Some ergodic theorems of linear systems of interacting diffusions


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1 Some ergodic theorems of linear systems of interacting diffusions 4], Â_ ŒÆ êæ ÆÆ Nov, 2009, uà ŒÆ
2 1 The ergodic theory of interacting systems has been studied for more than 30 years, by many probabilists, for example, Spitzer, Liggett, Shiga, Cox, Greven Generally speaking, a common feature of a lot of interacting systems is: there are infinitely many invariant probability measures
3 A natural problem: How to characterize all the invariant probability measures? In general, there are two methods to solve the above problem: duality theory coupling method
4 1.1. Duality theory If the interacting system has a dual process, one can study the invariant probability measures by the dual process. [7] Let η t, ξ t be two Markov Processes on X, Y, and let H(η, ξ) be a bounded measurable function on X Y. The processes η t, ξ t are said to be dual to one another w.r.t. to H if E η H(η t, ξ) = E ξ H(η, ξ t ) for all η X and ξ Y. Liggett, T.M. Interacting Particle Systems, 1985, Springer
5 Example (1.1.1) Liggett, Spitzer [5][6][13] obtain a complete characterization of all invariant probability measures for the simple exclusion model by using the dual process. Liggett, T.M., A characterization of the invariant measures for an infinite particle system with interactions. Trans. Amer. Math. Soc. 179 (1973), Liggett, T.M., A characterization of the invariant measures for an infinite particle system with interactions II. Trans. Amer. Math. Soc. 198 (1974), Spitzer, F., Recurrent random walk of an infinite particle system. Trans. Amer. Math. Soc. 198 (1974),
6 Let p(x, y) be a transition function for a symmetric, irreducible Markov chain on a countable set S. Oneparticle motion is determined by p = (p(x, y)) x,y S. Let I be the set of all invariant probability measures for the simple exclusion model and H be the set of all bounded pharmonic functions, i.e. H = {h( ) 0 h(x) 1, p(x, y)h(y) = h(x), x S}. y S One of main results in Liggett, Spitzer [5][6][13]µLet I ex be the set of all extremal elements of I, then there exists a onetoone correspondence between H and I ex
7 Example (1.1.2) Shiga 1980 [9][10] studies ergodic behaviors of a class of interacting diffusion systems, the socalled continuous time stepping stone model. dx i (t) = j S q ij X j (t)dt + 1 2N X i(t)(1 X i (t))db i (t), i S, where S is a countable set, {B i (t)} i S is a collection of independent onedimensional standard Brownian motions, and q ij 0, i j, j S q ij = 0, i S, sup i S q ii <. Background: Population genetics, X i [0, 1] is the frequency of A allele (gene). Shiga, T., An interacting system in population genetics. J. Math. Kyoto Univ. 20 (1980), , Shiga, T., An interacting system in population genetics II. J. Math. Kyoto Univ. 20 (1980),
8 More precisely let Q = (q ij ) and a t = e tq. We denote by ( X t, P i ) the continuous time Markov chain on S S associated with translation probability a t a t, where a t a t ( i, j) = a t (i 1, j 1 )a t (i 2, j 2 ) for each i = (i 1, i 2 ) S S and j = (j 1, j 2 ) S S. As pointed out in [9], there exists the following classification. Case I Case II Case III P i [ I ( X t )dt = ] = 1 for all i S S, (1) 0 P i [ I ( X t )dt < ] = 1 for all i S S, (2) 0 0 < P i [ I ( X t )dt = ] < 1 for all i S S, (3) where = {(i 1, i 2 ) S S i 1 = i 2 }. 0 Since Q is irreducible, Case I, Case II and Case III exhaust all possibilities
9 Shiga completely characterizes all invariant probability measures by an analogous approach of the dual process, and in [9][10] he obtains the following main resultsµ Assume Case I (1), then I ex = {δ 0, δ 1 }. Assume Case II (2), then I ex = {ν h h H}. Assume Case III (3), then I ex = {ν h h H }. H = {h( ) 0 h(i) 1, j S q ijh j = 0, subset of H. i S}, and H is a
10 1.2. Coupling method If the interacting system has no tractable dual process, usually, one can consider the interacting system in Z d translation invariant situation. It is possible to characterize all Z d translation invariant stationary distributions by using coupling method. Example (1.2.1) Liggett and Spitzer 1981 [8] obtain all Z d translation invariant stationary distributions for the coupled random walks based on a coupling method Liggett, T.M. Spitzer, F., Ergodic theorems for coupled random walks and other systems with locally interacting components. Z. Wahrsch. verw. Gebiete 56 (1981),
11 Example (1.2.2) Shiga 1992 [11] treats a class of interacting diffusion systems taking values in a suitable subspace of R Zd by the second moment calculations and a coupling technique. Shiga, T., Ergodic theorems and exponential decay of sample paths for certain interacting diffusion systems. Osaka J. Math. 29 (1992),
12 Example (1.2.3) Cox and Greven 1994 [2] study the ergodic theory of a class of interacting diffusion systems with the compact state space [0, 1] Zd by using a coupling technique and a dual comparison argument. dx i (t) = j Z d (a(i, j) δ(i, j))x j (t)dt + 2g(x i (t))db i (t), i Z d. (4) a(i, j) = a(0, j i), a(i, j) is an irreducible random walk kernel on Z d, and g : [0, 1] R + satisfies g > 0 on (0, 1), g(0) = g(1) = 0, g is Lipschitz Cox, J.T., Greven, A. Ergodic theorems for infinite systems of locally interacting diffusions. Ann. Probab. 22 (1994),
13 Let â(i, j) = a(i,j)+a(j,i) 2, Cox and Greven [2] obtain the following main resultsµ If â(i, j) is transient, then (I T ) ex = {ν θ 0 θ 1}. If â(i, j) is recurrent, then I ex = {δ 0, δ 1 }. T be the set of translation invariant measures on [0, 1] Zd
14 2 Linear systems of interacting diffusions
15 2.1. The model of a class of linear systems of interacting diffusions Let S be a countable set. Let X(t) = {X i (t)} i S be the system of interacting diffusions on [0, ), defined by the following stochastic differential equations: dx i (t) = (a(i, j) δ(i, j))x j (t)dt + cx i (t)db i (t), i S, (5) j S where c is a fixed positive constant and a = (a(i, j)) i,j S is an irreducible Markov chain translation probability matrix
16 2.2. Selfduality The linear system (5) has an important propertyµselfduality Let { X(t)} t 0 satisfy the following stochastic differential equations: d X(t) = (ã(i, j) δ(i, j)) X j (t)dt + c X i (t)db i (t), i S. (6) j S where ã(i, j) := a(j, i). Set Ξ F = {x [0, ) S x j = 0 for all but finitely many j S}. {X(t)} t 0 and { X(t)} t 0 are dual in the following sense. Given initial states x L 2 (γ) and x Ξ F, X(t), x = d x, X(t), (7) where X(t), x = i S X i(t) x i.
17 2.3. Some related work (1) Under the assumption that the translation kernel (a(i, j)) i,j S is doubly stochastic, Cox, Klenke, Perkins 1991 [3] use the system s selfduality property to prove the weak convergence of the linear system. Theorem 2.3 in [3] Assume that a is doubly stochastic. If L(X(0)) M θ, then L(X(t)) ν θ, where M θ = {µ sup x 2 i µ(dx) <, i S ( a t (i, k)x k θ ) 2 µ(dx) = 0, i S}, (8) lim t k S and ν θ is an invariant probability measure related to the constant initial state θ Cox, J.T., Klenke, A. Perkins, E.A., Convergence to equilibrium and linear systems duality. CMS Conf. Proc., 26, Amer. Math. Soc., Providence, RI., (2001),
18 (2) Greven and den Hollander 2006 [4] consider the following linear system of interacting diffusionsµ dx i (t) = j Z d (a(i, j) δ(i, j))x j (t)dt + bx i (t) 2 db i (t), i Z d, (9) where a(i, j) = a(0, j i). Greven and den Hollander obtain some very precise and complete results on the longtime behaviour of the system in Z d translation invariant situation by using selfduality and Metastability Greven, A., den Hollander, F., Phase transitions for the longtime behaviour of interacting diffusions. Ann. Probab. 35 (2007),
19 2.4. Our problem and results We are concerned with the longtime bahaviour of the linear system (5) on a countable set S, and we attempt to explore some relation between H, the set of all bounded aharmonic functions, and I ex the set of all extremal invariant probability measures for {X(t)} t 0. Let I be the set of all invariant probability measures for {X(t)} t 0. dx i (t) = j S (a(i, j) δ(i, j))x j (t)dt + cx i (t)db i (t), i S
20 Our main resultsµ Theorem 1 Assume that (a(i, j)) i,j S is doubly stochastic. If L[X(0)] M h, then L[X(t)] ν h t, where M h = {µ sup x 2 i µ(dx) <, i S ( a t (i, k)x k h(i) ) 2 µ(dx) = 0, i S}, lim t k S and ν h is an invariant probability measure related to the initial state h H. Comparing with Cox and Greven s result (8)
21 We set J 2 = {ν sup x 2 k ν(dx) < }. k Theorem 2 Assume that (a(i, j)) i,j S is transient and symmetric. Suppose that G := max i S Then {I J 2 } ex = {ν h h H}. 0 a s (i, i)ds <, c 2 G < 2. (10)
22 Theorem 3 Assume that (a(i, j)) i,j S is doubly stochastic and satisfies Case I (1). For any 0 θ <, if L[X(0)] M θ, then L[X(t)] δ 0 as t
23 2.5. Some conclusions Since Case I, Case II, Case III exhaust all possibilities, we want to consider the ergodic behaviors of the linear system (5) for Case I, Case II, Case III. In fact, For Case I (1) completely solved, see Theorem 3; For Case II (2) partly solved, we obtain some results on the ergodic behaviors of the linear system (5) under the condition that (a(i, j)) i,j S is transient and symmetric, see Theorem 2. For Case III (3) very complex, leave for future study
24 2.6. Proofs The proof of Theorem 1 is similar to that of Theorem 2.3 in [3]. Main idea of the proof of Theorem 2µ The first stepµto show h H, ν h (I J 2 ) ex. The second stepµto show (I J 2 ) ex {ν h h H}. Difficulty: (i) state space [0, ) S is not compact. (ii) S is a countable set without any algebraic and geometric structure. Cox, J.T., Klenke, A. Perkins, E.A., Convergence to equilibrium and linear systems duality. CMS Conf. Proc., 26, Amer.Math. Soc., Providence, RI., (2001),
25 Our methods We use FeynmanKac formula to prove ν h J 2 in the first step. In the second step, we define two probability measures µ 1 and µ 2 by truncationµ ( xi N ) / ( µ 1, f = f(x) 2N + δ xi N ) µ(dx) 2N + δ µ(dx), ( µ 2, f = f(x) 1 x i N ) / ( 2N δ µ(dx) 1 x i N ) 2N δ µ(dx). Then µ = λµ 1 + (1 λ)µ 2, λ = ( x i N 2N + δ)µ(dx) (0, 1)
26 The key points are to showµ 1 t t 0 T s µ 1 ds is tight. k a t(i, k)x k and l a t(j, l)x l are asymptotically uncorrelated, where X µ, µ (I J 2 ) ex. More precisely, we set h(i, j) = lim It suffices to prove where h(i) = x i µ(dx), i S. Actually, we can getµ lim n 1 t n tn 0 t h(i, j) = h(i)h(j), T s µ 1, x j ds = µ, x j = By calculation, we have h(i, j) = h(i)h(j). k,l a t(i, k)a t (j, l) x k x l µ(dx). x j µ(dx)
27 The proof of Theorem 3 is based on the comparison theorem (see Theorem 1 in [1]). The key pointµwe can find a version of the continuous time stepping stone model Y ε (t) = {Yi ε(t)} i S for comparison. dyi ε (t) = (a(i, j) δ(i, j))yj ε (t)dt+ g ε (Yi ε(t))db i(t), i S, t 0, j S (11) where g ε (v) = b ε (v ε) + (ε 1 v) +, v R Cox, J.T., Fleischmann, K., Greven, A. Comparison of interacting diffusions and an application to their ergodic theory. Probab. Theory Relat. Fields 92 (1996),
28 References [1] Cox, J.T., Fleischmann, K., Greven, A. Comparison of interacting diffusions and an application to their ergodic theory. Probab. Theory Relat. Fields 92 (1996), [2] Cox, J.T., Greven, A. Ergodic theorems for infinite systems of locally interacting diffusions. Ann. Probab. 22 (1994), [3] Cox, J.T., Klenke, A. Perkins, E.A., Convergence to equilibrium and linear systems duality. CMS Conf. Proc., 26, Amer.Math. Soc., Providence, RI., (2001), [4] Greven, A., den Hollander, F., Phase transitions for the longtime behaviour of interacting diffusions. Ann. Probab. 35 (2007), [5] Liggett, T.M., A characterization of the invariant measures for an infinite particle system with interactions. Trans. Amer. Math. Soc. 179 (1973), Trans. Amer. Math. Soc. 198 (1974), [6] Liggett, T.M., A characterization of the invariant measures for an infinite particle system with interactions II. Trans. Amer. Math. Soc. 198 (1974),
29 [7] Liggett, T.M., Interacting particle systems. Springer, Berlin, (1985). [8] Liggett, T.M. Spitzer, F., Ergodic theorems for coupled random walks and other systems with locally interacting components. Z. Wahrsch. verw. Gebiete 56 (1981), [9] Shiga, T., An interacting system in population genetics. J. Math. Kyoto Univ. 20 (1980), , [10] Shiga, T., An interacting system in population genetics II. J. Math. Kyoto Univ. 20 (1980), [11] Shiga, T., Ergodic theorems and exponential decay of sample paths for certain interacting diffusion systems. Osaka J. Math. 29 (1992), [12] Shiga, T., Shimizu, A. Infinite dimensional stochastic differential equations and their applications. J. Math. Kyoto Univ. 20 (1980), [13] Spitzer, F., Recurrent random walk of an infinite particle system. Trans. Amer. Math. Soc. 198 (1974),
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