Quasilinear Stochastic Cauchy Problem in Abstract Colombeau Spaces

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.

1. Introduction

The paper is devoted to construction of solutions to the abstract quasilinear Cauchy problem 𝑋′(𝑡)=𝐴𝑋(𝑡)+𝐹(𝑋)+𝐵𝑊(𝑡),𝑡≥0,𝑋(0)=𝑓, (1.1) 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 variables 𝑊(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 process 𝑊 (Ito integral), written as usually in Ito theory in the following form of differentials: 𝑑𝑋(𝑡)=𝐴𝑋(𝑡)𝑑𝑡+𝐹(𝑋)𝑑𝑡+𝐵𝑑𝑊(𝑡),𝑡≥0,𝑋(0)=𝑓. (1.2) 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,𝐻_𝑎) (see definition in Section 1) and extend the distribution approach to the algebra.

Let 𝐻_𝑎 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 𝐶^𝑘(ℝ), 𝑘=0,1,…. We consider the Cauchy problem (1.1) in the abstract stochastic Colombeau algebra 𝒢(Ω,𝐻_𝑎) supposing that 𝐹 is an infinitely differentiable mapping, 𝐵∈ℒ(ℍ,𝐻_𝑎), and 𝐻_𝑎⊂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 BW belonging to the algebra 𝒢(Ω,𝐻_𝑎).

As examples of 𝐴 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 each 𝑞∈ℕ₀ let 𝒜_𝑞 be the set of all 𝜑∈𝒟 such that _ℝ𝜑(𝑡)𝑑𝑡=1,_ℝ𝑡^𝑘𝜑(𝑡)𝑑𝑡=0,𝑘=1,…,𝑞. (2.1) For algebra 𝐻_𝑎 in a Hilbert space 𝐻 we define the space of functions 𝑢(𝜑)∶=𝑢(𝜑(𝑡),𝑡),𝜑∈𝒜₀,𝑡∈ℝ as follows: ℰ𝐻_𝑎∶=𝐶^∞ℝ;𝐻_𝑎^𝒜0=𝑢∶𝒜₀→𝐶^∞ℝ;𝐻_𝑎. (2.2) According to the definition 𝑢(𝜑) is an infinitely differentiable 𝐻_𝑎-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, 𝑢∶𝒜₀×ℝ→𝐻_𝑎∶𝑢=𝑢𝜑,𝑡,𝜑∈𝒜₀,𝑡∈ℝ. (2.3) and it is infinitely differentiable with respect to the second variable.

Differentiation and multiplication are defined as follows: ()𝜑𝜑𝑣𝜑,𝑢𝑢𝑣∶=𝑢_(𝑛)𝜑𝑑∶=^𝑛𝑑𝑡^𝑛𝑢𝜑,𝜑∈𝒜₀. (2.4) The space of the 𝐻_𝑎-valued distributions 𝒟′(𝐻_𝑎) being a subset of the abstract distributions space 𝒟′(𝐻) is embedded in ℰ(𝐻_𝑎) by the following mapping: 𝑖∶𝒟′𝐻_𝑎→ℰ𝐻_𝑎,(𝑖𝑤)𝜑∶=𝑤∗𝜑,𝑤∈𝒟′𝐻_𝑎,𝜑∈𝒜₀. (2.5) Now let us introduce the functions 𝜑_𝜀(𝑡)∶=(1/𝜀)𝜑(𝑡/𝜀),𝑡∈ℝ, 𝜀>0, 𝜑∈𝒜₀ and define the linear manyfold of moderate elements ℰ_𝑀(𝐻_𝑎) consisting of all 𝑢∈ℰ(𝐻_𝑎) satisfying the following condition:

(M) for each compact 𝐾⊂ℝ and each 𝑛∈ℕ₀ there exists 𝑞∈ℕ such that sup_𝑡∈𝐾𝑑^𝑛𝑑𝑡^𝑛𝑢(𝜑_𝜀,𝑡)_𝐻=𝒪_𝜀→0𝜀^−𝑞foreach𝜑∈𝒜_𝑞. (2.6) To complete the definition of the Colombeau algebra of abstract generalized functions we introduce 𝒩(𝐻_𝑎) consisting of all elements 𝑢∈ℰ(𝐻_𝑎) that satisfy the following condition:

(N) for each compact 𝐾⊂ℝ and each 𝑛∈ℕ₀ there exists 𝑝∈ℕ such that sup_𝑡∈𝐾𝑑^𝑛𝑑𝑡^𝑛𝑢(𝜑_𝜀,𝑡)_𝐻=𝒪_𝜀→0𝜀^𝑞−𝑝foreach𝜑∈𝒜_𝑞,𝑞≥𝑝. (2.7) Elements of the space ℰ_𝑀(𝐻_𝑎) form a differential algebra, and 𝒩(𝐻_𝑎) is the differential ideal in it. Now define the factor algebra 𝒢𝐻_𝑎ℰ∶=^𝑀𝐻_𝑎𝒩𝐻_𝑎. (2.8) Similarly to 𝒢(ℝ) (see, e.g., [6]), the algebra 𝒢(𝐻_𝑎) is an associative and commutative 𝐻_𝑎-valued differential one. Elements of 𝒢(𝐻_𝑎) are classes of mappings. We denote them by capitals 𝑈,𝑉,… and denote representatives of class 𝑈∈𝒢(𝐻_𝑎) 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′(𝐻_𝑎) into ℰ_𝑀(𝐻_𝑎) and 𝑖⁻¹(𝒩(𝐻_𝑎)) consists of the null element of 𝒟′(𝐻_𝑎). Thus, each element of 𝒟′(𝐻_𝑎) is imbedded in the corresponding class of 𝒢(𝐻_𝑎) by the mapping 𝑖.

Support of an element 𝑈∈𝒢(𝐻_𝑎) is defined as follows. We say that 𝑈 is equal to zero on an open set Λ⊂ℝ if its restriction to 𝒢_Λ(𝐻_𝑎) is equal to zero in 𝒢_Λ(𝐻_𝑎) (where algebra 𝒢_Λ(𝐻_𝑎) is defined in the same way as 𝒢_Λ(ℝ^𝑛) [6]). If 𝑤∈𝒟′(𝐻_𝑎), then, similarly to the case of 𝒟′(ℝ), support of 𝑖𝑤∈𝒢(𝐻_𝑎) coincides with that of 𝑤∈𝒟′(𝐻_𝑎).

Now define 𝒢(Ω,𝐻_𝑎), the algebra of 𝒢(𝐻_𝑎)-valued random variables {𝑈=𝑈(𝜔),𝜔∈(Ω,ℬ(Ω),𝜇)} as a mapping from (Ω,ℬ(Ω),𝜇) to 𝒢(𝐻_𝑎) 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^∞(𝐻_𝑎)) onto an element of ℬ(Ω), where the Borel 𝜎-algebra ℬ(𝐶^∞(𝐻_𝑎)) on the space 𝐶^∞(𝐻_𝑎) is generated by the system of neighborhoods in 𝐶^∞(𝐻_𝑎) defined by the system of seminorms 𝑝_𝑛,𝑘(𝑓)=sup_{𝑡∈[−𝑛,𝑛]}‖𝑓^(𝑘)(𝑡)‖_𝐻.

To complete the setting of the problem we define the generalized white noise process 𝑊=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,𝐻_𝑎).

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 _Ω𝑒^{𝑖(⟨𝜔,𝜃⟩,ℎ)ℍ}𝑑𝜇(𝜔)=𝑒^{−(1/2)‖𝜃‖2(𝑄ℎ,ℎ)},𝜃∈𝒮,ℎ∈ℍ. (2.9) (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: ⟨𝑊(⋅,𝜔),𝜃(⋅)⟩∶=⟨𝜔,𝜃⟩,𝜃∈𝒮. (2.10) 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: 𝑊₊(⋅,𝜔),𝜃(⋅)∶=(−1)^𝑘𝜒⋅𝑊^(−𝑘)(⋅),𝜃^(𝑘)(⋅),𝜃∈𝒮. (2.11) 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 generalized ℍ-valued white noise on a (Ω,ℬ(Ω),𝜇), more precisely 𝑄-white noise, is via derivative of ℍ-valued 𝑄-Wiener process {𝑊_𝑄(𝑡),𝑡≥0} [9] continued by zero on (−∞,0) as follows: 𝑊₊(⋅,𝜔),𝜃∶=−𝑊_𝑄(⋅,𝜔),𝜃′,𝜃∈𝒟. (2.12)

Finally, we map the defined generalized white noise process into the Colombeau algebra 𝒢(Ω,𝐻_𝑎) 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: 𝑤𝜑,𝑡,𝜔∶=⟨𝑊₊(⋅,𝜔),𝜑(𝑡−⋅)⟩,𝜑∈𝒜₀,𝑡∈ℝ. (2.13) So, 𝑤(𝜑,⋅,𝜔)∈𝐶^∞(ℝ,ℍ),𝜑∈𝒜₀,𝜔∈Ω a.s., and 𝑤(𝜑,⋅,⋅)∈𝐶^∞(ℝ,𝐿₂(Ω,ℍ)).

Let 𝐵∈ℒ(ℍ,𝐻_𝑎), and then applying 𝐵 to 𝑤 we obtain that 𝐵𝑤(𝜑,𝑡,𝜔)∈𝐻_𝑎 and the map 𝐵𝑤𝜑,⋅,⋅∶𝒜₀→𝐶^∞ℝ,𝐿₂Ω,𝐻_𝑎 (2.14) are representative of a class in 𝒢(Ω,𝐻_𝑎). 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,𝐻_𝑎) 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

Let 𝐻=ℍ=𝐿₂(ℝ) and the domain of 𝐴 lie in the set of continuous functions of 𝐿₂(ℝ). Let 𝐻_𝑎 be the set of finitely many times differentiable functions of 𝐿₂(ℝ) and 𝐻_𝑎⊆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,𝐻_𝑎). Since 𝐻_𝑎 is chosen as the set of finitely many times differentiable functions in 𝐿₂(ℝ), operator 𝐵∈ℒ(ℍ,𝐻_𝑎) 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 that 𝐴 generates a 𝐶₀-semigroup {𝑉(𝑡),𝑡≥0} in 𝐿₂(ℝ).

Consider the question whether there is a solution to the problem 𝑌′=𝐴𝑌+𝐹(𝑌)+𝐵𝑊,supp𝑌⊆[0,∞), (3.1) as an element of algebra 𝒢(Ω,𝐻_𝑎). 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,𝐻_𝑎) [7]. By definition of elements of 𝒢(Ω,𝐻_𝑎), for each fixed 𝜑∈𝒜₀, 𝐵𝑤(𝑡) is an infinitely differentiable function of 𝑡∈ℝ with values in 𝐻_𝑎 and measurable with respect to 𝜔∈Ω. Let us take an arbitrary 𝜑∈𝒜₀ and consider the problem 𝑦′(𝑡)=𝐴𝑦(𝑡)+𝐹𝑦(𝑡)+𝐵𝑤(𝑡),𝑡≥−𝜂,𝑦(𝑡)=0,𝑡≤−𝜂, (3.2) 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 𝑦(𝑡)=^𝑡_−𝜂𝑉(𝑡−𝑠)𝐹𝑦(𝑠)𝑑𝑠+^𝑡_−𝜂𝑉(𝑡−𝑠)𝐵𝑤(𝑠)𝑑𝑠=∶𝒬𝑦(𝑡),𝑡≥−𝜂. (3.3) The introduced operator 𝒬 is a Volterra type one. Using the differentiability of 𝐹 and boundedness of its derivative let us show that 𝒬^𝑘 (where 𝑘=𝑘(𝑇)) is a contraction on the segment [−𝜂;𝑇].

Since 𝐹 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 𝐹𝑦(𝑠)−𝐹(𝑧(𝑠))=𝐹^′(𝜉)𝑦(𝑠)−𝑧(𝑠),𝑠∈−𝜂;∞, (3.4) where 𝜉 is an appropriate point from (𝑦(𝑠);𝑧(𝑠)) and the following estimate holds: 𝐹𝑦(𝑠)−𝐹(𝑧(𝑠))≤𝐿𝑦(𝑠)−𝑧(𝑠),𝐿=max_𝜉∈ℝ𝐹^′(𝜉). (3.5) This and exponential boundedness of 𝐶₀ semigroups: ‖𝑉(𝑡)‖_{ℒ(𝐿2(ℝ))}≤𝐶𝑒^𝑎𝑡, (3.6) for each 𝑡∈[−𝜂;∞) imply that 𝒬𝑦(𝑡)−𝒬𝑧(𝑡)≤𝐶𝐿𝑒^{𝑎(𝑡+𝜂)}𝑡+𝜂max_{𝑠∈[−𝜂;𝑡]}𝑦(𝑠)−𝑧(𝑠). (3.7) For squares we have 𝒬²𝑦(𝑡)−𝒬²𝑧(𝑡)=^𝑡_−𝜂𝑉(𝑡−𝑠)𝐹𝒬𝑦(𝑠)𝑑𝑠−^𝑡_−𝜂𝑉()𝐹(𝒬(𝑧(𝑠=𝑡−𝑠)))𝑑𝑠_𝑡−𝜂𝑉(𝑡−𝑠)𝐹𝒬𝑦(𝑠)−𝐹(𝒬(𝑧(𝑠)))𝑑𝑠. (3.8) Then we have 𝒬²𝑦(𝑡)−𝒬²𝑧(𝑡)≤𝐶²𝐿²𝑒^{2𝑎(𝑡+𝜂)}𝑡+𝜂²2max_{𝑠∈[−𝜂;𝑡]}𝑦(𝑠)−𝑧(𝑠), (3.9) and for every 𝑘∈ℕ𝒬^𝑘𝑦(𝑡)−𝒬^𝑘𝑧(𝑡)≤𝐶^𝑘𝐿^𝑘𝑒^{𝑘𝑎(𝑡+𝜂)}𝑡+𝜂^𝑘𝑘!max_{𝑠∈[−𝜂;𝑡]}𝑦(𝑠)−𝑧(𝑠), (3.10) hence max_{𝑡∈[−𝜂;𝑇]}𝒬^𝑘𝑦(𝑡)−𝒬^𝑘𝑧(𝑡)≤𝐶^𝑘𝐿^𝑘𝑒^{𝑘𝑎(𝑇+𝜂)}𝑇+𝜂^𝑘𝑘!max_{𝑡∈[−𝜂;𝑇]}𝑦(𝑡)−𝑧(𝑡). (3.11) The constant in this estimate can be made less then unity by choosing 𝑘=𝑘(𝑇). Thus 𝒬^𝑘 is the contraction, and the sequence of approximations 𝑦_𝑛(𝑡)=𝒬^𝑛𝑘𝑦₀(𝑡) has the limit in 𝐻: 𝑦(𝑡)=lim_𝑛→∞𝒬^𝑛𝑘𝑦₀(𝑡), (3.12) uniform with respect to 𝑡∈[−𝜂;𝑇].

Note that if one takes an infinitely differentiable with respect to 𝑡 function 𝑦₀(⋅) as the first point for the approximating sequence, then function 𝑧(𝑡)=𝒬𝑦₀(𝑡)=^𝑡_−𝜂𝑉(𝑡−𝑠)𝐹𝑦₀(𝑠)𝑑𝑠+^𝑡_−𝜂𝑉(𝑡−𝑠)𝐵𝑤(𝑠)𝑑𝑠,𝑡≥−𝜂, (3.13) is also an infinitely differentiable with respect to 𝑡 function, and consequently 𝑦₁(⋅)=𝒬^𝑘𝑦₀(⋅) has the same property as well as all subsequent iterations 𝑦_𝑛(⋅).

It can be shown by the same arguments that the sequence 𝑦^′_𝑛(⋅) converges to its limit in 𝐻 uniformly with respect to 𝑡∈[−𝜂;𝑇]; hence 𝑦(⋅) is differentiable and 𝑦′(𝑡)=lim_𝑛→∞𝑦^′_𝑛(𝑡). Similarly it can be shown that 𝑦(⋅) is infinitely differentiable function with values in 𝐻.

Now we show that 𝑦_𝑛(𝑡)∈𝐻_𝑎 if 𝑦₀(𝑡)∈𝐻_𝑎, 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)∈𝐻_𝑎, 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 operators 𝑉(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 𝑉(𝑡−𝑠)𝐹𝑦₀(𝑠)∶𝐻_𝑎→𝐻_𝑎,𝑉(𝑡−𝑠)𝐵𝑤(𝑠)∈𝐻_𝑎, (3.14) hence 𝒬 acts in 𝐻_𝑎 and 𝑦₁(𝑡)=𝒬^𝑘𝑦₀(𝑡)∈𝐻_𝑎 as well as 𝑦_𝑛(𝑡) for any 𝑛∈ℕ.

Thus, we obtain 𝑦_𝑛(𝑡)∈𝐻_𝑎, but in the general case lim_𝑛→∞𝑦_𝑛(𝑡)=𝑦(𝑡) does not belong to 𝐻_𝑎 since algebra 𝐻_𝑎 is not closed in the sense of 𝐿₂(ℝ) convergence.

If 𝑦(𝑡)∈𝐻_𝑎, then we show that it is a representative of a class from 𝒢(Ω,𝐻_𝑎). 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 𝑢′(𝑡)=𝐴𝑢(𝑡)+ℱ(𝑡),𝑡≥0,𝑢(0)=0, (3.15) with 𝐴 generating a 𝐶₀ semigroup of operators {𝑉(𝑡),𝑡≥0} exists and is defined by the formulae 𝑢(𝑡)=∫^𝑡₀𝑉(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 of 𝐶₀ 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 that 𝑦 is a representative of a class 𝑌∈𝒢(Ω,𝐻_𝑎), 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 𝐹𝑦𝜑,𝑠,𝜔=𝐹𝑦𝜑,𝑠,𝜔−𝐹(0)=𝐹^′(𝜉)𝑦𝜑,𝑠,𝜔, (3.16) 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 max_𝑡∈𝐾𝑦𝜑_𝜀,𝑡,𝜔≤𝐶max_𝑡∈𝐾𝑒^{𝑎(𝑡+𝜂)}𝐵𝑤𝜑_𝜀,𝑡,𝜔+𝐶max_𝜉∈ℝ𝐹′(𝜉)max_𝑡∈𝐾𝑒^{𝑎(𝑡+𝜂)𝑡}_−𝜂𝑦𝜑_𝜀,𝑠,𝜔𝑑𝑠, (3.17) and due to boundedness of 𝐹′ we have max_𝑡∈𝐾𝑦𝜑_𝜀,𝑡,𝜔≤𝐶₁max_𝑡∈𝐾𝐵𝑤𝜑_𝜀,𝑡,𝜔+𝐶₂max_𝑡∈𝐾^𝑡_−𝜂𝑦𝜑_𝜀,𝑠,𝜔𝑑𝑠. (3.18) Since 𝐵𝑤 is a representative of class 𝐵𝑊 from 𝒢(Ω,𝐻_𝑎), the first term in the right-hand side of the inequality for every 𝜑∈𝒜_𝑞 increases as 𝜀→0 not faster than 𝜀^−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 𝑦(𝑡)≤𝑐+^𝑡_𝑡0𝑓(𝑠)𝑦(𝑠)𝑑𝑠,𝑐>0,𝑡>𝑡₀, (3.19) for positive continuous functions 𝑦 and 𝑓, than 𝑦(𝑡)≤𝑐𝑒^{∫𝑡𝑡0𝑓(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 supp 𝑌⊆[0;∞). To do this we consider two solutions of (3.2) 𝑦_𝜂1(⋅) and 𝑦_𝜂2(⋅) corresponding 𝜂₁≠𝜂₂ and verify that difference 𝑦_𝜂1(⋅)−𝑦_𝜂1(⋅) belongs to 𝒩(𝐻_𝑎). Note that 𝜂=max{𝜂₁,𝜂₂}. Then we have 𝑦^′_𝜂1(𝑡)−𝑦^′_𝜂2(𝑡)=𝐴𝑦_𝜂1(𝑡)−𝑦_𝜂2(𝑡)+𝐹𝑦_𝜂1(𝑡)−𝐹𝑦_𝜂2(𝑡)(𝑡)𝑦+𝑔,𝑡≥−𝜂,_𝜂1(𝑡)−𝑦_𝜂2(𝑡)=0,𝑡≤−𝜂, (3.20) where 𝑔∈𝒩(𝐻_𝑎) 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: max_𝑡∈𝐾𝑦_𝜂1𝜑_𝜀,𝑡,𝜔−𝑦_𝜂2𝜑_𝜀,𝑡,𝜔≤𝐶₁max_𝑡∈𝐾𝑔𝜑_𝜀,𝑡,𝜔+𝐶₂max_𝑡∈𝐾^𝑡_−𝜂𝑦_𝜂1𝜑_𝜀,𝑠,𝜔−𝑦_𝜂2𝜑_𝜀,𝑡,𝜔𝑑𝑠. (3.21) Since the first term satisfies to (N), Gronwall-Bellman lemma implies that 𝑦_𝜂1(𝑡)−𝑦_𝜂2(𝑡)∈𝒩(𝐻_𝑎), so supp𝑌⊆[0;∞).

Now we show that the solution of (3.1) is unique in the algebra 𝒢(Ω,𝐻_𝑎). Let 𝑌₁,𝑌₂∈𝒢(Ω,𝐻_𝑎) 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: 𝑦^′₁(𝑡)−𝑦^′₂(𝑡)=𝐴𝑦₁(𝑡)−𝑦₂(𝑡)+𝐹𝑦₁(𝑡)−𝐹𝑦₂(𝑡)(𝑡)𝑦+𝑔,𝑡≥−𝜂,₁(𝑡)−𝑦₂(𝑡)∈𝒩𝐻_𝑎,𝑡≤−𝜂, (3.22) where 𝑔 is an element of 𝒩(𝐻_𝑎). Then, as above, Gronwall-Bellman Lemma implies that 𝑦₁(𝑡)−𝑦₂(𝑡)∈𝒩(𝐻_𝑎), that is 𝑌₁−𝑌₂=0 in 𝒢(Ω,𝐻_𝑎).

Taking into account that the linear Cauchy problem, corresponding to (1.1) has the following form in spaces of distributions: 𝑋′,𝜑=𝛿,𝜑𝑓+𝐴𝑋,𝜑+𝐵𝑊,𝜑,𝜑∈𝒟, (3.23) and the required solution to the Cauchy problem (1.1) in 𝒢(Ω,𝐻_𝑎) is related with the obtained 𝑌 as follows: 𝑋(𝑡)=𝑌(𝑡)+(𝑉∗𝑖𝛿)(𝑡)𝑓, 𝑡≥0, 𝑓∈𝐻_𝑎.

In the general case since the limit of 𝑦_𝑛(𝑡)∉𝐻_𝑎, we obtain only the approximated solutions of (3.3)—the fundamental sequence {𝑦_𝑛} obtained by the following equalities 𝑦_𝑛(𝑡)=𝒬^𝑘𝑦_𝑛−1(𝑡),𝑡≤−𝜂. (3.24)

So, we get the following result.

Theorem 3.1. Let 𝐴 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),𝐻_𝑎) and 𝐵𝑊 be an element of 𝒢(Ω,𝐻_𝑎) 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;∞);𝐻_𝑎), then (3.1) has the unique solution in algebra 𝒢(Ω,𝐻_𝑎). In this case the solution to the Cauchy problem (1.1) in 𝒢(Ω,𝐻_𝑎) is 𝑋=𝑌+(𝑉∗𝑖𝛿)𝑓 for any 𝑓∈𝐻_𝑎.

Now consider the case of 𝐴 generating an integrated semigroup. If operator 𝐴 generates an exponentially bounded 𝑛-times integrated semigroup {𝑉_𝑛(𝑡),𝑡≥0}, then solution operators 𝑉(𝑡) of homogeneous Cauchy problem are defined as follows: ⟨𝑉,𝜑⟩=(−1)^𝑛⟨𝑉_𝑛,𝜑⟩, 𝜑∈𝒟, and instead of (3.3) we have the following equation: 𝑦𝜑,𝑡,𝜔=^𝑡_−𝜂𝑉_𝑛(𝑡−𝑠)𝐹^(𝑛)𝑦+𝜑,𝑠,𝜔𝑑𝑠_𝑡−𝜂𝑉_𝑛(𝑡−𝑠)(𝐵𝑤)^(𝑛)𝜑,𝑠,𝜔𝑑𝑠,𝜑∈𝒜₀,𝜔∈Ω,𝑡≥−𝜂. (3.25) 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 𝑦_𝑛 and the solution in 𝒢(Ω,𝐻_𝑎) if the limit of 𝑦_𝑛(𝑡) belongs to 𝐻_𝑎.

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 solutions 𝑦(𝜑_𝜀) as 𝜀→0, equations with generators of more general regularized semigroups, and equations in arbitrary Hilbert spaces.

Acknowledgments

This work was completed with the support of the Programme of the Ministry of Education and Science of Russian Federation 1.1016.2011 and RFBR 10-01-96003 Ural.