Stabilization of evolution equations by noise with

application to partial differential equations of parabolic type.^{1}

ANNA A. KWIECIŃSKA

IM UJ PREPRINT 1999/22

Abstract

We consider a deterministic equation of evolution X′(t) = AX(t)dt, in a separable Hilbert space. We prove that if A generates a C_{0}- semigroup, then this equation can be stabilized, in terms of Lyapunov exponents, by noise. Then we apply this abstract result to partial differential equations of parabolic type. We also compute the Lyapunov exponents of these PDEs, both deterministic and stochastic, as functions of the eigenvalues of the operator A.

1. Introduction.

The present paper is a development of [10], where we have constructed an example of a class of partial differential equations being stabilized (in terms of Lyapunov exponents) by noise. In this paper we provide sufficient conditions for exponential stabilization of abstract evolution equations. We apply these results to parabolic partial differential equations extending thus the example from [10].

1Research partially supported by KBN grant 2 P03A 016 16

First, in a separable Hilbert space H, we study an equation of evolution of the form

X′(t) = AX(t),

with the initial condition f (see Section 2). If f ∈ D(A) and A generates a C_{0}-semigroup, the general theory ensures the existence of a unique solution to the above problem, which we denote by x^{det}(f). For f ∈ D(A), f ≠ 0, the Lyapunov exponent of the deterministic equation is defined as:

λ^{det}(f) = lim sup_{t→∞} \frac{1}{t} log ∥x^{det}(t)∥.

We prove that λ^{det}(f) < ∞ (see Lemma 1). Next, we study the Dirichlet problem for a stochastic perturbation of the above equation, namely

dX = AXdt + σ ∑_{k=1}^{N} B_{k}Xdβ_{k},

with the initial condition f. If f ∈ D(A), A generates a C_{0}-semigroup and the operators B_{k} satisfy the assumptions (A1) (A3) and (A4), there exists a unique strong solution to the above stochastic problem, which we denote by x^{st}(f). For f ∈ D(A), f ≠ 0, the Lyapunov exponents of the stochastic equation are defined as

λ^{st}(f, ω) = lim sup_{t→∞} \frac{1}{t} log ∥x^{st}(t, ω)∥.

In Theorem 1 we prove an inequality relating the deterministic and stochastic Lyapunov exponents for a fixed initial condition f and almost all ω. Under our assumptions the stochastic Lyapunov exponents turn out to be smaller, almost surely, than the corresponding deterministic one. It means that the deterministic system is made more stable by adding a term with white noise. Moreover, there exists σ_{0} such that, for σ ≥ σ_{0} the stochastic Lyapunov exponent is strictly smaller than zero with probability one. This is especially interesting, because it means that for σ big enough the stochastic system is exponentially stable, although the deterministic one may be even exponentially unstable. Then taking B_{k} = b_{k}Id, k = 1, …, N, we prove the main result of our paper, that is Theorem 2. This theorem provides a sufficient condition for stabilization of deterministic evolution equations by noise (see Corollary 1), which is both fairly general and natural in the semigroup approach. We prove that, if the operator A generates a C_{0}-semigroup, then the equation of evolution can be made stable by noise, i.e. all the Lyapunov exponents become smaller than zero with probability one.

In Section 3, these abstract results are applied to partial differential equations of parabolic type. We study the Dirichlet problem for the equation

\frac{∂u}{∂t} = Au,

with the initial condition f, where A is a second order, symmetric differential operator satisfying suitable assumptions (for a formal setting see Section 3). We prove that the operator A generates a C_{0}-semigroup in the space L^{2}(O). Then we consider the Dirichlet problem for the following stochastic perturbation of the above equation

dv(t) = Av(t)dt + σ ∑_{k=1}^{N} b_{k}v(t)dβ_{k}(t),

with the initial condition f. It follows from the abstract theory that for σ big enough the stochastic system is exponentially stable. In addition, we obtain bounds on the Lyapunov exponents, both deterministic and stochastic, which turn out to be sharp (see Section 4).

In Section 4 we obtain stronger results concerning the Lyapunov exponents of the parabolic equations studied in Section 3 than those following from the abstract theory. We prove that these Lyapunov exponents exist as limits and in the stochastic case they are also non-random. What is more, in both cases they can be computed as functions of the eigenvalues of the operator A, see Theorem 4 and Theorem 5. Finally, we give an example, where the Lyapunov exponents can be computed explicitely.

The problem of stabilization of ordinary differential equations by noise, in terms of almost sure Lyapunov exponents, has been thoroughly studied, see for instance [2], [3], [4], [14] and [15]. However, to our knowledge, there are very few results concerning stochastic stabilization in the infinite-dimensional case. Apart from our previous paper, almost sure exponential stabilization has been studied very recently by Caraballo, Liu and Mao, see [5] and other types of stabilization have been studied by Leha, Maslowki and Ritter, see [12]. For a survey on stochastic stability see [13].

2. The abstract problem.

Let H be a separable Hilbert space and let A be a linear operator from D(A) ⊂ H into H. We consider the following deterministic equation of evolution

X′(t) = AX(t)  (1)

with the initial condition

X(0) = f ∈ D(A).  (2)

Following [16] by a solution to the above Cauchy problem we mean an H valued function x(t) such that x(t) is continuous for t ≥ 0, continuously differentiable, u(t) ∈ D(A) for t > 0 and (1)-(2) is satisfied. If A : D(A) ⊂ H → H generates a C_{0}-semigroup S(t) and if f ∈ D(A) the above Cauchy problem has a unique solution x(t) = S(t)f. The solution of the above problem will be denoted by x^{det}(t). For f ≠ 0, f ∈ D(A) the Lyapunov exponent of the problem (1)-(2) is defined as

λ^{det}(f) = lim sup_{t→∞} \frac{1}{t} log ∥x^{det}(t)∥.

Lemma 1 If the operator A generates a C_{0}-semigroup S(t), then for any initial value f

λ^{det}(f) ≤ λ,

where λ = const > 0 and does not depend on f.

Proof. It follows from the fact that since S(t) is a C_{0}-semigroup, there exist constants λ > 0 and and M ≥ 1 such that

∥S(t)∥ ≤ M exp(λt),

see [16], Theorem 2.2, p.4. □

Let us now consider the following stochastic problem

dX = AXdt + σ ∑_{k=1}^{N} B_{k}Xdβ_{k},  (3)
X(0) = f ∈ H,  (4)

where

A : D(A) ⊂ H → H, B_{k} : D(B_{k}) ⊂ H → H, k = 1, 2, …, N

and β_{1}(·), …, β_{N}(·), N < ∞, are independent, real Wiener processes. Let us assume the following:

(1) The operator A generates a semigroup S(t) and the operators σB_{1}, …, σB_{N} generate mutually commuting C_{0}-groups S_{k}(t), k = 1, …, N.

(2) For k = 1, …, N, D(σ^{2}B_{k}^{2}) ⊃ D(A) and ⋂_{k=1}^{N} D(((σB_{k})^{∗})^{2}) is a dense subset of H, where (σB_{k})^{∗} denotes the adjoint of σB_{k}.

(3) The operator C = A − \frac{1}{2}σ^{2} ∑_{k=1}^{N} B_{k}^{2}, D(C) = D(A) is the infinitesimal generator of a C_{0}-semigroup S_{0}(t), t ≥ 0.

(4) There exist λ_{0} ∈ R and bounded operators K_{1}, …, K_{N} such that

(λ_{0} − C)σB_{k}(λ_{0} − C)^{−1} = σB_{k} + K_{k}, k = 1, …, N.

Then, if f ∈ D(C), the above problem has a unique strong solution, as defined in [6] p.152, on any interval [0, T], where T > 0 – see [6], Proposition 6.29, p. 176 and Theorem 6.30, p. 177.

We will study next an evolution equation of the above form, and we assume that: (A1) The operators B_{k} are diagonalizable in the same basis and are bounded from above and below, i.e there exist {e_{j}} a basis of the Hilbert space H and constants d_{i}^{k}, where k = 1, …, N and i = 1, 2, …, such that

⟨B_{k}e_{i}, e_{j}⟩ = d_{k}^{i}δ_{ij}, k = 1, …, N; i, j = 1, 2, …,

where ⟨·, ·⟩ denotes the scalar product in H. The assumption that the operators B_{k} are bounded from above and below is then equivalent to the condition that there exist constants m_{k} > 0 and M_{k} > 0, k = 1, …, N, such that

m_{k} ≤ |d_{i}^{k}| ≤ M_{k}, k = 1, …, N; i = 1, 2, ….

Obviously the operators B_{k} also mutually commute. (A2) The operator A generates a C_{0}-semigroup. (A3) There exist λ_{0} ∈ R and bounded operators K_{1}, …, K_{N} such that

(λ_{0} − C)σB_{k}(λ_{0} − C)^{−1} = σB_{k} + K_{k}, k = 1, …, N,

where C = A − \frac{1}{2}σ^{2} ∑_{k=1}^{N} B_{k}^{2}. (A4) The operators A and B_{k}, k = 1, …, N, commute.

Proposition 1 The general assumptions (1)-(4) follow from (A1)-(A4).

To prove the above proposition we will need two lemmas.

Lemma 2 Let A and B be the infinitesimal generators of C_{0}-semigroups S(t) and T (t) respectively. If A and B commute, then S(t) and T (t) also commute.

Proof. Since A and B are generators of C_{0}-semigroups we have:

S(t)x = lim_{λ→∞} exp(tA_{λ})x, x ∈ H

and

T (t)x = lim_{γ→∞} exp(tB_{γ})x, x ∈ H,

where A_{λ} and B_{γ} are the Yosida approximations of A and B respectively, i.e. A_{λ} = λAR(λ : A) and B_{γ} = γBR(γ : B); see [16] Theorem 5.5, p.21. There exist λ̄ > 0 and γ̄ > 0 such that for λ > λ̄ and γ > γ̄ the operators A_{λ} and B_{γ} are well defined and bounded. In what follows we will consider the operators A_{λ} and B_{γ} with λ > λ̄ and γ > γ̄. As the operators A and B commute, the operators A_{λ} and B_{γ} also commute.

Let us fix arbitrary λ and γ. We put

S_{A_{λ}}^{N} = ∑_{n=1}^{N} \frac{(tA_{λ})^{n}}{n!}

and

S_{B_{γ}}^{N} = ∑_{n=1}^{N} \frac{(tB_{γ})^{n}}{n!}

A.A. Kwiecińska Stabilization of evolution equations 7

Since the operators A_{λ} and B_{γ} commute, we have

S_{A_{λ}}^{N}S_{B_{γ}}^{N} = S_{B_{γ}}^{N}S_{A_{λ}}^{N}

and passing to the limit with N we get

exp(tA_{λ}) exp(tB_{γ}) = exp(tB_{γ}) exp(tA_{λ}).

From the above equality we obtain for arbitrary x ∈ H:

lim_{λ→∞} exp(tA_{λ}) exp(tB_{γ})x = lim_{λ→∞} exp(tB_{γ}) exp(tA_{λ})x

and we deduce that

S(t) exp(tB_{γ})x = exp(tB_{γ})S(t)x.

Passing again to the limit with γ we deduce that

S(t)T (t)x = T (t)S(t)x.

□

Lemma 3 Let S(t) and T (t) be two commuting C_{0}-semigroups. We put R(t) = S(t)T (t). Then R(t) is also a C_{0}-semigroup. Proof. We have

R(0) = S(0)T (0) = Id

and since the semigroups S(t) and R(t) commute R(t+s) = S(t+s)T (t+s) = S(t)S(s)T (t)T (s) = S(t)T (t)S(s)T (s) = R(t)R(s). To prove that

lim_{t↓0} R(t)x = x, ∀x ∈ H,

we estimate

|T (t)S(t)x − x| = |T (t)S(t)x − T (t)x + T (t)x − x|

≤ |T (t)S(t)x − T (t)x| + |T (t)x − x| ≤ ∥T (t)∥|S(t)x − x| + |T (t)x − x|, where by | · | we have denoted the norm in the Hilbert space H and by ∥ · ∥ the operator norm. It follows from [16], Theorem 2.2, p.4 that there exist constants λ > 0 and M ≥ 1 such that for t ∈ [0, 1]

∥T (t)∥ ≤ M exp λ,

we can therefore conclude that

lim_{t↓0} |T (t)S(t)x − x| = 0

□

Proof of the Proposition. (1) Follows from Lemma 2 and (A1).

(2) We have D(B_{k}) = D(B_{k}^{∗}) = H, k = 1, …, N, which implies (2).

(3) Obviously D(C) = D(A), where C = A − \frac{1}{2}σ^{2} ∑_{k=1}^{N} B_{k}^{2}. The semigroup generated by the operator C is given by the following formula

S_{0}(t) = S(t) ∏_{k=1}^{N} exp(−\frac{1}{2}tσ^{2}B_{k}^{2}).

We have assumed that the operator A generates a C_{0}-semigroup. On the other hand the operators B_{k}^{2} are bounded and defined on the whole space H and therefore exp(−\frac{1}{2}tσ^{2}B_{k}^{2}), k = 1, …, N, define C_{0}-semigroups. Due to Lemma 2 these semigroups are mutually commuting since their generators are mutually commuting (the assumptions (A1) and (A4)), so Lemma 3 enables us to conclude that the operator C generates a C_{0}-semigroup.

(4) It is the assumption (A3). □

So Proposition 6.29 and Theorem 6.30 from [6] can be applied and if f ∈ D(C) = D(A) there exists a unique solution to the problem (3)-(4). This solution is given by the following formula

x^{st}(t) = S_{0}(t) ∏_{k=1}^{N} exp[σB_{k}β_{k}(t)]f.  (5)

For f ≠ 0, f ∈ D(A) the Lyapunov exponent of the system (3)-(4) is defined as:

λ^{st}(f, ω) = lim sup_{t→∞} \frac{1}{t} log ∥x^{st}(t, ω)∥.

From Lemma 1 and the next theorem it will follow that λ^{st}(f, ω) < ∞ a.s.

A.A. Kwiecińska Stabilization of evolution equations 9 Theorem 1 Let f ∈ D(A), f ≠ 0. Under the assumptions (A1)-(A4) the following inequality holds:

λ^{st}(f, ω) ≤ λ^{det}(f) − σ^{2}α a.s.  (6)

where

α = − lim sup_{t→∞} \frac{1}{t} log ∥ ∏_{k=1}^{N} exp[(−\frac{1}{2}B_{k}^{2})t]∥ > 0.

It follows that there exists σ_{0} such that for σ ≥ σ_{0}

λ^{st}(f, ω) < 0 a.s.  (7)

and σ_{0} does not depend on f.

Proof. We have

λ^{st}(f, ω) = lim sup_{t→∞} \frac{1}{t} log ∥S_{0}(t) ∏_{k=1}^{N} exp[σB_{k}β_{k}(t)]f∥

≤ lim sup_{t→∞} \frac{1}{t} log ∥ ∏_{k=1}^{N} exp[(−\frac{1}{2}σ^{2}B_{k}^{2})t] ∏_{k=1}^{N} exp[σB_{k}β_{k}(t)]∥∥S(t)f∥

≤ lim sup_{t→∞} \frac{1}{t} log ∥ ∏_{k=1}^{N} exp[(−\frac{1}{2}σ^{2}B_{k}^{2})t] ∏_{k=1}^{N} exp[σB_{k}β_{k}(t)]∥+lim sup_{t→∞} \frac{1}{t} log ∥S(t)f∥

≤ λ^{det}(f) + lim sup_{t→∞} \frac{1}{t} log ∥ ∏_{k=1}^{N} exp[(−\frac{1}{2}σ^{2}B_{k}^{2})t]∥∥ ∏_{k=1}^{N} exp[σB_{k}β_{k}(t)]∥

≤ λ^{det}(f)+lim sup_{t→∞} \frac{1}{t} log ∥ ∏_{k=1}^{N} exp[(−\frac{1}{2}σ^{2}B_{k}^{2})t]∥+lim sup_{t→∞} \frac{1}{t} log ∥ ∏_{k=1}^{N} exp[σB_{k}β_{k}(t)]∥.

Since the operators B_{k}, k = 1, …, N, are bounded from above by constants M_{k} > 0, k = 1, …, N, we have

∥ ∏_{k=1}^{N} exp[σB_{k}β_{k}(t)]∥ ≤ ∏_{k=1}^{N} exp[σM_{k}|β_{k}(t)|],

and hence

lim sup_{t→∞} \frac{1}{t} log ∥ ∏_{k=1}^{N} exp[σB_{k}β_{k}(t)]∥ ≤ 0,

since by the strong law of large numbers lim_{t→∞} β_{k}(t)/t = 0 with probability 1, k = 1, …, N (see [1], page 46).

On the other hand, since all the operators B_{k} are diagonal and there exist constants m_{k} > 0 and M_{k} > 0 such that

m_{k} ≤ ∥B_{k}∥ ≤ M_{k},

an easy computation gives the following bounds:

−\frac{1}{2}σ^{2}(∑_{k=1}^{N} M_{k}^{2}) ≤ \frac{1}{t} log ∥ ∏_{k=1}^{N} exp[−\frac{1}{2}σ^{2}(B_{k}^{2})t]∥ =

= σ^{2} \frac{1}{t} log ∥ ∏_{k=1}^{N} exp[−\frac{1}{2}(B_{k}^{2})t]∥ ≤ −\frac{1}{2}σ^{2}(∑_{k=1}^{N} m_{k}^{2})

and hence there exists α > 0 such that

lim sup_{t→∞} \frac{1}{t} log ∥ exp[(−\frac{1}{2}σ^{2} ∑_{k=1}^{N} B_{k}^{2})t]∥ = −σ^{2}α.

Now the first part of the statement of the theorem follows.

The second part follows directly from the first part and Lemma 1. □

Let us now consider the problem (3)-(4) with B_{k} = b_{k}Id, where b_{k} ≠ 0, k = 1, …, N, i.e.

dX(t) = AX(t)dt + σ ∑_{k=1}^{N} b_{k}X(t)dβ_{k}(t)  (8)
X(0) = f ∈ D(A)  (9)

We denote by λ^{st}(f, ω) the Lyapunov exponent of the stochastic problem (8)-(9) and by λ^{det}(f) the Lyapunov exponent of the deterministic problem (1)-(2).

Theorem 2 Let f ≠ 0, f ∈ D(A). If the operator A generates a C_{0}- semigroup, solutions to (1)-(2) and (8)-(9) exist, are unique and we have the following estimate:

λ^{st}(f, ω) ≤ λ^{det}(f) − \frac{1}{2}σ^{2} ∑_{k=1}^{N} b_{k}^{2} a.s.

Proof. The existence and uniqueness of solution to the deterministic problem (1)-(2) follow from the fact that A generates a C_{0}-semigroup. For the stochastic problem it is enough to check that the conditions (A1)-(A4) hold.

The assumptions (A1), (A2) and (A4) are obviously satisfied. We have to verify that (A3) holds. We compute:

(λ_{0}Id − C) = (λ_{0}Id − (A − \frac{1}{2}σ^{2} ∑_{k=1}^{N} b_{k}^{2}Id)) = ((λ_{0} − \frac{1}{2}σ^{2} ∑_{k=1}^{N} b_{k}^{2})Id − A).

Since A generates a C_{0}-semigroup the operator A is closed and there exists δ ≥ 0 such that the resolvent set ϱ(A) of A contains the ray (δ, ∞), see[16], Theorem 5.3, p.20, see also Section 3. It follows that if γ > δ , then R(γ : A) is well defined. Therefore, if we take λ_{0} satisfying

λ_{0} − \frac{1}{2}σ^{2} ∑_{k=1}^{N} b_{k}^{2} > δ

we have that

(λ_{0}Id − C)b_{k}Id(λ_{0}Id − C)^{−1} = b_{k}(λ_{0}Id − C)(λ_{0}Id − C)^{−1} = b_{k}Id.

It means that (A3) holds with K_{k} = 0, for k = 1, 2, …, N.

We can now apply Theorem 1. It is easy to check that α = \frac{1}{2} ∑_{k=1}^{N} b_{k}^{2}. □

Corollary 1 If the operator A generates a C_{0}-semigroup, the deterministic system (1)-(2) can be stabilized by noise.

Proof. Let us fix f ≠ 0, f ∈ D(A). Lemma 1 ensures the existence of a constant λ > 0 such that

λ^{det}(f) ≤ λ.

Let σ_{0} be such that

\frac{1}{2}σ_{0}^{2} ∑_{k=1}^{N} b_{k}^{2} > λ,

then, for σ ≥ σ_{0} we have λ^{st}(f, ω) < 0 for almost all ω. □

3. Application to partial differential equations of parabolic type.

Let O be a bounded domain in R^{d}: a ball or a set that can be mapped into a ball by a regular mapping of class C^{2}(Ō).

We will study the following parabolic equation:

\frac{∂u}{∂t} = Au,  (10)

where u = u(t, x), t ∈ R^{+} (where R^{+} denotes the interval [0, ∞)), x ∈ O. We set the initial condition

u(0, x) = f(x)  (11)

and the Dirichlet boundary condition

u(t, x) = 0, x ∈ ∂O.  (12)

The operator A is of the form

Au = ∑_{i,j} \frac{∂}{∂x_{i}} (a_{ij}(x)\frac{∂u}{∂x_{j}} ) + a(x)u,

where the coefficients satisfy the following assumptions: (C1) a_{ij} are differentiable (C2) a_{ij} = a_{ji} (C3) μ_{1}ξ^{2} ≤ ∑_{i,j} a_{ij}ξ_{i}ξ_{j} ≤ μ_{2}ξ^{2} (C4) |\frac{∂a_{ij}}{∂x_{k}} | ≤ μ_{3} (C5) |a| ≤ μ_{4}, where μ_{l} are constants and μ_{l} > 0, l = 1, …, 4.

It is easy to verify that the operator A is symmetric on the space H_{0}^{1}(O). The function f in (11) takes real values. In this section we consider only

deterministic real-valued initial conditions f ∈ H_{0}^{1}(O) ∩ H^{2}(O).

We will need the following subspace of the space H^{2}(O): W_{2,0}^{2}(O) - a subspace of H^{2}(O) generated by functions belonging to C^{2}(Ō) and vanishing on ∂O.

Later on we will prove that in our case W_{2,0}^{2}(O) = H_{0}^{1}(O) ∩ H^{2}(O).

There exists an orthonormal basis of L^{2}(O), {e_{j}}, j = 1, 2, …, consisting of eigenvectors of the operator A (see [11], p.181) such that

Ae_{j} = λ_{j}e_{j}, where λ_{j} ↘ −∞ (j → ∞)

and

[e_{k}, e_{l}] = (λ_{0} − λ_{k})^{2}δ_{k}^{l},

where [u, u]^{\frac{1}{2}} is equivalent to the norm in W_{2,0}^{2}(O). The constant λ_{0} is chosen such that λ_{k} < λ_{0}, k = 1, 2, . . ..

Lemma 4 Under our assumptions we have the following characterisation of the space W_{2,0}^{2}(O)

W_{2,0}^{2}(O) = {u = ∑_{j=1}^{∞} u_{j}e_{j}(x) ∈ L^{2}(O) : ∑_{j=1}^{∞} u_{j}^{2}(λ_{0} − λ_{j})^{2} < ∞}

Proof. Obviously \{\frac{1}{λ_{0}−λ_{j}} e_{j}\} is an orthonormal basis of W_{2,0}^{2}(O) with the scalar product [·, ·].

If u ∈ L^{2}(O) we can write

u = ∑_{j=1}^{∞} u_{j}e_{j} = ∑_{j=1}^{∞}(λ_{0} − λ_{j})u_{j} (\frac{1}{λ_{0} − λ_{j}} e_{j})

and the equality follows. □

Proposition 2 Under our assumptions the following equality holds:

W_{2,0}^{2}(O) = H_{0}^{1}(O) ∩ H^{2}(O).

Proof. Obviously we have

W_{2,0}^{2}(O) ⊂ H^{2}(O) (∗)

The inclusion

W_{2,0}^{2}(O) ⊂ H_{0}^{1}(O) (∗∗)

follows from the trace theorem (see [17], p.48):

Theorem (trace theorem): Let O be a bounded set of class C^{1}. There exists a linear continuous operator γ_{0} ∈ L(H^{1}(O), L^{2}(∂O)) such that:

γ_{0}u = u|_{∂O}, ∀u ∈ C^{1}(Ō).

We have

u ∈ W_{2,0}^{2}(O) ⟺ ∃{u_{m}} ∈ C^{2}(Ō), u_{m}|_{∂O} = 0, u_{m} → u in H^{2}(O).

Since γ_{0}u_{m} = 0, γ_{0} is continuous in H^{1}(O) and u_{m} → u in H^{1}(O), we deduce that

γ_{0}u = u|_{∂O} = 0.

In the case of bounded regions with smooth boundary u ∈ H_{0}^{1}(O) is equivalent to the conditions u ∈ H^{1}(O) and u|_{∂O} = 0, see [8], p.122. Therefore we conclude that u ∈ H_{0}^{1}(O), which proves (∗∗).

The inclusions (∗) and (∗∗) give

W_{2,0}^{2}(O) ⊂ H_{0}^{1}(O) ∩ H^{2}(O).

Let us now take u ∈ H_{0}^{1}(O) ∩ H^{2}(O). The operator A is symmetric on the space H_{0}^{1}(O) and therefore is closable, see [9], p. 269. We have

Au = Āu = Ā ∑_{j=1}^{∞} u_{j}e_{j}(x) = ∑_{j=1}^{∞} u_{j}Āe_{j}(x) = ∑_{j=1}^{∞} u_{j}Ae_{j}(x)

= ∑_{j=1}^{∞} u_{j}λ_{j}e_{j}(x) ∈ L^{2}(O).

Note that ∑_{j=1}^{∞} u_{j}λ_{j}e_{j}(x) ∈ L^{2}(O) if and only if ∑_{j=1}^{∞} u_{j}^{2}λ_{j}^{2} < ∞, which implies ∑_{j=1}^{∞} u_{j}^{2}(λ_{0} − λ_{j})^{2}, and by Lemma 4 it gives u ∈ W_{2,0}^{2}(O). Hence the inclusion

H_{0}^{1}(O) ∩ H^{2}(O) ⊂ W_{2,0}^{2}(O)

follows. □

If the coefficients of the operator A are sufficiently smooth, then Theorem 2.7, p.211, from [16], ensures that the operator A generates an analytic semigroup of operators on the space L^{2}(O). However our assumptions (C1)- (C5) do not guarantee the smoothness of the coefficients of A and therefore we present a separate proof of the fact that the operator A generates a C_{0}- semigroup.

Theorem 3 The operator A with D(A) = H_{0}^{1}(O) ∩ H^{2}(O) generates a C_{0}- semigroup S(t) such that ∥S(t)∥ ≤ exp(λ_{1}t).

In the proof of the above theorem we will make use of a more general result from [16] – Theorem 5.3, p.20.

Theorem A linear operator A : D(A) ⊂ X → X, where X is a Banach space, is the infinitesimal generator of a C_{0}-semigroup T (t) satisfying ∥T (t)∥ ≤ M exp(ωt), if and only if

(i) A is closed and D(A) is dense in X (ii) The resolvent set ϱ(A) of A contains the ray (δ, ∞) and ∥R(λ : A)^{n}∥ ≤ M/(λ − δ)^{n}, for λ > δ, n = 1, 2, . . ..

Proof of Theorem 3. We will first prove that condition (i) is satisfied. The space H_{0}^{1}(O) ∩ H^{2}(O) is dense in L^{2}(O). We have already remarked that the operator A is closable (see the proof of Proposition 2). Let us take x_{n} ∈ D(A) such that x_{n} → x and Ax_{n} → y. We can write

x_{n} = ∑_{j=1}^{∞} x_{j}^{n}e_{j}(x), x = ∑_{j=1}^{∞} x_{j}e_{j}(x) and y = ∑_{j=1}^{∞} y_{j}e_{j}(x).

We compute

Ax_{n} = Āx_{n} = Ā ∑_{j=1}^{∞} x_{j}^{n}e_{j}(x) = ∑_{j=1}^{∞} x_{j}^{n}Āe_{j}(x) = ∑_{j=1}^{∞} x_{j}^{n}Ae_{j}(x) =

= ∑_{j=1}^{∞} x_{j}^{n}λ_{j}e_{j}(x) (n → ∞)

Since x_{n} → x (n → ∞), we have that x_{j}^{n} → x_{j} (n → ∞) for each j. On the other hand since Ax_{n} → y (n → ∞), we deduce that x_{j}^{n}λ_{j} → y_{j} (n → ∞) for each j. Hence we obtain:

y_{j} = λ_{j}x_{j}

and

x = ∑_{j=1}^{∞} \frac{y_{j}}{λ_{j}} e_{j}(x).

Therefore by Lemma 4 and Proposition 2, x ∈ D(A) and y = Ax, which implies that the operator A is closed.

To prove that (ii) holds let us fix an arbitrary constant λ > λ_{1} (this implies λ > λ_{k} for k = 1, 2, …). We have

(λI − A)u = ∑_{j=1}^{∞}(λ − λ_{j})u_{j}e_{j}(x)

and

(λI − A)^{−1}v = ∑_{j=1}^{∞} \frac{1}{λ − λ_{j}} v_{j}e_{j}(x).

The second operator is well defined for any v ∈ L^{2}(O). We can write

R(λ : A)^{n} = ∑_{j=1}^{∞} (\frac{1}{λ − λ_{j}} )^{n} v_{j}e_{j}(x).

We easily deduce that

∥R(λ : A)^{n}∥ = \frac{1}{(λ − λ_{1})^{n}}

and we conclude that condition (ii) is satisfied with M = 1 and δ = λ_{1}. □

This semigroup is given by the following formula

S(t)f = ∑_{j=1}^{∞} exp(tλ_{j})f_{j}e_{j}(x)  (13)

where f_{j} =< f, e_{j} >.

Corollary 2 We have the following estimate:

λ^{det}(f) ≤ λ_{1}.

Proof. It follows directly from Theorem 3.

Remark 1 In fact we will prove in Section 4 that this estimate cannot be improved.

The results from Section 2 guarantee that the deterministic equation (10) considered with the initial condition (11) and the Dirichlet boundary condition (12) can be stabilized by noise. More precisely, let us consider the following stochastic perturbation of the deterministic problem

dv(t) = Av(t)dt + σ ∑_{k=1}^{N} b_{k}v(t)dβ_{k}(t),  (14)

where b_{k} ≠ 0, k = 1, …, N, with the initial condition

v(0, x, ω) = f(x), x ∈ O  (15)

and the Dirichlet boundary condtion

v(t, x, ω) = 0, x ∈ ∂O  (16)

Conditions (15) and (16) hold for a.a. ω ∈ Ω.

The estimate from Theorem 2 together with Theorem 3 give us the following conclusion

Corollary 3 The stochastic system (14)-(16) is stable for σ satisfying

\frac{1}{2}σ^{2} ∑_{k=1}^{N} b_{k}^{2} > λ_{1}.

In Section 4 we will prove that this result cannot be improved.

4. Computation of Lyapunov exponents.

For parabolic equations studied in Section 3, both deterministic and stochastic, we can obtain stronger results concerning their Lyapunov exponents, than those following from the general theory. In fact we can prove that these Lyapunov exponents exist as limits and in the stochastic case they are also non-random. Moreover, in both cases we can compute the Lyapunov exponents as functions of the eigenvalues of the operator A.

We consider the Dirichlet problem (10)-(12), where the domain O and the operator A are as in Section 3. Also as in the previous section the initial condition f ∈ H_{0}^{1}(O) ∩ H^{2}(O) and is real valued.

If f ∈ L^{2}(O), we can write

f = ∑_{j=1}^{∞} f_{j}e_{j}, where f_{j} =< f, e_{j} >  (17)

It follows from (13) that the unique solution to the problem (10)-(12) is given by the following formula

u(t) = S(t)f = ∑_{j=1}^{∞} exp(tλ_{j})f_{j}e_{j}  (18)

Theorem 4 Let f ≠ 0 and let j_{0} be the smallest integer j ≥ 1 in the expansion (17) of f such that f_{j_{0}} ≠ 0.

Then the Lyapunov exponent of (10) – (12) exists as a limit and is given by

λ^{det}(f) = λ_{j_{0}}.

Therefore the top Lyapunov exponent is equal to λ_{1}.

Proof. On the one hand,

\frac{1}{t} log ∥ ∑_{j=0}^{∞} exp(tλ_{j})f_{j}e_{j}∥ ≤ \frac{1}{t} log( ∑_{j=j_{0}}^{∞} | exp(tλ_{j_{0}})f_{j}|^{2})^{\frac{1}{2}} = λ_{j_{0}} + \frac{1}{t} log ∥f∥

while on the other hand,

\frac{1}{t} log ∥ ∑_{j=0}^{∞} exp(tλ_{j})f_{j}e_{j}∥ ≥ \frac{1}{t} log | exp(tλ_{j_{0}})f_{j_{0}}| = λ_{j_{0}} + \frac{1}{t} log |f_{j_{0}}|.

The existence of the limit and the equality follow. □

The formula (5) implies that the solution to the stochastic problem (14)- (16) is equal to

v(t) = exp(∑_{k=1}^{N} σb_{k}β_{k}(t)) exp[(−\frac{1}{2}σ^{2} ∑_{k=1}^{N} b_{k}^{2})t]u(t),

where u(t) is the solution to the deterministic problem. It is now easy to prove the following theorem using the above formula and the fact that lim_{t→∞} β_{k}(t)/t = 0 a.s. for k = 1, 2, …, N.

Theorem 5 The Lyapunov exponent of the system (14)-(16) almost surely exists as a limit, is non-random and the following formula holds:

λ^{st}(f, ω) = λ^{det}(f) − \frac{1}{2}σ^{2} ∑_{k=1}^{N} b_{k}^{2} = λ_{j_{0}} − \frac{1}{2}σ^{2} ∑_{k=1}^{N} b_{k}^{2} a.s.

Therefore the top Lyapunov exponent is equal to λ_{1} − \frac{1}{2}σ^{2} ∑_{k=1}^{N} b_{k}^{2}.

Example Let us take

O = (a, b), where a < b

and A = △ + αId, where α is a constant.

Then the eigenvalues and the eigenfunctions of the operator A can be computed explicitely and are equal to, respectively:

α_{j} = −\frac{j^{2}π^{2}}{(b − a)^{2}} + α

and

e_{j}(x) = (\frac{2}{b − a})^{\frac{1}{2}} sin(\frac{jπ(x − a)}{(b − a)}),

j = 1, 2, …, see [7] Example 1.2.3, p.10.

We deduce that the Lyapunov exponents of the deterministic system and the stochastic system are equal to, respectively:

λ^{det}(f) = −\frac{j_{0}^{2}π^{2}}{(b − a)^{2}} + α

and

λ^{st}(f) = −\frac{j_{0}^{2}π^{2}}{(b − a)^{2}} + α − \frac{1}{2}σ^{2} ∑_{k=1}^{N} b_{k}^{2}.

We conclude that the stochastic system is stable if and only if its top Lyapunov exponent satisfies

λ = −\frac{π^{2}}{(b − a)^{2}} + α − \frac{1}{2}σ^{2} ∑_{k=1}^{N} b_{k}^{2} < 0.

Ackonowledgements The author wishes to thank Professor M. Capiński for valuable guidance. The author is also indebted to E. Motyl and S. Peszat for helpful discussions.

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Institute of Mathematics, Jagiellonian University, ul. Reymonta 4, 30-059 Kraków, Poland. email: akwiecin@im.uj.edu.pl