# Quasilinear Stochastic Cauchy Problem in Abstract Colombeau Spaces

Authors: Irina V. Melnikova; Uljana A. Alekseeva
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

Generalized solutions to the abstract Cauchy problem for a quasilinear equation with the generator of an integrated semigroup and with terms reflecting nonlinear perturbations and white noise type perturbations are under consideration. An abstract stochastic Colombeau algebra is constructed, and solutions in the algebra are studied.


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## Body

## 1. Introduction

The paper is devoted to construction of solutions to the abstract quasilinear Cauchy problem(1.1)X'(t)=AX(t)+F(X)+BW(t),t≥0,X(0)=f,
where A is the generator of a C0 semigroup or an integrated semigroup in a Hilbert space H, F is a nonlinear mapping from H to H, B is a linear bounded operator from a Hilbert space ℍ to H, and W={W(t),t≥0} is a stochastic process of white noise type with values in ℍ: W(t)=W(t,ω),ω∈(Ω,ℬ(Ω),μ).Irregularity of white noise caused by independence of random variablesW(t1) and W(t2) for t1≠t2 and by infinite variance necessitates to define the white noise in such a way that the problem (1.1) makes certain sense.One of the well-known ways to do this is to consider the corresponding integral equation replacing the white noise term by the integral with respect to a Wiener processW (Ito integral), written as usually in Ito theory in the following form of differentials:
(1.2)dX(t)=AX(t)dt+F(X)dt+BdW(t),t≥0,X(0)=f.
For the Cauchy problem (1.2) with generators of semigroups of class C0, with Lipschitz nonlinearities under some growth conditions mild solutions are constructed (see, i.e., [1, 2]). In this approach questions of whether the solutions obtained are differentiable, whether they satisfy the problem (1.1), and whether the techniques can be applied for A generating regularized semigroups remain open.Another approach, which we are going to use and generalize, is to consider (1.1) in spaces of abstract distributions, but here, due to nonlinearity of F in the equation, the problem of distribution products arises. A novel approach is to define an abstract stochastic Colombeau algebra 𝒢(Ω,Ha) (see definition in Section 1) and extend the distribution approach to the algebra.LetHa be an algebra in a Hilbert space H, in particular, the subspace of continuous or a finitely many times differentiable functions in L2(ℝ) closed under the topology of Ck(ℝ), k=0,1,…. We consider the Cauchy problem (1.1) in the abstract stochastic Colombeau algebra 𝒢(Ω,Ha) supposing that F is an infinitely differentiable mapping, B∈ℒ(ℍ,Ha), and Ha⊂domA. We define white noise W as an element in 𝒟+′(ℍ), the space of abstract distributions with values in ℍ and supports in [0,∞), then by convolution with functions from 𝒟 we transform W∈𝒟+′(ℍ) to infinitely differentiable with respect to t functions and as a result we obtain an element BWbelonging to the algebra 𝒢(Ω,Ha).As examples ofA satisfying the conditions and generating different integrated semigroups one can take many of differential operators A=P(i(∂/∂x)) of correct in the sense of Petrovskiy systems [3]. The operators may be disturbed by bounded ones of any nature. For more examples see in [4, 5].In the paper we combine the multiplication theory in Colombeau algebras which has found applications in solving differential equations, mainly hyperbolic ones (see, e.g., [6, 7]) with the theory of regularized semigroups and the theory of stochastic processes in spaces of abstract distributions (see, e.g., [8, 9]). This makes possible to solve nonlinear abstract stochastic equations with different types of white noise.


## 2. Definition of Abstract Colombeau Algebras

At the beginning we introduce Colombeau space of abstract (Hilbert space valued) generalized functions. For eachq∈ℕ0 let 𝒜q be the set of all φ∈𝒟 such that
(2.1)∫ℝ‍φ(t)dt=1,∫ℝ‍tkφ(t)dt=0,k=1,…,q.
For algebra Ha in a Hilbert space H we define the space of functions u(φ):=u(φ(t),t),φ∈𝒜0,t∈ℝ as follows:
(2.2)ℰ(Ha):=(C∞(ℝ;Ha))𝒜0={u:𝒜0→C∞(ℝ;Ha)}.
According to the definition u(φ) is an infinitely differentiable Ha-valued function of real argument t∈ℝ for each φ∈𝒜0. Thus u can be considered as a function of two variables: φ∈𝒜0 and t∈ℝ, that is,
(2.3)u:𝒜0×ℝ→Ha:u=u(φ,t),φ∈𝒜0,t∈ℝ.
and it is infinitely differentiable with respect to the second variable.Differentiation and multiplication are defined as follows:(2.4)(uv)(φ):=u(φ)v(φ),u(n)(φ):=dndtnu(φ),φ∈𝒜0.
The space of the Ha-valued distributions 𝒟'(Ha) being a subset of the abstract distributions space 𝒟'(H) is embedded in ℰ(Ha) by the following mapping:
(2.5)i:𝒟'(Ha)→ℰ(Ha),(iw)(φ):=w*φ,w∈𝒟'(Ha),φ∈𝒜0.
Now let us introduce the functions φɛ(t):=(1/ɛ)φ(t/ɛ),t∈ℝ, ɛ>0, φ∈𝒜0 and define the linear manyfold of moderate elements ℰM(Ha) consisting of all u∈ℰ(Ha) satisfying the following condition:(M) for each compactK⊂ℝ and each n∈ℕ0 there exists q∈ℕ such that
(2.6)supt∈K∥dndtnu(φɛ,t)∥H=𝒪ɛ→0(ɛ-q)foreachφ∈𝒜q.
To complete the definition of the Colombeau algebra of abstract generalized functions we introduce 𝒩(Ha) consisting of all elements u∈ℰ(Ha) that satisfy the following condition:(N) for each compactK⊂ℝ and each n∈ℕ0 there exists p∈ℕ such that
(2.7)supt∈K∥dndtnu(φɛ,t)∥H=𝒪ɛ→0(ɛq-p)foreachφ∈𝒜q,q≥p.
Elements of the space ℰM(Ha) form a differential algebra, and 𝒩(Ha) is the differential ideal in it. Now define the factor algebra
(2.8)𝒢(Ha):=ℰM(Ha)𝒩(Ha).
Similarly to 𝒢(ℝ) (see, e.g., [6]), the algebra 𝒢(Ha) is an associative and commutative Ha-valued differential one. Elements of 𝒢(Ha) are classes of mappings. We denote them by capitals U,V,… and denote representatives of class U∈𝒢(Ha) by corresponding small letter u.Due to the structure theorems for abstract distributions [10], similarly to the ℝ-valued case, we obtain that i maps the elements of 𝒟'(Ha) into ℰM(Ha) and i-1(𝒩(Ha)) consists of the null element of 𝒟'(Ha). Thus, each element of 𝒟'(Ha) is imbedded in the corresponding class of 𝒢(Ha) by the mapping i.Support of an elementU∈𝒢(Ha) is defined as follows. We say that U is equal to zero on an open set Λ⊂ℝ if its restriction to 𝒢Λ(Ha) is equal to zero in 𝒢Λ(Ha) (where algebra 𝒢Λ(Ha) is defined in the same way as 𝒢Λ(ℝn) [6]). If w∈𝒟'(Ha), then, similarly to the case of 𝒟'(ℝ), support of iw∈𝒢(Ha) coincides with that of w∈𝒟'(Ha).Now define𝒢(Ω,Ha), the algebra of 𝒢(Ha)-valued random variables {U=U(ω),ω∈(Ω,ℬ(Ω),μ)} as a mapping from (Ω,ℬ(Ω),μ) to 𝒢(Ha) measurable in the following sense: there exists a representative u∈U such that for any φ∈𝒜0u-1(φ,·) maps any Borel subset of ℬ(C∞(Ha)) onto an element of ℬ(Ω), where the Borel σ-algebra ℬ(C∞(Ha)) on the space C∞(Ha) is generated by the system of neighborhoods in C∞(Ha) defined by the system of seminorms pn,k(f)=supt∈[-n,n]∥f(k)(t)∥H.To complete the setting of the problem we define the generalized white noise processW=W(·,ω) for each ω∈Ω, as an element in 𝒟+′(ℍ), and the space of abstract distributions with values in ℍ and supports in [0,∞), and then transform it into an element of 𝒢(Ω,Ha).One way to do this is based on the ideas of abstract stochastic distributions (see, e.g., [8, 9]). Let 𝒮=𝒮(ℝ) be the space of rapidly decreasing test functions. Denote by 𝒮'(ℍ) the space of ℍ-valued distributions over 𝒮 and consider a Borel σ-algebra ℬ(Ω) generated by the weak topology of Ω:=𝒮'(ℍ). Then by the generalization of the Bochner–Minlos theorem to the case of Hilbert space valued generalized functions [11], there exist a unique probability measure μ on ℬ(Ω) and a trace class operator Q satisfying the condition as follows
(2.9)∫Ω‍ei(〈ω,θ〉,h)ℍdμ(ω)=e-(1/2)∥θ∥2(Qh,h),θ∈𝒮,h∈ℍ.
(Here and below, if it is not pointed out especially, ∥·∥ denotes the norm of L2(ℝ)).It makes possible to define the white noise process on(Ω,ℬ(Ω),μ) with values in 𝒮'(ℍ)⊂𝒟'(ℍ) by the identical mapping as follows:
(2.10)〈W(·,ω),θ(·)〉:=〈ω,θ〉,θ∈𝒮.
The above-defined process is the generalization of the corresponding real-valued Gaussian process [12], and it has zero mean and Cov〈W,θ〉=∥θ∥2Q. Define W+ with support in [0,∞) as follows:
(2.11)〈W+(·,ω),θ(·)〉:=(-1)k〈(χ·W(-k))(·),θ(k)(·)〉,θ∈𝒮.
Here W(-k) is a continuous function that according to the structure theorem is a primitive of W of an order k and χ is the Heaviside function.Another way to define a generalizedℍ-valued white noise on a (Ω,ℬ(Ω),μ), more precisely Q-white noise, is via derivative of ℍ-valued Q-Wiener process {WQ(t),t≥0} [9] continued by zero on (-∞,0) as follows:
(2.12)〈W+(·,ω),θ〉:=-〈WQ(·,ω),θ'〉,θ∈𝒟.Finally, we map the defined generalized white noise process into the Colombeau algebra𝒢(Ω,Ha) in the following manner. By convolution with a function from 𝒜0 we transform W+(·,ω)∈𝒟'(ℍ) into the infinitely differential with respect to t∈ℝ and measurable with respect to ω∈Ω function w as follows:
(2.13)w(φ,t,ω):=〈W+(·,ω),φ(t-·)〉,φ∈𝒜0,t∈ℝ.
So, w(φ,·,ω)∈C∞(ℝ,ℍ),φ∈𝒜0,ω∈Ω a.s., and w(φ,·,·)∈C∞(ℝ,L2(Ω,ℍ)).LetB∈ℒ(ℍ,Ha), and then applying B to w we obtain that Bw(φ,t,ω)∈Ha and the map
(2.14)Bw(φ,·,·):𝒜0→C∞(ℝ,L2(Ω,Ha))
are representative of a class in 𝒢(Ω,Ha). The corresponding class we denote by BW. Since the support of W+∈𝒟'(ℍ) is [0;∞), by definition of support of an element of 𝒢(Ω,Ha) we have suppBW=[0;∞). That is the sense we attach to the stochastic term in (1.1).


## 3. Solutions to the Cauchy Problem with Infinitely Differentiable Nonlinearities

LetH=ℍ=L2(ℝ) and the domain of A lie in the set of continuous functions of L2(ℝ). Let Ha be the set of finitely many times differentiable functions of L2(ℝ) and Ha⊆domA. Then multiplication of elements of L2(ℝ) is well defined on the set L2(ℝ)∩domA as pointwise continuous functions multiplication.In this section for the problem (1.1), where nonlinearity F is infinitely differentiable, bounded with all its derivatives, and has the property F(0)=0 and where stochastic term BW is constructed above, we will search a solution as an element of the abstract stochastic Colombeau algebra 𝒢(Ω,Ha). Since Ha is chosen as the set of finitely many times differentiable functions in L2(ℝ), operator B∈ℒ(ℍ,Ha) can be taken, for example, as convolution with a finitely many times differentiable function from L2(ℝ) and with condition of the convolution existence.Suppose at the beginning thatA generates a C0-semigroup {V(t),t≥0} in L2(ℝ).Consider the question whether there is a solution to the problem(3.1)Y'=AY+F(Y)+BW,suppY⊆[0,∞),
as an element of algebra 𝒢(Ω,Ha). To do this, for an arbitrary η>0, we consider Bw(t):=Bw(φ,t,ω), φ∈𝒜0, ω∈Ω, with support in [-η,∞) as a representative of the white noise term BW∈𝒢(Ω,Ha) [7]. By definition of elements of 𝒢(Ω,Ha), for each fixed φ∈𝒜0, Bw(t) is an infinitely differentiable function of t∈ℝ with values in Ha and measurable with respect to ω∈Ω. Let us take an arbitrary φ∈𝒜0 and consider the problem
(3.2)y'(t)=Ay(t)+F(y(t))+Bw(t),t≥-η,y(t)=0,t≤-η,
where y(t)=y(φ,t,ω),φ∈𝒜0,ω∈Ω. We will search a solution of this problem belonging to C∞([-η;∞);domA) for ω a.s.Consider the equation(3.3)y(t)=∫-ηt‍V(t-s)F(y(s))ds+∫-ηt‍V(t-s)Bw(s)ds=:𝒬y(t),t≥-η.
The introduced operator 𝒬 is a Volterra type one. Using the differentiability of F and boundedness of its derivative let us show that 𝒬k (where k=k(T)) is a contraction on the segment [-η;T].SinceF is differentiable, we have F(μ)-F(λ)=F'(ξ)(μ-λ),ξ∈(μ;λ), for any μ,λ∈ℝ. Then for any y(·) and z(·) we get the pointwise equality
(3.4)F(y(s))-F(z(s))=F′(ξ)(y(s)-z(s)),s∈[-η;∞),
where ξ is an appropriate point from (y(s);z(s)) and the following estimate holds:
(3.5)∥F(y(s))-F(z(s))∥≤L∥y(s)-z(s)∥,L=maxξ∈ℝ|F′(ξ)|.
This and exponential boundedness of C0 semigroups:
(3.6)∥V(t)∥ℒ(L2(ℝ))≤Ceat,
for each t∈[-η;∞) imply that
(3.7)∥𝒬y(t)-𝒬z(t)∥≤CLea(t+η)(t+η)maxs∈[-η;t]∥y(s)-z(s)∥.
For squares we have
(3.8)𝒬2y(t)-𝒬2z(t)=∫-ηt‍V(t-s)F(𝒬(y(s)))ds-∫-ηt‍V(t-s)F(𝒬(z(s)))ds=∫-ηt‍V(t-s)[F(𝒬(y(s)))-F(𝒬(z(s)))]ds.
Then we have
(3.9)∥𝒬2y(t)-𝒬2z(t)∥≤C2L2e2a(t+η)(t+η)22maxs∈[-η;t]∥y(s)-z(s)∥,
and for every k∈ℕ(3.10)∥𝒬ky(t)-𝒬kz(t)∥≤CkLkeka(t+η)(t+η)kk!maxs∈[-η;t]∥y(s)-z(s)∥,
hence
(3.11)maxt∈[-η;T]∥𝒬ky(t)-𝒬kz(t)∥≤CkLkeka(T+η)(T+η)kk!maxt∈[-η;T]∥y(t)-z(t)∥.
The constant in this estimate can be made less then unity by choosing k=k(T). Thus 𝒬k is the contraction, and the sequence of approximations yn(t)=𝒬nky0(t) has the limit in H:
(3.12)y(t)=limn→∞𝒬nky0(t),
uniform with respect to t∈[-η;T].Note that if one takes an infinitely differentiable with respect tot function y0(·) as the first point for the approximating sequence, then function
(3.13)z(t)=𝒬y0(t)=∫-ηt‍V(t-s)F(y0(s))ds+∫-ηt‍V(t-s)Bw(s)ds,t≥-η,
is also an infinitely differentiable with respect to t function, and consequently y1(·)=𝒬ky0(·) has the same property as well as all subsequent iterations yn(·).It can be shown by the same arguments that the sequenceyn′(·) converges to its limit in H uniformly with respect to t∈[-η;T]; hence y(·) is differentiable and y'(t)=limn→∞yn′(t). Similarly it can be shown that y(·) is infinitely differentiable function with values in H.Now we show thatyn(t)∈Ha if y0(t)∈Ha, t≥0. Let t≥0 be fixed. Note firstly that F(α)=𝒪(α) as α→0. Really, due to the infinite differentiability of F and property F(0)=0F(α) can be repersented by the Taylor series with first term proportional to α. Then, since y0(t)∈Ha, it is differentiable with respect to variable of L2(ℝ), and y0(t)→0 as variable of L2(ℝ) tends to infinity. Thus, F(y0(t))=𝒪(y0(t)) as variable of L2(ℝ) tends to infinity.Further, semigroup operatorsV(t) map L2(ℝ) into L2(ℝ), and, moreover, they map differentiable functions (with respect to variable of L2(ℝ)) into the set of differentiable ones due to their boundness. It follows that
(3.14)V(t-s)F(y0(s)):Ha→Ha,V(t-s)Bw(s)∈Ha,
hence 𝒬 acts in Ha and y1(t)=𝒬ky0(t)∈Ha as well as yn(t) for any n∈ℕ.Thus, we obtainyn(t)∈Ha, but in the general case limn→∞yn(t)=y(t) does not belong to Ha since algebra Ha is not closed in the sense of L2(ℝ) convergence.Ify(t)∈Ha, then we show that it is a representative of a class from 𝒢(Ω,Ha). As it is known (see, e.g., [4]), if ℱ(·) is a differentiable function or ℱ(t)∈domA for any t≥0, then solution of the inhomogeneous abstract Cauchy problem
(3.15)u'(t)=Au(t)+ℱ(t),t≥0,u(0)=0,
with A generating a C0 semigroup of operators {V(t),t≥0} exists and is defined by the formulae u(t)=∫0t‍V(t-s)ℱ(s)ds. Since in the case under consideration F(y(t)) as well as any representative Bw(t) of white noise process are L2(ℝ)-valued infinitely differentiable with respect to t functions, the solution of (3.3) is a solution to the problem (3.2).Due to the property ofC0 semigroup V(t)=0 as t<0, it follows from (3.3) that y(t)=0 as t≤-η; that is, support of the obtained solution lies in [-η;∞).Now we show thaty is a representative of a class Y∈𝒢(Ω,Ha), that means that it satisfies the condition (M) almost everywhere. It follows from differentiability of F and the condition F(0)=0 that for each s∈[-η;t] the following equality
(3.16)F(y(φ,s,ω))=F(y(φ,s,ω))-F(0)=F′(ξ)y(φ,s,ω),
where ξ∈(0;y(φ,s,ω)), φ∈𝒜0, ω∈Ω, holds. Now for an arbitrary compact K⊂ℝ from (3.3) we obtain that
(3.17)maxt∈K∥y(φɛ,t,ω)∥≤Cmaxt∈Kea(t+η)∥Bw(φɛ,t,ω)∥+Cmaxξ∈ℝ∥F'(ξ)∥maxt∈Kea(t+η)∫-ηt‍∥y(φɛ,s,ω)∥ds,
and due to boundedness of F' we have
(3.18)maxt∈K∥y(φɛ,t,ω)∥≤C1maxt∈K∥Bw(φɛ,t,ω)∥+C2maxt∈K∫-ηt‍∥y(φɛ,s,ω)∥ds.
Since Bw is a representative of class BW from 𝒢(Ω,Ha), the first term in the right-hand side of the inequality for every φ∈𝒜q increases as ɛ→0 not faster than ɛ-q for some q∈ℕ. Then, due to Gronwall-Bellmann inequality the left-hand side behaves in the same way, that proves the condition (M) with n=0. Let us remind that the Gronwall-Bellmann inequality states that if
(3.19)y(t)≤c+∫t0t‍f(s)y(s)ds,c>0,t>t0,
for positive continuous functions y and f, than y(t)≤ce∫t0tf(s)ds. Behavior of derivatives of y(·) can be checked up in the same manner using that F is infinitely differentiable and its derivatives are bounded.Now let us show that suppY⊆[0;∞). To do this we consider two solutions of (3.2) yη1(·) and yη2(·) corresponding η1≠η2 and verify that difference yη1(·)-yη1(·) belongs to 𝒩(Ha). Note that η=max{η1,η2}. Then we have
(3.20)yη1′(t)-yη2′(t)=A(yη1(t)-yη2(t))+F(yη1(t))-F(yη2(t))+g(t),t≥-η,yη1(t)-yη2(t)=0,t≤-η,
where g∈𝒩(Ha) as the difference of two representatives of the stochastic term BW whose support is in [0;∞). Then, similar to (3.18), we obtain the following estimation:
(3.21)maxt∈K∥yη1(φɛ,t,ω)-yη2(φɛ,t,ω)∥≤C1maxt∈K∥g(φɛ,t,ω)∥+C2maxt∈K∫-ηt‍∥yη1(φɛ,s,ω)-yη2(φɛ,t,ω)∥ds.
Since the first term satisfies to (N), Gronwall-Bellman lemma implies that yη1(t)-yη2(t)∈𝒩(Ha), so suppY⊆[0;∞).Now we show that the solution of (3.1) is unique in the algebra 𝒢(Ω,Ha). Let Y1,Y2∈𝒢(Ω,Ha) with support in [0;∞) be two solutions of (3.1). Then for any representatives y1, y2 of these classes and each η>0 the following relations hold:
(3.22)y1′(t)-y2′(t)=A(y1(t)-y2(t))+F(y1(t))-F(y2(t))+g(t),t≥-η,y1(t)-y2(t)∈𝒩(Ha),t≤-η,
where g is an element of 𝒩(Ha). Then, as above, Gronwall-Bellman Lemma implies that y1(t)-y2(t)∈𝒩(Ha), that is Y1-Y2=0 in 𝒢(Ω,Ha).Taking into account that the linear Cauchy problem, corresponding to (1.1) has the following form in spaces of distributions:
(3.23)〈X',φ〉=〈δ,φ〉f+〈AX,φ〉+〈BW,φ〉,φ∈𝒟,
and the required solution to the Cauchy problem (1.1) in 𝒢(Ω,Ha) is related with the obtained Y as follows: X(t)=Y(t)+(V*iδ)(t)f, t≥0, f∈Ha.In the general case since the limit ofyn(t)∉Ha, we obtain only the approximated solutions of (3.3)—the fundamental sequence {yn} obtained by the following equalities
(3.24)yn(t)=𝒬kyn-1(t),t≤-η.So, we get the following result.Theorem 3.1.
LetA be the generator of a C0-semigroup {V(t),t≥0} in L2(ℝ). Let F be an infinitely differentiable function in ℝ, bounded with all its derivatives and F(0)=0. Let B∈ℒ(L2(ℝ),Ha) and BW be an element of 𝒢(Ω,Ha) with representative Bw defined by (2.13). Then for any η>0 and φ∈𝒜0 there exists the unique solution of (3.2) y∈C∞([-η;∞);H). If y∈C∞([-η;∞);Ha), then (3.1) has the unique solution in algebra 𝒢(Ω,Ha). In this case the solution to the Cauchy problem (1.1) in 𝒢(Ω,Ha) is X=Y+(V*iδ)f for any f∈Ha.Now consider the case ofA generating an integrated semigroup. If operator A generates an exponentially bounded n-times integrated semigroup {Vn(t),t≥0}, then solution operators V(t) of homogeneous Cauchy problem are defined as follows: 〈V,φ〉=(-1)n〈Vn,φ〉, φ∈𝒟, and instead of (3.3) we have the following equation:
(3.25)y(φ,t,ω)=∫-ηt‍Vn(t-s)F(n)(y(φ,s,ω))ds+∫-ηt‍Vn(t-s)(Bw)(n)(φ,s,ω)ds,φ∈𝒜0,ω∈Ω,t≥-η.
Here all derivatives (in t) exist due to infinite differentiability of F and Bw. Using the equality, similarly to the case of semigroups of class C0, we obtain the corresponding approximations yn and the solution in 𝒢(Ω,Ha) if the limit of yn(t) belongs to Ha.


## 4. Conclusions

In conclusion we note that the present paper is only the beginning of researches of abstract stochastic equations with nonlinearities in Colombeau algebras. Among important questions that remain open and are supposed to be investigated in future are convergence of solutionsy(φɛ) as ɛ→0, equations with generators of more general regularized semigroups, and equations in arbitrary Hilbert spaces.



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