l approche des grandes déviations

Transcription

1 Épuisement de ressources partagées en environnement Markovien : l approche des grandes déviations Joint work with François Delarue (Paris Diderot) et René Schott (Nancy 1) Journées de Probabilité à Lille 1-5 Septembre 2008

2 François Delarue et René Schott

3 Algorithm 1 Resource sharing problem between d customers Resource money, memory Maximum need for a customer m Total amount of available resource L m < d m, L > 0 2 Modelization by a process R n = (R 1 n,...,rd n ) 3 R i n = resource allocated to the customer i {1,..., d} at time n 4 Questions of interest Dynamics for (R n ) n 0? Behavior for m large? Deadlock = breakdown of the system

4 Modelization 1 Markovian dynamics for (R n ) n 0

5 Modelization 1 Markovian dynamics for (R n ) n 0 Constant transition probabilities (Ellis 77, Flajolet 86, Louchard-Schott 91) Influence of the spatial position (Maier 91) Probabilities governed by a dynamical system (Guillotin-Schott 02) q(t n x, ±e 1 ),..., q(t n x, ±e d ) 2 CDS: Transitions governed by a Markovian environment q(ξ n, ±e 1 ),..., q(ξ n, ±e d ) 3 Properties of the deadlock time? Behavior for m large: scaling properties? Expectation? Law? (After scaling) 4 Influence of the environment?

6 Typical Behavior (Constant Transition Probabilities) 1 If the drift is directed to the absorption area 2 Absorption time m due to typical paths

7 Typical Behavior (Constant Transition Probabilities) 1 If the drift is null ( diffusive case ) 2 Absorption time m 2 due to normal fluctuations

8 Typical Behavior (Constant Transition Probabilities) 1 If the drift is directed to the origin ( stable case ) 2 Absorption time exp(cm) due to large deviations

9 Part I- Framework and assumptions Environment: Markov chain (ξ n ) n 0 Finite state space: transition matrix P Irreducible chain: invariant probab. measure µ (H1) Transition probabilities of the non-reflected walk (S n ) n 0 p(s n /m,ξ n, ±e j ), j {1,...,d} p(y,i, ±e j ) Lipschitz in y Ellipticity: p(y,i, ±e j ) > 0 (H2) Transition probabilities of the reflected walk (R n ) n 1 Reflection q(r n /m,ξ n, ±e j ), j {1,...,d} { q(y,i, ej ) = 0 for y j = 0 q(y,i,e j ) = 0 for y j = 1

10 Part I- Framework and assumptions Environment: Markov chain (ξ n ) n 0 Finite state space: transition matrix P Irreducible chain: invariant probab. measure µ (H1) Transition probabilities of the non-reflected walk (S n ) n 0 p(s n /m,ξ n, ±e j ), j {1,...,d} p(y,i, ±e j ) Lipschitz in y Ellipticity: p(y,i, ±e j ) > 0 (H2) Transition probabilities of the reflected walk (R n ) n 1 q(y,i, ±e j ) = p(y,i, ±e j ) for 0 < y j < 1 q(y,i,e j ) = p(y,i,e j ) + p(y,i, e j ) for y j = 0 q(y,i, e j ) = p(y,i,e j ) + p(y,i, e j ) for y j = 1

11 Averaging 1 Local drift f (y,i) = [p(y,i,u)u], u=±e j y = location of the walk, i = state of the environment 2 Averaged drift f (y) = µ(i)f (y,i) i E 3 Decomposition of the non-reflected walk n 1 S n = S 0 + f (m 1 S k,ξ k ) + Mart. k=0

12 Averaging 1 Local drift 2 Averaged drift f (y,i) = [p(y,i,u)u], u=±e j f (y) = µ(i)f (y,i) i E 3 Decomposition of the non-reflected walk S (m) t mt 1 = m 1 (m) S mt = S 0 +m 1 f (m 1 S k,ξ k )+m 1 Mart. k=0

13 Averaging 1 Local drift 2 Averaged drift f (y,i) = [p(y,i,u)u], u=±e j f (y) = µ(i)f (y,i) i E 3 Decomposition of the non-reflected walk S (m) t = m 1 S mt t (m) S f ( S s (m) )ds + m 1 Mart.

14 Averaging 1 Local drift 2 Averaged drift f (y,i) = [p(y,i,u)u], u=±e j f (y) = µ(i)f (y,i) i E 3 Decomposition of the non-reflected walk Hyperbolic Scaling ( S (m) t = m 1 S mt ) t 0 proba (s t ) t 0 ODE ds t dt = f (s t )

15 Hyperbolic Scaling 1 Non-reflected ( S (m) t = m 1 S mt ) t 0 proba (s t ) t 0 ODE ds t dt = f (s t ) 2 Reflected ( R (m) t Reflected DE 3 (k t ) t 0 pushing process dk i t dt = = m 1 R mt ) t 0 proba (r t ) t 0 0 if 0 < r i t < 1 0 if r i t = 0 0 if r i t = 1 dr t dt = f (r t ) dk t dt

16 Reflection: formal definition 1 Skorohod problem: w : t [0,+ ) w t R d (w 0 [0,1] d ) continuous,!(x,k ) : t [0,+ ) (x t,k t ) [0,1] d R d with k locally bounded variation, such that k t = t 0 w t = x t + k t, n s d k s, k t = t 0 1 {xs [0,1] d }d k s, with n s unit outward normals to [0,1] d at x s. Let Ψ Sk the Skorohod map w x.

17 Reflection: formal definition 1 Skorohod problem: w : t [0,+ ) w t R d (w 0 [0,1] d ) continuous,!(x,k ) : t [0,+ ) (x t,k t ) [0,1] d R d with k locally bounded variation, such that k t = t 0 w t = x t + k t, n s d k s, k t = t 0 1 {xs [0,1] d }d k s, with n s unit outward normals to [0,1] d at x s. Let Ψ Sk the Skorohod map w x. 2 Reflected DE dr t dt = f (r t ) dk t dt = unique solution of r = Ψ Sk (r ) f (r t )dt

18 Part II- Diffusive case ( f = 0) CDS 07 1 Law of Large Numbers for Reflected walk ( R (m) t = m 1 R mt ) t 0 proba (r t ) t 0 dr t Ref.DE dt = f (r t ) dk t dt 2 Assume f = 0. Then, the limit is trivial! Ref. DE dr t dt = 0. Theorem [CDS, Rand.Struct.Algo. 07] Diffusive scaling: (ˆR (m) t = m 1 R m 2 t ) t 0 law (r t ) t 0 Stoch. Ref. DE dr t = b(r t )dt + σ(r t )db t dk t (k t ) t 0 of bounded variation

19 Deadlock Time under Diffusive case ( f = 0) 1 Prove convergence to dr t = b(r t )dt + σ(r t )db t dk t by the corrector method: 2 v solution of the Poisson equation (P I) v(y, ) = f (y, ) }{{} Generator 3 Itô s like formula kill f m 1 S m 2 t + m 1 v(ξ m 2 t,s m 2 t /m) m2 t 1 = Init.Cond. + m 2 k=0 Corollary (Diffusive case) ( f = 0) b(ξ k,s k /m) + m 1 Mart. m 2 t 1 T (m) L = inf{n 0, R n 1 Lm}, T L = inf{t 0, r t 1 L} m 2 T (m) L law T L

20 Part III- Deadlock Time in Stable Case 1 Hyperbolic scaling: ( R (m) t = m 1 R mt ) t 0 proba (r t ) t 0 Ref. DE 2 Stable case 3 Dimension 2 dr t dt = f (r t ) dk t dt. f i (x) < 0, x i (0,1] (H3) 4 Behaviour of RDE r t 0 as t +

21 Part III- Deadlock Time in Stable Case 1 Hyperbolic scaling: ( R (m) t = m 1 R mt ) t 0 proba (r t ) t 0 Ref. DE 2 Stable case dr t dt = f (r t ) dk t dt. f i (x) < 0, x i (0,1] (H3) 3 Conclusion T L = inf { t 0, {T (m) L d (r t ) i L } = + i=1 < C} is a rare event for m large

22 Freidlin and Wentzell Theory 1 Non-reflected walk (S n ) n 0 p(s n /m,ξ n, ±e j ), j {1,...,d} 2 Large Deviation Principle for ( S (m) t = m 1 S mt ) 0 t T lim inf m m 1 ln P { ( S (m) t ) 0 t T A } inf φ A,φ 0 =x 0 I(φ) lim supm 1 ln P { ( S (m) t ) 0 t T A } inf I(φ) m φ Ā,φ 0=x 0 with action functional I(φ) = T 0 L(φ s, φ s )ds LDP for averaging: Freidlin-Wenzell 84, Gulinsky-Veretennikov 93, Feng-Kurtz 05 LDP for reflected processes: Dupuis 88 if φ abs. cont.

23 Freidlin and Wentzell Theory 1 Non-reflected walk (S n ) n 0 p(s n /m,ξ n, ±e j ), j {1,...,d} 2 Large Deviation Principle for ( S (m) t = m 1 S mt ) 0 t T I(φ) = T 0 L(φ s, φ s )ds if φ abs. cont. 3 Hamiltonian [ n H(x,α) = lim n 1 [ ln E exp( α,v )p(x,ξ k,v) ] ] n 0 k=1 v=±e j }{{} Laplace trans. of jumps for freezed position [ ] 4 Lagrangian L(x,v) = sup α,v H(x,α) α

24 Freidlin and Wentzell Theory 1 Non-reflected walk (S n ) n 0 p(s n /m,ξ n, ±e j ), j {1,...,d} 2 Large Deviation Principle for ( S (m) t = m 1 S mt ) 0 t T I(φ) = T 0 L(φ s, φ s )ds if φ abs. cont. 3 Hamiltonian ] H(x, α) = Perron Froebenius [P i,j exp( α,v )p(x,i,v) v=±e k [ ] 4 Lagrangian L(x,v) = sup α,v H(x,α) α

25 Lagrangian in the Reflected Setting 1 Lagrangian is a local cost 2 Boundary x (0,1) d L ref (x,v) = L(x,v) a L ref (x, v) = L(x, v) b L ref (x, v) = + c L ref (x, v) = inf β>0 L(x, v + βn) Continuity, convexity lost!

26 LDP in the Reflected Setting Theorem [CDS preprint 07] R (m) obeys Large Deviation Principle with this Lagrangian. Sketch: represent the walk as random recurrent sequence, F(x,i,U) p(x,i, ) U k i.i.d. uniform, and projection Π : R d [0,1] d, R (m) k+1 = (2Π Id) ( R(m) k + 1 (m) F( R k,ξ k,u k ) ) m m m m (reflected walk). Introduce Y, Z (unprojected, projected) Y (m) k+1 = Y (m) k + 1 (m) mf( R k,ξ k,u k ) m m m (m) F( R,ξ k,u k ) ) Then, Z (m) k+1 m = Π ( Z (m) k m Z (m) = Ψ Sk (Y (m) ), + 1 m Z (m) t k m R (m) t 1 m

27 LDP in the Reflected Setting Theorem [CDS preprint 07] R (m) obeys Large Deviation Principle with this Lagrangian. Sketch: represent the walk as random recurrent sequence, F(x,i,U) p(x,i, ) U k i.i.d. uniform, and projection Π : R d [0,1] d, R (m) k+1 = (2Π Id) ( R(m) k + 1 (m) F( R k,ξ k,u k ) ) m m m m (reflected walk). Introduce Y, Z (unprojected, projected) Y (m) k+1 = Y (m) k + 1 (m) mf( R k,ξ k,u k ) m m m (m) F( R,ξ k,u k ) ) Then, Z (m) k+1 m = Π ( Z (m) k m Z (m) = Ψ Sk (Y (m) ), + 1 m Z (m) t k m R (m) t 1 m

28 LDP Reflected: Sketch of Proof and Y (m) k+1 m = Y (m) k m Z (m) = Ψ Sk (Y (m) ), + 1 m (m) F( R k,ξ k,u k ) m Z (m) t R (m) t 1 m 1 Y (m) has LDP with J Y (θ) = T 0 L(Ψ Sk(θ t ), θ t )dt 2 Z (m) has LDP with J Z (φ) = inf{ T 0 L(φ t, θ t )dt;ψ Sk (θ) = φ}. [Hence, R (m) too] 3 identify J Z = I with our Lagrangian.

29 Deadlock Phenomenon 1 Quasi-potential : global cost V (x) = inf [ T L ref (φ s, φ s )ds; φ 0 = 0, φ T = x, T > 0 ] 0 2 Deadlock phenomenon first success in a Bernoulli scheme 3 Probability of success exp [ m V ], V = inf(v (x); x 1 = L) 4 Length of an excursion = O(m)

30 Deadlock Phenomenon 1 Quasi-potential : global cost V (x) = inf [ T L ref (φ s, φ s )ds; φ 0 = 0, φ T = x, T > 0 ] 0 2 Deadlock phenomenon first success in a Bernoulli scheme 3 Probability of success exp [ m V ], V = inf(v (x); x 1 = L) Theorem [CDS preprint 07] E(T (m) L ) = exp [ m( V + o(1) )] P { exp [ m( V δ) ] T (m) L exp [ m ( V + δ )]} 1 R T (m) L converges in probability towards the set of minimizers of V T (m) L /σ m Exp.(1) for some σ m = e m( V+o(1)) (+ assumpt.) law

31 Deadlock Phenomenon 1 Quasi-potential : global cost V (x) = inf [ T L ref (φ s, φ s )ds; φ 0 = 0, φ T = x, T > 0 ] 0 2 Deadlock phenomenon first success in a Bernoulli scheme 3 Probability of success exp [ m V ], V = inf(v (x); x 1 = L) Theorem [CDS preprint 07] E(T (m) L ) = exp [ m( V + o(1) )] P { exp [ m( V δ) ] T (m) L exp [ m ( V + δ )]} 1 R T (m) L converges in probability towards the set of minimizers of V T (m) L /σ m Exp.(1) for some σ m = e m( V+o(1)) (+ assumpt.) law

32 Deadlock Phenomenon 1 Quasi-potential : global cost V (x) = inf [ T L ref (φ s, φ s )ds; φ 0 = 0, φ T = x, T > 0 ] 0 2 Deadlock phenomenon first success in a Bernoulli scheme 3 Probability of success exp [ m V ], V = inf(v (x); x 1 = L) Theorem [CDS preprint 07] E(T (m) L ) = exp [ m( V + o(1) )] P { exp [ m( V δ) ] T (m) L exp [ m ( V + δ )]} 1 R T (m) L converges in probability towards the set of minimizers of V T (m) L /σ m Exp.(1) for some σ m = e m( V+o(1)) (+ assumpt.) law

33 Deadlock Phenomenon 1 Quasi-potential : global cost V (x) = inf [ T L ref (φ s, φ s )ds; φ 0 = 0, φ T = x, T > 0 ] 0 2 Deadlock phenomenon first success in a Bernoulli scheme 3 Probability of success exp [ m V ], V = inf(v (x); x 1 = L) Theorem [CDS preprint 07] E(T (m) L ) = exp [ m( V + o(1) )] P { exp [ m( V δ) ] T (m) L exp [ m ( V + δ )]} 1 R T (m) L converges in probability towards the set of minimizers of V T (m) L /σ m Exp.(1) for some σ m = e m( V+o(1)) (+ assumpt.) law

34 Quasi-potential 1 If V C 1 inside the domain H(x, V(x)) = 0 Hamilton Jacobi 2 Boundary condition Reflection (Weak) Neumann boundary condition 3 Quasi-potential increases along optimal paths a) V/ x 1 (0, x 2 ) < 0 b) V/ x 1 (0, x 2 ) > 0

35 Quasi-potential 1 If V C 1 inside the domain H(x, V(x)) = 0 Hamilton Jacobi 2 Boundary condition 3 Quasi-potential increases along optimal paths a) V/ x 1 (0, x 2 ) < 0, H ( (0, x 2 ), V(0,x 2 ) ) = 0 b) V/ x 1 (0, x 2 ) > 0, H ( (0, x 2 ), (0, [ V/ x 2 ](0,x 2 )) ) = 0

36 Quasi-potential 1 If V C 1 inside the domain H(x, V(x)) = 0 Hamilton Jacobi 2 Boundary condition H(x, + V(x)) = 0 min( V (x),0) if x i = 0 x i ( + V (x)) i = max( V (x),0) if x i = 1 x i V (x) if x i (0,1) x i

37 Identification of the Quasi-potential Proposition Let W be C 1 on [0,1] d \ {0} with W (0) = 0 and 1 H(x, W (x)) = 0 2 Boundary condition H(x, + W(x)) = 0 α H(x, + W(x)), n 0 for n outward normal at x Then, W is the quasi-potential, and optimal path is the backward RDE φ = α H(φ t, + W (φ t )) k t with φ = 0,φ 0 = x ( typical deadlock path )

38 Identification of the Quasi-potential Proposition Let W be C 1 on [0,1] d \ {0} with W (0) = 0 and 1 H(x, W (x)) = 0 2 Boundary condition H(x, + W(x)) = 0 α H(x, + W(x)), n 0 for n outward normal at x Then, W is the quasi-potential, and optimal path is the backward RDE φ = α H(φ t, + W (φ t )) k t with φ = 0,φ 0 = x ( typical deadlock path )

39 Two-Stacks Model Example. Generalization of Maier 91 1 Environment with two states = {1, 2}, d = 2 p(x,i,v) = 1 2 l i[1 g 1 (x 1 )] v = e l i[1 + g 1 (x 1 )] v = e (1 l i)[1 g 2 (x 2 )] v = e (1 l i)[1 + g 2 (x 2 )] v = e 2 l i (0,1) 2 g 1,g 2 : [0,1] [0,1) Lipschitz, g j (z) > 0 for z > 0 3 Mean trend ( f (x) = lg 1 (x 1 ) (1 l)g 2 (x 2 ) ) l = i {1,2} l i µ(i) x1 x2 4 W (x) = 2 tanh 1 (g 1 (y))dy + 2 tanh 1 (g 2 (y))dy 0 0

40 Merci!