Non-Essential Uses of Probability in Analysis Part IV Efficient Markovian Couplings. Krzysztof Burdzy University of Washington

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1 Non-Essential Uses of Probability in Analysis Part IV Efficient Markovian Couplings Krzysztof Burdzy University of Washington 1

2 Review See B and Kendall (2000) for more details. See also the unpublished monograph of Aldous and Fill (1999). X positive-recurrent Markov process symmetric with respect to some reference measure m, p(t, x, y) density relative to m, p(t, x, y) = c+g(x, y)e µ 2t +R(t, x, y), (1) first eigenvalue = 0, first eigenfunction = c, µ 2 = second eigenvalue, g(x, y) = ϕ ϕ(x)ϕ(y) where the ϕ are orthogonal eigenfunctions with eigenvalue µ 2, R(t, x, y) 0 faster than e µ 2t as t 2

3 A coupling is a pair (X 1, X 2 ) of (typically dependent) copies of the Markov process X. Good couplings are characterized by small coupling time τ = inf{t 0 : X 1 t = X 2 t }. Typically, X 1 and X 2 are constructed so that X 1 t = X 2 t for all t τ. Suppose that (X 1 0, X 2 0 ) = (x 1, x 2 ). 3

4 The eigenfunction representation (1) gives p(t, x 1, y) p(t, x 2, y) (2) = (g(x 1, y) g(x 2, y))e µ 2t + R(t, x 1, y) R(t, x 2, y) while the coupling yields p(t, x 1, y)dy p(t, x 2, y)dy (3) = P (Xt 1 dy X0 1 = x 1 ) P (Xt 2 dy X0 2 = x 2 ) P (Xt 1 dy, t < τ X0 1 = x 1 ) + P (Xt 2 dy, t < τ X0 2 = x 2 ). Suppose that one can prove that P (t < τ Xt 1 = x 1, Xt 2 = x 2 ) e µ t. (4) (2), (3) and (4) imply that µ µ 2. We will call µ the coupling exponent. 4

5 Informal Definition of Efficiency We will call a coupling (X 1, X 2 ) an efficient Markovian coupling if (X 1, X 2 ) is a Markov process and µ = µ 2. Informal Efficient Coupling Heuristic A coupling (X 1, X 2 ) is efficient if and only if, for all t, and given {t < τ}, the conditional distributions of (Xt 1, Xt 2 ) and (Xt 2, Xt 1 ) are singular with respect to each other. The above heuristic is NOT true in a rigorous sense but it works in sufficiently many circumstances to make it almost true. 5

6 Symmetric Markov chains with finite state space X = {X t : t 0} a continuous time symmetric Markov process with a finite state space D and transition probabilities p(t, y, x) = p(t, x, y) = P (X s+t = y X s = x). (X 1, X 2 ) a Markovian coupling for the process X: {(Xt 1, Xt 2 ) : t 0}, {Xt 1 : t 0} and {Xt 2 : t 0} are Markov with respect to the filtration generated jointly by X 1 and X 2, and X 1 and X 2 have the same transition probabilities as X. τ = inf{t 0 : Xt 1 = Xt 2 } Xt 1 = Xt 2 for all t τ. 6

7 We will say that D 2 is irreducible with respect to a given coupling if every state (y 1, y 2 ) with y 1 y 2 is accessible from any other state (x 1, x 2 ) with x 1 x 2. Theorem (i) If D 2 is irreducible for a coupling then this coupling is not efficient. (ii) Suppose that for every pair of distinct points x 1, x 2 D there exists a function f : D R with the property that P (x 1,x 2 ) (f(x 1 t ) f(x 2 t ) > 0 t < τ) = 1 Then the coupling is efficient. 7

8 D 8

9 Example Suppose the state space D is a finite subinterval of the integers Z and X can jump from x only to x 1 or x + 1, for every x. Assume that (Xt 1, Xt 2 ) almost surely never jumps to (Xt, 2 Xt ). 1 Then (X 1, X 2 ) is an efficient coupling. In particular, the coupling is efficient if X 1 and X 2 have independent jumps until the coupling time τ. 9

10 Example Consider a Markov process with jump rates indicated in the figure. There are no efficient couplings for this process

11 Example Ehrenfest model Consider two urns with n marked balls distributed among them. At every arrival time for a Poisson process, a ball is randomly chosen from among all balls in both urns and moved to the other urn. D = {(i 1, i 2,..., i n ) : i k = 0 or 1} U k i.i.d. exponential, mean 1 T k = U U k N k i.i.d. uniform on {1, 2,..., n} {J k } i.i.d. with P (J k = 0) = P (J k = 1) = 1/2 X Tk = (j 1, j 2,..., j Nk,..., j n ) X Tk = (j 1, j 2,..., J k,..., j n ) 11

12 (X 1, X 2 ): use one family {T k, N k, J k } k 1 (XT 1 k, XT 2 k ) = ((j1, 1..., jn 1 k,..., jn), 1 (j1, 2..., jn 2 k,..., jn)) 2 (XT 1 k, XT 2 k ) = ((j1, 1..., J k,..., jn), 1 (j1, 2..., J k,..., jn)) 2 12

13 Suppose that X0 1 = (j1, 1 j2, 1..., jn). 1 Let f(i 1, i 2,..., i n ) be the number of k such that i k = jk 1. If X1 0 X0 2 then f(xt 1 ) f(xt 2 ) > 0 for all t < τ. Hence the coupling is efficient. Elementary estimates yield µ µ 2 = 1/n. = 1/n, so However, the mean time to coupling is not of order 1/µ = n. Eτ n log n. See Liggett (1985) for the use of monotonicity in the context of particle system couplings. 13

14 Reflected Brownian motion Synchronous couplings D Lipschitz domain in the plane B planar Brownian motion X t = x 0 + B t + Y t = y 0 + B t + t 0 t 0 n(x s )dl X s, n(y s )dl Y s. 14

15 Cranston and Le Jan (1990): Synchronous couplings never meet in strictly convex domains. Generic behavior: τ := inf{t 0 : X t = Y t } =. p(t, x, y) transition densities for X, p(t, x, y) = c 1 +ϕ 2 (x)ϕ 2 (y)e µ 2t +R(t, x, y), R(t, x, y) 0 faster than e µ 2t as t Lemma Suppose that for some µ 0 and x, y D, E (x,y) X t Y t c(x, y)e µt for t 0. If ϕ 2 (x) ϕ 2 (y) then µ µ 2. 15

16 Proof The function ϕ 2 is not identically equal to 0 so, exp(c 1 z 1 + c 2 z 2 )ϕ 2 (z 1, z 2 )dz 1 dz 2 > 0, D for some c 1, c 2 0. Since D is bounded there exists c 3 > 0 such that c 1 3 is a Lipschitz constant for exp(c 1 x 1 + c 2 x 2 ), and so E X t Y t c 3 (E[exp(c 1 Xt 1 + c 2 Xt 2 )] ) E[exp(c 1 Yt 1 + c 2 Yt 2 )]. 16

17 Then, c(x, y)e µt E (x,y) X t Y t c 3 (E exp(c 1 Xt 1 + c 2 Xt 2 ) ) E exp(c 1 Yt 1 + c 2 Yt 2 ) = c 3 exp(c 1 z 1 + c 2 z 2 )p(t, x, z)dz D c 3 exp(c 1 z 1 + c 2 z 2 )p(t, y, z)dz D = c 3 [ϕ 2 (x) ϕ 2 (y)] exp(c 1 z 1 + c 2 z 2 ) D ϕ 2 (z)e µ 2t dz + R(t, x, y) = c 4 (x, y)e µ 2t + R(t, x, y). Hence µ µ 2. 17

18 Definition We will call a synchronous coupling (X, Y ) of reflected Brownian motions in D efficient if for some x and y with ϕ 2 (x) ϕ 2 (y), we have µ = µ 2. Theorem If D is a triangle with an angle strictly greater than π/2 then the synchronous coupling for the reflected Brownian motion in D is efficient. Conjecture Synchronous couplings are not efficient in triangles with all angles less than π/2. 18

19 Mirror couplings Generic behavior: τ := inf{t 0 : X t = Y t } <. 19

20 Lemma Suppose for some x, y D, P (x,y) (τ > t) c(x, y)e µt for t 0. If ϕ 2 (x) ϕ 2 (y) then µ µ 2. Definition A mirror coupling (X, Y ) of reflected Brownian motions in D is said to be efficient if µ = µ 2 for some x and y with ϕ 2 (x) ϕ 2 (y). Theorem (i) If D is a triangle with an angle strictly greater than π/2 then the mirror coupling for the reflected Brownian motion in D is efficient. (ii) If all angles of the triangle D are distinct from each other and strictly less than π/2 then the mirror coupling for the reflected Brownian motion in D is not efficient. 20

21 Open problem Does there exist an efficient coupling of reflected Brownian motions in a triangle with acute angles? 21

22 Convergence of synchronous couplings (X, Y ) synchronous coupling of reflected Brownian motions in a planar domain D Theorem (B and Chen 2002) X t Y t 0 as t if (i) D consists of polygonal lines or (ii) D is a lip domain. D D 22

23 Theorem (B, Chen and Jones 2006) Suppose that D has smooth boundary. Let Λ(D) = ν(x)dx D + log cos α(x, y) ω x (dy)dx D D = 2π(1 # of holes in D) + log cos α(x, y) ω x (dy)dx. D D If Λ(D) > 0 then lim t log X t Y t t = Λ(D) 2 D. x α y 23

24 Corollary X t Y t 0 as t if D is smooth and D has at most one hole. X t Y t e tλ(d)/(2 D ) Open problems (i) Does there exist a bounded planar domain D such that X t Y t 0? (ii) If D is the exterior of an ellipse then Λ(D) = 0. Does there exist a closed curve such that if D is its exterior then Λ(D) < 0? 24

25 REFERENCES 1. D. Aldous and J. Fill (1999) Reversible Markov Chains and Random Walks on Graphs (book in preparation, available online: / aldous/rwg/book.html) 2. K. Burdzy and Z. Chen (2002) Coalescence of synchronous couplings Probab. Theory Rel. Fields 123, K. Burdzy, Z. Chen and P. Jones (2006) Synchronous couplings of reflected Brownian motions in smooth domains Illinois. J. Math., Doob Volume (to appear) 4. K. Burdzy and W. Kendall (2000) Efficient Markovian couplings: examples and counterexamples Ann. Appl. Probab. 10,

26 5. M. Cranston and Y. Le Jan (1990) Noncoalescence for the Skorohod equation in a convex domain of R 2 Probability Theory and Related Fields 87, T.M. Liggett (1985) Interacting Particle Systems Springer-Verlag, New York. 26