Markov chains model reduction

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, 216 1 / 21

Model Reduction N η N (t) random walk C. Landim Markov chains model reduction March 17, 216 2 / 21

Model Reduction Identification of the slow variables Derive their asymptotic behavior C. Landim Markov chains model reduction March 17, 216 3 / 21

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, 216 3 / 21

Model Reduction Σ N A A 3 A 1 A 2 C. Landim Markov chains model reduction March 17, 216 4 / 21

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, 216 4 / 21

Model Reduction X N (t) not Markov Hidden Markov Chain 3 2 1 N d N d N d t C. Landim Markov chains model reduction March 17, 216 5 / 21

Model Reduction X N (t) not Markov Hidden Markov Chain 3 2 1 N d N d N d t No convergence X N (t) in Skorohod topology C. Landim Markov chains model reduction March 17, 216 5 / 21

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, 216 6 / 21

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, 216 7 / 21

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, 216 7 / 21

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, 216 8 / 21

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, 216 9 / 21

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, 216 9 / 21

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, 216 9 / 21

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, 216 1 / 21

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, 216 1 / 21

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, 216 1 / 21

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, 216 1 / 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 C. Landim Markov chains model reduction March 17, 216 11 / 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, 216 11 / 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 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, 216 11 / 21

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, 216 12 / 21

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, 216 12 / 21

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, 216 12 / 21

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, 216 13 / 21

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, 216 14 / 21

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, 216 15 / 21

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, 216 15 / 21

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, 216 16 / 21

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, 216 16 / 21

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, 216 16 / 21

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, 216 16 / 21

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, 216 17 / 21

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, 216 18 / 21

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, 216 19 / 21

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, 216 19 / 21

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, 216 2 / 21

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, 216 21 / 21

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, 216 21 / 21

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, 216 21 / 21

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, 216 21 / 21