear non-Lipschitz stochastic Cauchy problems have a unique solution. So this paper completes our research. We find some results similar to those of [8, 9] about the solutions in theory of Colombeau algebra.The paper is organized as follows. In Section2, we give some definitions and references for stochastic analysis, (C,E,P)-algebras, and algebras of generalized stochastic processes. In Section 3, we study the following Cauchy problems formally written as (1)P:∂2U∂x∂y=FU+W,Uγ=A,∂U∂yγ=B,P′:∂2U∂x∂y=FUW,Uγ=A,∂U∂yγ=B,where γ is a monotonic curve of equation y=f(x), γ is not a characteristic curve, A and B are generalized stochastic processes on R, and W is a generalized process on R2. That is, A, B, and W are weakly measurable maps of some probability space (Ω,Σ,μ) with values in the Schwartz distribution space D′R, respectively, D′R2. The function F is smooth; it can be non-Lipschitz (in U) but F and all derivatives have polynomial growth.Forω fixed, ω∈Ω, we replace problem P (resp., P′) by a generalized one well-formulated (P(ω)gen) (resp., (P′(ω)gen)) in a convenient algebra. To do this, we use regularizations and cutoff techniques. We use two parameters. The first parameter regularizes the data and the second one replaces the problem by a family of Lipschitz problems. We show that problem P (resp., P′) has a unique solution in some algebras of generalized stochastic processes. Moreover, if problem P (resp., P′) admits a solution satisfying appropriate growth estimates on some open subset O of R2, then this solution and the generalized one are equal in a meaning given in Theorem 16.In Section4, we are interested in a nonlinear stochastic Cauchy problem with the white noise as initial data: (2)P1:∂2U∂x∂y=FU,Uγ=W,∂U∂yγ=0,where γ is the curve of equation y=f(x), γ is not a characteristic curve, and W is the white noise on R. The function F is smooth; it can be non-Lipschitz but F and all derivatives have polynomial growth. We treat problem P1 in the same way as the previous ones and we study the limiting behavior of the generalized solution.


## 2. Algebra of Generalized Stochastic Processes

### 2.1. Stochastic Analysis

We refer the reader to [8, 10, 11], for some basic facts from stochastic analysis as construction of white noise and the relation between the white noise and Wiener process on Rd.Let(Ω,Σ,μ) be a probability space. A weakly measurable map (3)X:Ω⟶D′Rdis called a generalized stochastic process on Rd.For each fixed test functionφ∈DRd, the map Ω→R; ω↦〈X(ω),φ〉 is a random variable. The space of generalized stochastic processes is denoted by DΩ′Rd.White noiseW˙ on Rd is constructed as follows.The probability space is the space of tempered distributionsΩ=S′Rd and Σ is the Borel σ-algebra generated by the weak topology. According to Bochner-Minlos theorem, there is a unique probability measure μ on (Ω,Σ) such that (4)∫eiXω,φdμω=e-1/2φL2Rd2 for φ∈SRd.The white noiseW˙:Ω→D′Rd can be define as the identity mapping 〈X(ω),φ〉=〈ω,φ〉, φ∈DRd. Remark that W˙ is a generalized Gaussian process with mean zero and variance E(W˙(φ)2)=φL2(Rd)2, where E denotes mathematical expectation. Its covariance is (5)EW˙φW˙ψ=∫Rdφyψydy. The white noise on Rd is realized as the d-fold distributional derivative of the Wiener process.


### 2.2. Algebras of Generalized Functions

#### 2.2.1. The Presheaves of(C,E,P)-Algebras

We recall briefly some notions that form the basis of our study [7, 12]. We refer the reader to the references. Take K=R or C. Consider(1)Λ a set of indices, A a solid subring of the ring KΛ (that is to say, for any (sλ)λ,(rλ)λ∈KΛ×A, with A=(rλ)λ:(rλ)λ∈A, if (sλ)λ≤(rλ)λ (i.e., for any λ, sλ≤rλ), then (sλ)λ∈A), and IA a solid ideal of A;(2)E a sheaf of K-topological algebras on a topological space X, such that, for any open set O in X, the algebra E(O) is endowed with a family P(O)=(pi)i∈I(O) of seminorms satisfying (6)∀i∈IO,∃j,k,C∈IO×IO×R+∗,∀f,g∈EO:pifg≤Cpjfpkg.Assume that, for any open subsetsO1, O2 of X such that O1⊂O2, we have I(O1)⊂I(O2) and if ρ12 is the restriction operator E(O2)→E(O1), then, for each pi∈P(O1), the seminorm p~i=pi∘ρ12 extends pi to P(O2).Assume that, for any familyO=(Oh)h∈H of open subsets of X if O=∪h∈H⁡Oh, then, for each pi∈P(O), i∈I(O), there exists a finite subfamily O1,…,Oni of O and corresponding seminorms pnj∈P(Onj), 1≤j≤i, such that, for any u∈E(O), piu≤∑j=1ni‍pnj(uOnj).SetC=A/IA, (7)XA,E,PO=uλλ∈EOΛ:∀i∈IO,piuλλ∈A,NIA,E,PO=uλλ∈EOΛ:∀i∈IO,piuλλ∈IA.X(A,E,P) is a sheaf of subalgebras of the sheaf EΛ and N(IA,E,P) is a sheaf of ideals of X(A,E,P) [13]. The constant sheaf X(A,K,·)/N(IA,K,·) is the sheaf C=A/IA. We call presheaf of (C,E,P)-algebra the factor presheaf of algebras A=X(A,E,P)/N(IA,E,P).We writeuλ the class in A(O) of (uλ)λ∈Λ∈X(A,E,P)(O). We simplify the notations by writing X(O) (resp., N(O)) instead of X(A,E,P)(O) (resp., N(IA,E,P)(O)).Overgenerated Rings. Let Bp=rn,λλ∈(R+∗)Λ:n=1,…,p and B be the subset of (R+∗)Λ obtained as rational functions with coefficients in R+∗, of elements in Bp as variables. Define (8)A=aλλ∈KΛ∣∃bλλ∈B,∃λ0∈Λ,∀λ≺λ0:aλ≤bλ.We say that A is overgenerated by Bp (A is a solid subring of KΛ). If IA is some solid ideal of A, we also say that C=A/IA is overgenerated by Bp.Relationship with Distribution Theory. Let O be an open subset of Rn. If φεε∈0,1 is a family of mollifiers φεx=1/εnφx/ε, x∈Rn, ∫φxdx=1 and if T∈D′Rn, the convolution product family T∗φεε is a family of smooth functions slowly increasing in 1/ε. So, the space of distributions D′(O) can be embedded into A(O). Taking ε as component of the multi-index λ∈Λ, we shall choose the subring A overgenerated by some Bp of (R+∗)Λ containing the family ελ.Association Process. Consider O an open subset of X, E a sheaf of topological K-vector spaces containing E as a subsheaf, and a a map from Λ to K such that (aλ)λ=(aλ)λ is an element of A. Assume that (9)NO⊂uλλ∈XO:limEO,Λ⁡uλ=0. We say that u=uλ and v=vλ∈E(O) are a-E associated if, for each neighborhood V of 0 for the E-topology, there exists λ0∈Λ such that λ≺λ0⇒aλ(uλ-vλ)∈V. We write u~E(O)av if limE(O),Λaλ(uλ-vλ)=0. We define an association process between u=uλ and T∈E(O) by u~T⇔limE(O),Λuλ=T.D ′ -Singular Support. Assume that (10)ND′AO=uλλ∈XO:limD′O,λ→0⁡uλ=0⊃NO. Let ODA′(u) be the set of all x having a neighborhood V on which u is associated with a distribution. The D′-singular support of u∈A(O) is the set singsuppD′(u)=SDA′A(u)=Ω∖ODA′(u).


#### 2.2.2. Generalized Operator and Generalized Restriction Mapping

We denote byλ=ε,η an element of Λ=Λ1×Λ2, Λ1=Λ2=0,1.LetO be an open subset of R2 and Fη∈C∞(O×R,R). We say that the algebra A(O) is stable under the family Fηλ if for all uλλ∈X(O) and iλλ∈N(O) we have Fη(·,·,uλ)λ∈X(O), Fη(·,·,uλ+iλ)-Fη(·,·,uλ)λ∈N(O).IfAO is stable under Fηλ, for u=uλ∈AO, Fη(·,·,uλ) is a well defined element of AO (i.e., not depending on the representative uλλ of u).The stability condition is verified if the familyFηλ is smoothly tempered, that is to say, if the following two conditions are satisfied:(i) For eachK⋐R2, l∈N, and uλλ∈X(O), there is a positive finite sequence C0,…,Cl, such that PK,l(Fη(·,·,uλ))≤∑i=0lCiPK,li(uλ).(ii) For eachK⋐R2, l∈N, vλλ, and uλλ∈X(O), there is a positive finite sequence D1,…,Dl such that (11)PK,lFη·,·,vλ-Fη·,·,uλ≤∑j=1lDjPK,ljvλ-uλ. Let F∈C∞(O×R,R) and fηη∈C∞(R)Λ1; we define (12)Fηx,y,z=Fx,y,zfηz. If AO is stable under Fηλ, the operator (13)F:AO⟶AO,u=uλ⟼Fη·,·,uλ is called the generalized operator associated with F via the family fηη.Considerf∈C∞R. We say that f is compatible with second-side restriction if (14)uλ·,f·λ∈XR,∀uλλ∈XR2,iλ·,f·λ∈NR,∀iλλ∈NR2. Clearly, if u=uλ∈A(R2), then uλ·,f(·) is a well defined element of A(R).If the functionf is compatible with second-side restriction, the mapping (15)Rf:AR2⟶AR,u=uλ⟼uλ·,f· is called the generalized second-side restriction mapping associated with f.


### 2.3. The AlgebrasA(O) and Ap(O),p∈N∗

SetE=C∞ and X=Rd for d=1,2 and E=D′ and set Λ as a set of indices, λ∈Λ. For any open set O, in Rd, E(O) is endowed with the P(O) topology defined by the family of the seminorms: (16)PK,luλ=supα≤l⁡PK,αuλ,withPK,αuλ=DαuλxL∞K,K⋐O and Dα=∂α1+⋯+αd/∂z1α1⋯∂zdαd for z=(z1,…,zd)∈O, l∈N, α=(α1,…,αd)∈Nd, and K⋐O means that K is a compact subset of O. Let A be a subring of the ring RΛ. We consider a solid idealIA of A. Put (17)XO=uλλ∈C∞OΛ:∀K⋐O,∀l∈N,PK,luλλ∈A,NO=uλλ∈C∞OΛ:∀K⋐O,∀l∈N,PK,luλλ∈IA. The generalized derivation Dα:u(=uε)↦Dαu=Dαuε provides A(O)=X(O)/N(O) with a differential algebraic structure.Takep∈N∗. For any open set O, in Rd, E(O)=C∞(O) is endowed with the Pp(O) topology defined by the family of the seminorms (18)NK,lpuλ=supα≤l⁡NK,αpuλ,withNK,αpuλ=DαuλxLpK,K⋐O, and l∈N, α∈Nd. Put (19)XpO=uλλ∈C∞OΛ:∀K⋐O,∀l∈N,NK,lpuλλ∈A,NpO=uλλ∈C∞OΛ:∀K⋐O,∀l∈N,NK,lpuλλ∈IA,ApO=XpONpO.The generalized derivation Dα:u(=uε)↦Dαu=Dαuε provides Ap(O) with a differential algebraic structure.Remark 1.
TheNK,l2 (p=2) norms are bounded by the PK,l norms. We have A(O)⊂A2(O).


### 2.4. Algebras of Generalized Stochastic Processes

LetO be an open set in Rd.Definition 2.
AAO-generalized stochastic process on a probability space (Ω,Σ,μ) is a mapping U:Ω→AO such that there is a representing function (20)u=RU:Λ×O×Ω⟶R with the following properties:(i)
For fixedλ∈Λ, the map (x,ω)↦u(λ,x,ω) is jointly measurable on O×Ω.(ii)
Almost surely inω∈Ω, the map λ↦u(λ,·,ω) belongs to X(O) and it is a representative of U(ω); that is, almost surely in ω∈Ω, U(ω)λλ=u(λ,·,ω)λ∈X(O).
The algebra of generalized stochastic processes is denoted byAΩ(O).Definition 3.
AA2(0)-generalized stochastic process on a probability space (Ω,Σ,μ) is a map U:Ω→A2O such that there is a representing function (21)u=RU:Λ×O×Ω⟶R with the following properties:(i)
For fixedλ∈Λ, the map (x,ω)↦u(λ,x,ω) is jointly measurable on O×Ω.(ii)
Almost surely inω∈Ω, the map λ↦u(λ,·,ω) belongs to X2(O) and it is a representative of U(ω); that is, almost surely in ω∈Ω, U(ω)λλ=u(λ,·,ω)λ∈X2(O).
The algebra of generalized stochastic processes is denoted byA2Ω(O).Remark 4.
Letχ∈DR having the properties(22)∫χsds=1;∫spχsds=0,1≤p≤2;define φ by φx,y=χxχy. LetV∈DΩ′Rd be a generalized stochastic process. If λ∈Λ, then Vω∗φλ is measurable with respect to ω∈Ω and smooth with respect to x∈Rd, hence jointly measurable. Also, (Vω∗φλ)λ belongs to X(Rd). Thus, (23)RVλ,x,ω=Vω∗φλx=Vωλxqualifies as a representative for a random generalized function. We obtain an imbedding τ:DΩ′Rd↪AΩ(Rd).


## 2.1. Stochastic Analysis

We refer the reader to [8, 10, 11], for some basic facts from stochastic analysis as construction of white noise and the relation between the white noise and Wiener process on Rd.Let(Ω,Σ,μ) be a probability space. A weakly measurable map (3)X:Ω⟶D′Rdis called a generalized stochastic process on Rd.For each fixed test functionφ∈DRd, the map Ω→R; ω↦〈X(ω),φ〉 is a random variable. The space of generalized stochastic processes is denoted by DΩ′Rd.White noiseW˙ on Rd is constructed as follows.The probability space is the space of tempered distributionsΩ=S′Rd and Σ is the Borel σ-algebra generated by the weak topology. According to Bochner-Minlos theorem, there is a unique probability measure μ on (Ω,Σ) such that (4)∫eiXω,φdμω=e-1/2φL2Rd2 for φ∈SRd.The white noiseW˙:Ω→D′Rd can be define as the identity mapping 〈X(ω),φ〉=〈ω,φ〉, φ∈DRd. Remark that W˙ is a generalized Gaussian process with mean zero and variance E(W˙(φ)2)=φL2(Rd)2, where E denotes mathematical expectation. Its covariance is (5)EW˙φW˙ψ=∫Rdφyψydy. The white noise on Rd is realized as the d-fold distributional derivative of the Wiener process.


## 2.2. Algebras of Generalized Functions

### 2.2.1. The Presheaves of(C,E,P)-Algebras

We recall briefly some notions that form the basis of our study [7, 12]. We refer the reader to the references. Take K=R or C. Consider(1)Λ a set of indices, A a solid subring of the ring KΛ (that is to say, for any (sλ)λ,(rλ)λ∈KΛ×A, with A=(rλ)λ:(rλ)λ∈A, if (sλ)λ≤(rλ)λ (i.e., for any λ, sλ≤rλ), then (sλ)λ∈A), and IA a solid ideal of A;(2)E a sheaf of K-topological algebras on a topological space X, such that, for any open set O in X, the algebra E(O) is endowed with a family P(O)=(pi)i∈I(O) of seminorms satisfying (6)∀i∈IO,∃j,k,C∈IO×IO×R+∗,∀f,g∈EO:pifg≤Cpjfpkg.Assume that, for any open subsetsO1, O2 of X such that O1⊂O2, we have I(O1)⊂I(O2) and if ρ12 is the restriction operator E(O2)→E(O1), then, for each pi∈P(O1), the seminorm p~i=pi∘ρ12 extends pi to P(O2).Assume that, for any familyO=(Oh)h∈H of open subsets of X if O=∪h∈H⁡Oh, then, for each pi∈P(O), i∈I(O), there exists a finite subfamily O1,…,Oni of O and corresponding seminorms pnj∈P(Onj), 1≤j≤i, such that, for any u∈E(O), piu≤∑j=1ni‍pnj(uOnj).SetC=A/IA, (7)XA,E,PO=uλλ∈EOΛ:∀i∈IO,piuλλ∈A,NIA,E,PO=uλλ∈EOΛ:∀i∈IO,piuλλ∈IA.X(A,E,P) is a sheaf of subalgebras of the sheaf EΛ and N(IA,E,P) is a sheaf of ideals of X(A,E,P) [13]. The constant sheaf X(A,K,·)/N(IA,K,·) is the sheaf C=A/IA. We call presheaf of (C,E,P)-algebra the factor presheaf of algebras A=X(A,E,P)/N(IA,E,P).We writeuλ the class in A(O) of (uλ)λ∈Λ∈X(A,E,P)(O). We simplify the notations by writing X(O) (resp., N(O)) instead of X(A,E,P)(O) (resp., N(IA,E,P)(O)).Overgenerated Rings. Let Bp=rn,λλ∈(R+∗)Λ:n=1,…,p and B be the subset of (R+∗)Λ obtained as rational functions with coefficients in R+∗, of elements in Bp as variables. Define (8)A=aλλ∈KΛ∣∃bλλ∈B,∃λ0∈Λ,∀λ≺λ0:aλ≤bλ.We say that A is overgenerated by Bp (A is a solid subring of KΛ). If IA is some solid ideal of A, we also say that C=A/IA is overgenerated by Bp.Relationship with Distribution Theory. Let O be an open subset of Rn. If φεε∈0,1 is a family of mollifiers φεx=1/εnφx/ε, x∈Rn, ∫φxdx=1 and if T∈D′Rn, the convolution product family T∗φεε is a family of smooth functions slowly increasing in 1/ε. So, the space of distributions D′(O) can be embedded into A(O). Taking ε as component of the multi-index λ∈Λ, we shall choose the subring A overgenerated by some Bp of (R+∗)Λ containing the family ελ.Association Process. Consider O an open subset of X, E a sheaf of topological K-vector spaces containing E as a subsheaf, and a a map from Λ to K such that (aλ)λ=(aλ)λ is an element of A. Assume that (9)NO⊂uλλ∈XO:limEO,Λ⁡uλ=0. We say that u=uλ and v=vλ∈E(O) are a-E associated if, for each neighborhood V of 0 for the E-topology, there exists λ0∈Λ such that λ≺λ0⇒aλ(uλ-vλ)∈V. We write u~E(O)av if limE(O),Λaλ(uλ-vλ)=0. We define an association process between u=uλ and T∈E(O) by u~T⇔limE(O),Λuλ=T.D ′ -Singular Support. Assume that (10)ND′AO=uλλ∈XO:limD′O,λ→0⁡uλ=0⊃NO. Let ODA′(u) be the set of all x having a neighborhood V on which u is associated with a distribution. The D′-singular support of u∈A(O) is the set singsuppD′(u)=SDA′A(u)=Ω∖ODA′(u).


### 2.2.2. Generalized Operator and Generalized Restriction Mapping

We denote byλ=ε,η an element of Λ=Λ1×Λ2, Λ1=Λ2=0,1.LetO be an open subset of R2 and Fη∈C∞(O×R,R). We say that the algebra A(O) is stable under the family Fηλ if for all uλλ∈X(O) and iλλ∈N(O) we have Fη(·,·,uλ)λ∈X(O), Fη(·,·,uλ+iλ)-Fη(·,·,uλ)λ∈N(O).IfAO is stable under Fηλ, for u=uλ∈AO, Fη(·,·,uλ) is a well defined element of AO (i.e., not depending on the representative uλλ of u).The stability condition is verified if the familyFηλ is smoothly tempered, that is to say, if the following two conditions are satisfied:(i) For eachK⋐R2, l∈N, and uλλ∈X(O), there is a positive finite sequence C0,…,Cl, such that PK,l(Fη(·,·,uλ))≤∑i=0lCiPK,li(uλ).(ii) For eachK⋐R2, l∈N, vλλ, and uλλ∈X(O), there is a positive finite sequence D1,…,Dl such that (11)PK,lFη·,·,vλ-Fη·,·,uλ≤∑j=1lDjPK,ljvλ-uλ. Let F∈C∞(O×R,R) and fηη∈C∞(R)Λ1; we define (12)Fηx,y,z=Fx,y,zfηz. If AO is stable under Fηλ, the operator (13)F:AO⟶AO,u=uλ⟼Fη·,·,uλ is called the generalized operator associated with F via the family fηη.Considerf∈C∞R. We say that f is compatible with second-side restriction if (14)uλ·,f·λ∈XR,∀uλλ∈XR2,iλ·,f·λ∈NR,∀iλλ∈NR2. Clearly, if u=uλ∈A(R2), then uλ·,f(·) is a well defined element of A(R).If the functionf is compatible with second-side restriction, the mapping (15)Rf:AR2⟶AR,u=uλ⟼uλ·,f· is called the generalized second-side restriction mapping associated with f.


## 2.2.1. The Presheaves of(C,E,P)-Algebras

We recall briefly some notions that form the basis of our study [7, 12]. We refer the reader to the references. Take K=R or C. Consider(1)Λ a set of indices, A a solid subring of the ring KΛ (that is to say, for any (sλ)λ,(rλ)λ∈KΛ×A, with A=(rλ)λ:(rλ)λ∈A, if (sλ)λ≤(rλ)λ (i.e., for any λ, sλ≤rλ), then (sλ)λ∈A), and IA a solid ideal of A;(2)E a sheaf of K-topological algebras on a topological space X, such that, for any open set O in X, the algebra E(O) is endowed with a family P(O)=(pi)i∈I(O) of seminorms satisfying (6)∀i∈IO,∃j,k,C∈IO×IO×R+∗,∀f,g∈EO:pifg≤Cpjfpkg.Assume that, for any open subsetsO1, O2 of X such that O1⊂O2, we have I(O1)⊂I(O2) and if ρ12 is the restriction operator E(O2)→E(O1), then, for each pi∈P(O1), the seminorm p~i=pi∘ρ12 extends pi to P(O2).Assume that, for any familyO=(Oh)h∈H of open subsets of X if O=∪h∈H⁡Oh, then, for each pi∈P(O), i∈I(O), there exists a finite subfamily O1,…,Oni of O and corresponding seminorms pnj∈P(Onj), 1≤j≤i, such that, for any u∈E(O), piu≤∑j=1ni‍pnj(uOnj).SetC=A/IA, (7)XA,E,PO=uλλ∈EOΛ:∀i∈IO,piuλλ∈A,NIA,E,PO=uλλ∈EOΛ:∀i∈IO,piuλλ∈IA.X(A,E,P) is a sheaf of subalgebras of the sheaf EΛ and N(IA,E,P) is a sheaf of ideals of X(A,E,P) [13]. The constant sheaf X(A,K,·)/N(IA,K,·) is the sheaf C=A/IA. We call presheaf of (C,E,P)-algebra the factor presheaf of algebras A=X(A,E,P)/N(IA,E,P).We writeuλ the class in A(O) of (uλ)λ∈Λ∈X(A,E,P)(O). We simplify the notations by writing X(O) (resp., N(O)) instead of X(A,E,P)(O) (resp., N(IA,E,P)(O)).Overgenerated Rings. Let Bp=rn,λλ∈(R+∗)Λ:n=1,…,p and B be the subset of (R+∗)Λ obtained as rational functions with coefficients in R+∗, of elements in Bp as variables. Define (8)A=aλλ∈KΛ∣∃bλλ∈B,∃λ0∈Λ,∀λ≺λ0:aλ≤bλ.We say that A is overgenerated by Bp (A is a solid subring of KΛ). If IA is some solid ideal of A, we also say that C=A/IA is overgenerated by Bp.Relationship with Distribution Theory. Let O be an open subset of Rn. If φεε∈0,1 is a family of mollifiers φεx=1/εnφx/ε, x∈Rn, ∫φxdx=1 and if T∈D′Rn, the convolution product family T∗φεε is a family of smooth functions slowly increasing in 1/ε. So, the space of distributions D′(O) can be embedded into A(O). Taking ε as component of the multi-index λ∈Λ, we shall choose the subring A overgenerated by some Bp of (R+∗)Λ containing the family ελ.Association Process. Consider O an open subset of X, E a sheaf of topological K-vector spaces containing E as a subsheaf, and a a map from Λ to K such that (aλ)λ=(aλ)λ is an element of A. Assume that (9)NO⊂uλλ∈XO:limEO,Λ⁡uλ=0. We say that u=uλ and v=vλ∈E(O) are a-E associated if, for each neighborhood V of 0 for the E-topology, there exists λ0∈Λ such that λ≺λ0⇒aλ(uλ-vλ)∈V. We write u~E(O)av if limE(O),Λaλ(uλ-vλ)=0. We define an association process between u=uλ and T∈E(O) by u~T⇔limE(O),Λuλ=T.D ′ -Singular Support. Assume that (10)ND′AO=uλλ∈XO:limD′O,λ→0⁡uλ=0⊃NO. Let ODA′(u) be the set of all x having a neighborhood V on which u is associated with a distribution. The D′-singular support of u∈A(O) is the set singsuppD′(u)=SDA′A(u)=Ω∖ODA′(u).


## 2.2.2. Generalized Operator and Generalized Restriction Mapping

We denote byλ=ε,η an element of Λ=Λ1×Λ2, Λ1=Λ2=0,1.LetO be an open subset of R2 and Fη∈C∞(O×R,R). We say that the algebra A(O) is stable under the family Fηλ if for all uλλ∈X(O) and iλλ∈N(O) we have Fη(·,·,uλ)λ∈X(O), Fη(·,·,uλ+iλ)-Fη(·,·,uλ)λ∈N(O).IfAO is stable under Fηλ, for u=uλ∈AO, Fη(·,·,uλ) is a well defined element of AO (i.e., not depending on the representative uλλ of u).The stability condition is verified if the familyFηλ is smoothly tempered, that is to say, if the following two conditions are satisfied:(i) For eachK⋐R2, l∈N, and uλλ∈X(O), there is a positive finite sequence C0,…,Cl, such that PK,l(Fη(·,·,uλ))≤∑i=0lCiPK,li(uλ).(ii) For eachK⋐R2, l∈N, vλλ, and uλλ∈X(O), there is a positive finite sequence D1,…,Dl such that (11)PK,lFη·,·,vλ-Fη·,·,uλ≤∑j=1lDjPK,ljvλ-uλ. Let F∈C∞(O×R,R) and fηη∈C∞(R)Λ1; we define (12)Fηx,y,z=Fx,y,zfηz. If AO is stable under Fηλ, the operator (13)F:AO⟶AO,u=uλ⟼Fη·,·,uλ is called the generalized operator associated with F via the family fηη.Considerf∈C∞R. We say that f is compatible with second-side restriction if (14)uλ·,f·λ∈XR,∀uλλ∈XR2,iλ·,f·λ∈NR,∀iλλ∈NR2. Clearly, if u=uλ∈A(R2), then uλ·,f(·) is a well defined element of A(R).If the functionf is compatible with second-side restriction, the mapping (15)Rf:AR2⟶AR,u=uλ⟼uλ·,f· is called the generalized second-side restriction mapping associated with f.


## 2.3. The AlgebrasA(O) and Ap(O),p∈N∗

SetE=C∞ and X=Rd for d=1,2 and E=D′ and set Λ as a set of indices, λ∈Λ. For any open set O, in Rd, E(O) is endowed with the P(O) topology defined by the family of the seminorms: (16)PK,luλ=supα≤l⁡PK,αuλ,withPK,αuλ=DαuλxL∞K,K⋐O and Dα=∂α1+⋯+αd/∂z1α1⋯∂zdαd for z=(z1,…,zd)∈O, l∈N, α=(α1,…,αd)∈Nd, and K⋐O means that K is a compact subset of O. Let A be a subring of the ring RΛ. We consider a solid idealIA of A. Put (17)XO=uλλ∈C∞OΛ:∀K⋐O,∀l∈N,PK,luλλ∈A,NO=uλλ∈C∞OΛ:∀K⋐O,∀l∈N,PK,luλλ∈IA. The generalized derivation Dα:u(=uε)↦Dαu=Dαuε provides A(O)=X(O)/N(O) with a differential algebraic structure.Takep∈N∗. For any open set O, in Rd, E(O)=C∞(O) is endowed with the Pp(O) topology defined by the family of the seminorms (18)NK,lpuλ=supα≤l⁡NK,αpuλ,withNK,αpuλ=DαuλxLpK,K⋐O, and l∈N, α∈Nd. Put (19)XpO=uλλ∈C∞OΛ:∀K⋐O,∀l∈N,NK,lpuλλ∈A,NpO=uλλ∈C∞OΛ:∀K⋐O,∀l∈N,NK,lpuλλ∈IA,ApO=XpONpO.The generalized derivation Dα:u(=uε)↦Dαu=Dαuε provides Ap(O) with a differential algebraic structure.Remark 1.
TheNK,l2 (p=2) norms are bounded by the PK,l norms. We have A(O)⊂A2(O).


## 2.4. Algebras of Generalized Stochastic Processes

LetO be an open set in Rd.Definition 2.
AAO-generalized stochastic process on a probability space (Ω,Σ,μ) is a mapping U:Ω→AO such that there is a representing function (20)u=RU:Λ×O×Ω⟶R with the following properties:(i)
For fixedλ∈Λ, the map (x,ω)↦u(λ,x,ω) is jointly measurable on O×Ω.(ii)
Almost surely inω∈Ω, the map λ↦u(λ,·,ω) belongs to X(O) and it is a representative of U(ω); that is, almost surely in ω∈Ω, U(ω)λλ=u(λ,·,ω)λ∈X(O).
The algebra of generalized stochastic processes is denoted byAΩ(O).Definition 3.
AA2(0)-generalized stochastic process on a probability space (Ω,Σ,μ) is a map U:Ω→A2O such that there is a representing function (21)u=RU:Λ×O×Ω⟶R with the following properties:(i)
For fixedλ∈Λ, the map (x,ω)↦u(λ,x,ω) is jointly measurable on O×Ω.(ii)
Almost surely inω∈Ω, the map λ↦u(λ,·,ω) belongs to X2(O) and it is a representative of U(ω); that is, almost surely in ω∈Ω, U(ω)λλ=u(λ,·,ω)λ∈X2(O).
The algebra of generalized stochastic processes is denoted byA2Ω(O).Remark 4.
Letχ∈DR having the properties(22)∫χsds=1;∫spχsds=0,1≤p≤2;define φ by φx,y=χxχy. LetV∈DΩ′Rd be a generalized stochastic process. If λ∈Λ, then Vω∗φλ is measurable with respect to ω∈Ω and smooth with respect to x∈Rd, hence jointly measurable. Also, (Vω∗φλ)λ belongs to X(Rd). Thus, (23)RVλ,x,ω=Vω∗φλx=Vωλxqualifies as a representative for a random generalized function. We obtain an imbedding τ:DΩ′Rd↪AΩ(Rd).


## 3. Nonlinear Stochastic Problems

### 3.1. A Nonlinear Stochastic Problem with Additive Generalized Stochastic Process

We consider the Cauchy problem formally written as(24)P:∂2U∂x∂y=FU+W,Uγ=A,∂U∂yγ=B,where γ is a monotonic curve of equation y=f(x), γ is not a characteristic curve, A, B∈AΩ(R), and W∈AΩ(R2) is a AR2-generalized stochastic process on a probability space (Ω,Σ,μ). The function F is smooth; it can be non-Lipschitz but F and all derivatives have polynomial growth. Assume that F(0)=0. We look for a solution U:Ω→A2R2∈A2Ω(R2) (e.g., we can take F(U)=-U-U3 or F(U)=-U3).Then,U is a solution to problem (P) if and only if, for any ω∈Ω, U(ω) is solution to the problem P(ω):(25)Pω:∂2Uω∂x∂y=FUω+Wω,Uωγ=Aω,∂Uω∂yγ=Bω.


### 3.2. A Nonlinear Stochastic Problem with Multiplicative Generalized Stochastic Process

We consider the Cauchy problem formally written as(26)P′:∂2U∂x∂y=FUW,Uγ=A,∂U∂yγ=B,where γ is a monotonic curve of equation y=f(x), γ is not a characteristic curve, A, B∈AΩ(R), and W∈AΩ(R2) is a AR2-generalized stochastic process on a probability space (Ω,Σ,μ). The function F is smooth; it can be non-Lipschitz but F and all derivatives have polynomial growth. Assume that F(0)=0. We look for a solution U:Ω→A2R2∈A2Ω(R2) (e.g., we can take F(U)=-U3).Then,U is a solution to problem (P′) if and only if, for any ω∈Ω, U(ω) is solution to the problem P′(ω):(27)P′ω:∂2Uω∂x∂y=FUωWω,Uωγ=Aω,∂Uω∂yγ=Bω.


### 3.3. Cutoff Procedure

Letrηη be in R∗0,1 such that rη>0 and limη→0rη=+∞. Set Eη=-rη,rη.Consider a family of smooth one-variable functionshηη such thatHsupz∈Iη⁡hηz=1,hηz=0,ifz≥rη1,if-rη+1≤z≤rη-1. Assume that ∂nhη/∂zn is bounded on Eηfor any integer n, n>0. Set (28)supz∈Eη⁡∂nhη∂znz=Mn. Let ϕη(z)=zhη(z). We approximate the function Fby the family of functions Fηη defined by (29)Fηz=Fϕηz=Fzhηz.Assume thatF(0)=0. Here, function F is smooth; it can be non-Lipschitz but F and all derivatives have polynomial growth. More precisely, we assume the existence of p∈N such that (30)∀l∈N,∃cl>0,supz∈R⁡DlFz≤cl1+zp. Thus, (31)∀l∈N,∃μl>0,supz∈R;α≤l⁡DαFηz=supz≤rη;α≤l⁡DαFϕηz≤al1+rηp. So, according to [7, 12], ARis stable under the family Fηη.


### 3.4. Construction ofAR2

PutU(ω)=U(ω)ε,η and (32)Wωε,ηx,y=ϕηWω∗φεx,y.Define φ by φ(x,y)=χ(x)χ(y) with χ∈DR having the properties(33)∫χsds=1;∫spχsds=0,1≤p≤2.Consider (χε)ε a family of mollifiers such that ϰε(y)=1/εχ(y/ε); then, φε(x,y)=ϰε(x)ϰε(y). Put Aωε=Aω∗χε and Bωε=Bω∗χε.We make the following assumptions to generate a convenient(C,E,P)-algebra adapted to our problem:H1
∃p>0, ∀l∈N, ∃al>0, supz∈R;α≤l⁡DαFη(z)≤al(1+rη)p.H2
∀K⋐R2, ∀l∈N, ∃ρl>0, PK,l(W(ω)ε,η)≤ρK,l(1+rη)p.H3
C=A/IA is overgenerated by the following elements of R∗0,1:  εε,η,ηε,η,rηε,η,(exp⁡rη)ε,η.H4
AR2=X(R2)/N(R2) is built on C with (E,P)=C∞(R2),PK,lK⋐R2,l∈N.H5
A2R2=X2(R2)/N2(R2) is built on C with (E,P′)=C∞(R2),NK,l2K⋐R2,l∈N.


### 3.5. Generalized Differential Problems Associated with the Formal Ones

Our goal is to give a meaning to the problems formally written asP and P′.


#### 3.5.1. Generalized Differential Problem Associated withP

Forω fixed, the problem associated with P(ω) can be written as the well-formulated problem (34)Pωgen:∂2Uω∂x∂y=FUω+Wωε,η,RfUω=Aωε,Rf∂Uω∂y=Bωε; then, (35)∂2Uω∂x∂y=FηUω+Wωε,η,Uωγ=Aω∗χε,∂Uω∂yγ=Bω∗χε. In terms of representatives, and thanks to the stability and restriction hypothesis, if we can find U(ω)ε,η∈C∞R2 verifying (36)Pωε,η:∂2Uωε,η∂x∂yx,y=FηUωε,ηx,y+Wωε,ηx,y,Uωε,ηx,fx=Aωεx=Aω∗χεx,∂Uωε,η∂yx,fx=Bωεx=Bω∗χεx and if we can prove that (Uωε,η)ε,η∈X2(R2), then U(ω)=[Uωε,η] is a solution of P(ω)gen.LetV(ω)=V(ω)ε,η be another solution to P(ω)gen. The uniqueness of the solution to Pgen(ω) will be the consequence of V(ω)ε,η-U(ω)ε,ηε,η∈N(R2).Remark 5.
Dependence from some regularizing family. The problemP(ω)gen itself, and so a solution of it, a priori depends on the family of cutoff functions and, in the case of irregular data, on the family of mollifiers [7].Remark 6.
F ( U ) is such that (37)FU:Ω⟶AR2,ω⟼FηUωε,η,FηUωε,η:R2⟶R,x,y⟼FηUωε,ηx,y. Moreover, (38)RU=u:Λ×R2×Ω⟶R;λ,x,y,ω⟼Uωε,ηx,y=uλ,x,y,ω, with λ=ε,η.


#### 3.5.2. Generalized Differential Problem Associated withP′

Forω fixed, the problem associated with P′(ω) can be written as the well-formulated problem (39)Pgen′ω:∂2Uω∂x∂y=FUωWωε,η,RfUω=Aωε,Rf∂Uω∂y=Bωε; then, (40)∂2Uω∂x∂y=FηUωWωε,η,Uωγ=Aω∗χε,∂Uω∂yγ=Bω∗χε. In terms of representatives, and thanks to the stability and restriction hypothesis, if we can find U(ω)ε,η∈C∞R2 verifying (41)P′ωε,η:∂2Uωε,η∂x∂yx,y=FηUωε,ηx,yWωε,ηx,y,Uωε,ηx,fx=Aωεx=Aω∗χεx,∂Uωε,η∂yx,fx=Bωεx=Bω∗χεx and if we can prove that (Uωε,η)ε,η∈X2(R2), then U(ω)=[Uωε,η] is a solution of (P′(ω)gen).


### 3.6. Generalized Problems

#### 3.6.1. Solution to the Parametrized Regular Problems

Forω fixed, we consider the families of regularized problems (Pωε,η) and (P′ωε,η). We are going to prove that (Pωε,η) and (P′ωε,η) have a unique smooth solution under assumptions H1 and H2 and the assumptionHε,η
(a)
f∈C∞(R),  f′>0,  f(R)=R,(b)
Fη∈C∞(R,R),(c)
Aωε,Bωε∈C∞(R). Following [6], one can prove that (Pωε,η) is equivalent to the integral formulation (42)Uωε,ηx,y=Uω0,ε,ηx,y-∬Dx,y,fFηUωε,ηξ,ζ+Wωε,ηξ,ζdξdζ and (P′ωε,η) is equivalent to the integral formulation (43)Uωε,ηx,y=Uω0,ε,ηx,y-∬Dx,y,fFηUωε,ηξ,ζ⁡Wωε,ηξ,ζdξdζ,where U(ω)0,ε,η(x,y)=Υωεy-Υωε(f(x))+Aωε(x) and Υωε denotes a primitive of Bωε∘f-1, with (44)Dx,y,f=ξ,ζ:f-1y≤ξ≤x,y≤ζ≤fξify≤fx,ξ,ζ:x≤ξ≤f-1y,fξ≤ζ≤yify≥fx.Theorem 7.
Under AssumptionsHε,η, H1, and H2, problem (Pωε,η)(resp., (P′ωε,η)) has a unique solution, U(ω)ε,η, in C∞(R2).See [6, 12] for a detailed proof; replace uε,η by U(ω)ε,ηand Fη(x,y,uε,η(x,y)) by Fη(U(ω)ε,η(x,y))+W(ω)ε,ηx,y (resp., Fη(U(ω)ε,η(x,y))W(ω)ε,ηx,y).


#### 3.6.2. Solution to the Problems

Theorem 8.
Assume thatU(ω)ε,η is the solution to problem (Pωε,η)(resp., (P′ωε,η)); then, problem (P(ω)gen)(resp., (P′ωgen)) has a unique solution U(ω)=U(ω)ε,η in AR2.U ( ω ) is the solution to (Pωε,η) (resp., (P′ωε,η)) if U(ω)ε,ηε,η∈X(R2); that is, (45)PK,lUωε,ηε,η∈A,∀K⋐R2,∀l∈N. The proof follows the same steps as the existence results which can be found in [12], replacing uε,η by U(ω)ε,ηand Fη(x,y,uε,η(x,y)) by Fη(U(ω)ε,η(x,y))+W(ω)ε,ηx,y (resp., Fη(U(ω)ε,η(x,y))W(ω)ε,ηx,y). An induction process on the order of the successive derivatives shows that U(ω)ε,ηε,η belongs to X(R2). The Gronwall lemma is essential tool for the uniqueness.Theorem 9.
Assume thatU(ω)ε,η is the solution to problem (Pωε,η)(resp., (P′ωε,η)); then, problem (P(ω)gen)(resp., (P′ωgen)) has a unique solution U(ω)=U(ω)ε,η in A2R2.Proof.
U ( ω ) is the solution to (Pωε,η) if U(ω)ε,ηε,η∈X2(R2). We shall prove that (46)NK,l2Uωε,ηε,η∈A,∀K⋐R2,∀l∈N. But (47)DαUωε,ηL2K≤μK1/2DαUωε,ηL∞K and, as U(ω)ε,ηε,η∈X(R2), we have Dα(U(ω)ε,η)∞ε,η∈A. Thus, (48)DαUωε,ηL2Kε,η=NK,l2Uωε,ηε,η∈A. So U(ω)∈A2R2 and it is the solution to problem (P(ω)gen) in A2R2. Set (49)U:Ω⟶A2R2,ω⟼Uω. Then, U∈A2Ω(R2).Theorem 10.
The mappingU is the solution to problem (P)(resp., (P′)) and it is almost surely unique in A2Ω(R2).Proof.
SinceU(ω) is the unique solution to problem (P(ω)gen) in A2R2, then almost surely in ω∈Ω, the map λ↦RU(λ,·,·,ω)=U(ω)λ, (λ=ε,η), belongs to X2(R2) and it is a representative of U(ω) (i.e., U(ω)=U(ω)λ). For fixed λ=ε,η∈Λ, the map (50)x,y,ω⟼RUλ,x,y,ω=Uωλx,y=uλx,y,ω is jointly measurable on R2×Ω. So U is the solution to problem (P) almost surely unique in A2Ω(R2).


### 3.7. The Regularized and the Nonregularized Problems

Remark 11.
The generalized function represented by the family of solutions to the regularized problems(Pωε,η) (resp., (P′ωε,η)) is defined from the integral representation. Thus, we are going to study the relationship between this generalized function and the classical solutions to (P(ω)gen) (resp., (P′ωε,η)), when they exist, on a domain O such that ∀x,y∈O, D(x,y,f)⊂O. This justifies choosing O=a,b×f(a),f(b) when a,b∈R2 with a<b.
If problem(P(ω)gen) (resp., (P′ωgen)) has a smooth solution v on O, then, necessarily, we have O⊂R2∖singsuppU(ω).Let us recall that there exists a canonical sheaf embedding ofC∞ into A, through the morphism of algebra (51)σO:C∞O⟶AO,g⟼gε,where O is any open subset of R2 and gε=g. The presheaf A allows defining restriction and, as usually, we denote by uO the restriction on O of U∈AR2.Theorem 12.
LetO be an open subset of R2 such that (52)O⊂R2∖singsuppUω. Assume that O=⋃ηOη when Oηη is an increasing family of open subsets of R2 such that Oη=aη,bη×f(aη),f(bη) when (aη,bη)∈R2 with aη<bη. Assume that the nonregularized problem (P(ω)gen)(resp., (P′ωgen)) has a smooth solution V(ω) on O such that sup(x,y)∈OηV(ω)(x,y)<rη-1 for any η. Let U(ω)∈AR2 be the generalized function represented by the family U(ω)ε,ηε,η of solutions to (Pωε,η)(resp., (P′ωε,η)). Then, σOV(ω)=U(ω)O.We refer the reader to [12] for a detailed proof.We can say that the solutionU=U(ω) to the regularized problem coincides with the solution to the nonregularized equation on O.


### 3.8. A Special Case

We consider the Cauchy problem formally written as(53)S:∂2U∂x∂y=W,Uγ=A,∂U∂yγ=B,where γ is a monotonic curve of equation y=f(x), γ is not a characteristic curve, A, B∈AΩ(R), and W∈AΩ(R2) is a AR2-generalized stochastic process on a probability space (Ω,Σ,μ).This problem coincides with problem(P) for F=0 and with problem (P′) for F=1. We obtain the solution U to problem S. U(ω)=U(ω)ε,η in A2R2 is defined, with the previous notations, by (54)Uωε,ηx,y=Uω0,ε,ηx,y-∬Dx,y,fWωε,ηξ,ζdξdζ.


## 3.1. A Nonlinear Stochastic Problem with Additive Generalized Stochastic Process

We consider the Cauchy problem formally written as(24)P:∂2U∂x∂y=FU+W,Uγ=A,∂U∂yγ=B,where γ is a monotonic curve of equation y=f(x), γ is not a characteristic curve, A, B∈AΩ(R), and W∈AΩ(R2) is a AR2-generalized stochastic process on a probability space (Ω,Σ,μ). The function F is smooth; it can be non-Lipschitz but F and all derivatives have polynomial growth. Assume that F(0)=0. We look for a solution U:Ω→A2R2∈A2Ω(R2) (e.g., we can take F(U)=-U-U3 or F(U)=-U3).Then,U is a solution to problem (P) if and only if, for any ω∈Ω, U(ω) is solution to the problem P(ω):(25)Pω:∂2Uω∂x∂y=FUω+Wω,Uωγ=Aω,∂Uω∂yγ=Bω.


## 3.2. A Nonlinear Stochastic Problem with Multiplicative Generalized Stochastic Process

We consider the Cauchy problem formally written as(26)P′:∂2U∂x∂y=FUW,Uγ=A,∂U∂yγ=B,where γ is a monotonic curve of equation y=f(x), γ is not a characteristic curve, A, B∈AΩ(R), and W∈AΩ(R2) is a AR2-generalized stochastic process on a probability space (Ω,Σ,μ). The function F is smooth; it can be non-Lipschitz but F and all derivatives have polynomial growth. Assume that F(0)=0. We look for a solution U:Ω→A2R2∈A2Ω(R2) (e.g., we can take F(U)=-U3).Then,U is a solution to problem (P′) if and only if, for any ω∈Ω, U(ω) is solution to the problem P′(ω):(27)P′ω:∂2Uω∂x∂y=FUωWω,Uωγ=Aω,∂Uω∂yγ=Bω.


## 3.3. Cutoff Procedure

Letrηη be in R∗0,1 such that rη>0 and limη→0rη=+∞. Set Eη=-rη,rη.Consider a family of smooth one-variable functionshηη such thatHsupz∈Iη⁡hηz=1,hηz=0,ifz≥rη1,if-rη+1≤z≤rη-1. Assume that ∂nhη/∂zn is bounded on Eηfor any integer n, n>0. Set (28)supz∈Eη⁡∂nhη∂znz=Mn. Let ϕη(z)=zhη(z). We approximate the function Fby the family of functions Fηη defined by (29)Fηz=Fϕηz=Fzhηz.Assume thatF(0)=0. Here, function F is smooth; it can be non-Lipschitz but F and all derivatives have polynomial growth. More precisely, we assume the existence of p∈N such that (30)∀l∈N,∃cl>0,supz∈R⁡DlFz≤cl1+zp. Thus, (31)∀l∈N,∃μl>0,supz∈R;α≤l⁡DαFηz=supz≤rη;α≤l⁡DαFϕηz≤al1+rηp. So, according to [7, 12], ARis stable under the family Fηη.


## 3.4. Construction ofAR2

PutU(ω)=U(ω)ε,η and (32)Wωε,ηx,y=ϕηWω∗φεx,y.Define φ by φ(x,y)=χ(x)χ(y) with χ∈DR having the properties(33)∫χsds=1;∫spχsds=0,1≤p≤2.Consider (χε)ε a family of mollifiers such that ϰε(y)=1/εχ(y/ε); then, φε(x,y)=ϰε(x)ϰε(y). Put Aωε=Aω∗χε and Bωε=Bω∗χε.We make the following assumptions to generate a convenient(C,E,P)-algebra adapted to our problem:H1
∃p>0, ∀l∈N, ∃al>0, supz∈R;α≤l⁡DαFη(z)≤al(1+rη)p.H2
∀K⋐R2, ∀l∈N, ∃ρl>0, PK,l(W(ω)ε,η)≤ρK,l(1+rη)p.H3
C=A/IA is overgenerated by the following elements of R∗0,1:  εε,η,ηε,η,rηε,η,(exp⁡rη)ε,η.H4
AR2=X(R2)/N(R2) is built on C with (E,P)=C∞(R2),PK,lK⋐R2,l∈N.H5
A2R2=X2(R2)/N2(R2) is built on C with (E,P′)=C∞(R2),NK,l2K⋐R2,l∈N.


## 3.5. Generalized Differential Problems Associated with the Formal Ones

Our goal is to give a meaning to the problems formally written asP and P′.


### 3.5.1. Generalized Differential Problem Associated withP

Forω fixed, the problem associated with P(ω) can be written as the well-formulated problem (34)Pωgen:∂2Uω∂x∂y=FUω+Wωε,η,RfUω=Aωε,Rf∂Uω∂y=Bωε; then, (35)∂2Uω∂x∂y=FηUω+Wωε,η,Uωγ=Aω∗χε,∂Uω∂yγ=Bω∗χε. In terms of representatives, and thanks to the stability and restriction hypothesis, if we can find U(ω)ε,η∈C∞R2 verifying (36)Pωε,η:∂2Uωε,η∂x∂yx,y=FηUωε,ηx,y+Wωε,ηx,y,Uωε,ηx,fx=Aωεx=Aω∗χεx,∂Uωε,η∂yx,fx=Bωεx=Bω∗χεx and if we can prove that (Uωε,η)ε,η∈X2(R2), then U(ω)=[Uωε,η] is a solution of P(ω)gen.LetV(ω)=V(ω)ε,η be another solution to P(ω)gen. The uniqueness of the solution to Pgen(ω) will be the consequence of V(ω)ε,η-U(ω)ε,ηε,η∈N(R2).Remark 5.
Dependence from some regularizing family. The problemP(ω)gen itself, and so a solution of it, a priori depends on the family of cutoff functions and, in the case of irregular data, on the family of mollifiers [7].Remark 6.
F ( U ) is such that (37)FU:Ω⟶AR2,ω⟼FηUωε,η,FηUωε,η:R2⟶R,x,y⟼FηUωε,ηx,y. Moreover, (38)RU=u:Λ×R2×Ω⟶R;λ,x,y,ω⟼Uωε,ηx,y=uλ,x,y,ω, with λ=ε,η.


### 3.5.2. Generalized Differential Problem Associated withP′

Forω fixed, the problem associated with P′(ω) can be written as the well-formulated problem (39)Pgen′ω:∂2Uω∂x∂y=FUωWωε,η,RfUω=Aωε,Rf∂Uω∂y=Bωε; then, (40)∂2Uω∂x∂y=FηUωWωε,η,Uωγ=Aω∗χε,∂Uω∂yγ=Bω∗χε. In terms of representatives, and thanks to the stability and restriction hypothesis, if we can find U(ω)ε,η∈C∞R2 verifying (41)P′ωε,η:∂2Uωε,η∂x∂yx,y=FηUωε,ηx,yWωε,ηx,y,Uωε,ηx,fx=Aωεx=Aω∗χεx,∂Uωε,η∂yx,fx=Bωεx=Bω∗χεx and if we can prove that (Uωε,η)ε,η∈X2(R2), then U(ω)=[Uωε,η] is a solution of (P′(ω)gen).


## 3.5.1. Generalized Differential Problem Associated withP

Forω fixed, the problem associated with P(ω) can be written as the well-formulated problem (34)Pωgen:∂2Uω∂x∂y=FUω+Wωε,η,RfUω=Aωε,Rf∂Uω∂y=Bωε; then, (35)∂2Uω∂x∂y=FηUω+Wωε,η,Uωγ=Aω∗χε,∂Uω∂yγ=Bω∗χε. In terms of representatives, and thanks to the stability and restriction hypothesis, if we can find U(ω)ε,η∈C∞R2 verifying (36)Pωε,η:∂2Uωε,η∂x∂yx,y=FηUωε,ηx,y+Wωε,ηx,y,Uωε,ηx,fx=Aωεx=Aω∗χεx,∂Uωε,η∂yx,fx=Bωεx=Bω∗χεx and if we can prove that (Uωε,η)ε,η∈X2(R2), then U(ω)=[Uωε,η] is a solution of P(ω)gen.LetV(ω)=V(ω)ε,η be another solution to P(ω)gen. The uniqueness of the solution to Pgen(ω) will be the consequence of V(ω)ε,η-U(ω)ε,ηε,η∈N(R2).Remark 5.
Dependence from some regularizing family. The problemP(ω)gen itself, and so a solution of it, a priori depends on the family of cutoff functions and, in the case of irregular data, on the family of mollifiers [7].Remark 6.
F ( U ) is such that (37)FU:Ω⟶AR2,ω⟼FηUωε,η,FηUωε,η:R2⟶R,x,y⟼FηUωε,ηx,y. Moreover, (38)RU=u:Λ×R2×Ω⟶R;λ,x,y,ω⟼Uωε,ηx,y=uλ,x,y,ω, with λ=ε,η.


## 3.5.2. Generalized Differential Problem Associated withP′

Forω fixed, the problem associated with P′(ω) can be written as the well-formulated problem (39)Pgen′ω:∂2Uω∂x∂y=FUωWωε,η,RfUω=Aωε,Rf∂Uω∂y=Bωε; then, (40)∂2Uω∂x∂y=FηUωWωε,η,Uωγ=Aω∗χε,∂Uω∂yγ=Bω∗χε. In terms of representatives, and thanks to the stability and restriction hypothesis, if we can find U(ω)ε,η∈C∞R2 verifying (41)P′ωε,η:∂2Uωε,η∂x∂yx,y=FηUωε,ηx,yWωε,ηx,y,Uωε,ηx,fx=Aωεx=Aω∗χεx,∂Uωε,η∂yx,fx=Bωεx=Bω∗χεx and if we can prove that (Uωε,η)ε,η∈X2(R2), then U(ω)=[Uωε,η] is a solution of (P′(ω)gen).


## 3.6. Generalized Problems

### 3.6.1. Solution to the Parametrized Regular Problems

Forω fixed, we consider the families of regularized problems (Pωε,η) and (P′ωε,η). We are going to prove that (Pωε,η) and (P′ωε,η) have a unique smooth solution under assumptions H1 and H2 and the assumptionHε,η
(a)
f∈C∞(R),  f′>0,  f(R)=R,(b)
Fη∈C∞(R,R),(c)
Aωε,Bωε∈C∞(R). Following [6], one can prove that (Pωε,η) is equivalent to the integral formulation (42)Uωε,ηx,y=Uω0,ε,ηx,y-∬Dx,y,fFηUωε,ηξ,ζ+Wωε,ηξ,ζdξdζ and (P′ωε,η) is equivalent to the integral formulation (43)Uωε,ηx,y=Uω0,ε,ηx,y-∬Dx,y,fFηUωε,ηξ,ζ⁡Wωε,ηξ,ζdξdζ,where U(ω)0,ε,η(x,y)=Υωεy-Υωε(f(x))+Aωε(x) and Υωε denotes a primitive of Bωε∘f-1, with (44)Dx,y,f=ξ,ζ:f-1y≤ξ≤x,y≤ζ≤fξify≤fx,ξ,ζ:x≤ξ≤f-1y,fξ≤ζ≤yify≥fx.Theorem 7.
Under AssumptionsHε,η, H1, and H2, problem (Pωε,η)(resp., (P′ωε,η)) has a unique solution, U(ω)ε,η, in C∞(R2).See [6, 12] for a detailed proof; replace uε,η by U(ω)ε,ηand Fη(x,y,uε,η(x,y)) by Fη(U(ω)ε,η(x,y))+W(ω)ε,ηx,y (resp., Fη(U(ω)ε,η(x,y))W(ω)ε,ηx,y).


### 3.6.2. Solution to the Problems

Theorem 8.
Assume thatU(ω)ε,η is the solution to problem (Pωε,η)(resp., (P′ωε,η)); then, problem (P(ω)gen)(resp., (P′ωgen)) has a unique solution U(ω)=U(ω)ε,η in AR2.U ( ω ) is the solution to (Pωε,η) (resp., (P′ωε,η)) if U(ω)ε,ηε,η∈X(R2); that is, (45)PK,lUωε,ηε,η∈A,∀K⋐R2,∀l∈N. The proof follows the same steps as the existence results which can be found in [12], replacing uε,η by U(ω)ε,ηand Fη(x,y,uε,η(x,y)) by Fη(U(ω)ε,η(x,y))+W(ω)ε,ηx,y (resp., Fη(U(ω)ε,η(x,y))W(ω)ε,ηx,y). An induction process on the order of the successive derivatives shows that U(ω)ε,ηε,η belongs to X(R2). The Gronwall lemma is essential tool for the uniqueness.Theorem 9.
Assume thatU(ω)ε,η is the solution to problem (Pωε,η)(resp., (P′ωε,η)); then, problem (P(ω)gen)(resp., (P′ωgen)) has a unique solution U(ω)=U(ω)ε,η in A2R2.Proof.
U ( ω ) is the solution to (Pωε,η) if U(ω)ε,ηε,η∈X2(R2). We shall prove that (46)NK,l2Uωε,ηε,η∈A,∀K⋐R2,∀l∈N. But (47)DαUωε,ηL2K≤μK1/2DαUωε,ηL∞K and, as U(ω)ε,ηε,η∈X(R2), we have Dα(U(ω)ε,η)∞ε,η∈A. Thus, (48)DαUωε,ηL2Kε,η=NK,l2Uωε,ηε,η∈A. So U(ω)∈A2R2 and it is the solution to problem (P(ω)gen) in A2R2. Set (49)U:Ω⟶A2R2,ω⟼Uω. Then, U∈A2Ω(R2).Theorem 10.
The mappingU is the solution to problem (P)(resp., (P′)) and it is almost surely unique in A2Ω(R2).Proof.
SinceU(ω) is the unique solution to problem (P(ω)gen) in A2R2, then almost surely in ω∈Ω, the map λ↦RU(λ,·,·,ω)=U(ω)λ, (λ=ε,η), belongs to X2(R2) and it is a representative of U(ω) (i.e., U(ω)=U(ω)λ). For fixed λ=ε,η∈Λ, the map (50)x,y,ω⟼RUλ,x,y,ω=Uωλx,y=uλx,y,ω is jointly measurable on R2×Ω. So U is the solution to problem (P) almost surely unique in A2Ω(R2).


## 3.6.1. Solution to the Parametrized Regular Problems

Forω fixed, we consider the families of regularized problems (Pωε,η) and (P′ωε,η). We are going to prove that (Pωε,η) and (P′ωε,η) have a unique smooth solution under assumptions H1 and H2 and the assumptionHε,η
(a)
f∈C∞(R),  f′>0,  f(R)=R,(b)
Fη∈C∞(R,R),(c)
Aωε,Bωε∈C∞(R). Following [6], one can prove that (Pωε,η) is equivalent to the integral formulation (42)Uωε,ηx,y=Uω0,ε,ηx,y-∬Dx,y,fFηUωε,ηξ,ζ+Wωε,ηξ,ζdξdζ and (P′ωε,η) is equivalent to the integral formulation (43)Uωε,ηx,y=Uω0,ε,ηx,y-∬Dx,y,fFηUωε,ηξ,ζ⁡Wωε,ηξ,ζdξdζ,where U(ω)0,ε,η(x,y)=Υωεy-Υωε(f(x))+Aωε(x) and Υωε denotes a primitive of Bωε∘f-1, with (44)Dx,y,f=ξ,ζ:f-1y≤ξ≤x,y≤ζ≤fξify≤fx,ξ,ζ:x≤ξ≤f-1y,fξ≤ζ≤yify≥fx.Theorem 7.
Under AssumptionsHε,η, H1, and H2, problem (Pωε,η)(resp., (P′ωε,η)) has a unique solution, U(ω)ε,η, in C∞(R2).See [6, 12] for a detailed proof; replace uε,η by U(ω)ε,ηand Fη(x,y,uε,η(x,y)) by Fη(U(ω)ε,η(x,y))+W(ω)ε,ηx,y (resp., Fη(U(ω)ε,η(x,y))W(ω)ε,ηx,y).


## 3.6.2. Solution to the Problems

Theorem 8.
Assume thatU(ω)ε,η is the solution to problem (Pωε,η)(resp., (P′ωε,η)); then, problem (P(ω)gen)(resp., (P′ωgen)) has a unique solution U(ω)=U(ω)ε,η in AR2.U ( ω ) is the solution to (Pωε,η) (resp., (P′ωε,η)) if U(ω)ε,ηε,η∈X(R2); that is, (45)PK,lUωε,ηε,η∈A,∀K⋐R2,∀l∈N. The proof follows the same steps as the existence results which can be found in [12], replacing uε,η by U(ω)ε,ηand Fη(x,y,uε,η(x,y)) by Fη(U(ω)ε,η(x,y))+W(ω)ε,ηx,y (resp., Fη(U(ω)ε,η(x,y))W(ω)ε,ηx,y). An induction process on the order of the successive derivatives shows that U(ω)ε,ηε,η belongs to X(R2). The Gronwall lemma is essential tool for the uniqueness.Theorem 9.
Assume thatU(ω)ε,η is the solution to problem (Pωε,η)(resp., (P′ωε,η)); then, problem (P(ω)gen)(resp., (P′ωgen)) has a unique solution U(ω)=U(ω)ε,η in A2R2.Proof.
U ( ω ) is the solution to (Pωε,η) if U(ω)ε,ηε,η∈X2(R2). We shall prove that (46)NK,l2Uωε,ηε,η∈A,∀K⋐R2,∀l∈N. But (47)DαUωε,ηL2K≤μK1/2DαUωε,ηL∞K and, as U(ω)ε,ηε,η∈X(R2), we have Dα(U(ω)ε,η)∞ε,η∈A. Thus, (48)DαUωε,ηL2Kε,η=NK,l2Uωε,ηε,η∈A. So U(ω)∈A2R2 and it is the solution to problem (P(ω)gen) in A2R2. Set (49)U:Ω⟶A2R2,ω⟼Uω. Then, U∈A2Ω(R2).Theorem 10.
The mappingU is the solution to problem (P)(resp., (P′)) and it is almost surely unique in A2Ω(R2).Proof.
SinceU(ω) is the unique solution to problem (P(ω)gen) in A2R2, then almost surely in ω∈Ω, the map λ↦RU(λ,·,·,ω)=U(ω)λ, (λ=ε,η), belongs to X2(R2) and it is a representative of U(ω) (i.e., U(ω)=U(ω)λ). For fixed λ=ε,η∈Λ, the map (50)x,y,ω⟼RUλ,x,y,ω=Uωλx,y=uλx,y,ω is jointly measurable on R2×Ω. So U is the solution to problem (P) almost surely unique in A2Ω(R2).


## 3.7. The Regularized and the Nonregularized Problems

Remark 11.
The generalized function represented by the family of solutions to the regularized problems(Pωε,η) (resp., (P′ωε,η)) is defined from the integral representation. Thus, we are going to study the relationship between this generalized function and the classical solutions to (P(ω)gen) (resp., (P′ωε,η)), when they exist, on a domain O such that ∀x,y∈O, D(x,y,f)⊂O. This justifies choosing O=a,b×f(a),f(b) when a,b∈R2 with a<b.
If problem(P(ω)gen) (resp., (P′ωgen)) has a smooth solution v on O, then, necessarily, we have O⊂R2∖singsuppU(ω).Let us recall that there exists a canonical sheaf embedding ofC∞ into A, through the morphism of algebra (51)σO:C∞O⟶AO,g⟼gε,where O is any open subset of R2 and gε=g. The presheaf A allows defining restriction and, as usually, we denote by uO the restriction on O of U∈AR2.Theorem 12.
LetO be an open subset of R2 such that (52)O⊂R2∖singsuppUω. Assume that O=⋃ηOη when Oηη is an increasing family of open subsets of R2 such that Oη=aη,bη×f(aη),f(bη) when (aη,bη)∈R2 with aη<bη. Assume that the nonregularized problem (P(ω)gen)(resp., (P′ωgen)) has a smooth solution V(ω) on O such that sup(x,y)∈OηV(ω)(x,y)<rη-1 for any η. Let U(ω)∈AR2 be the generalized function represented by the family U(ω)ε,ηε,η of solutions to (Pωε,η)(resp., (P′ωε,η)). Then, σOV(ω)=U(ω)O.We refer the reader to [12] for a detailed proof.We can say that the solutionU=U(ω) to the regularized problem coincides with the solution to the nonregularized equation on O.


## 3.8. A Special Case

We consider the Cauchy problem formally written as(53)S:∂2U∂x∂y=W,Uγ=A,∂U∂yγ=B,where γ is a monotonic curve of equation y=f(x), γ is not a characteristic curve, A, B∈AΩ(R), and W∈AΩ(R2) is a AR2-generalized stochastic process on a probability space (Ω,Σ,μ).This problem coincides with problem(P) for F=0 and with problem (P′) for F=1. We obtain the solution U to problem S. U(ω)=U(ω)ε,η in A2R2 is defined, with the previous notations, by (54)Uωε,ηx,y=Uω0,ε,ηx,y-∬Dx,y,fWωε,ηξ,ζdξdζ.


## 4. A Nonlinear Stochastic Cauchy Problem with the White Noise as Data

We consider the Cauchy problems formally written as(55)P1:∂2U∂x∂y=FU,Uγ=W,∂U∂yγ=0,P2:∂2V∂x∂y=0,Vγ=W,∂V∂yγ=0,where γ is a monotonic curve of equation y=f(x), γ is not a characteristic curve, and W∈AΩ(R) is the white noise on R. The function F is smooth; it can be non-Lipschitz but F and all derivatives have polynomial growth. Assume that F(0)=0. We look for a solution U:Ω→AR2∈AΩ(R2) and V:Ω→AR2∈AΩ(R2).U is a solution to problem (P1) if and only if, for any ω∈Ω, U(ω) is a solution to the formally problem(56)P1ω:∂2Uω∂x∂y=FUω,Uωγ=Wω,∂Uω∂yγ=0.V is a solution to problem (P2) if and only if, for any ω∈Ω, V(ω) is a solution to the formally problem (57)P2ω:∂2Vω∂x∂y=0,Vωγ=Wω,∂Vω∂yγ=0. We consider the same hypotheses and we take the same spaces AR2 and A2R2 built for problems P and P′.


### 4.1. A Generalized Differential Problem Associated with the Formal One

Our goal is to give a meaning to the problem formally written asP1(ω). For ω fixed, the problem associated with P1(ω) can be written as the well-formulated problem (58)P1genω:∂2Uω∂x∂y=FUω,RfUω=Wωε,Rf∂Uω∂y=0; then, (59)∂2Uω∂x∂y=FηUω,Uωγ=Wω∗χε,∂Uω∂yγ=0. The problem associated with P2(ω) can be written as the well-formulated problem (60)P2genω:∂2Vω∂x∂y=0,RfVω=Wωε,Rf∂Vω∂y=0; then, (61)∂2Vω∂x∂y=0,Vωγ=Wω∗χε,∂Vω∂yγ=0. In terms of representatives, and thanks to the stability and restriction hypothesis, if we can find U(ω)ε,η∈C∞R2 verifying (62)P1ωε,η:∂2Uωε,η∂x∂yx,y=FηUωε,ηx,y,Uωε,ηx,fx=Wω∗χεx,∂Uωε,η∂yx,fx=0 and if we can prove that Uωε,ηε,η∈X2(R2), then U(ω)=Uωε,η is a solution of P1(ω)gen.We have[V(ω)ε,η]=Wω∗χε; then, V(ω)ε,η(x,y)=Wω∗χεx.


### 4.2. Generalized Problem

#### 4.2.1. Solution to the Parametrized Regular Problem

Forω fixed, we consider the family of regularized problems P1(ω)ε,η. We are going to prove that P1(ω)ε,η has a unique smooth solution under the following assumptions:Hε,η
⁡
(a)
f∈C∞(R),  f′>0,  f(R)=R,(b)
Fη∈C∞(R,R),(c)
Wωε∈C∞(R).H1
∃p>0,  ∀l∈N,  ∃al>0, supz∈R;α≤l⁡DαFη(z)≤al1+rηp.Following [6], one can prove that (P1(ω)ε,η) is equivalent to the integral formulation (63)Uωε,ηx,y=Uω0,ε,ηx,y-∬Dx,y,fFηUωε,ηξ,ζdξdζ,where U(ω)0,ε,η(x,y)=Wω∗χεx, with (64)Dx,y,f=ξ,ζ:f-1y≤ξ≤x,y≤ζ≤fξify≤fx,ξ,ζ:x≤ξ≤f-1y,fξ≤ζ≤yify≥fx.Theorem 13.
Under AssumptionsHε,η and H1, problem (P1ωε,η) has a unique solution, U(ω)ε,η, in C∞(R2).We refer the reader to [6, 12] for a detailed proof (replace uε,η by U(ω)ε,ηand Fη(x,y,uε,η(x,y)) by Fη(U(ω)ε,η(x,y))).


#### 4.2.2. Solution toP1

Theorem 14.
Assume thatU(ω)ε,η is the solution to problem (P1ωε,η); then, problem (P1(ω)gen) has a unique solution U(ω)=U(ω)ε,η in AR2.See [12] for a detailed proof (replace uε,η by U(ω)ε,ηand Fη(x,y,uε,η(x,y)) by Fη(U(ω)ε,η(x,y))).Theorem 15.
Assume thatU(ω)ε,η is the solution to problem (P1ωε,η); then, problem (P1(ω)gen) has a unique solution U(ω)=U(ω)ε,η in A2R2.Proof.
U ( ω ) is the solution to (P1ωε,η) if U(ω)ε,ηε,η∈X2(R2). We shall prove that (65)NK,l2Uωε,ηε,η∈A,∀K⋐R2,∀l∈N.But (66)DαUωε,ηL2K≤μK1/2DαUωε,ηL∞Kand, as U(ω)ε,ηε,η∈X(R2), we have Dα(U(ω)ε,η)∞ε,η∈A. Thus, (67)DαUωε,ηL2Kε,η=NK,l2Uωε,ηε,η∈A.So U(ω)∈A2R2 and it is the solution to problem P1(ω)gen in A2R2. Set (68)U:Ω⟶A2R2,ω⟼Uω.Then, U∈A2Ω(R2).Theorem 16.
The mappingU is the solution to problem (P1) and it is almost surely unique in A2Ω(R2).Proof.
SinceU(ω) is the unique solution to problem P1(ω)gen in A2R2, then almost surely in ω∈Ω, the map λ↦RU(λ,·,·,ω)=U(ω)λ, (λ=ε,η), belongs to X2(R2) and it is a representative of U(ω) (i.e., U(ω)=U(ω)λ). For fixed λ=ε,η∈Λ, the map (69)x,y,ω⟼RUλ,x,y,ω=Uωλx,yis jointly measurable on R2×Ω. So U is the solution to problem (P1) almost surely unique in A2Ω(R2).Theorem 17.
LetO be an open subset of R2 such that (70)O⊂R2∖singsuppUω.Assume that O=⋃ηOη when Oηη is an increasing family of open subsets of R2 such that Oη=aη,bη×f(aη),f(bη) when (aη,bη)∈R2 with aη<bη. Assume that the nonregularized problem has a smooth solution V(ω) on O such that sup(x,y)∈OηV(ω)(x,y)<rη-1  for any η. Let U(ω)∈AR2 be the generalized function represented by the family U(ω)ε,ηε,η of solutions to (P1ωε,η). Then, σOV(ω)=U(ω)O.We refer the reader to [12].


### 4.3. Limiting Behavior of the Solution (See [8, 10])

TakeWε=Wω∗χε. We have E(Wε)=0 and V(Wε)=σε2=χεL2(R)2. Then, the variance of Wε tends to infinity as ε tends to 0. That implies the following.Theorem 18.
There is a subsequenceεk→0 such that, μ-almost surely in ω∈Ω, (71)limk→0⁡RVεk,η,x,y,ω=limk→0⁡Vωεk,ηx,y=∞for almost all (x,y)∈R2.Proof.
See [8] Corollary 1 and [10].Assume thatlimz→∞F(z)=L. Define the function M:R2→R by M(x,y)=xyL.Theorem 19.
Under the assumptions above, every subsequence ofε→0 has a subsequence εk→0 such that for all compact set K⋐R2(72)limk→0⁡RUεk,η,·,·,ω-RVεk,η,·,·,ω-ML1K=0μ-almost surely.That is,(73)limk→0⁡Uωεk,η-Vωεk,η-ML1K=0μ-almost surely.Proof.
Takeλ=ε,η. We have (74)Uωλ-Vωλ-M=∂2Uωλ∂x∂y-∂2Vωλ+M∂x∂y,∂2Uωλ∂x∂y-∂2Vωλ+M∂x∂y=FUωλ-FVωλ+M+FVωλ+M-L=Uωλ-Vωλ-M∫01∂F∂zUωλ+σUωλ-Vωλ-Mdσ+FVωλ+M-L.So (75)Uωλ-Vωλ-ML1K≤Uωλ-Vωλ-ML1K∂F∂zL∞R+FVωλ+M-LL1K.By Theorem 18, there is a subsequence εk→0 such that μ-almost surely in ω∈Ω almost everywhere (x,y∈R2), limk→0V(ω)εk,η(x,y)=∞, as limz→∞F(z)=L we deduce that(76)limk→0⁡FVωεk,ηλ+M-LL1K=0.almost everywhere. Hence, by Lebesgue’s theorem and Gronwall’s lemma, the assertion follows.Theorem 20.
LetV∈DΩ′R2 be the distributional solution to the free equation P2. Then, the representative U(ω)ε,η of the generalized solution to the nonlinear problem P1 converge to V+M with respect to the strong topology of D′R2, in probability as ε→0.Proof.
Letq be one of the defining seminorms of the strong topology of D′R2. According to Theorem 19, every subsequence of ε→0 has a subsequence εk→0 such that for all compact set K⋐R2(77)qRUεk,η,·,·,ω-RVεk,η,·,·,ω-M⟶0almost surely. This is equivalent to convergence in probability.


## 4.1. A Generalized Differential Problem Associated with the Formal One

Our goal is to give a meaning to the problem formally written asP1(ω). For ω fixed, the problem associated with P1(ω) can be written as the well-formulated problem (58)P1genω:∂2Uω∂x∂y=FUω,RfUω=Wωε,Rf∂Uω∂y=0; then, (59)∂2Uω∂x∂y=FηUω,Uωγ=Wω∗χε,∂Uω∂yγ=0. The problem associated with P2(ω) can be written as the well-formulated problem (60)P2genω:∂2Vω∂x∂y=0,RfVω=Wωε,Rf∂Vω∂y=0; then, (61)∂2Vω∂x∂y=0,Vωγ=Wω∗χε,∂Vω∂yγ=0. In terms of representatives, and thanks to the stability and restriction hypothesis, if we can find U(ω)ε,η∈C∞R2 verifying (62)P1ωε,η:∂2Uωε,η∂x∂yx,y=FηUωε,ηx,y,Uωε,ηx,fx=Wω∗χεx,∂Uωε,η∂yx,fx=0 and if we can prove that Uωε,ηε,η∈X2(R2), then U(ω)=Uωε,η is a solution of P1(ω)gen.We have[V(ω)ε,η]=Wω∗χε; then, V(ω)ε,η(x,y)=Wω∗χεx.


## 4.2. Generalized Problem

### 4.2.1. Solution to the Parametrized Regular Problem

Forω fixed, we consider the family of regularized problems P1(ω)ε,η. We are going to prove that P1(ω)ε,η has a unique smooth solution under the following assumptions:Hε,η
⁡
(a)
f∈C∞(R),  f′>0,  f(R)=R,(b)
Fη∈C∞(R,R),(c)
Wωε∈C∞(R).H1
∃p>0,  ∀l∈N,  ∃al>0, supz∈R;α≤l⁡DαFη(z)≤al1+rηp.Following [6], one can prove that (P1(ω)ε,η) is equivalent to the integral formulation (63)Uωε,ηx,y=Uω0,ε,ηx,y-∬Dx,y,fFηUωε,ηξ,ζdξdζ,where U(ω)0,ε,η(x,y)=Wω∗χεx, with (64)Dx,y,f=ξ,ζ:f-1y≤ξ≤x,y≤ζ≤fξify≤fx,ξ,ζ:x≤ξ≤f-1y,fξ≤ζ≤yify≥fx.Theorem 13.
Under AssumptionsHε,η and H1, problem (P1ωε,η) has a unique solution, U(ω)ε,η, in C∞(R2).We refer the reader to [6, 12] for a detailed proof (replace uε,η by U(ω)ε,ηand Fη(x,y,uε,η(x,y)) by Fη(U(ω)ε,η(x,y))).


### 4.2.2. Solution toP1

Theorem 14.
Assume thatU(ω)ε,η is the solution to problem (P1ωε,η); then, problem (P1(ω)gen) has a unique solution U(ω)=U(ω)ε,η in AR2.See [12] for a detailed proof (replace uε,η by U(ω)ε,ηand Fη(x,y,uε,η(x,y)) by Fη(U(ω)ε,η(x,y))).Theorem 15.
Assume thatU(ω)ε,η is the solution to problem (P1ωε,η); then, problem (P1(ω)gen) has a unique solution U(ω)=U(ω)ε,η in A2R2.Proof.
U ( ω ) is the solution to (P1ωε,η) if U(ω)ε,ηε,η∈X2(R2). We shall prove that (65)NK,l2Uωε,ηε,η∈A,∀K⋐R2,∀l∈N.But (66)DαUωε,ηL2K≤μK1/2DαUωε,ηL∞Kand, as U(ω)ε,ηε,η∈X(R2), we have Dα(U(ω)ε,η)∞ε,η∈A. Thus, (67)DαUωε,ηL2Kε,η=NK,l2Uωε,ηε,η∈A.So U(ω)∈A2R2 and it is the solution to problem P1(ω)gen in A2R2. Set (68)U:Ω⟶A2R2,ω⟼Uω.Then, U∈A2Ω(R2).Theorem 16.
The mappingU is the solution to problem (P1) and it is almost surely unique in A2Ω(R2).Proof.
SinceU(ω) is the unique solution to problem P1(ω)gen in A2R2, then almost surely in ω∈Ω, the map λ↦RU(λ,·,·,ω)=U(ω)λ, (λ=ε,η), belongs to X2(R2) and it is a representative of U(ω) (i.e., U(ω)=U(ω)λ). For fixed λ=ε,η∈Λ, the map (69)x,y,ω⟼RUλ,x,y,ω=Uωλx,yis jointly measurable on R2×Ω. So U is the solution to problem (P1) almost surely unique in A2Ω(R2).Theorem 17.
LetO be an open subset of R2 such that (70)O⊂R2∖singsuppUω.Assume that O=⋃ηOη when Oηη is an increasing family of open subsets of R2 such that Oη=aη,bη×f(aη),f(bη) when (aη,bη)∈R2 with aη<bη. Assume that the nonregularized problem has a smooth solution V(ω) on O such that sup(x,y)∈OηV(ω)(x,y)<rη-1  for any η. Let U(ω)∈AR2 be the generalized function represented by the family U(ω)ε,ηε,η of solutions to (P1ωε,η). Then, σOV(ω)=U(ω)O.We refer the reader to [12].


## 4.2.1. Solution to the Parametrized Regular Problem

Forω fixed, we consider the family of regularized problems P1(ω)ε,η. We are going to prove that P1(ω)ε,η has a unique smooth solution under the following assumptions:Hε,η
⁡
(a)
f∈C∞(R),  f′>0,  f(R)=R,(b)
Fη∈C∞(R,R),(c)
Wωε∈C∞(R).H1
∃p>0,  ∀l∈N,  ∃al>0, supz∈R;α≤l⁡DαFη(z)≤al1+rηp.Following [6], one can prove that (P1(ω)ε,η) is equivalent to the integral formulation (63)Uωε,ηx,y=Uω0,ε,ηx,y-∬Dx,y,fFηUωε,ηξ,ζdξdζ,where U(ω)0,ε,η(x,y)=Wω∗χεx, with (64)Dx,y,f=ξ,ζ:f-1y≤ξ≤x,y≤ζ≤fξify≤fx,ξ,ζ:x≤ξ≤f-1y,fξ≤ζ≤yify≥fx.Theorem 13.
Under AssumptionsHε,η and H1, problem (P1ωε,η) has a unique solution, U(ω)ε,η, in C∞(R2).We refer the reader to [6, 12] for a detailed proof (replace uε,η by U(ω)ε,ηand Fη(x,y,uε,η(x,y)) by Fη(U(ω)ε,η(x,y))).


## 4.2.2. Solution toP1

Theorem 14.
Assume thatU(ω)ε,η is the solution to problem (P1ωε,η); then, problem (P1(ω)gen) has a unique solution U(ω)=U(ω)ε,η in AR2.See [12] for a detailed proof (replace uε,η by U(ω)ε,ηand Fη(x,y,uε,η(x,y)) by Fη(U(ω)ε,η(x,y))).Theorem 15.
Assume thatU(ω)ε,η is the solution to problem (P1ωε,η); then, problem (P1(ω)gen) has a unique solution U(ω)=U(ω)ε,η in A2R2.Proof.
U ( ω ) is the solution to (P1ωε,η) if U(ω)ε,ηε,η∈X2(R2). We shall prove that (65)NK,l2Uωε,ηε,η∈A,∀K⋐R2,∀l∈N.But (66)DαUωε,ηL2K≤μK1/2DαUωε,ηL∞Kand, as U(ω)ε,ηε,η∈X(R2), we have Dα(U(ω)ε,η)∞ε,η∈A. Thus, (67)DαUωε,ηL2Kε,η=NK,l2Uωε,ηε,η∈A.So U(ω)∈A2R2 and it is the solution to problem P1(ω)gen in A2R2. Set (68)U:Ω⟶A2R2,ω⟼Uω.Then, U∈A2Ω(R2).Theorem 16.
The mappingU is the solution to problem (P1) and it is almost surely unique in A2Ω(R2).Proof.
SinceU(ω) is the unique solution to problem P1(ω)gen in A2R2, then almost surely in ω∈Ω, the map λ↦RU(λ,·,·,ω)=U(ω)λ, (λ=ε,η), belongs to X2(R2) and it is a representative of U(ω) (i.e., U(ω)=U(ω)λ). For fixed λ=ε,η∈Λ, the map (69)x,y,ω⟼RUλ,x,y,ω=Uωλx,yis jointly measurable on R2×Ω. So U is the solution to problem (P1) almost surely unique in A2Ω(R2).Theorem 17.
LetO be an open subset of R2 such that (70)O⊂R2∖singsuppUω.Assume that O=⋃ηOη when Oηη is an increasing family of open subsets of R2 such that Oη=aη,bη×f(aη),f(bη) when (aη,bη)∈R2 with aη<bη. Assume that the nonregularized problem has a smooth solution V(ω) on O such that sup(x,y)∈OηV(ω)(x,y)<rη-1  for any η. Let U(ω)∈AR2 be the generalized function represented by the family U(ω)ε,ηε,η of solutions to (P1ωε,η). Then, σOV(ω)=U(ω)O.We refer the reader to [12].


## 4.3. Limiting Behavior of the Solution (See [8, 10])

TakeWε=Wω∗χε. We have E(Wε)=0 and V(Wε)=σε2=χεL2(R)2. Then, the variance of Wε tends to infinity as ε tends to 0. That implies the following.Theorem 18.
There is a subsequenceεk→0 such that, μ-almost surely in ω∈Ω, (71)limk→0⁡RVεk,η,x,y,ω=limk→0⁡Vωεk,ηx,y=∞for almost all (x,y)∈R2.Proof.
See [8] Corollary 1 and [10].Assume thatlimz→∞F(z)=L. Define the function M:R2→R by M(x,y)=xyL.Theorem 19.
Under the assumptions above, every subsequence ofε→0 has a subsequence εk→0 such that for all compact set K⋐R2(72)limk→0⁡RUεk,η,·,·,ω-RVεk,η,·,·,ω-ML1K=0μ-almost surely.That is,(73)limk→0⁡Uωεk,η-Vωεk,η-ML1K=0μ-almost surely.Proof.
Takeλ=ε,η. We have (74)Uωλ-Vωλ-M=∂2Uωλ∂x∂y-∂2Vωλ+M∂x∂y,∂2Uωλ∂x∂y-∂2Vωλ+M∂x∂y=FUωλ-FVωλ+M+FVωλ+M-L=Uωλ-Vωλ-M∫01∂F∂zUωλ+σUωλ-Vωλ-Mdσ+FVωλ+M-L.So (75)Uωλ-Vωλ-ML1K≤Uωλ-Vωλ-ML1K∂F∂zL∞R+FVωλ+M-LL1K.By Theorem 18, there is a subsequence εk→0 such that μ-almost surely in ω∈Ω almost everywhere (x,y∈R2), limk→0V(ω)εk,η(x,y)=∞, as limz→∞F(z)=L we deduce that(76)limk→0⁡FVωεk,ηλ+M-LL1K=0.almost everywhere. Hence, by Lebesgue’s theorem and Gronwall’s lemma, the assertion follows.Theorem 20.
LetV∈DΩ′R2 be the distributional solution to the free equation P2. Then, the representative U(ω)ε,η of the generalized solution to the nonlinear problem P1 converge to V+M with respect to the strong topology of D′R2, in probability as ε→0.Proof.
Letq be one of the defining seminorms of the strong topology of D′R2. According to Theorem 19, every subsequence of ε→0 has a subsequence εk→0 such that for all compact set K⋐R2(77)qRUεk,η,·,·,ω-RVεk,η,·,·,ω-M⟶0almost surely. This is equivalent to convergence in probability.


## 5. Conclusion

We can use efficiently the(C,E,P)-algebras in studying stochastic differential equations and the white noise. We hope that this study will convince the reader that the (C,E,P)-algebras are a very good tool to study stochastic generalized processes.



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*Source: 5963420-2016-10-18.xml*