Large deviations in presence of small noise for delay differential equations at an instability
aa r X i v : . [ m a t h . P R ] J un LARGE DEVIATIONS IN PRESENCE OF SMALL NOISE FOR DELAYDIFFERENTIAL EQUATIONS AT AN INSTABILITY
NISHANTH LINGALA
Abstract.
We consider delay differential equations (DDE) that are on the verge of an insta-bility, i.e. the characteristic equation for the linearized equation has one root as zero and allother roots have negative real parts. In presence of small mean-zero noise, we study the largedeviations from the corresponding deterministic system. Using spectral theory for DDE it iseasy to see that, the projection on to the one dimensional space corresponding to the zero rootis exponentially equivalent with the original process. For the one-dimensional process we makethe observation that the results of Freidlin-Wentzell apply. Introduction
We consider R n valued processes governed by delay differential equations (DDE) of the form˙ x ( t ) = L (Π t x ) + εG (Π t x ) + εF (Π t x ) σ ( ξ t ) , (1)where • Π t is the segment extractor defined by (Π t f )( θ ) = f ( t + θ ) for θ ∈ [ − r,
0] where r > t : C ([ − r, ∞ ); R n ) → C := C ([ − r, R n ) • L , G, F : C → R n , with L being linear, and G, F being bounded with bounded deriva-tives • σ is a bounded mean zero R -valued function of the Markov noise ξ • ε ≪ µ : [ − r, → R n × n , continuousfrom the left on the interval ( − r,
0) and normalized with µ (0) = 0 n × n , such that L η = Z [ − r, dµ ( θ ) η ( θ ) , ∀ η ∈ C . (2)This is not a restriction: every continuous linear operator L has such a representation.We make the following assumption on L to reflect an instability scenario: Assumption 1.1.
Define ∆( λ ) = λI n × n − Z [ − r, dµ ( θ ) e λθ , where I is the identity matrix. The characteristic equation det (∆( λ )) = 0 , λ ∈ C (3) has one solution as zero and all other solutions have negative real parts. Roughly speaking, after the initial transients have decayed, significant changes in x occuron times of order O (1 /ε ) due to the effect of G . Since σ is mean-zero, large deviations fromthe corresponding deterministic system are rare on times of order O (1 /ε ). We obtain the ratefunction governing the large deviations. Spectral theory for DDE
Under assumption 1.1 the space C can be split as C = P ⊕ Q such that for the unperturbedsystem ˙ x = L (Π t x ), the projection of Π t x onto P does not change at all, and the norm ofthe projection of Π t x onto Q decays exponentially fast. When the perturbations are presentas in (1), the P projection evolves slowly and the Q projection stays small. The P space isone-dimensional. We find a one dimensional evolution equation which is exponential equivalentto the P projection and for which the results from chapter 7 of [3] applies yielding the largedeviations rate function.Here we show, given an η ∈ C , how to find the projection onto the space P . For details, seechapter 7 of [1] and chapter 4 of [2].We use R n ∗ to distinguish the set of 1 × n vectors, from R n which is the set of n × h· , ·i : C ([0 , r ]; R n ∗ ) × C ([ − r, , R n ) → R , given by(4) h ψ, η i := ψ (0) η (0) − Z − r Z θ ψ ( s − θ ) dµ ( θ ) η ( s ) ds. Choose d such that ∆(0) d = 0 n × and d such that d ∆(0) = 0 × n . Define Φ ∈ C by theconstant Φ( • ) = d and Ψ ∈ C ([0 , r ]; R n ∗ ) by Ψ( • ) = cd where the constant c is choosen so that h Ψ , Φ i = 1 for the bilinear form in (4). The space C can be split as C = P ⊕ Q where P isthe space spanned by the constant function Φ. The projection operator is π : C → P given by π ( η ) = Φ h Ψ , η i . The space Q can be written as { η ∈ C : π ( η ) = 0 } . We find use for ˆΨ def = Ψ(0).The solution to the unperturbed system˙ x ( t ) = L (Π t x )(5)can be written as Π t x = π Π t x + ( I − π )Π t x = Φ z t + y t where z t = h Ψ , Π t x i and y t = Π t x − Φ z ( t ). Note that z ∈ R is a scalar, and Φ z t ∈ P and y t ∈ Q . It can be shown that for the unperturbed system (5), ˙ z = 0, i.e., z is a constant in time.Further, it can be shown that || y t || decreases to zero exponentially fast (because the dynamicson Q is governed by eigenvalues with negative real parts). Let { T ( t ) } t ≥ , T ( t ) : C → C be thesemigroup generated by the DDE (5), i.e. for η ∈ C , T ( t ) η is the solution to (5) with the initialcondition η . Then, for η ∈ P , T ( t ) η = η and, ∃ K, κ > || T ( t ) η || ≤ Ke − κt || η || , ∀ η ∈ Q. (6)Solution to the perturbed equation (1) can be written in terms of the semigroup T . For thispurpose, let { } : [ − r, → R n × n be defined as { } ( θ ) = 0 n × n for θ < { } (0) = I n × n .The solution to (1) with initial condition Π x = η can be written asΠ t x ( θ ) = T ( t ) η ( θ ) + Z t T ( t − s ) { } ( θ ) (cid:18) G (Π s x ) + F (Π s x ) σ ( ξ s ) (cid:19) ds, θ ∈ [ − r, . The j th column of T ( t − s ) { } is the solution of (5) with the initial condition as the j th columnof { } . Though ( { } ) .j does not belong to C , the bilinear form (4) still makes sense and wehave π (( { } ) .j ) = Φ ˆΨ( { } ) .j . We still have the exponential decay || T ( t )( I − π )( { } ) .j || ≤ Ke − κt || ( I − π )( { } ) .j || = Ce − κt .Using the fact that T commutes with π we have the equations dz t = ε ˆΨ G (Φ z t + y t ) dt + ε ˆΨ F (Φ z t + y t ) σ ( ξ t ) dt, (7) y t = T ( t ) y + ε Z t T ( t − s )( I − π ) { } (cid:18) G (Φ z s + y s ) + F (Φ z s + y s ) σ ( ξ s ) (cid:19) ds. DP FOR DDE AT AN INSTABILITY 3
Using the exponential decay of || T ( t − s )( I − π ) { } || and the boundedness of F, G, σ we havethat || y t − T ( t ) y || < Cε (8)for some C >
0. 3.
An exponentially equivalent process
Let the scalar process z be defined by d z t = ε ˆΨ G (Φ z t ) dt + ε ˆΨ F (Φ z t ) σ ( ξ t ) dt, z = z . (9)Using the bounded derivatives of F, G and boundedness of σ , and then using (8) and the expo-nential decay (6) we have | z t − z t | ≤ Cε Z t ( | z s − z s | + || y s || ) ds ≤ Cε Z t ( | z s − z s | + || T ( s ) y || + Cε ) ds ≤ Cε t + Cε (1 − e − κt ) + Cε Z t | z s − z s | ds. Using Gronwall inequality we have that ∃ C > | z t − z t | ≤ Cε, t ∈ [0 , T /ε ](10)for some fixed T, ε > ε > ε . It is easy to see from (9) that significant changes for z happens on time of order O (1 /ε ), and because σ is mean-zero function, significant deviationsfrom the deterministic system d z t = ε ˆΨ G (Φ z t ) dt would be rare on times of order O (1 /ε ).By (10) analogous statement holds for z t . So we define z εt = z t/ε and study the rate functiongoverning the large deviations of z ε from the correpsonding determinstic system for t ∈ [0 , T ].Define z εt = z t/ε . Then, by (10), z ε and z ε are exponentially equivalent, and so the rate functionfor z ε and z ε are same.Note that z εt is governed by d z εt = ˆΨ G (Φ z εt ) dt + ˆΨ F (Φ z εt ) σ ( ξ εt ) dt, z ε = z , (11)where ξ εt = ξ t/ε . The results of Freidlin-Wentzell (chapter 7 of [3]) apply for the large deviationsof z εt from the deterministic system ˙ z t = ˆΨ G (Φ z t ).4. Large deviations of z εt Theorem 7.4.1 in [3] gives the following result.
Theorem 4.1.
Let the process z ε be governed by (9) . Assume the noise ξ is homogenous markovprocess such that for any z , α ∈ R lim T →∞ T ln E ξ exp (cid:18) α Z T ˆΨ F (Φ z ) σ ( ξ s ) ds (cid:19) = H F ( z , α ) uniformly in the initial condition ξ and the function H F be differentiable with respect to α . Let H ( z , α ) = α ˆΨ G (Φ z ) + H F ( z , α ) . Let L ( z , β ) := sup α [ αβ − H ( z , α )] . On C ([0 , T ]; R ) introducethe functional S T ( ϕ ) = (R T L ( ϕ s , ˙ ϕ s ) ds, ϕ is absolutely continuous ∞ otherwise . The functional S T is the normalized action functional in C ([0 , T ]; R ) for the family of processes z ε as ε → , the normalizing coefficient being /ε . NISHANTH LINGALA
Remark 4.1.
Writing x ε ( t ) = x ( t/ε ) we have x ε ( t ) = Φ(0) z εt + y εt (0) and so | x ε ( t ) − Φ(0) z εt − T ( t/ε ) y (0) | ≤ C | z εt − z εt | + C | y εt − T ( t/ε ) y | ≤ Cε.
Recalling that || T ( t/ε ) y || ≤ Ke − κt/ε || y || ; if || y || is small enough, we can approximate the exitrates of x ε by exit rates of Φ z ε . Remark 4.2.
For the case of noise ξ being N -state continuous time Markov chain, theorem7.4.2 of [3] shows that H F ( z , α ) is the largest eigenvalue of the N × N matrix Q α, z defined by ( Q α, z − Q ) ij = δ ij σ i α ˆΨ F (Φ z ) where Q is the generator of the Markov chain and σ i is the valueof σ for the i th state. Remark 4.3.
Let ξ be a two-state symmetric markov chain with switching rate g/ , i.e. lim t ↓ t P → ( t ) = g/ t ↓ t P → ( t )(12) where P i → j ( t ) is the probability of transition from state i to state j in time t . Let σ ( ξ = 1) = − σ ( ξ = 2) = σ . In this case, the functional S T can be explicitly evaluated as S T ( ϕ ) = Z T g − vuut − ˙ ϕ s − ˆΨ G (Φ ϕ s ) σ ˆΨ F (Φ ϕ s ) ! ds for ϕ absolutely continuous with | ˙ ϕ s − ˆΨ G (Φ ϕ s ) | ≤ | σ ˆΨ F (Φ ϕ s ) | for s ∈ [0 , T ] and ∞ for allother ϕ . The following function would be useful in studying exit related problems: V ( t, a, b ) = inf ϕ = a,ϕ t = b S t ( ϕ ) . The solution can be written as V ( t, a, b ) = inf ˙ ϕ s = ˆΨ G (Φ ϕ s )+ σ ˆΨ F (Φ ϕ s ) u s , | u s |≤ , ϕ = a, ϕ t = b Z t g (cid:16) − p − u s (cid:17) ds. Linear delay equations with fast markov perturbations
In this section we make an independent observation regarding processes of the form˙ x ε ( t ) = L (Π t x ε ) + σ ( ξ εt ) , Π x ε = η ∈ C (13)where ξ εt = ξ t/ε with ξ being a homogenous markov process and σ being a mean-zero R n -valuedfunction of the noise. Assume that for any α ∈ R n lim T →∞ T ln E ξ exp (cid:18)Z T α ∗ σ ( ξ s ) ds (cid:19) = H ( α )uniformly in the initial condition ξ and the function H be differentiable with respect to α . Let L ( β ) := sup α [ α ∗ β − H ( α )]. On C ([0 , T ]; R n ) introduce the functional S σ T ( ϕ ) = (R T L ( ˙ ϕ s ) ds, ϕ is absolutely continuous ∞ otherwise . The functional S σ T is the normalized action functional in C ([0 , T ]; R n ) for the family of processes R · σ ( ξ εs ) ds as ε →
0, the normalizing coefficient being 1 /ε .Define the map B η : C ([0 , T ]; R n ) → C ([0 , T ]; R n ) by B η ψ = v where v is the solution of v ( t ) = η (0) + Z t L (Π s v ) ds + ψ ( t ) . with the understanding that Π v = η . More explicit representation of v can be given by thevariation-of-constants formula. The map B η has inverse given by ( B − η v )( t ) = v ( t ) − η (0) − DP FOR DDE AT AN INSTABILITY 5 R t L (Π s v ) ds . It can be shown using Gronwall inequality that B η is Lipschitz. By contractionprinciple we have that the action functional for x ε is given by S T ( ϕ ) = (R T L ( ˙ ϕ s − L (Π s ϕ )) ds, ϕ is absolutely continuous ∞ otherwise , with the understanding that Π ϕ is the initial condition.Consider the case of x being R -valued, and ξ being a two-state markov chain as in the remark4.3. The following function would be useful in studying exit related problems: V ( t, η, b ) = inf Π ϕ = η, ϕ ( t )= b S t ( ϕ ) . The solution can be written as V ( t, η, b ) = inf ˙ ϕ s = L (Π s ϕ )+ σ u s , | u s |≤ , Π ϕ = η, ϕ ( t )= b Z t g (cid:16) − p − u s (cid:17) ds. Let f : [ − r, ∞ ) → R be defined by f ( t ) = 0 for t < f (0) = 1, and for t > f satisfies˙ f ( t ) = L (Π t f ). Let { T ( t ) } t ≥ be the solution semigroup as defined in section 2. Then thesolution to ˙ ϕ s = L (Π s ϕ ) + σ u s , Π ϕ = η, can be represented using the variation-of-constants formula as ϕ ( t ) = T ( t ) η (0) + Z t f ( t − s ) σ u s ds. Hence we have V ( t, η, b ) = inf R t f ( t − s ) σ u s ds = b − T ( t ) η (0) | u s |≤ Z t g (cid:16) − p − u s (cid:17) ds. The RHS above can be computed explicity using calculus of variations. We have for the opti-mality, u s = − ρf ( t − s ) √ ρ f ( t − s ) with the Lagrange multiplier ρ obtained using R t f ( t − s ) σ u s ds = b − T ( t ) η (0).Note that ε R · σ ( ξ εs ) ds converges weakly as ε → σ √ g W · where W is a Wiener process.However, the large deviations principle for ˙ x ε ( t ) = L (Π t x ε ) + ε ( ε σ ( ξ εt )) is different from thelarge deviations principle for dx ε ( t ) = L (Π t x ε ) dt + εσ √ gdW t .Large deviations for DDE with noise as Wiener process is considered in [4]. References [1] J.K Hale, S.M Verduyn Lunel.
Introduction to functional differential equations . Springer Verlag, 1993.[2] O Diekmann, S.A van Gils, S.M Verduyn Lunel, H.O Walther.
Delay equations . Springer Verlag, 1995.[3] M.I Freidlin, A.D Wentzell.
Random perturbations of dynamical systems . Springer, 3 rd ed., 2012.[4] S-E.A. Mohammed, T. Zhang. Large deviations for stochastic systems with memory. Discrete and ContinuousDynamical Systems-B ,6