Dynamics of reaction-diffusion equations with small Lévy noise

Transcription

1 Dynamics of reaction-diffusion equations with small Lévy noise A. Debussche J. Gairing C. Hein M. Högele P. Imkeller I. Pavlyukevich ENS Cachan, U de los Andes Bogota, HU Berlin, U Jena Banff, January 20, 2015 Debussche, A., Högele, M., Imkeller, P. The dynamics of nonlinear reaction-diffusion equations with small Lévy noise. Lecture Notes in Mathematics Springer: Berlin 2013.

2 DYNAMICS OF REACTION-DIFFUSION EQUATIONS WITH SMALL LÉVY NOISE 1 1. Dynamical model: reaction-diffusion equation with Lévy noise Example: EBM; energy balance: reaction term diffusion term: heat transport from equator to poles; noise: Lévy, includes Gaussian Goal: study meta-stability properties of reaction-diffusion systems with α-stable Lévy noise of small intensity state of the system in example: heat distribution X ε (t, ζ): temperature at time t observed at ζ [0, 1] (interval of global latitudes), subject to noise, intensity ε; idealized version: reaction term U(u) = λ( u4 2 ζ 2, noise: L Lévy process in H = H 1 0([0, 1]) 4 u2 2 ) (λ > 0), diffusion operator dx ε t (ζ) = [ 2 ζ 2Xε t (ζ) U (X ε t (ζ))]dt+εdl t (ζ), X ε t (0) = X ε t (1) = 0, t > 0, X ε 0(ζ) = x(ζ), with x H, ζ [0, 1].

3 DYNAMICS OF REACTION-DIFFUSION EQUATIONS WITH SMALL LÉVY NOISE 2 2. Deterministic part: Chafee-Infante equation For π 2 < λ (kπ) 2, k N, consider du t = 2 ζ 2 u t U (u t ), t [0, T ], u t (0) = u t (1) = 0, t [0, T ], u 0 (ζ) = x(ζ), ζ [0, 1]. For initial values x H 1 0([0, 1]) there is a unique continuous solution in H = H 1 0([0, 1]), e.g. Chafee-Infante 74, Henry 83, Carr, Pego 89 There are two stable states {ϕ +, ϕ } with resp. domains of attraction D + and D and a smooth separatrix S = H \ (D + D ).

4 DYNAMICS OF REACTION-DIFFUSION EQUATIONS WITH SMALL LÉVY NOISE 3 3. α-stable processes in Hilbert space (H,,, ) (L t ) t 0 càdlàg version of symmetric α-stable process in H, Lévy measure ν(dy) = σ(ds) dr r 1+α, r = y, s = y/ y, y 0, σ finite measure on B 1(0) Lévy-Hinčin representation of characteristic function E [ ] ( e i u,l t = exp t H (e i u,y 1 i u, y 1 { y 1} ) ) ν(dy), u H. for ν : B(H) [0, ] satisfying H ν(a) = ν( A), A B(H). ( 1 y 2 ) ν(dy) <, In contrast to (Q-)Brownian motion in H: no scalar decomposition L t λ k L k t e k for ONB (e n ) H, (L k t ) t 0 independent k=1 L is heavy-tailed and need not have any moments.

5 DYNAMICS OF REACTION-DIFFUSION EQUATIONS WITH SMALL LÉVY NOISE 4 4. Stochastic Chafee-Infante equation For ε > 0 and U (y) = λ(y 3 y) dx ε t (ζ) = [ 2 ζ 2Xε t (ζ) U (X ε t (ζ))]dt+εdl t (ζ), X ε t (0) = X ε t (1) = 0, t > 0, X ε 0(ζ) = x(ζ), with x H, ζ [0, 1]. There is a unique càdlàg mild solution in H (Peszat, Zabczyk 07, Marinelli, Röckner 09), satisfying the strong Markov property X ε t = S(t)x t 0 S(t s)f(x ε s) ds + ε t 0 S(t s) dl s. Pb: Let σ ε x = σ ε be exit time of X ε starting in x from D +. Describe the law of σ ε asymptotically as ε 0.

6 DYNAMICS OF REACTION-DIFFUSION EQUATIONS WITH SMALL LÉVY NOISE 5 5. Probabilistic approach of exit times Decompose noise into small ( ε 1 2) and large (> ε 1 2) jumps. L t = ξ ε t + η ε t (η ε t ) t 0 compound Poisson process with jump measure ν ε ν ε (A) = dν ( ) A ε 1 2B1(0) c β ε, A B(H), β ε = ν ( ) ε 1 2 B c 1 (0) ε α 2.

7 DYNAMICS OF REACTION-DIFFUSION EQUATIONS WITH SMALL LÉVY NOISE 6 5. Probabilistic approach of exit times η ε t = T k t W k, L(W k ) = ν ε T k = k τ i L(τ i ) = EXP (β ε ). i=1 (ξ ε t ) t 0 is a martingale with bounded jumps. Denote the stochastic convolution ξ t = t 0 S(t s) dξ ε s, and the small jump part Y ε t = S(t)x t 0 S(t s)f(y ε s ) ds + εξ t.

8 DYNAMICS OF REACTION-DIFFUSION EQUATIONS WITH SMALL LÉVY NOISE 7 6. Deviation in interjump intervals Now T1 0 S(T 1 s) dl t = T1 0 S(T 1 s) dl s + T1 L= ξ (T 1 )+W 1. Decompose Yt ε (x) u t (x) = Rt(x) ε + εξt. Using properties of u prove that Rt(x) ε 0+ if ξt 0+. Control small jumps convolution by P ( ) sup εξt cε γ c ξ T ε 1 2γ+α 2. t [0,T ] Use this to estimate small jumps deviation: there is ϑ > α 2, γ > 0 such that P ( x D + (ε γ ) : ) sup Yt ε (x) u t (x) (1/2)ε γ ε ϑ. t [0,T 1 ]

9 DYNAMICS OF REACTION-DIFFUSION EQUATIONS WITH SMALL LÉVY NOISE 8 7. Relaxation in interjump intervals Thm 1 (Temam 92): All trajectories enter a large ball in H 1 0([0, 1]) in uniform relaxation time T rec. Thm 2 For γ (0, 1) one can define a reduced domain of attraction D + (ε γ ), ε > 0, such that all trajectories starting in D + (ε γ ) enter B(ϕ +, ε γ ) before time R ε = c γ ln ε. one-dimensional case, ϕ + = 0, x B 1 (0): U (u) Mu for u near 0. Hence R ε = x ε γ 1 U (u) du x 1 ε γ Mu du Const + γ ln ε. M

10 DYNAMICS OF REACTION-DIFFUSION EQUATIONS WITH SMALL LÉVY NOISE 9 8. Deviation/relaxation in interjump intervals For small ε > 0 ET 1 = 1/β ε ε α 2 > Trec + c γ ln ε. We can expect to make large jump from close to stable state! Y t ε ε γ R( ε) γ T 1 Thus large jumps εη ε should dominate exit behavior for small ε > 0, exit times of polynomial order. Denote D + 0 (εγ ) = D + (ε γ ) ϕ + the shifted reduced domain. Consider the mass the reshifted domain of attraction λ + (ε) = ν ( (1/ε) ( D + 0 (εγ ) ) c).

11 DYNAMICS OF REACTION-DIFFUSION EQUATIONS WITH SMALL LÉVY NOISE Typical behavior a 0.6 X ε t 0.4 εw εw b εw 1 τ τ τ Expected jump time Eτ k = 1 β ε = α 2 ε α/2 polynomial in ε Relaxation time R ε = O( ln ε ) logarithmic in ε Between big jumps X ε driven by small jumps εξ ε Deviation probability small Typically, X ε jumps from a neighborhood of 0 by εw k at T k. Typically, X ε exits I = [ b, a] by jumping at times T k.

12 DYNAMICS OF REACTION-DIFFUSION EQUATIONS WITH SMALL LÉVY NOISE Exit time law: heuristic proof T k = τ 1 + τ τ k, ET 1 = 1/β ε P(εW 1 / [ b, a]) = 1 β ε ( a ε + b ε ) dy εα = y1+α αβ ε [ 1 a α + 1 ] b α E x σ ε = = ET k P(σ ε = T k ) k=1 k ET 1 P(εW 1 I,..., εw k 1 I, εw k / I) k=1 k ET 1 [1 P(εW 1 / I)] k 1 P(εW 1 / I) k=1 = P(εW 1 / I) β ε 1 P(εW 1 / I) 2 = α ε α [ 1 a α + 1 ] 1 b α

13 DYNAMICS OF REACTION-DIFFUSION EQUATIONS WITH SMALL LÉVY NOISE Asymptotic first exit time from a domain of attraction Thm 3 For any x D + and θ > 1 [ lim E x e θλ+ (ε)σ ε] = 1 ε θ = EXP (1). Cor 1: For all x D + Moreover For small ε > 0 L ( λ + (ε)σ ε x) EXP (1), ε 0 +. lim ε 0+ λ+ (ε)e x [σ ε ] = 1. E x [σ ε ] = 1/λ + (ε) c/ε α. Thus exit times have polynomial growth asymptotically!

14 DYNAMICS OF REACTION-DIFFUSION EQUATIONS WITH SMALL LÉVY NOISE comparison: Gaussian and Lévy dynamics, dim 1 ˆσ ε = inf{t 0 : ˆXε (t) / [ b, a]} σ ε = inf{t 0 : X ε (t) / [ b, a]} h = U( b) < U(a) U(x) h h = U( b) < U(a) U(x) h b ˆX ε (t) = x t 0 U ( ˆX ε (t)) ds + εw (t) Thm 4 (Freidlin-Wentzell): P x (e (2h δ)/ε2 < ˆσ ε < e (2h+δ)/ε2 ) 1 Kramers law ( 40, Williams, Bovier et al.): E xˆσ ε ε π U ( b) U (0) e2h/ε2 Exponential law (Day, Bovier et al.) P x ( ˆσε E xˆσ ε > u) exp ( u) a x b X ε (t) = x t 0 U (X ε (s )) ds + εl(t) Thm 5 P x ( 1 ε α δ < σ ε < 1 ε α+δ ) 1 E x σ ε 1 ε α ( R\[ b,a] a x ) dy 1 y 1+α P x ( σε E x σ ε > u) exp ( u)

15 DYNAMICS OF REACTION-DIFFUSION EQUATIONS WITH SMALL LÉVY NOISE Paleo-climatic time series temperature indicators: 18 O, 16 O, methane, calcium etc. GRIP ice core data: 20 abrupt changes in climate of Greenland during last ice age ( to y) (D/O events) T Age (thousands of years before present) rapid warming by 5-10 C within one decade subsequent slower cooling within a few centuries fast return to stable cold ground state simulations: Ganopolsky/Rahmstorf, Potsdam Institute for Climate Impact Research

16 DYNAMICS OF REACTION-DIFFUSION EQUATIONS WITH SMALL LÉVY NOISE Dansgaard-Oeschger events. Statistical analysis Calcium signal from GRIP: about samples for y waiting times between D/O events: multiples of 1470 years. What triggers the transitions? modeling by Langevin equation ( reaction): dx(t) = U (t, X(t))dt+NOISE U multi well potential, depicts energy balance P. Ditlevsen (Geophys. Res. Lett. 1999): power spectrum analysis of time series: NOISE contains strong α-stable component with α 1.75.

17 DYNAMICS OF REACTION-DIFFUSION EQUATIONS WITH SMALL LÉVY NOISE p-variation as test statistic Which model of noise fits best with time series: estimate, test parameter Ditlevsen s analysis: power spectrum of residua of time series Problem: Stationarity? Aim: better test statistics than peaks of power spectrum. Model assumption: with some U interpret data as X ε (t) = x t 0 U (X ε (s ))ds +εl(t)= Y ε (t)+l ε (t) L Lévy process containing α-stable component with unknown α, Y ε of bounded variation; estimate, test α Idea: p-variation characteristic for fluctuation behavior of noise processes. V p,n t (X) = [nt] i=1 X( i n ) X(i 1 n ) p, V p t = lim V p,n t n

18 DYNAMICS OF REACTION-DIFFUSION EQUATIONS WITH SMALL LÉVY NOISE α-stable Lévy Processes L Lévy process with characteristics (d, γ, ν) iff E(e iul(t) ) = e t( 1 2 du2 +iγu+ R [eiuy 1 iuy1 { y 1} ]ν(dy)), u R, t 0, ν measure on Borel sets in R with ν({0}) = 0, R [ y 2 1]ν(dy) <. L α-stable symmetric Lévy process if E(e iul(t) ) = e c(α)t u α, ν(dy) = 1 y α+1dy, u, y R α = 0.75 α = 1.75

19 DYNAMICS OF REACTION-DIFFUSION EQUATIONS WITH SMALL LÉVY NOISE p-variation and the Blumenthal-Getoor Index L α-stable process with jump measure ν; then p-variation identified by Blumenthal-Getoor index β L = inf{s 0 : { y 1} y s ν(dy) < } γ L = inf{p > 0 : V p 1 (L) < } Thm 1 L symmetric α-stable. Then γ L = β L = α. Problem: How to read γ L = α off the sequence (V p,n t (L)) n N? Calls for results about the asymptotic behavior of the sequence.

20 DYNAMICS OF REACTION-DIFFUSION EQUATIONS WITH SMALL LÉVY NOISE The case α = 2: Brownian motion For n N V p,n 1 (W ) consists of n independent increments and Thm 2 (LLN type) E(V p,n 1 (W )) = n 1 p 2 E( W (1) p ) n 1+p 2 V p,n t (W ) te( W (1) p ) in probability, Y of bounded variation. Then also n 1+p 2 V p,n t (W + Y ) te( W (1) p ) in probability. Thm 3 (CLT type) (n 1 2 [n 1+ p 2 V p,n t (W ) te( W (1) p )]) t 0 = (n 1 2 +p 2 V p,n t (W ) n 1 2 te( W (1) p )) t 0 ((var( W (1) p )) 1 2 W (t))t 0 weakly with respect to the Skorokhod metric, and an independent Brownian motion W.

21 DYNAMICS OF REACTION-DIFFUSION EQUATIONS WITH SMALL LÉVY NOISE The case α < 2 (Lit: Corcuera, Nualart, Wörner 07; case p < α for LLN type, p < α 2 type) for CLT Problem: p < α 2 < 1 not satisfactory for paleo-climatic data! Beyond α 2 no CLT type result available, no asymptotic normality, but asymptotically of different type. Thm 4 (LT type) L α-stable with α ]0, 2[. Then (V p,n t (L) Bt n (α, p)) t 0 L weakly with respect to the Skorokhod metric, and an independent α p -stable process L. Here B n t (α, p) = n 1 αte( L(1) p p α ), 2 < p < α, nt 2 E(sin((nt) 1 L(1) p )), p = α, 0, α < p. Same result with L + Y instead of L if Y is of finite p-variation and α 2 < p < 1 or p > α.

22 DYNAMICS OF REACTION-DIFFUSION EQUATIONS WITH SMALL LÉVY NOISE Test for α with real and simulated data Thm 5: law of V 2α,n (X) converges to 1 2-stable law if data of time series X have α-stable residuals Kolmogorov-Smirnov statistics: distance between empirical law of V 2p,n (X) and 1 2-stable law, as a function of p; minimum of curve: right α simulated time series of a 0.6-stable Levy process, n = real time series from the Greenland ice, n = 200

23 DYNAMICS OF REACTION-DIFFUSION EQUATIONS WITH SMALL LÉVY NOISE Gaussian and Lévy dynamics: one dimension W Wiener process L symmetric α-stable Lévy process Tail behavior P ( W (1) x) exp( cx 2 ) P ( L(1) x) c 1 x α, x ˆX ε (t) = x t 0 U ( ˆX ε (t)) ds + εw (t) X ε (t) = x t 0 U (X ε (s )) ds + εl(t) ˆτ(ε) = inf{t 0 : ˆXε (t) / [ b, a]} τ(ε) = inf{t 0 : X ε (t) / [ b, a]} h = U( b) < U(a) U(x) h h = U( b) < U(a) U(x) h b E xˆτ(ε) a ε π U ( b) U (0) exp(2h ε 2 ) x b E x τ(ε) 1 ε α ( R\[ b,a] a x ) dy 1 y 1+α

24 DYNAMICS OF REACTION-DIFFUSION EQUATIONS WITH SMALL LÉVY NOISE Gaussian and Lévy dynamics: one dimension Conjecture: make tails of Lévy process exponentially light to recover Gaussian exit behavior. Tail behavior L Lévy process with jump measure having tails P ( L(1) x) exp( cx α ), x sub-exponential tails: α < 1 super-exponential tails: α > 1 Consider X ε (t) = x t 0 U (X ε (s )) ds+εl(t) τ(ε) = inf{t 0 : X ε (t) / [ 1, 1]} Conjecture: E x (τ(ε)) ε 0 exp( c ε2) as α 2.

25 DYNAMICS OF REACTION-DIFFUSION EQUATIONS WITH SMALL LÉVY NOISE The phase transition at α = 1 Thm 10 [sub-exponential tails] For δ > 0 there is ε 0 > 0 such that for all 0 < ε ε 0, t 0: (1 δ) exp( Cε 1 δ t) sup P x (τ(ε) > t) exp( 1 2 C εt), x 1 with C ε := 2ν([ 1 ε, )). Hence for x < 1 lim ε 0 εα ln E x τ(ε) = 1. Thm 11 [super-exponential tails] q ε ε-quantile of jump measure ν. Then for δ > 0 there is ε 0 > 0 such that for all 0 < ε ε 0, t 0: (1 δ) exp( Dε 1 δ t) sup x 1 P x (τ(ε) > t) (1 + δ) exp( Dε 1+δ t), where D ε = exp( d α ln ε εq ε ) and d α = α(α 1) 1 α 1. Hence for x < 1 d 1 α lim ε ln ε α 1 1 ln E x τ(ε) = 1. ε 0

26 DYNAMICS OF REACTION-DIFFUSION EQUATIONS WITH SMALL LÉVY NOISE α-stable towards Gaussian. The phase transition at α = 1 Comparison of regimes for mean exit time Power tails jump tails ν([x, )) = x r, x 1 for some r > 0. Then 2 lim ε 0 ε r E x τ(ε) = 1. Sub-exponential tails jump tails ν([u, )) = exp( u α ), u 1, α < 1. Then lim ε 0 εα ln E x τ(ε) = 1. Super-exponential tails jump tails ν([u, )) = exp( u α ), u 1, α > 1. Then d 1 α lim ε ln ε α 1 1 ln E x τ(ε) = 1. ε 0 Gaussian diffusion no jumps, L one-dimensional Brownian motion. Then 1 2 (U( 1) U(1)) 1 lim ε 0 ε 2 ln E x τ(ε) = 1.

27 DYNAMICS OF REACTION-DIFFUSION EQUATIONS WITH SMALL LÉVY NOISE Heuristics of exits: climbing versus jumping The Brownian case LD theory: diffusion has to climb potential in order to exit at lowest saddle point The power tail case for ε > 0 split L = η ε + ξ ε, compound Poisson pure jump part η ε with jumps of height larger than 1 ε ; small jump and Gaussian part ξ ε with jumps not exceeding 1 ε ; exit asymptotically due to one big jump, as shown in first talk The case of exponential tails for ε > 0 split L = η ε + ξ ε, compound Poisson pure jump part η ε with jumps of height larger than g ε ; small jump and Gaussian part ξ ε with jumps not exceeding this bound; choose g ε individually according to sub- and super-exponential tails show that exit before time T while not returning to an interval around stable fixed point 0 of radius δ > 0 requires that either increments of ξ ε exceed certain bounds (for which probability is small enough), or sum of large jumps before time T exceeds bound 1 δ in any case large jumps responsible for exits

28 DYNAMICS OF REACTION-DIFFUSION EQUATIONS WITH SMALL LÉVY NOISE Heuristics of exits: climbing versus jumping N T random number of large jumps before time T W i jump n o i, i N. N T Poisson with expectation β ε T, where β ε = ν([ g ε, g ε ] c ) = 2 exp( x α ) For n fixed, probability that sum of large jumps exceeds bound 1 δ estimated by P (N T > n) + n k P (N T = k)p ( εw i > 1 δ) k=1 i=1 Idea for estimation: for n N P (N T > n) (1 + δ) exp( n ln n) (Stirling s formula) choose n = n ε suitably

29 DYNAMICS OF REACTION-DIFFUSION EQUATIONS WITH SMALL LÉVY NOISE Heuristics of exits: climbing versus jumping essential term to estimate for 1 k n ε P ( k εw i > 1 δ) i=1 law of i.i.d. random variables ( W i ) i N : βε 1 2ν [gε, [ hence P ( k i=1 εw i > 1 δ) β k ε = β k ε k exp( inf{ x α i : i=1 k exp( inf{ x α i : i=1 k i=1 k i=1 x i 1 δ ε, x i [g ε, [}) x i = 1 δ ε, x i [g ε, [}) minimization problem in the exponent of this estimate causes phase transition

30 DYNAMICS OF REACTION-DIFFUSION EQUATIONS WITH SMALL LÉVY NOISE Heuristics of exits: climbing versus jumping By suitable choice of g ε : lower boundary for x i in inf can be taken 0. sub-exponential tails k inf{ x α i : i=1 k x i = 1, x i 0} = 1 i=1 The minimum is taken on the boundary of the simplex, and x i = 1 n, 1 i n, corresponds to maximum of the function (x 1,, x n ) n i=1 x α i

31 DYNAMICS OF REACTION-DIFFUSION EQUATIONS WITH SMALL LÉVY NOISE Heuristics of exits: climbing versus jumping Super-exponential tails k inf{ x α i : i=1 k x i = 1, x i 0} = n( 1 n )α i=1 The minimum is taken for x i = 1 n, 1 i n, the unique local minimum of the function n (x 1,, x n ) Bifurcation in the asymptotic behavior: phase transition due to switch from concavity to convexity at α = 1 of i=1 x x α, x 0, x α i big jumps of the Lévy process govern asymptotic behavior α < 1: biggest jump responsible for exit α > 1: cumulative action of several large jumps responsible for exit

32 DYNAMICS OF REACTION-DIFFUSION EQUATIONS WITH SMALL LÉVY NOISE Paleoclimatic time series Lisiecki, Raymo, Paleoceanography 2005 concentration variation of 18 O to 16 O taken from marine sediments at 57 globally distributed sites (e.g. Brunhes, Matuyama, Jaramillo): global average temperature time series basic feature: from 0 to -1 Myr periodicity y from -1 Myr to -1.8 Myr periodicity y Milankovich cycles: axial tilt ( y) and eccentricity ( y)