Formulas for generalized principal Lyapunov exponent for parabolic PDEs

An integral formula is given representing the generalized principal Lyapunov estimate for random linear parabolic PDEs. As an application, an upper estimate of the exponent is obtained.


Introduction
In [8] the current authors presented the theory of generalized principal Lyapunov exponents for linear random parabolic partial differential equations (PDEs) of the form where D ⊂ R N , endowed with boundary conditions of either Dirichlet or Robin type, driven by an ergodic flow (θ t ) t∈R on Ω. The second-and first-order coefficients of the equation are assumed to be bounded uniformly in ω ∈ Ω, whereas as concerns zero-order coefficients some integral-type conditions are required.
That generalizes the theory in [6], where it was assumed that all the coefficients are bounded. The generalized principal Lyapunov exponent is defined as the logarithmic growth rate of some distinguished solutions (see Theorem 2.1(i)-(ii)). It comes out that the generalized principal Lyapunov exponent is just equal to the top Lyapunov exponent (Theorem 2.1(iv)). Moreover, for any nontrivial nonnegative solution its logarithmic growth rate equals the generalized principal Lyapunov exponent (Theorem 2.1(iii)).
Bearing in mind that Lyapunov exponents have relevance for establishing (in)stability of nonlinear PDEs of parabolic type, it is very important to find ways of estimating them.
It is the purpose of the present paper to give some estimates of formulas for the generalized principal Lyapunov exponent.
The paper is organized as follows. In Section 2 some preliminaries are given. In particular, the standing assumptions are established, and necessary results from [8] are presented. Section 3 is devoted to the proof of the integral formula of the generalized principal Lyapunov exponent (Theorem 3.1). In Section 4 an upper estimate is presented of the generalized principal Lyapunov exponent in terms of the principal eigenvalues of the (elliptic) equations with "frozen" coefficients (Theorem 4.1).

Preliminaries
In the present section we introduce the main concepts and assumptions, and formulate the main results on generalized principal Lyapunov exponents, as presented in [8].
By a measurable space we understand a pair (P, P), where P is a set and P is a σ-algebra of subsets of P . For measurable spaces (P, P) and (R, R), a function f : P → R is (P, R)-measurable if for any A ∈ R its preimage, f −1 (A), belongs to P.
By a measure space we understand a triple (P, P, µ), where (P, P) is a measurable space and µ is a measure defined on P. When µ is finite we speak of a finite measure space, and when µ(P ) = 1 we speak of a probability space.
For a metrizable space X, B(X) denotes the σ-algebra of Borel sets of X.
Sometimes we write simply (θ t ) t∈R for a metric flow.
Throughout the paper the standing assumption is that ((Ω, F, P), (θ t ) t∈R ) is a metric flow which is moreover ergodic: If F ∈ F is such that θ t (F ) = F for all t ∈ R (in other words, F is invariant ), then either P(F ) = 0 or P(F ) = 1. Furthermore, the probability measure P is assumed to be complete.
Consider a family, indexed by ω ∈ Ω, of linear second order partial differential equations where s ∈ R is an initial time and D ⊂ R N is a bounded domain with boundary ∂D, complemented with boundary condition Above, ν = (ν 1 , . . . , ν N ) denotes the unit normal vector pointing out of ∂D.
Throughout the present paper, · stands for the standard norm in L 2 (D) or for the standard norm in L(L 2 (D)) (= the Banach space of bounded linear operators from L 2 (D) into L 2 (D)), depending on the context. L 2 (D) + denotes the set of those functions in L 2 (D) + that are nonnegative Lebesgue-a.e. on D.
We make the following assumptions (α is a positive constant).
(iv) (Local regularity of zero order terms) For each ω ∈ Ω and each T > 0 the restriction (A2)(i)-(iii) are just the second, third and fourth items in assumption (R)(ii) in [8].) Moreover, the following result holds, which we state here for further reference.

Proof. By Lemma
is nonnegative, we can apply Tonelli's theorem to conclude that But θ t P = P for any t ∈ R, so it follows that c (+) 0 ∈ L 1 ((Ω, F, P)).
We give now one of the main results formulated and proved in [8].

Integral Formula for Generalized Principal Lyapunov Exponent
In this section we give a representation of the generalized Lyapunov exponent as the integral over Ω of some function connected with the Dirichlet form. We assume that (A0) through (A4) are satisfied. Let V denote the Banach space as in [8,Sect. 3] (in the Dirichlet or Neumann cases V is a closed subspace of the Sobolev space W 1 2 (D)). For ω ∈ Ω the Dirichlet form B ω = B ω (·, ·) is a bilinear form on V is defined as (we use summation convention) in the Dirichlet and Neumann boundary condition cases, and (3.2) in the Robin boundary condition case (H N −1 denotes (N −1)-dimensional Hausdorff measure, which is, under (A0), the same as surface measure). From the fact that any solution is classical it follows that w(ω) belongs to C 1 (D), hence the function κ : Ω → R, is well defined.
The following is the main result of this section.
In the case of bounded zero-order terms an analog of Theorem 3.1 was proved in [6] (cf. [6,Thm. 3.5.3]). For analogs of the formula for other types of equations, see the survey paper [5].
Before giving the proof of Theorem 3.1 we formulate and prove a couple of auxiliary results. and Proof. Fix ω ∈Ω 0 . By Theorem 2.1(i), we have Proceeding along the lines of the proof of [6, Proposition 2.5.1] one obtains that the mapping [−1, 1] ∋ t → U θ−2ω (t + 2)w(θ −2 ω) ∈ C 2+α (D) is continuous, so a fortiori that mapping with C 2+α (D) replaced by C 1 (D). We have thus proved that the restriction of the mapping t → w(θ t ω) to [−1, 1] is continuous. Since ω ∈Ω 0 is arbitrary, the mapping is continuous on its whole domain (−∞, ∞). The continuity of the mapping t → κ(θ t ω) is a consequence of the continuity of the first mapping and (A2).
for all t ∈ R; (ii)
Proof. It follows from [6, Proposition 2.1.4] and the definition of κ that for any 0 ≤ s < t. As, by Lemma 3.1, the integrand on the right-hand side above is continuous in τ , the statement (i) follows by standard calculus (for a similar reasoning, see [6, Lemma 3.5.3]). Part (ii) is straightforward.
Indication of proof. We give only the first step of the proof. Namely, we prove that the mapping is (F, B(C([0, T ]×D)))-measurable. In view of Proposition 2.1 this is equivalent to showing that for t ∈ [0, T ] and x ∈D fixed the mapping is (F, B(R))-measurable, which follows in turn from the fact that, for each n ∈ N, is (F, B(R))-measurable and that . We apply the above reasoning now to the derivatives of C ω 0 in t and x i , of suitable orders.
In view of Lemma 3.3, for each M > 0 the setΩ M belongs to F. So it suffices to prove the measurability of w restricted toΩ M , for each M > 0. In order to do this, observe first that (A2) implies that the closure,Ŷ M , of which is continuous, since it follows from the parabolic strong maximum principle thatû(0;â, u 0 ) = 0 for u 0 ∈ L 2 (D) + \ {0}. It then follows that w : Proof. By Proposition 2.1 and (A2), the mapping Ω ∋ ω → a ω ij (0, ·) ∈ C(D) is (F, B(C(D)))-measurable. As the mapping (F, B(R))-measurable. The measurability of the remaining summands is proved in a similar way. Lemma 3.6. κ + ∈ L 1 ((Ω, F, P)).

Estimates from above
In the present section we consider symmetric problems of the form To emphasize that (4.1) is considered for some (fixed) ω ∈ Ω we write (4.1) ω . We assume (A0) through (A4). The Dirichlet form B ω (·, ·) takes the form: in the Dirichlet and Neumann cases, and It is well known (see, e.g., [3]) that, for fixed ω ∈ Ω, the largest (necessarily real) eigenvalue of the (elliptic) boundary value problem We will denote this quantity (called the principal eigenvalue of (4.1) ω ) by λ princ (ω). Moreover, there exists v ∈ V , v = 1, such that v(x) > 0 for all x ∈ D and the maximum in (4.5) is attained precisely at v and −v. Such a v is called the normalized principal eigenfunction of (4.1) ω .
Proof. Since V is separable, we can take a countable set { u k ∈ V : k ∈ N } such that u k = 1 for each k ∈ N and span Q { u k : k ∈ N } is dense in V . As V \ {0} ∋ u → −B ω (u, u)/ u 2 ∈ R is, for each ω ∈ Ω, continuous, we have that λ princ (ω) = max −B ω (u, u) u 2 : u ∈ V, u = 0 = sup −B ω (u, u) u 2 : u ∈ span Q { u k : k ∈ N }, u = 0 . In view of the above it suffices to prove, repeating reasoning as in the proof of Lemma 3.5, that for each nonzero u ∈ span Q { u k : k ∈ N } the mapping Ω ∋ ω → −B ω (u, u)/ u 2 ∈ R is (F, B(R))-measurable.