APPROXIMATION OF SOLUTIONS

OF STOCHASTIC EVOLUTION EQUATIONS

BY ROTHE’S METHOD

H. Breckner and W. Grecksch

ABSTRACT

The purpose of this paper is the investigation of Rothe’s method for an abstract stochastic evolution equation, considering Lipschitz continuous functions and a nonlinear maximal monotone operator, which satisfies a growth condition. By using time discretization there is constructed a sequence of step functions that converges strongly to the solution of the evolution equation.

Key words: stochastic partial differential equations, Rothe’s method

AMS subject classification: 60H15, 60H30, 35R60

1 Introduction

In the investigation of stochastic partial differential equations the functional analytical formulations with generalized solution concepts are very useful. Important results were obtained by I. Gyöngy [3], [4], W. Grecksch [2], N. V. Krylov, B. L. Rozovskij [6], C. Tudor [2] and show different methods in the research of stochastic partial differential equations; for example, one can consider infinitesimal operators of a semigroup or can use monotone operators over Sobolev spaces in order to obtain assertions about the existence and uniqueness of the solutions of the considered equations.

Methods as Galerkin’s method or Rothe’s method, inspirated from the research of deterministic partial differential equations (see, for instance, [8a], [7]) have been extended to the stochastic case. In [5] H. Liske investigated a special stochastic linear parabolic Ito equation by applying for the first time Rothe’s method in the stochastic case. W. Grecksch and C. Tudor developed in [2] this method for an abstract stochastic parabolic evolution equation, replacing the linearity condition of the considered operator by the monotonicity and coerciveness condition.

The purpose of this paper is to present Rothe’s method for an abstract stochastic parabolic evolution equation (in the sense of Ito), assuming for the considered operator maximal monotonicity and a growth condition. We do not use any coerciveness or linearity condition.

We establish the existence and uniqueness (with probability 1) of the solution of the following abstract stochastic evolution equation:

dX(ω, t) = −AX(ω, t)dt + f(t, X(ω, t))dt + g(t, X(ω, t))dw(ω, t) (1)

with the initial condition X(ω, 0) = X_{0}(ω), (2)

where ω ∈ Ω (a.e.) and t ∈ [0, T] ⊂ IR.

Regarding this problem we assume that the following hypotheses are fulfilled:

(H_{1}) (Ω, F, P) is a probability space, {F_{t} | t ∈ [0, T]} is a filtration (contained in F) with respect to a given real Wiener process (w(t))_{t∈[0,T]} ^{1};

(H_{2}) (V, H, V^{∗}) is an evolution triple (see [8a], p.416), where (V, (·, ·)_{V}) and (H, (·, ·)) are real separable Hilbert spaces;

(H_{3}) A : V → V^{∗} is a maximal monotone operator (see [8b], p.820) satisfying the following growth condition: For each r > 0 there exists a c_{r} > 0 such that for any function

v : Ω × [0, T] → V with E∫_{0}^{T}∥v(t)∥_{V}^{2} dt < r it follows that E∫_{0}^{T}∥Av(t)∥_{V^{∗}}^{2}dt < c_{r};

(H_{4}) f, g : [0, T] × H → H are mappings satisfying the following inequalities for all t_{1}, t_{2} ∈ [0, T] and for all functions v_{1}, v_{2} : Ω → H with E∥v_{1}∥^{2} < ∞, E∥v_{2}∥^{2} < ∞:

E∥f(t_{1}, v_{1}) − f(t_{1}, v_{2})∥^{2} ≤ α|t_{1} − t_{2}|^{2} + βE∥v_{1} − v_{2}∥^{2},
E∥g(t_{1}, v_{1}) − g(t_{1}, v_{2})∥^{2} ≤ α|t_{1} − t_{2}|^{2} + βE∥v_{1} − v_{2}∥^{2};

(H_{5}) X_{0} : Ω → H is F_{0}-measurable and E∥X_{0}∥^{2} < ∞.

An adapted process (X(t))_{t∈[0,T]} with X(ω, t) ∈ V (for a.e. ω ∈ Ω and all t ∈ [0, T])

and with E∥X(t)∥^{2} < ∞ for all t ∈ [0, T] and also E∫_{0}^{T}∥X(t)∥_{V}^{2} dt < ∞, is called solution of problem (P_{1}) if it satisfies equation (1) and the initial condition (2) in the following sense:

1As usual in the notation of random variables or stochastic processes we omit the assignment ω ∈ Ω.

(X(t) − X_{0}, v) = − ∫_{0}^{t}⟨AX(s), v⟩ds + ∫_{0}^{t}(f(s, X(s)), v)ds + ∫_{0}^{t}(g(s, X(s)), v)dw(s)

for a.e. ω ∈ Ω and for all v ∈ V, t ∈ [0, T], and if X(0) = X_{0} for a.e. ω ∈ Ω.

2 Notations

Throughout this paper we denote by: B_{[0,T]}^{1} the σ-algebra of all Borel sets of the interval [0, T] ⊂ IR; S^{∗} the dual space of a Banach space S; E(·) the mathematical expectation; R(F) the range of a function F.

All vector spaces that will occur in our paper are considered over the field of real numbers.

Let (S, ∥ · ∥_{S}) be a Banach space. We denote by L_{S}^{2}(Ω) the linear space of all functions v : Ω → S which are F-measurable and for which E∥v(ω, t)∥_{S}^{2} < ∞. Further we denote by L_{S}^{2}(Ω × [0, T]) the linear space of all functions v : Ω × [0, T] → S which are

F × B_{[0,T]}^{1}-measurable and for which E∫_{0}^{T}∥v(ω, t)∥_{S}^{2} < ∞. Equipped with the norms

(E∥ · ∥_{S}^{2})^{\frac{1}{2}} , (E∫_{0}^{T}∥ · ∥_{S}^{2}dt)^{\frac{1}{2}} , respectively, these spaces become Banach spaces. If, in ad-

dition, S is a Hilbert space, then these spaces even become Hilbert spaces.

⟨·, ·⟩ denotes the duality between the space V and his dual V^{∗}. Instead of the norm ∥ · ∥_{V^{∗}} we write ∥ · ∥_{∗}.

The weak convergence is denoted by ⇀.

3 Main Result

Let a be a real positive number, fixed for the moment. To this number we assign the following stochastic evolution equation

d(e^{−at}X(t)) =−e^{−at}(AX(t) + aJ X(t))dt + e^{−at}f(t, X(t))dt + e^{−at}g(t, X(t))dw(t) (3)

with ω ∈ Ω and t ∈ [0, T], and the initial condition (2).

We denote by (P_{2}) the problem of finding in the space L_{V}^{2} (Ω×[0, T]) an adapted process (X(t))_{t∈[0,T]}, which satisfies equation (3) and the initial condition (2).

Using the Ito formula, it can be shown that X is a solution of (P_{1}) if and only if X is a solution of (P_{2}). The advantage of problem (P_{2}) is the possibility of a favorable choice of the constant a (see relation (33)), which gives us some useful properties.

Let 0 = t_{0} < t_{1} < … < t_{N} = T be an equidistant partition of [0, T] with h_{N} = t_{n} − t_{n−1} = \frac{T}{N} , n ∈ \{1, 2, …, N\}.

We set w_{n} = w(t_{n}), for each n ∈ {0, 1, …, N}.

We discretize problem (P_{2}) in the following way:

\{ \begin{matrix}e^{−at_{n}}x_{n} + e^{−at_{n}}(Ax_{n} + aJ x_{n})h_{N} = e^{−at_{n−1}}y_{n−1} + e^{−at_{n−1}}f(t_{n−1}, x_{n−1})h_{N} \\ n = 1, 2, …, N, \end{matrix} (4)

\{ \begin{matrix}y_{n} = x_{n} + g(t_{n−1}, x_{n})(w_{n} − w_{n−1}) \\ n = 1, 2, …, N, \end{matrix} (5)

x_{0} = y_{0} = X_{0}. (6)

Equations (4) and (5) are understood as follows

\{ \begin{matrix}(e^{−at_{n}}x_{n}, v) + ⟨e^{−at_{n}}(Ax_{n} + aJ x_{n}), v⟩h_{N} = \\ = (e^{−at_{n−1}}y_{n−1}, v) + (e^{−at_{n−1}}f(t_{n−1}, x_{n−1}), v)h_{N} \\ n = 1, 2, …, N, \end{matrix} (7)

for a.e. ω ∈ Ω and for all v ∈ V,

\{ \begin{matrix}(y_{n}, v) = (x_{n}, v) + (g(t_{n−1}, x_{n}), v)(w_{n} − w_{n−1}) \\ n = 1, 2, …, N, \end{matrix} (8)

for a.e. ω ∈ Ω and for v ∈ V.

Lemma 3.1 For each n ∈ {1, 2, …, N} equation (1) has with probability 1 a unique V -valued and F_{t_{n−1}}-measurable solution x_{n} ∈ L_{V}^{2} (Ω).

In order to prove Lemma 3.1 one can apply a corollary of the theorem by Browder and Minty about monotone operators (see [8b], Corollary 32.26, p.868).

For a.e. ω ∈ Ω and all t ∈ [0, T) we set

x̂_{N}(t) = ∑_{n=1}^{N} I_{[t_{n−1},t_{n})}(t)x_{n}

and for t = T we set x̂_{N}(T) = x_{N}.

The main result of this paper is the following theorem:

Theorem 3.2 There exists a unique (with probability 1) solution (X(t))_{t∈[0,T]} of problem (P_{1}) such that the following assertions hold:

(i) (X(t))_{t∈[0,T]} considered as a H-valued process is continuous; (ii) the sequence (x̂_{N}) converges strongly to X in the space L_{V}^{2} (Ω × [0, T]); (iii) (x̂_{N}(T)) converges strongly to X(T) in the space L_{H}^{2}(Ω).

4 Proof of the Main Result

At first we introduce some notations. For each n ∈ {1, 2, …, N} we denote

v_{n} = e^{−at_{n}}x_{n} and u_{n} = e^{−at_{n}}y_{n},

and for all t ∈ [0, T] and n ≠ N we set

I_{n}(t) = I_{[t_{n−1},t_{n})}(t)

and for n = N

I_{N}(t) = I_{[t_{N−1},t_{N}]}(t).

Other notations are the following:

v̂_{N}(t) = ∑_{n=1}^{N} I_{n}(t)v_{n},  û_{N}(t) = ∑_{n=1}^{N} I_{n}(t)u_{n},

F_{N}(t) = ∑_{n=1}^{N} I_{n}(t)e^{−at_{n−1}}f(t_{n−1}, x_{n−1}), G_{N}(t) = ∑_{n=1}^{N} I_{n}(t)e^{−at_{n}}g(t_{n−1}, x_{n}),

B_{N}(t) = ∑_{n=1}^{N} I_{n}(t)e^{−at_{n}}(Ax_{n} + aJ x_{n}).

The following lemma contains some a-priori estimates, which are used to prove weak convergence properties, contained in Lemma 4.2, 4.3 and 4.4.

Lemma 4.1 There exist two constants c_{1}, c_{2} > 0 (which do not depend on N) such that E∥v_{n}∥^{2} ≤ c_{1}, (9) E∥u_{n}∥^{2} ≤ c_{1}, (10) for all n ∈ {1, 2, …, N} and

E∫_{0}^{T}∥v̂_{N}(t)∥_{V}^{2} dt ≤ c_{2}, (11)

E ∫_{0}^{T} ∥x̂_{N}(t)∥_{V}^{2} dt ≤ c_{2}, (12)

for all natural numbers N. Proof. Let n ∈ {1, 2, …, N}. In (7) we put particulary v = v_{n} and obtain (13)

∥v_{n}∥^{2} + ⟨e^{−at_{n}}(Ax_{n} + aJ x_{n}), v_{n}⟩h_{N} = (u_{n−1}, v_{n}) + (e^{−at_{n−1}}f(t_{n−1}, x_{n−1}), v_{n})h_{N} for a.e. ω ∈ Ω.

By using the monotonicity of the operator A and the Schwarz inequality, we get

⟨e^{−at_{n}}(Ax_{n} + aJ x_{n}), v_{n}⟩ ≥ b∥v_{n}∥_{V}^{2} − k_{1},

where b = a − \frac{1}{2} and k_{1} are positive constants (we choose a > \frac{1}{2}). So (13) becomes

∥v_{n}∥^{2} + b∥v_{n}∥_{V}^{2} h_{N} − k_{1}h_{N} ≤ (u_{n−1}, v_{n}) + e^{−at_{n−1}}(f(t_{n−1}, x_{n−1}), v_{n})h_{N}.

Now by applying the Schwarz inequality and elementary calculus we obtain (14)

\frac{1}{2}∥v_{n}∥^{2} + b∥v_{n}∥_{V}^{2} h_{N} − k_{1}h_{N} ≤ \frac{1}{2}∥u_{n−1}∥^{2} + \frac{1}{2}e^{−at_{n−1}}∥f(t_{n−1}, x_{n−1})∥^{2}h_{N} + \frac{1}{2}∥v_{n}∥^{2}h_{N}.

From (8) we get

E∥y_{n−1}∥^{2} = E∥x_{n−1}∥^{2} + E∥g(t_{n−2}, x_{n−1})(w_{n−1} − w_{n−2})∥^{2}
+ 2E((g(t_{n−2}, x_{n−1}), x_{n−1})(w_{n−1} − w_{n−2}))

for n ≥ 2. Taking into consideration the properties of the conditional expectation and of a Wiener process, on the one hand, and the F_{t_{n−2}}-measurability of x_{n−1}, on the other hand, we obtain E∥u_{n−1}∥^{2} = E∥v_{n−1}∥^{2} + h_{N}e^{−2at_{n−1}}E∥g(t_{n−2}, x_{n−1})∥^{2}. (15) Since f and g are Lipschitz continuous, one can show that e^{−2at_{n−1}}E∥f(t_{n−1}, x_{n−1})∥^{2} ≤ k_{2}(1 + E∥v_{n−1}∥^{2}) (16) e^{−2at_{n−1}}E∥g(t_{n−2}, x_{n−1})∥^{2} ≤ k_{2}(1 + E∥v_{n−1}∥^{2}), (17) where k_{2} is a positive constant, which does not depend on N.

From (14), (15), (16) and (17) we obtain

\frac{1}{2}E∥v_{n}∥^{2} + bh_{n}E∥v_{n}∥_{V}^{2} ≤ \frac{1}{2}E∥v_{n−1}∥^{2} + k_{3}h_{N} + k_{4}E∥v_{n−1}∥^{2} + k_{5}h_{N}E∥v_{n}∥^{2},

where k_{3}, k_{4} and k_{5} are positive constants which do not depend on N.

Summing up from n = 1 to n = p (where p ∈ {1, 2, …, N}), we get

\frac{1}{2}E∥v_{p}∥^{2} + bh_{N} ∑_{n=1}^{p} E∥v_{n}∥_{V}^{2} ≤ \frac{1}{2}E∥v_{0}∥^{2} + k_{6} + k_{7}h_{N} ∑_{n=1}^{p} E∥v_{n}∥^{2}, (18)

where k_{6} and k_{7} are positive constants (which do not depend on p or N). Hence it is true that

\frac{1}{2}E∥v_{p}∥^{2} ≤ \frac{1}{2}E∥X_{0}∥^{2} + k_{6} + k_{7}h_{N} ∑_{n=1}^{p} E∥v_{n}∥^{2}.

By applying Gronwall’s Lemma (see [1], p.198) we obtain

E∥v_{p}∥^{2} ≤ (E∥X_{0}∥^{2} + k_{6})e^{k_{7}T} .

So we have shown that

E∥v_{p}∥^{2} ≤ k_{8} for all p ∈ {1, …, N}, (19)

where k_{8} is a positive constant (which does not depend on N). These inequalities, (15) and (17) imply the existence of a positive constant c_{1} (which does not depend on N) such that (9) and (10) holds.

From (18) (for p = N) and (9) it follows that

bh_{n} ∑_{n=1}^{p} E∥v_{n}∥_{V}^{2} ≤ \frac{1}{2}E∥v_{0}∥^{2} + k_{6} + c_{1}k_{7}T.

Hence there exists a positive constant k_{9} (which does not depend on N) such that

E∫_{0}^{T}∥v̂_{N}(t)∥_{V}^{2} dt ≤ k_{9},

for all natural numbers N and hence by the definition of v̂_{N} we have

E∫_{0}^{T}∥x̂_{N}(t)∥_{V}^{2} dt ≤ e^{2aT} E∫_{0}^{T}∥v̂_{N}(t)∥_{V}^{2} dt ≤ e^{2aT} k_{9}.

So, there exists a positive constant c_{2} (which does not depend on N) such that (11) and (12) hold for all natural numbers N. □

Lemma 4.2 The following assertions hold:

(i) There exist v̂ ∈ L_{V}^{2} (Ω × [0, T]) and a subsequence (v̂_{N′}) of (v̂_{N}) such that

v̂_{N′} ⇀ v̂  in  L_{V}^{2} (Ω × [0, T])

and

û_{N′} ⇀ v̂  in  L_{H}^{2}(Ω × [0, T]).

(ii) If x̂ : Ω × [0, T] → V satisfies

x̂(t) = e^{at}v̂(t) for a.e. (ω, t) ∈ Ω × [0, T],

then

x̂_{N′} ⇀ x̂  in  L_{V}^{2} (Ω × [0, T]).

(iii) There exist û ∈ L_{H}^{2}(Ω) and a subsequence (û_{N′′}(T)) of (û_{N′}(T)) such that

û_{N′′}(T) ⇀ û  in  L_{H}^{2}(Ω)
v̂_{N′′}(T) ⇀ û  in  L_{H}^{2}(Ω).

Proof. (i) By Lemma 4.1 and by Proposition B.1 from the appendix there exist v̂ ∈ L_{V}^{2} (Ω × [0, T]) and a subsequence (v̂_{N′}) of (v̂_{N}) such that v̂_{N′} ⇀ v̂ in L_{V}^{2} (Ω × [0, T]).

Just like (15) it can be shown that

E∥u_{n} − v_{n}∥^{2} = h_{N}e^{−2at_{n}}E∥g(t_{n−1}, x_{n})∥^{2}, for all n ∈ {1, 2, …, N}. (20)

By using these relations and (17) we get

E∫_{0}^{T}∥û_{N}(t) − v̂_{N}(t)∥^{2}dt ≤ k_{2}h_{N}^{2} ∑_{n=1}^{N}(1 + E∥v_{n}∥^{2}).

Together with (9) this inequality yields

E∫_{0}^{T}∥û_{N}(t) − v̂_{N}(t)∥^{2}dt → 0.

Consequently û_{N′} ⇀ v̂ in the space L_{H}^{2}(Ω × [0, T]).

(ii) For all s, δ ≥ 0 the following inequality holds:

0 ≤ (1 − e^{−δs})^{2} ≤ δ^{2}s^{2}. (21)

By using this inequality we can write

E∫_{0}^{T}∥x̂_{N}(t) − e^{at}v̂_{N}(t)∥_{V}^{2} dt ≤ a^{2}e^{2aT} h_{N}^{2}E∫_{0}^{T}∥v̂_{N}(t)∥_{V}^{2} dt.

Taking into consideration (11) we obtain

E∫_{0}^{T} ∥x̂_{N}(t) − e^{at}v̂_{N}(t)∥_{V}^{2} dt → 0. (22)

From v̂_{N′} ⇀ v̂ (in L_{V}^{2} (Ω × [0, T])) we conclude that x̂_{N′} ⇀ x̂, where

x̂(ω, t) = e^{at}v̂(ω, t) for a.e. (ω, t) ∈ Ω × [0, T].

(iii) In view of (10) we have

E∥û_{N}(T)∥^{2} = E∥u_{N}∥^{2} ≤ c_{1}

for each natural number N. By Proposition B.1 there exist û ∈ L_{H}^{2}(Ω) and a subsequence (û_{N′′}) of (û_{N′}) such that (û_{N′′}(T)) converges weakly to û in the space L_{H}^{2}(Ω). From (20) it follows that E∥û_{N}(T) − v̂_{N}(T)∥^{2} → 0. In consequence we have

v̂_{N′′}(T) ⇀ û  in L_{H}^{2}(Ω).  □

Lemma 4.3 There exist B ∈ L_{V^{∗}}^{2}(Ω × [0, T]) and a subsequence (B_{N′′′}) of (B_{N′′}) such that

B_{N′′′} ⇀ B in L_{V^{∗}}^{2}(Ω × [0, T]),

∫_{0}^{t} B_{N′′′}(s)ds ⇀ ∫_{0}^{t} B(s)ds in L_{V^{∗}}^{2}(Ω × [0, T]),

e^{−at}(Ax̂_{N′′′}(t) + aJ x̂_{N′′′}(t)) ⇀ B(t) in L_{V^{∗}}^{2}(Ω × [0, T]),

∫_{0}^{T} B_{N′′′}(s)ds ⇀ ∫_{0}^{T} B(s)ds in L_{V^{∗}}^{2}(Ω).

Proof. We can write

E∫_{0}^{T}∥B_{N}(t)∥_{∗}^{2}dt ≤ 2E∫_{0}^{T}(∥Ax̂_{N}(t)∥_{∗}^{2} + a^{2}∥x̂_{N}(t)∥_{V}^{2} )dt.

From hypothesis (H_{3}) and from (12) it follows that (B_{N}) is a bounded sequence in the space L_{V^{∗}}^{2}(Ω × [0, T]). By Proposition B.1 there exist B ∈ L_{V^{∗}}^{2}(Ω × [0, T]) and a subsequence (B_{N′′′}) of (B_{N′′}) which converges weakly to B. By apllying Corollary B.5 we get

∫_{0}^{t} B_{N′′′}(s)ds ⇀ ∫_{0}^{t} B(s)ds in L_{V^{∗}}^{2}(Ω × [0, T]),

∫_{0}^{T} B_{N′′′}(s)ds ⇀ ∫_{0}^{T} B(s)ds in L_{V^{∗}}^{2}(Ω).

We also have

E∫_{0}^{T}∥B_{N}(t)−e^{−at}(Ax̂_{N}(t)+aJ x̂_{N}(t))∥_{∗}^{2}dt ≤ 2a^{2}e^{2aT} h_{N}^{2}E ∫_{0}^{T} ∥(Ax̂_{N}(t)∥_{∗}^{2} +a^{2}∥x̂_{N}(t)∥_{V}^{2} )dt.

From hypothesis (H_{3}) and from (12) we obtain

E∫_{0}^{T}∥B_{N}(s) − e^{−at}(Ax̂_{N}(t) + aJ x̂_{N}(t))∥_{∗}^{2}dt → 0.

Thus

e^{−at}(Ax̂_{N′′′}(t) + aJ x̂_{N′′′}(t)) ⇀ B(t) in L_{V^{∗}}^{2}(Ω × [0, T]). □

Notation. Further we will denote the subsequence of indices (N′′′) obtained in Lemma 4.3 also by (N′).

Lemma 4.4 There exist F ∈ L_{H}^{2}(Ω × [0, T]) and a subsequence (F_{N′′}) of (F_{N′}) such that

F_{N′′} ⇀ F in L_{H}^{2}(Ω × [0, T]),

e^{−at}f(t, x̂_{N′′}(t)) ⇀ F (t) in L_{H}^{2}(Ω × [0, T]),

∫_{0}^{t} F_{N′′}(s)ds ⇀ ∫_{0}^{t} F (s)ds in L_{H}^{2}(Ω × [0, T]),

∫_{0}^{T} F_{N′′}(s)ds ⇀ ∫_{0}^{T} F (s)ds in L_{H}^{2}(Ω).

Proof. Since V is continuously embedded in H, there exists a positive constant c such that

∥v∥ ≤ c∥v∥_{V} for all v ∈ V. (23)

By using (16), (11) and (23) we obtain

E∫_{0}^{T}∥F_{N}(t)∥^{2}dt ≤ k_{2}(T + E∥X_{0}∥^{2}) + ck_{2}E∫_{0}^{T}∥v̂_{N}(t)∥_{V}^{2} dt ≤ k_{10},

where k_{10} is a positive constant which does not depend on N.

In view of Proposition B.1 there exist F ∈ L_{H}^{2}(Ω × [0, T]) and a subsequence (F_{N′′}) of (F_{N′}) such that (F_{N′′}) converges weakly to F.

From (4) and (5) we obtain

x_{n} − x_{n−1} = (1 − e^{−ah_{N}})x_{n} − e^{−ah_{N}} (Ax_{n} + aJ x_{n})h_{N} + f(t_{n−1}, x_{n−1})h_{N}
+ g(t_{n−2}, x_{n−1})(w_{n−1} − w_{n−2}) for all n ∈ {2, 3, … N}.

Taking into account Lemma 4.1 and (16), (17), (9) and (21), it follows that

E∥x_{n} − x_{n−1}∥^{2} ≤ h_{N}k_{11} + h_{N}∥Ax_{n} + aJ x_{n}∥_{∗}^{2}, for all n ∈ {1, 2, …, N}

(k_{11} is a positive constant which does not depend on N). Now by using (16), (9), (21) and (H_{4}) we obtain

E∫_{0}^{T} ∥F_{N}(t) − e^{−at}f(t, x̂_{N}(t)∥^{2} ≤ h_{N}k_{12} + 2h_{N}βE∫_{0}^{T}∥Ax̂_{N}(t) + aJ x̂_{N}(t)∥_{∗}^{2}dt

(k_{12} is a positive constant which does not depend on N). Taking into consideration (12) and hypothesis (H_{3}) we conclude that

E∫_{0}^{T}∥F_{N}(t) − e^{−at}f(t, x̂_{N}(t)∥^{2}dt → 0

and hence

e^{−at}f(t, x̂_{N′′}(t)) ⇀ F (t).

The other conclusions of the lemma are obtained by applying Corollary B.5. □

Lemma 4.5 There exist G ∈ L_{H}^{2}(Ω × [0, T]) and a subsequence (G_{N′′′}) of (G_{N′′}) such that

G_{N′′′} ⇀ G in L_{H}^{2}(Ω × [0, T]),
e^{−at}g(t, x̂_{N′′′}(t)) ⇀ G(t) in L_{H}^{2}(Ω × [0, T]),

∫_{0}^{t} G_{N′′′}(s)dw(s) ⇀ ∫_{0}^{t} G(s)dw(s) in L_{H}^{2}(Ω × [0, T]),

∫_{0}^{T} G_{N′′′}(s)dw(s) ⇀ ∫_{0}^{T} G(s)dw(s) in L_{H}^{2}(Ω).

Proof. By using (17) and (11) we see that

E∫_{0}^{T}∥G_{N}(t)∥^{2}dt ≤ k_{2}T + k_{2}E∫_{0}^{T}∥v̂_{N}(t)∥^{2} ≤ k_{2}T + cc_{2}k_{2}.

In virtue of Proposition B.1 there exist G ∈ L_{H}^{2}(Ω × [0, T]) and a subsequence (G_{N′′′}) of (G_{N′′}) such that (G_{N′′′}) converges weakly to G. Taking into consideration (17), (11), (21) and (H_{4}), we have

E∫_{0}^{T} ∥G_{N}(t) − e^{−at}g(t, x̂_{N}(t)∥^{2}dt ≤ 2a^{2}h_{N}^{2}k_{2}E∫_{0}^{T}∥v̂_{N}(t)∥^{2}dt + h_{N}^{2}k_{13}

for each N (k_{13} is a positive constant which does not depend on N). So we get

E∫_{0}^{T}∥G_{N}(t) − e^{−at}g(t, x̂_{N}(t)∥^{2}dt → 0.

and hence

e^{−at}g(t, x̂_{N′′′}(t)) ⇀ G(t).

The other conclusions of the lemma are obtained by applying Corollary B.5. □

The results from Theorem 3.2 are contained in the following theorem, which gives us the proof of the main result of the paper.

Theorem 4.6 There exists a F_{t}-measurable process (X(t))_{t∈[0,T]} with

X = x̂  in  L_{V}^{2} (Ω × [0, T]),

such that the following assertions are true:

(i) For all t ∈ [0, T] , all v ∈ V and a.e. ω ∈ Ω it holds that

(e^{−at}X(t), v) = (X_{0}, v) − ∫_{0}^{t}⟨B(s), v⟩ds + ∫_{0}^{t}(F (s), v)ds + ∫_{0}^{t}(G(s), v)dw(s) (24)

and (X(t))_{t∈[0,T]} considered as a H-valued process is continuous. (ii) There exists a subsequence of (x̂_{N}) that converges strongly to X in L_{V}^{2} (Ω × [0, T]).

(iii) There exists a subsequence of (x̂_{N}(T)) that converges strongly to X(T) in L_{H}^{2}(Ω). (iv) The function X is with probability 1 the unique solution of problem (P_{1}).

(v) The whole sequence (x̂_{N}) converges strongly to X in L_{V}^{2} (Ω×[0, T]) and the whole sequence (x̂_{N}(T)) converges strongly to X(T) in L_{H}^{2}(Ω).

Proof. (i) We will denote the subsequence of indices (N′′′) obtained from the previous lemmas also by (N). From (4) and (5) we conclude that

u_{n} + e^{−at_{n}}(Ax_{n} + aJ x_{n})h_{N} = u_{n−1} + e^{−at_{n−1}}f(t_{n−1}, x_{n−1})h_{N}
+ e^{−at_{n}}g(t_{n−1}, x_{n})(w_{n} − w_{n−1})

for each n ∈ {1, 2, …, N}.

Let t ∈]0, T] be arbitrarily choosen. Then there exists a q ∈ {1, 2, …, N} such that t ∈]t_{q−1}, t_{q}]. Summing up from n = 1 to n = q we get

û_{N}(t) + ∫_{0}^{t_{q}} B_{N}(s)ds = X_{0} + ∫_{0}^{t_{q}} F_{N}(s)ds + ∫_{0}^{t_{q}} G_{N}(s)dw(s).

This equation can be rewritten as

(25)  û_{N}(t) = X_{0} − ∫_{0}^{t} B_{N}(s)ds + ∫_{0}^{t} F_{N}(s)ds + ∫_{0}^{t} G_{N}(s)dw(s)
+ r_{N}^{1}(t) + r_{N}^{2}(t) + r_{N}^{3}(t),

where r_{N}^{1} : Ω × [0, T] → V^{∗}, r_{N}^{2}, r_{N}^{3} : Ω × [0, T] → H are defined by

r_{N}^{1}(t) = − ∑_{n=1}^{N} I_{]t_{n−1},t_{n}]}(t) ∫_{t}^{t_{n}}B_{N}(s)ds,

r_{N}^{2}(t) = ∑_{n=1}^{N} I_{]t_{n−1},t_{n}]}(t) ∫_{t}^{t_{n}}F_{N}(s)ds,

r_{N}^{3}(t) = ∑_{n=1}^{N} I_{]t_{n−1},t_{n}]}(t) ∫_{t}^{t_{n}}G_{N}(s)dw(s).

We note that

E∫_{0}^{T}∥r_{N}^{1}(t)∥_{∗}^{2}dt ≤ h_{N}^{2}E∫_{0}^{T}∥Ax̂_{N}(t) + aJ x̂_{N}(t)∥_{∗}^{2}dt,

E∫_{0}^{T}∥r_{N}^{2}(t)∥^{2}dt ≤ h_{N}^{2}E∫_{0}^{T}∥F_{N}(t)∥^{2}dt,

E∫_{0}^{T}∥r_{N}^{3}(t)∥^{2}dt ≤ h_{N}^{2}E∫_{0}^{T}∥G_{N}(t)∥^{2}dt.

Therefore we have esspecialy r_{N}^{1} ⇀ 0 in L_{V^{∗}}^{2}(Ω × [0, T]), as well as r_{N}^{2} ⇀ 0 and r_{N}^{3} ⇀ 0 in L_{H}^{2}(Ω × [0, T]).

Taking into consideration the weak convergences in the Lemmas 4.2, 4.3, 4.4 and 4.5 and passing to the limit in (25) when N → ∞, we obtain

e^{−at}x̂(t) = X_{0} − ∫_{0}^{t} B(s)ds + ∫_{0}^{t} F (s)ds + ∫_{0}^{t} G(s)dw(s) for a.e. (ω, t) ∈ Ω × [0, T].

From the Ito formula for ∥ · ∥^{2}, applied to processes over (V, H, V^{∗}), it follows that there exists a F_{t}-measurable process (X(t))_{t∈[0,T]} such that

e^{−at}X(t) = X_{0} − ∫_{0}^{t} B(s)ds + ∫_{0}^{t} F (s)ds + ∫_{0}^{t} G(s)dw(s) (26)

for all t ∈ [0, T] and a.e. ω ∈ Ω, and for which

x̂(ω, t) = X(ω, t) for a.e. (ω, t) ∈ Ω × [0, T], in L_{V}^{2} (Ω × [0, T]);

(X(t))_{t∈[0,T]} considered as a H-valued process is continuous (see [6], Theorem 3.1, p.88). (ii) By applying the Ito formula to (26) we get

(27)  E∥e^{−aT} X(T)∥^{2} = E∥X_{0}∥^{2} − 2E ∫_{0}^{T}⟨B(s), e^{−as}X(s)⟩ds
+ 2E ∫_{0}^{T}(F (s), e^{−as}X(s))ds + E∫_{0}^{T}∥G(s)∥^{2}dw(s).

Next we set t = T in (25). Thus

û_{N}(T) = X_{0} − ∫_{0}^{T} B_{N}(s)ds + ∫_{0}^{T} F_{N}(s)ds + ∫_{0}^{T} G_{N}(s)dw(s).

By using the results from the Lemmas 4.2, 4.3 and 4.5 and passing to the limit when N → ∞, we obtain

û = X_{0} − ∫_{0}^{T} B(s)ds + ∫_{0}^{T} F (s)ds + ∫_{0}^{T} G(s)dw(s) for a.e. ω ∈ Ω.

Taking into consideration (26) we get û(ω) = e^{−aT} X(ω, T) for a.e. ω ∈ Ω. (28) From Lemma 4.2 we conclude that v̂_{N}(T) ⇀ û in L_{H}^{2}(Ω). Therefore it results from Proposition B.2 that

E∥û∥^{2} ≤ lim inf_{N→∞} E∥v̂_{N}(T)∥^{2}. (29)

Put

δ = lim sup_{N→∞} E∥v̂_{N}(T)∥^{2} − E∥û∥^{2}.

From (27), (28) and (29) we obtain

(30)  δ + E∥X_{0}∥^{2} − 2E∫_{0}^{T}⟨B(s), e^{−as}X(s)⟩ds + 2E∫_{0}^{T}(F (s), e^{−as}X(s))ds
+ E∫_{0}^{T}∥G(s)∥^{2}ds ≤ lim sup_{N→∞} E∥v̂_{N}(T)∥^{2}.

From (5), (13) and (15) we get

E∥v_{n}∥^{2} + 2E⟨e^{−at_{n}}(A(x_{n}) + aJ (x_{n})), v_{n}⟩h_{N} ≤ E∥v_{n−1}∥^{2}
+ 2E(e^{−at_{n−1}}f(t_{n−1}, x_{n−1}), v_{n})h_{N} + E∥e^{−at_{n−1}}g(t_{n−2}, x_{n−1})∥^{2}h_{N}

for each n ∈ {2, 3, …, N}, and in the case n = 1

E∥v_{1}∥^{2} + 2E⟨e^{−at_{1}}(A(x_{1}) + aJ (x_{1})), v_{1}⟩h_{N} ≤ E∥X_{0}∥^{2} + 2E(e^{−at_{0}}f(t_{0}, x_{0}), v_{1})h_{N}. Summing up these relations from n = 1 to n = N, we obtain

(31)  E∥v̂_{N}(T)∥^{2} ≤ E∥X_{0}∥^{2} − 2E ∫_{0}^{T}⟨B_{N}(s), v̂_{N}(s)⟩ds
+ 2E ∫_{0}^{T}(F_{N}(s), v̂_{N}(s))ds + E∫_{0}^{T}∥G_{N}(s)∥^{2}ds.

In the proofs of Lemma 4.3, 4.4 and 4.5 we have seen that

E∫_{0}^{T}∥B_{N}(t) − e^{−at}(Ax̂_{N}(t) + aJ x̂_{N}(t))∥_{∗}^{2}dt → 0,

E∫_{0}^{T}∥F_{N}(t) − e^{−at}f(t, x̂_{N}(t))∥^{2}dt → 0,

E∫_{0}^{T}∥G_{N}(t) − e^{−at}g(t, x̂_{N}(t))∥^{2}dt → 0

for N → ∞. Consequently, it results that

lim_{N→∞} E∫_{0}^{T}⟨B_{N}(t), v̂_{N}(t)⟩dt = lim_{N→∞} E∫_{0}^{T}⟨e^{−at}(Ax̂_{N}(t) + aJ x̂_{N}(t)), v̂_{N}(t)⟩dt,

lim_{N→∞} E∫_{0}^{T}(F_{N}(t), v̂_{N}(t))dt = lim_{N→∞} E∫_{0}^{T}⟨e^{−at}f(t, x̂_{N}(t)), v̂_{N}(t)⟩dt,

lim_{N→∞} E∫_{0}^{T}∥G_{N}(t)∥^{2}dt = lim_{N→∞} E∫_{0}^{T}∥e^{−at}g(t, x̂_{N}(t))∥^{2}dt

In this equalities we omitted to write lim sup_{N→∞}, because all the above sequences are bounded and thus always a subsequence can be found which is convergent and for which we use the same notation.

From (30) and (31) it follows that

(32) δ − 2E∫_{0}^{T}⟨B(t), e^{−at}X(t)⟩dt + 2E∫_{0}^{T}(F (t), e^{−at}X(t))dt + E∫_{0}^{T}∥G(t)∥^{2}dt ≤
≤ −2 lim_{N→∞} E∫_{0}^{T}⟨e^{−at}Ax̂_{N}(t) + aJ x̂_{N}(t), v̂_{N}(t)⟩dt
+ 2 lim_{N→∞} E∫_{0}^{T}⟨e^{−at}f(t, x̂_{N}(t)), v̂_{N}(t)⟩dt + lim_{N→∞} E∫_{0}^{T}∥e^{−at}g(t, x̂_{N}(t))∥^{2}dt.

Let L be the right part of this inequality. Further, for each natural number N we put

L_{N} = −2E∫_{0}^{T}e^{−2at}⟨Ax̂_{N}(t) − AX(t) + aJ (x̂_{N}(t) − X(t)), x̂_{N}(t) − X(t)⟩dt
+ 2E∫_{0}^{T}e^{−2at}(f(t, x̂_{N}(t)) − f(t, X(t)), x̂_{N}(t) − X(t))dt
+ E∫_{0}^{T}e^{−2at}∥g(t, x̂_{N}(t)) − g(t, X(t))∥^{2}dt
+ E∫_{0}^{T}e^{−2at}∥x̂_{N}(t) − X(t)∥^{2}dt.

We claim that

(33)  L_{N} ≤ 0  for all natural numbers N.

Indeed, since the operator A is monotone, we have

−2E∫_{0}^{T}e^{−2at}⟨Ax̂_{N}(t) − AX(t) + aJ (x̂_{N}(t) − X(t)), x̂_{N}(t) − X(t)⟩dt ≤
− 2aE∫_{0}^{T} e^{−2at}∥x̂_{N}(t) − X(t)∥_{V}^{2} dt.

The Lipschitz continuity of f and g implies that

2E∫_{0}^{T}e^{−2at}(f(t, x̂_{N}(t)) − f(t, X(t)), x̂_{N}(t) − X(t))dt ≤ (1 + β)E∫_{0}^{T} e^{−2at}∥x̂_{N}(t) − X(t)∥^{2}dt,

E∫_{0}^{T}e^{−2at}∥g(t, x̂_{N}(t)) − g(t, X(t))∥^{2}dt ≤ βE∫_{0}^{T} e^{−2at}∥x̂_{N}(t) − X(t)∥^{2}dt.

Consequently, we have

L_{N} ≤ (2βc + c + 1 − 2a)E∫_{0}^{T} e^{−2at}∥x̂_{N}(t) − X(t)∥^{2}dt,

(where c is the positive constant that appears in (23)). For a convenient choice of a (such that 2βc + c + 1 − 2a < 0 and a > \frac{1}{2}), we obtain L_{N} ≤ 0.

Next we note that

lim_{N→∞} L_{N} = L + 2E∫_{0}^{T}⟨B(t), e^{−at}X(t)⟩dt − 2E∫_{0}^{T}(F (t), e^{−at}X(t))dt
+ E∫_{0}^{T}(e^{−at}g(t, X(t)) − 2G(t), e^{−at}g(t, X(t)))dt
+ lim_{N→∞} E∫_{0}^{T}e^{−2at}∥x̂_{N}(t) − X(t)∥_{V}^{2} dt.

From (32), (33) and (34) it follows

δ + E∫_{0}^{T}∥G(t) − e^{−at}g(t, X(t)∥^{2} + lim_{N→∞} E∫_{0}^{T}e^{−2at}∥x̂_{N}(t) − X(t)∥_{V}^{2} dt ≤ 0.

This inequality implies δ = 0,

e^{−at}g(t, X(t)) = G(t) for a.e. (ω, t) ∈ Ω × [0, T], (34)

and

lim_{N→∞} E∫_{0}^{T}e^{−2at}∥x̂_{N}(t) − X(t)∥_{V}^{2} dt = 0. (35)

From (35) it follows that x̂_{N} → X in L_{V}^{2} (Ω × [0, T]). We mention that (x̂_{N}) denotes here a subsequence of the original sequence (x̂_{N}).

(iii) Since δ = 0, we have

E∥v̂_{N}(T)∥^{2} → E∥û∥^{2}.

By using Lemma 4.2 together with relation (28) and Proposition B.4, it follows that v̂_{N}(T) → e^{−aT} X(T) in L_{H}^{2}(Ω). Therefore x̂_{N}(T) → X(T); (x̂_{N}(T)) denotes here a subsequence of the original sequence (x̂_{N}(T)).

(iv) Taking into consideration the conclusion (ii) of this theorem and the Lipschitz continuity of f, we have

e^{−at}f(t, x̂_{N}(t)) → e^{−at}f(t, X(t)) in L_{H}^{2}(Ω × [0, T]).

From Lemma 4.4 it follows that

e^{−at}f(t, X(t)) = F (t) for a.e. (ω, t) ∈ Ω × [0, T]. (36)

Next we show that

e^{−at}(AX(t) + aJ X(t)) = B(t) for a.e. (ω, t) ∈ Ω × [0, T]. (37)

In order to prove (37) we define Â: L_{V}^{2} (Ω × [0, T]) → (L_{V}^{2} (Ω × [0, T]))^{∗} by

⟨Â(y), u⟩_{∗} = E∫_{0}^{T}⟨e^{−at}(Ay(t) + aJ y(t)), e^{−at}u(t)⟩dt for all u ∈ L_{V}^{2} (Ω × [0, T]),

where ⟨·, ·⟩_{∗} shows the duality between L_{V}^{2} (Ω × [0, T]) and (L_{V}^{2} (Ω × [0, T]))^{∗}. Let b^{∗} ∈ (L_{V}^{2} (Ω × [0, T]))^{∗} be defined by

⟨b^{∗}, u⟩_{∗} = E∫_{0}^{T}⟨B(t), e^{−at}u(t)⟩dt for all u ∈ L_{V}^{2} (Ω × [0, T]).

In view of the monotonicity of A we have

E∫_{0}^{T}e^{−2at}⟨(A + aJ)(x̂_{N}(t)) − (A + aJ)(y(t)), x̂_{N}(t) − y(t)⟩dt ≥ 0

for all y ∈ L_{V}^{2} (Ω × [0, T]). Passing to the limit when N → ∞ and using Lemma 4.3 and conclusion (ii) of this theorem, we get

E∫_{0}^{T}⟨B(t) − e^{−at}(Ay(t) + aJ y(t)), e^{−at}(X(t) − y(t))⟩dt ≥ 0 (38)

for all y ∈ L_{V}^{2} (Ω × [0, T]).

From (38) it follows that

⟨Â(y) − b^{∗}, y − X⟩_{∗} ≥ 0 for all y ∈ L_{V}^{2} (Ω × [0, T]). (39)

The monotonicity of A implies the monotonicity of Â, i.e.

⟨Â(y_{1}) − Â(y_{2}), y_{1} − y_{2}⟩_{∗} ≥ 0 for all y_{1}, y_{2} ∈ L_{V}^{2} (Ω × [0, T]).

In order to deduce the maximal monotonicity of Â we prove that

R(Â + Ĵ) = (L_{V}^{2} (Ω × [0, T]))^{∗},

where Ĵ is the duality map between L_{V}^{2} (Ω × [0, T]) and (L_{V}^{2} (Ω × [0, T]))^{∗}. We choose an arbitrary b̂ ∈ (L_{V}^{2} (Ω × [0, T]))^{∗}. From Riesz’s Representation Theorem it follows that for b̂ ∈ (L_{V}^{2} (Ω × [0, T]))^{∗} there exists a b ∈ L_{V}^{2} (Ω × [0, T]) such that

⟨b̂, u⟩_{∗} = E∫_{0}^{T}(e^{−at}b(t), e^{−at}u(t))_{V}dt for all u ∈ L_{V}^{2} (Ω × [0, T]).

From Proposition A.2 it follows that R(A + (1 + a)J ) = V^{∗}. Next we take into consideration that J b(t) ∈ V^{∗}, for all t ∈ [0, T] and a.e. ω ∈ Ω and that A + (1 + a)J has a continuous inverse (see Proposition A.2). Therefore we can find y ∈ L_{V}^{2} (Ω × [0, T]) such that

Ay(t) + (1 + a)J y(t) = J b(t) for all t ∈ [0, T] and a.e. ω ∈ Ω.

This means that

E∫_{0}^{T}⟨e^{−at}(Ay(t) + (1 + a)J y(t)), e^{−at}u(t)⟩dt = E∫_{0}^{T}⟨e^{−at}J b(t), e^{−at}u(t)⟩dt

for all u ∈ L_{V}^{2} (Ω × [0, T]). Hence ⟨(Â + Ĵ)(y), u⟩_{∗} = ⟨b̂, u⟩_{∗}. So we have proved that

R(Â + Ĵ) = (L_{V}^{2} (Ω × [0, T]))^{∗}.

From (39) and the maximal monotonicity of Â it follows that ÂX = b^{∗}. Hence

E∫_{0}^{T}⟨e^{−at}(AX(t) + aJ X(t)), e^{−at}u(t)⟩dt = E∫_{0}^{T}⟨B(t), e^{−at}u(t)⟩dt

for all u ∈ L_{V}^{2} (Ω × [0, T]). Therefore (37) holds. Finally, from (34), (36), (37) and (24) it results that

e^{−at}X(t) = X_{0} − ∫_{0}^{t} e^{−as}(AX(s) + aJ X(s))ds + ∫_{0}^{t} e^{−as}f(s, X(s))ds
+ ∫_{0}^{t} e^{−as}g(s, X(s))dw(s)

for all t ∈ [0, T] and a.e. ω ∈ Ω. By the equivalence between the problems (P_{1}) and (P_{2}) it follows that X is solution of (P_{1}).

In order to show the uniqueness of the solution of (P_{1}), we assume that (X(t))_{t∈[0,T]} and (Y (t))_{t∈[0,T]} are two solutions of (P_{1}). Let t ∈ [0, T] be fixed for the moment. Then

X(t) − Y (t) = − ∫_{0}^{t} (AX(s) − AY (s))ds + ∫_{0}^{t} (f(s, X(s)) − f(s, Y (s)))ds
+ ∫_{0}^{t} (g(s, X(s)) − g(s, Y (s)))dw(s)

for a.e. ω ∈ Ω. By applying the Ito formula it follows that

E∥X(t) − Y (t)∥^{2} = −2E∫_{0}^{t}⟨AX(s) − AY (s), X(s) − Y (s)⟩ds
+ 2E∫_{0}^{t}(f(s, X(s)) − f(s, Y (s)), X(s) − Y (s))ds
+ E∫_{0}^{t}∥g(s, X(s)) − g(s, Y (s))∥^{2}ds.

Taking into consideration that A is monotone and f, g are Lipschitz continuous, we get

E∥X(t) − Y (t)∥^{2} ≤ (2β + 1)E∫_{0}^{t} ∥X(t) − Y (t)∥^{2}dt.

By applying Gronwall’s Lemma it follows that

E∥X(t) − Y (t)∥^{2} = 0,

therefore

X(t) = Y (t) for a.e. ω ∈ Ω.

But X and Y have continuous trajectories in H and consequently

P ( sup_{t∈[0,T]} ∥X(t) − Y (t)∥^{2} = 0) = 1.

This means that (P_{1}) has with probability 1 a unique solution.

(v) Let (x̂_{N′}) be an arbitrary subsequence of the original sequence (x̂_{N}), such that (x̂_{N′}) converges weakly to a function x ∈ L_{V}^{2} (Ω×[0, T]). Repeating the argumentation of all results contained in Lemma 4.2 till Theorem 4.6-(iv) for this subsequence, we conclude that x is a solution of (P_{1}). From the uniqueness of the solution of problem (P_{1}) we obtain x = X. Thus all weakly convergent subsequences of (x̂_{N}) have the same limit X. By Proposition B.1 it follows that (x̂_{N}) converges weakly to X.

Let (x̂_{N′}) be a subsequence of the original sequence (x̂_{N}). All results contained in Lemma 4.2 till Theorem 4.6-(ii) (applied for (x̂_{N′}) in the space L_{V}^{2} (Ω×[0, T])) imply the existence of a subsequence of (x̂_{N′}) which converges strongly to X. So each subsequence of (x̂_{N}) has, on his part, a subsequence which converges strongly to X. By using Proposition B.2 it results that the original sequence (x̂_{N}) converges strongly to X.

Reasoning just like in the proof of the strong convergence of (x̂_{N}) to X in the space L_{V}^{2} (Ω × [0, T]), it can be shown that the whole sequence (x̂_{N}(T)) converges strongly to X(T) in L_{H}^{2}(Ω). □

Appendix

A Some Properties of Maximal Monotone Operators

Proposition A.1 ([8b], Theorem 32.F, p.881). Let S be a Hilbert space and let A : S → S^{∗} be a monotone operator. Then the following assertions are equivalent:

(1) A is maximal monotone.

(2) R(A + J) = S^{∗}, where J is the duality map of S.

Proposition A.2 ([8b], Proposition 31.5, p.821). Let S be a Hilbert space and let A : S → S^{∗} be a maximal monotone operator. For all λ > 0 the operator A + λJ is invertable and its inverse is continuous.

B Some Convergence Principles

Proposition B.1 ([8a], Proposition 21.23, p.258). If (x_{n}) is a bounded sequence in a reflexive Banach space, then (x_{n}) has a weakly convergent subsequence. If, in addition, each weakly convergent subsequence of (x_{n}) has the same limit x ∈ S, then (x_{n}) converges weakly to x.

Proposition B.2 ([8a], Proposition 21.23, p.258). Let (x_{n}) be a sequence in a Banach space S. Then the following assertions hold:

(i) If (x_{n}) converges weakly to x (where x ∈ S), then (x_{n}) is bounded and

∥x∥_{S} ≤ lim inf_{n→∞} ∥x_{n}∥_{S}.

(ii) If every subsequence of (x_{n}) has, on his part, a subsequence which converges strongly to the same limit x (where x ∈ S), then x_{n} → x.

Proposition B.3 ([8a], Proposition 21.23, p.258). If (x_{n}) is a sequence in a locally uniformly convex Banach space S satisfying x_{n} ⇀ x and ∥x_{n}∥_{S} → ∥x∥_{S} (where x ∈ S), then x_{n} → x.

Proposition B.4 ([8a], Proposition 21.27, p.261). Let S_{1} and S_{2} be Banach spaces and let L : S_{1} → S_{2} be a continuous linear operator. If (x_{n}) is a sequence in S_{1} such that x_{n} ⇀ x (where x ∈ S_{1}), then L(x_{n}) ⇀ L(x).

By applying Proposition B.4 we obtain the following corollary.

Corollary B.5 If S is a Hilbert space and if (x_{n}) is a sequence from L_{S}^{2}(Ω × [0, T]) which converges weakly to x ∈ L_{S}^{2}(Ω × [0, T]), then the following assertions are true:

(i)  ∫_{0}^{t} x_{n}(s)ds ⇀ ∫_{0}^{t} x(s)ds in L_{S}^{2}(Ω × [0, T]);

(ii)  ∫_{0}^{t} x_{n}(s)dw(s) ⇀ ∫_{0}^{t} x(s)dw(s) in L_{S}^{2}(Ω × [0, T]);

(iii)  ∫_{0}^{T} x_{n}(s)ds ⇀ ∫_{0}^{T} x(s)ds in L_{S}^{2}(Ω);

(iv)  ∫_{0}^{T} x_{n}(s)dw(s) ⇀ ∫_{0}^{T} x(s)dw(s) in L_{S}^{2}(Ω).

References

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[4] I. Gyöngy: On the approximation of stochastic partial differential equations II. Stochastics 26 (1989), no. \underline{3}, p. 129-164.

[5] H. Liske: Res̆enie nac̆al’no-kraevoj zadac̆i dlja stochastic̆eskogo uravnenija parabolic̆eskogo tipa metodom poludiskretizacija. Teor. sluc̆. proc. \underline{13} (1985), p. 51-56.

[6] N.V. Krylov, B.L. Rozovskij: Ob evoljucionnych stochastic̆eskich uravnenijach. V: Gamkrelidze, R.V. (Glavn. Red.): Itogi nauki i techniki, serija sovremennych problemy matematiki, tom \underline{14}, VINITI, Moskva 1979.

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[8] E. Zeidler: Nonlinear functional analysis and it’s applications. Springer Verlag, New York - Berlin, 1990.

[8a] Vol. II/A: Linear Monotone Operators.

[8b] Vol. II/B: Nonlinear Monotone Operators.