Markov chains model reduction

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Νάρκισσος Χατζηιωάννου
6 χρόνια πριν
Προβολές:

1 Markov chains model reduction C. Landim Seminar on Stochastic Processes 216 Department of Mathematics University of Maryland, College Park, MD C. Landim Markov chains model reduction March 17, / 21

2 Model Reduction N η N (t) random walk C. Landim Markov chains model reduction March 17, / 21

3 Model Reduction Identification of the slow variables Derive their asymptotic behavior C. Landim Markov chains model reduction March 17, / 21

4 Model Reduction d = 2 Mixing time N 2 Hitting time N 2 log N d 3 Mixing time N 2 Hitting time N d C. Landim Markov chains model reduction March 17, / 21

5 Model Reduction Σ N A A 3 A 1 A 2 C. Landim Markov chains model reduction March 17, / 21

6 Model Reduction Σ N A A 3 A 1 A 2 Ψ N : Σ N {, 1, 2, 3} Ψ N (x) = X N (t) = Ψ N (η N (t)) 3 k 1{x A k } k= C. Landim Markov chains model reduction March 17, / 21

7 Model Reduction X N (t) not Markov Hidden Markov Chain N d N d N d t C. Landim Markov chains model reduction March 17, / 21

8 Model Reduction X N (t) not Markov Hidden Markov Chain N d N d N d t No convergence X N (t) in Skorohod topology C. Landim Markov chains model reduction March 17, / 21

9 Trace process Σ N E E 3 E 1 E 2 N = Σ N \ [E E 1 E 2 E 3 ] C. Landim Markov chains model reduction March 17, / 21

10 Trace of a Markov chain η(t) E-valued Markov chain F E η F (t) trace of η(t) on F F = {a, b} E b T 3 b T 3 a T 1 T 2 T 1 + T 2 a t t C. Landim Markov chains model reduction March 17, / 21

11 Trace of a Markov chain η(t) E-valued Markov chain F E η F (t) trace of η(t) on F F = {a, b} E b T 3 b T 3 a T 1 T 2 T 1 + T 2 a t t η F (t) Markov chain r F (x, y) = λ(x) P x [H F \{x} = H y ] C. Landim Markov chains model reduction March 17, / 21

12 Trace process Σ N E E 3 E 1 E 2 N = Σ N \ [E E 1 E 2 E 3 ] r F (x, y) = λ(x) P x [H F \{x} = H y ] long jumps C. Landim Markov chains model reduction March 17, / 21

13 Problem η E (t) trace of η(t) on E = E E 1 E 2 E 3 3 Ψ N : E {, 1, 2, 3} Ψ N (x) = k 1{x E k } k= X N (t) = Ψ N (η E (t)) Hidden M. C. C. Landim Markov chains model reduction March 17, / 21

14 Problem η E (t) trace of η(t) on E = E E 1 E 2 E 3 3 Ψ N : E {, 1, 2, 3} Ψ N (x) = k 1{x E k } k= X N (t) = Ψ N (η E (t)) Hidden M. C. X N (θ N t) X (t) [ max E x 1{η(θ N s) E} ds ] x E C. Landim Markov chains model reduction March 17, / 21

15 Problem η E (t) trace of η(t) on E = E E 1 E 2 E 3 3 Ψ N : E {, 1, 2, 3} Ψ N (x) = k 1{x E k } k= X N (t) = Ψ N (η E (t)) Hidden M. C. X N (θ N t) X (t) [ max E x 1{η(θ N s) E} ds ] x E η(θ N t) X (t) soft topology C. Landim Markov chains model reduction March 17, / 21

16 Martingale approach Beltrán, L. (21-15) X N (t) = Ψ N (η E (tθ N )) X (t) C. Landim Markov chains model reduction March 17, / 21

17 Martingale approach X N (t) = Ψ N (η E (tθ N )) X (t) Tightness X N (t) C. Landim Markov chains model reduction March 17, / 21

18 Martingale approach X N (t) = Ψ N (η E (tθ N )) X (t) Tightness X N (t) X (t) solves martingale problem: F (X t ) F (X ) Exists L (LF )(X s ) ds martingale F : {,..., 3} R C. Landim Markov chains model reduction March 17, / 21

19 Martingale approach X N (t) = Ψ N (η E (tθ N )) X (t) Tightness X N (t) X (t) solves martingale problem: F (X t ) F (X ) F (X t ) F (X ) Exists L (LF )(X s ) ds martingale F : {,..., 3} R 3 r(x s, k) [F (k) F (X s )] ds k= C. Landim Markov chains model reduction March 17, / 21

20 Martingale approach Ψ N (η E (θ N t)) X (t) F (X t ) F (X ) 3 r(x s, k) [F (k) F (X s )] ds k= martingale C. Landim Markov chains model reduction March 17, / 21

21 Martingale approach Ψ N (η E (θ N t)) X (t) F (X t ) F (X ) 3 r(x s, k) [F (k) F (X s )] ds k= martingale H : E R H(η E (θ N t)) H(η E ()) θn [L E H](η) = ξ E R E (η, ξ) {H(ξ) H(η)} [L E H] (η E (s)) ds C. Landim Markov chains model reduction March 17, / 21

22 Martingale approach Ψ N (η E (θ N t)) X (t) F (X t ) F (X ) 3 r(x s, k) [F (k) F (X s )] ds k= martingale H : E R H(η E (θ N t)) H(η E ()) θn [L E H](η) = ξ E R E (η, ξ) {H(ξ) H(η)} [L E H] (η E (s)) ds H = F Ψ N F (X N (t)) F (X N ()) θ N [L E (F Ψ)](η E (θ N s)) ds C. Landim Markov chains model reduction March 17, / 21

23 Closing the equation θ N [L E (F Ψ)](η E (θ N s)) ds 3 k= r(x s, k) [F (k) F (X s )] ds C. Landim Markov chains model reduction March 17, / 21

24 Closing the equation θ N [L E (F Ψ)](η E (θ N s)) ds 3 k= r(x s, k) [F (k) F (X s )] ds [L E (F Ψ)](η) = ξ E R E (η, ξ) {(F Ψ)(ξ) (F Ψ)(η)} = = 3 [F (k) F (X N )] R E (η, ξ) ξ E k k= 3 [F (k) F (X N )] R E (η, E k ) k= C. Landim Markov chains model reduction March 17, / 21

25 Closing the equation θ N [L E (F Ψ)](η E (θ N s)) ds 3 k= r(x s, k) [F (k) F (X s )] ds [L E (F Ψ)](η) = ξ E R E (η, ξ) {(F Ψ)(ξ) (F Ψ)(η)} = = 3 [F (k) F (X N )] R E (η, ξ) ξ E k k= 3 [F (k) F (X N )] R E (η, E k ) k= θ N R E (η E (θ N s), E k ) ds r(x N s, k) ds C. Landim Markov chains model reduction March 17, / 21

26 Local ergodicity θ N R E (η E (θ N s), E k ) 1{X N s k}ds r(x N s, k) 1{X N s k} ds Σ N E E 3 E 1 E 2 C. Landim Markov chains model reduction March 17, / 21

27 Local ergodicity θ N R E (η E (θ N s), E k ) 1{X N s k}ds r(x N s, k) 1{X N s k} ds G : E R Ĝ = E πn [G(η) E,..., E 3 ] π N ss 1 Ĝ(η) = π N (η)g(η) η E j π N (E j ) η E j Ĝ(η) = Ĝ(Ψ N(η)) { } G(η E (θ N s)) Ĝ(ηE (θ N s)) ds (C1) C. Landim Markov chains model reduction March 17, / 21

28 Consequence θ N R E (η E (θ N s), E k ) 1{X N s θ N R E (η E (θ N s), E k ) 1{X N s k}ds = j}ds r(xs N, k) 1{Xs N k} ds r(j, k) 1{X N s = j} ds C. Landim Markov chains model reduction March 17, / 21

29 Consequence θ N R E (η E (θ N s), E k ) 1{X N s θ N R E (η E (θ N s), E k ) 1{X N s k}ds = j}ds r(x N s, k) 1{X N s r(j, k) 1{X N s θ N π N (η)r E (η, E k ) =: θ N r N (j, k) π N (E j ) η E j 1 θ N π N (η)r E (η, E k ) r(j, k) π N (E j ) η E j k} ds = j} ds (C2) θ N [L E (F Ψ)](η E (θ N s)) ds (LF )(X s ) ds C. Landim Markov chains model reduction March 17, / 21

30 Summary Ĝ(η) = E πn [G(η) E,..., E 3 ] { } G(η E (θ N s)) Ĝ(ηE (θ N s)) ds (C1) C. Landim Markov chains model reduction March 17, / 21

31 Summary Ĝ(η) = E πn [G(η) E,..., E 3 ] { } G(η E (θ N s)) Ĝ(ηE (θ N s)) ds θ N π N (η)r E (η, E k ) = θ N r N (j, k) r(j, k) π N (E j ) η E j (C1) (C2) C. Landim Markov chains model reduction March 17, / 21

32 Summary Ĝ(η) = E πn [G(η) E,..., E 3 ] { } G(η E (θ N s)) Ĝ(ηE (θ N s)) ds θ N π N (η)r E (η, E k ) = θ N r N (j, k) r(j, k) π N (E j ) η E j (C1) (C2) X N (t) = Ψ N (η E (tθ N )) X t C. Landim Markov chains model reduction March 17, / 21

33 Summary Ĝ(η) = E πn [G(η) E,..., E 3 ] { } G(η E (θ N s)) Ĝ(ηE (θ N s)) ds θ N π N (η)r E (η, E k ) = θ N r N (j, k) r(j, k) π N (E j ) η E j (C1) (C2) X N (t) = Ψ N (η E (tθ N )) X t [ max E x 1{η(θ N s) E} ds ] x E (C3) Ψ N (η(tθ N )) X t soft topology C. Landim Markov chains model reduction March 17, / 21

34 Applications Random walks Random walks random traps Random walks in potential field Spin models Kawasaki dynamics in d = 2 Blume-Capel dynamics Mean field Potts models Interacting particle systems Condensing zero-range processes ABC model Inclusion-exclusion dynamics Random polymers In interaction with substrates repulsive wall Reversible and non-reversible dynamics Logarithmic barriers C. Landim Markov chains model reduction March 17, / 21

35 Tools Processes which visit points: General: Potential theory (C1) (C3) formulated in terms of capacities Ergodic properties Spectral gap C. Landim Markov chains model reduction March 17, / 21

36 Condensing zero-range processes Model: T L = {1,..., L} periodic boundary conditions State space N T L configurations η = {η x : x T L } C. Landim Markov chains model reduction March 17, / 21

37 Condensing zero-range processes Model: T L = {1,..., L} periodic boundary conditions State space N T L configurations η = {η x : x T L } Dynamics: g : N R + g() = g(k) > < p 1 x x + 1 at rate pg(η x ) x x 1 at rate (1 p)g(η x ) g(k) = k independent random walks g(k) = 1{k 1} queues and servers C. Landim Markov chains model reduction March 17, / 21

38 Condensation Evans (25), Armendariz, Beltran, Chleboun, Ferrari, Godrèche, Grosskinsky, Loulakis, Schuetz, Sisko, Spohn ( k ) α g(1) = 1 g(k) = k 2 α > k 1 N number of particles E L,N = {η N T L : x T L η x = N} {η(t) : t } irreducible Phase transition µ L,N α > 2 T removes the site with largest number of particles {N L : L 1} N L /L ρ > ρ µ L,N T 1 ν ρ L(ρ ρ ) α > 1 L fixed 1 l N N lim N µ L,N {max 1 x L η x N l N } = 1 C. Landim Markov chains model reduction March 17, / 21

39 Model Reduction, Coarse graining Fix 1 l N N EN x = {η : η x N l N } 1 x L L E N = EN x E L,N = E N N x=1 C. Landim Markov chains model reduction March 17, / 21

40 Model Reduction, Coarse graining Fix 1 l N N EN x = {η : η x N l N } 1 x L L E N = EN x E L,N = E N N x=1 E L,N E 4 N E 3 N N E 1 N E 2 N C. Landim Markov chains model reduction March 17, / 21

41 Model Reduction, Coarse graining Fix 1 l N N EN x = {η : η x N l N } 1 x L L E N = EN x E L,N = E N N x=1 η E N (t) trace of {η(t) : t } on E N = L x=1 Ex N Ψ N : E N {1,..., L} Ψ N (η) = x iff η E x N X N (t) = Ψ N (η E N (N 1+α t)) X (t) C. Landim Markov chains model reduction March 17, / 21

42 Model Reduction, Coarse graining Fix 1 l N N EN x = {η : η x N l N } 1 x L L E N = EN x E L,N = E N N x=1 η E N (t) trace of {η(t) : t } on E N = L x=1 Ex N Ψ N : E N {1,..., L} Ψ N (η) = x iff η E x N X N (t) = Ψ N (η E N (N 1+α t)) X (t) θ N = N 1+α Reversible: r(x, y) = C(α) L cap(x, y) Non-reversible p = 1 C. Landim Markov chains model reduction March 17, / 21

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