Spatial Ergodicity of the Harris Flows

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Communications on Stochastic Analysis Volume 11 Number 2 Article 6 6-217 Spatial Ergodicity of the Harris Flows E.V. Glinyanaya Institute of Mathematics NAS of Ukraine, glinkate@gmail.

1 Communications on Stochastic Analysis Volume 11 Number 2 Article Spatial Ergodicity of the Harris Flows E.V. Glinyanaya Institute of Mathematics NAS of Ukraine, glinkate@gmail.com Follow this and additional works at: Part of the Analysis Commons, and the Other Mathematics Commons Recommended Citation Glinyanaya, E.V. 217 "Spatial Ergodicity of the Harris Flows," Communications on Stochastic Analysis: Vol. 11 : No. 2, Article 6. Available at: This Article is brought to you for free and open access by the Journals at LSU Digital Commons. It has been accepted for inclusion in Communications on Stochastic Analysis by an authorized editor of LSU Digital Commons. For more information, please contact gcoste1@lsu.edu.

2 Communications on Stochastic Analysis Vol. 11, No Serials Publications SPATIAL ERGODICITY OF THE HARRIS FLOWS E. V. GLINYANAYA Abstract. In the paper we consider the Harris flow of Brownian particles at fixed moment of time as a random process on R and investigate mixing property for it. 1. Intorduction In this article we study spatial properties of the Harris flows of Brownian particles on the real line. Such flows arose originally in the paper of T.E. Harris [8] as a model of continuum system of ordered Brownian motions with correlation that depends only on the distance between the particles. Let Γ be a real continuous positive definite function on R such that Γ = 1 and Γ is Lipshits outside any neighborhood of zero. Definition 1.1. The Harris flow with the local characteristic Γ is a family {xu,, u R} of Brownian martingales with respect to the joint filtration such that 1 for every u R 2 for every u 1 u 2 and t xu, = u; xu 1, t xu 2, t; 3 for every u 1, u 2 R the joint characteristic is d xu 1,, xu 2, t = Γxu 1, t xu 2, tdt. It was proved in [8] that such family exists and, moreover the function Γ defines its distribution uniquely. It is known that depending on the correlation function Γ Harris flow can consists of the continuous or step functions with respect to a spatial variable [11]. Different properties of the Harris flows were studied in [1, 1, 6, 7, 12, 5, 4]. Since the correlation depends only on a distance between two particles, then its distribution is invariant with respect to spatial shifts. In the present paper we prove that in the most interesting cases the Harris flow has even mixing property with respect to a spatial variable. Under the condition on local characteristic Γ: 1 u 1 Γu du < Received ; Communicated by A. A. Dorogovtsev. 21 Mathematics Subject Classification. Primary 6G1, 6G48; Secondary 6K35. Key words and phrases. Stochastic flows, mixing property. 223

3 224 E. V. GLINYANAYA the coalescence occurs in the Harris flow [11]. If for fixed time t the function x, t is a step function then we can consider a point process on R: N = δ xuj,t, u j D where D is the set of jumps of the function x, t see, for example, [3]. Then the questions about ergodicity and mixing properties of the point process N can be reduced to the questions about ergodicity and mixing properties of the Harris flow with respect to spatial variable. The article is organized as follows. In the first part we present the necessary definitions and reduce the studying of the mixing property of the Harris flow to the investigation of its n poing motions. In the second part we prove the uniqueness of the corresponding martingale problem and obtain the main result about the mixing property. The last part contains examples and applications. Note that invariant flows under a spatial transform in one- and multidimensional case was studied by C. Zirbel [15]. 2. n point Motions and Mixing Property Here we will consider the flow as a family of random mappings x, t from R to R. The second property in the definition of the Harris flow causes that x, t is nondecreasing function. Define the transformation T h, h R on a set of mappings from R to R by the rule: T h f = f + h h. As a consequence of the third condition from Definition 1.1 one can see that for every h R the family {T h xu, t, u R, t } has the same distribution as {xu, t, u R, t }. Precisely, we denote by MR a set of nondecreasing mappings from R to R with cylindric σ field C and let µ x be a measure on MR produced by the process x, t, i.e. for any k 1 and i BR, i {1,..., k} : µ x {f MR : fu i i, i {1,..., k}} = P{xu i, t i, i {1,..., k}}. In this terms the group of transformation {T h } preserves the measure µ x, i.e. µ x T 1 h = µ x, h R. By definition see, for example [2], the group of transformations {T h } h R has a mixing property with respect to the measure µ x if for any F 1, F 2 L 2 M, C, µ x lim h M F 1 T h ff 2 fµ x df = M F 1 fµ x df M F 2 fµ x df. 2.1 We note that it is enough to check 2.1 for any functions F 1, F 2 from the class E = {F : F f = exp{i n i=1 λ ifu i }, λ i, u i R, i = 1,..., m; m N}. For a point u R n the n dimensional process x u, t = xu 1, t, xu 2, t,..., xu n, t is called by an n point motion of the Harris flow. It follows from the above remark that it will suffice to consider an n point motions of the Harris flow to prove a mixing property. To this end we fix a point u R n+m with u i < u i+1, i = 1, 2,..., n + m and consider the n + m point motion of the Harris flow: Z h t = xu 1, t,..., xu n, t, xu n+1 + h, t h,..., xu n+m + h, t h.

4 SPATIAL ERGODICITY OF THE HARRIS FLOWS 225 From the definition of the Harris flow the process Z h characterizes by three properties: 1 Z h = u 1,..., u n+m ; 2 Zh i t Zi+1 h t; 3 for any i {1,..., n + m} Zh i is a Brownian martingale and for i, j {1,..., n} or i, j {n + 1,..., n + m} t Zh, i Z j h t = ΓZhs i Z j h sds, and for i {1,..., n} and j {n + 1,..., n + m} Z i h, Z j h t = t ΓZhs i Z j h s + hds. To obtain the mixing property of the Harris flow we will show that Z h has a weak limit as h and find out the properties of the limit. Theorem 2.1. Let Γ be such that Γu as u. Then Z h weakly converges in C[, 1] n+m to some martingale Z as h. Moreover, for any i {1,..., n + m} Zh i is a Brownian martingale and for i, j {1,..., n} or i, j {n + 1,..., n + m} Z i, Z j t = t and for i {1,..., n} and j {n + 1,..., n + m} ΓZs i Z j sds, 2.2 Z, i Z j t =. 2.3 Proof. Firstly, we note that the family {Z h } h> is weakly precompact set in C[, 1] n+m since each coordinate Zh i is a Wiener process. The next step is to prove that any limit point of the set {Z h } h is a martingale with quadratic characteristics 2.2 and 2.3. Since the process {Zh i t; t } is a martingale for any h, we have that for any k 1, for any continuous and bounded function f from R k to R and t 1 < t 2 <... < t k < s Ef Z i ht 1,..., Z i ht k Z i hs Z i ht k =. Let Z be some limit point, i.e. for h n Z hn Z in C[, 1] n+m, then Ef Zi t 1,..., Z t i k Zi s Z t i k = lim Ef Zh i n n t 1,..., Zh i n t n Zh i n s Zh i n t k =. The last passage to the limit is justified since Z i h s Zi h t k has the same distribution for any h with exponential moments. So we have the martingale property for any limit process of the set {Z h } h>. Next, for i, j {1,..., n} or i, j {n + 1,..., n + m} a process t h t = Zi htz j h t ΓZhs i Z j h sds m i,j

5 226 E. V. GLINYANAYA is a martingale. Since the mapping from C[, 1] n to C[, 1] Z Z i Z j ΓZ i s Z j sds is continuous, then {m i,j h } h> is weakly precompact in C[, 1] and has a weak limit m i,j. By the same arguments as it was done for Z, m i,j is a martingale and, moreover, m i,j t t = Z t i Z j t Γ Z s i Z j sds, where Z is some limit point of {Z h } h>. For the case when i {1,..., n} and j {n + 1,..., n + m} a process m i,j h = Zi htz j h t t ΓZhs i Z j h s + hds, is a martingale. Since Γu as u then we obtain that a weak limit of the set {m i,j h } h> is a martingale of the form m i,j = Z t i Z j t. So we get the properties 2.2 and 2.3 for any limit process of the set {Z h } h>. Theorem is proved. We characterized a weak limit of the set {Z h } h> as a martingale with characteristics given by 2.2 and 2.3. One can obtain a martingale with such properties if we take two independent Harris flows x, x with the same Γ and put Z t = xu 1, t,..., xu n, t, x u n+1, t,..., x u n+m, t. In the next section we prove that the process Z is the unique weak limit point of the set {Z h } h>. From this fact the mixing property for the Harris flow is follows. 3. Uniqueness of Solution to Martingale Problem We consider a limit point Z of the set {Z h } h> as a solution to a generalized martingale problem and obtain uniqueness of such solution. We give here some notations and results from [13] devoted to a generalized martingale problem. Let an operator L = 1 2 d a ij x + x i x j i,j=1 d i=1 b i x x i be defined on certain domain D R d where a is a function from D to the space of real symmetric non-negative definite d d matrices and b is a function from D to R d. Suppose that a and b are locally bounded on D. Let ˆD = D { } denote one-point compactification of D and let B ˆD denote the Borel subsets of ˆD [13]. Denote Ω D = {ω C[,, D} Let {D n } n=1 be an increasing sequence of bounded domains such that D n D n+1 and n=1 D n = D. Define τ Dn ω = inf{t : ωt / D n } τ n may be equal to infinity and let τ D = lim n τ Dn. Now define ˆΩ D = {ω C[,, ˆD : either τ D = or τ D < and ωτ D + t =, t > }

6 SPATIAL ERGODICITY OF THE HARRIS FLOWS 227 We consider ˆΩ D as a closed subset of C[,, ˆD, so it is a complete separable metric space. Let ˆF D denote the Borel σ algebra on ˆΩ D and define the filtration ˆF t D = σws, s t. Definition 3.1. [13], p. 42 A family of probability measures {P x } x ˆD on ˆΩ D, ˆF D such that for the process Xt, ω = ωt 1 P x X = x = 1; 2 fxt τ Dn t τ D n LfXsds is a martingale with respect to ˆΩ D, ˆF D, Fˆ D t, P x for all n = 1, 2,... and all f C 2 D is a solution to the generalized martingale problem for the operator L on the domain D. We use next result about existence and uniqueness of a solution to a generalized martingale problem. Theorem 3.2 [13], Theorem Let the coefficients a and b be locally bounded and measurable on D and assume that a is continuous on D and that d a i,j xv i v j > i,j=1 for x D and v R d \ {}. Then there exists a unique solution {P x, x ˆD} to the generalized martingale problem on D. Moreover, let a n = ψ n a + 1 ψ n I, b n = ψ n b, where ψ n : R d R is a C function satisfying ψ n x = 1 for x D n, ψ n x = for x / D n+1 and ψ n 1. Then P x ˆF D = P n x FτDn for n = 1, 2,... and x D, where τd n {P n x, x R d } denotes the unique solution to the martingale problem on R d for L n with coefficients a n and b n. The familty {P x, x D} posseses the Feller property and the family {P x, x ˆD} posesses the strong Markov property. The result of the next lemma follows from the previous theorem. Consider the operator L n,m = 1 n Γu i u j + 1 n+m Γu i u j 2 u i u j 2 u i u j i,j=1 i,j=n+1 on the domain D = {u R n+m : u 1 < u 2 <... < u n, u n+1 < u n+2 <... < u n+m }. Denote D k = {u R n+m : u k, u i < u i+1 1 k, i = 1, 2,..., n 1, n + 1,..., n + m}. Lemma 3.3. Let X 1 and X 2 be solution to generalized martingale problem for L n,m on D and put τ i = inf{x i t D}. Then τ 1, X s 1, s < τ 1 and τ 2, X s 2, s < τ 2 have the equal distribution. Proof. Statement of the lemma follows from the theorem 3.2. Denote by C a subset of ˆΩ D such that ω i t = ω i+1 t for i {1,..., n+m}\{n} implies ω i t+s = ω i+1 t+s for all s. The next lemma gives us the uniqueness

7 228 E. V. GLINYANAYA to the martingale problem for L n,m in R n+m and the proof is similar to lemma 3.2 in [8]. Lemma 3.4. For any u R n+m there exists a unique solution to the martingale problem for the operator L n+m in R n+m, such that P u C = 1. Proof. We will use the method of mathematical induction with respect to n and m. For n = m = 1 we have the operator L 1,1 = The two-dimensional u 2 1 u 2 2 standard Wiener process is the unique solution for the martingale problem for L 1,1 in R 2. Assume that the statement of the theorem is true for n and m 1 and prove it for n and m. Let us fix a point u = u 1,..., u n, u n+1,..., u n+m, where u i u i+1 for i {1,..., n + m} \ {n}. Firstly we assume that the point u such that u i = u i+1 for some i {n + 1,..., n + m}. Note that L n,m fu 1,..., u n,..., u n+m = L n,m 1 gu 1,..., u n, u n+1,..., u i, u i+2,..., u n+m, where g : R n+m 1 R is defined as follows: Indeed, gx 1,..., x n, x n+1,..., x n+m 1 = fx 1,..., x n, x n+1,..., x i, x i, x i+1,..., x n+m 1. L n,m 1 gu 1,..., u n, u n+1,..., u i, u i+2,..., u n+m = 1 n Γu l u j g u l u j 2 = l,j=1 l,j {n+1,...,n+m}\{i+1} n Γu l u j fu 1,..., u i, u i, u i+2,..., u n+m u l u j l,j=1 l,j {n+1,...,n+m}\{i,i+1} + 1 f ii + f i+1,i+1 + f i+1,i + f 2 i+1,i = L n,m fu 1,..., u i, u i, u i+2,..., u n+m. Γu l u j g u l u j Γu l u j fu 1,..., u i, u i, u i+2,..., u n+m u l u j This implies that a process X 1 t,..., X n t,..., X n+m t is a solution for the martingale problem for L n,m with start point u if and only if a process X 1 t,..., X n t,..., X n+m 1 t is a solution for the martingale problem for L n,m 1 with start point u 1,..., u n,..., u i, u i+2,..., u n+m. Uniqueness for solution follows from inductive hypothesis. Now we assume that the point u = u 1,..., u n, u n+1,..., u n+m such that u i < u i+1 for all i {1,..., n + m} \ {n}. Denote H = { v R n+m : v i v i+1, i {1,..., n + m} \ {n}, j {1,..., n + m} \ {n} : v j = v j+1 }. Let P u be a solution for the martingale problem for L n,m on D with start point u and let {X t, t } be a corresponding process. Put τ = inf{t : X t H}.

8 SPATIAL ERGODICITY OF THE HARRIS FLOWS 229 Since X τ H, by previous case there exists unique law P X τ that solve the martingale problem for L n,m with start point X τ. Now we denote as P u a distribution of the process X that has the same law as X up to the moment τ and {Xτ + s, s }, conditioned by σ field F τ is given by the law P Xτ. Then from [14], Theorem 6.1.2, P u solve the martingale problem for L n,m with start from u and P u C = 1. From previous lemma, any solution X starting from u is governed by P u until reaching H. The distribution of X τ + s, s, conditioned on F τ solves the martingale problem with start from X τ, and hence has the law P X τ. It follows that X has the same law as X. Lemma is proved. 4. Mixing Coefficients for Harris Flow Let us consider the Harris flow {xu, t, u R, t [, 1]} with local characteristic Γ with suppγ [ c, c] for some constant c >. For the random process {xu,, u R} in C[, 1] we will find out an estimation for the strong mixing coefficient which is defined as αh = sup{ PA B PAPB, A F u, B F u+h, u R}, where F v u = σ{xw,, w [u, v]}. Denote D c = {u R n+m : u i < u i+1, i = 1, 2,..., n + m, u n+1 u n > 2c} and for u D c consider next two processes: X 1 t = xu 1, t, xu 2, t,..., xu n+m, t, X 2 t = x u 1, t, x u 2, t,..., x u n, t, x u n+1, t, x u n+2, t..., x u n+m, t and denote τ 1 = inf{t : xu n+1, t xu n, t = c} and τ 2 = inf{t : x u n+1, t x u n, t = c}. Lemma 4.1. The processes τ 1, X 1 t, t < τ 1 and τ 1, X 2 t, t < τ 2 have the equal distribution. Proof. It is easy to see that the processes X 1 and X 2 satisfy the generalized martingale problem for the operator L n,m = 1 n Γu i u j + 1 n+m Γu i u j 2 u i u j 2 u i u j i,j=1 i,j=n+1 in the domain D c. The proof of this lemma is the same as the proof of lemma 3.4. Lemma 4.2. Let suppγ [ c, c]. Then for h > c 2 αh 2 e x2 /2 dx π Proof. Let h > c. For the proof it is sufficient to consider sets A F u, B F u+h of the form: A = {xu i, t i I i, i = 1, 2,..., n} h c

9 23 E. V. GLINYANAYA and B = {xv j, s j J j, j = 1,..., m}, where u 1 <... < u n < u, u + h < v 1 <... < v m, and t i, s j [, 1], I i, J i BR for i = 1,..., n, j = 1,..., m. We denote by τ = inf{s : x u n, s = x v 1, s + c}. For such sets A, B, using the previous lemma we get: PA B = P A B {max t i, s j < τ} + P A B {max t i, s j τ} i,j i,j = P x u i, t i I i, i = 1,..., n, x v j, s j J j, j = 1,..., m, max t i, s j < τ i,j + P A B {max t i, s j τ} i,j = P x u i, t i I i, i = 1,..., n, x v j, s j J j, j = 1,..., m P x u i, t i I i, i = 1,..., n, x v j, s j J j, j = 1,..., m, max t i, s j τ i,j + P A B {max t i, s j τ}. i,j Using independence between processes x, x : PA B PAPB 2P{max t i, s j τ} i,j = Pτ e x2 /2 dx. π Since ergodicity follows from the mixing property we can deduce asymptotic properties of the Harris flow with respect to spatial parameter. If we assume that for fixed time t the set {xu, t,u [, 1]} is finite with probability 1, we can conclude from ergodicity that lim U h c ν [,U] U = Eν [,1] 1, where ν [,U] = #{xu, t,u [, U]} is the number of clusters in the Harris flow at the time t = 1. For the Arratia flow, i.e. for the flow with Γ = 1I {} [1] we obtain: ν [,U] 2 lim U U = π. References 1. Arratia, R.: Brownian motions of the real line, PhD diss., Univ. Wisconsin, Cornfeld, I. P., Fomin, S. V., and Sinai, Ya. G.: Ergodic Theory, Springer, New York, Daley, D. J., Vere-Jones, D.: An Introduction to the Theory of Point Processes, Volume II: General Theory and Structure Springer, New York, Dorogovtsev, A. A., Fomichov, V. V.: The rate of week convergence of the n-point motions of Harris flows, Dynam. Syst. Appl , no. 3,

10 SPATIAL ERGODICITY OF THE HARRIS FLOWS Dorogovtsev, A. A., Gnedin, A. V., and Vovchanskii, M. B.: Iterated logarithm law for sizes of clusters in Arratia flow, Theory of Stochastic Processes, , no. 2, Dorogovtsev, A. A. and Nishchenko, I. I.: An analysis of stochastic flows, Com. Stoch. Analysis 8214, no. 3, Dorogovtsev, A. A. and Ostapenko, O..: Large deviations for flows of interacting Brownian motions, Stoch and Dynam. 1 21, no. 3, Harris, T.E.: Coalescing and noncoalescing stochastic flows in R 1, Stoch. Process and their Applic , Kallenberg, O.: Foundation of Modern Probability, Springer, New York, Le Jan, Y. and Raimond, O.: Flows, coalescence and noise, Ann. Probab , Matsumoto, H.: Coalescting stochastic flows on the real line, Osaka J. Math , no. 1, Nishchenko, I. I.: Discrete time approximation of coalescing stochastic flows on the real line, Theory of Stochastic Processes , no. 1, Pinsky, R. G.: Positive Harmonic Functions and Diffusion, Cambridge Univ. Press, Stroock, D., Varadhan, S. R. S.: Multidimesional Diffusion Processes, Springer, Berlin, New York, Zirbel, C. L.: Stochastic flows: dispersion of a mass distribution and lagrangian observations of a random field, PhD dissertation, Princeton university, E. V. Glinyanaya: Department of the Theory of Stochastic Processes, Institute of Mathematics NAS of Ukraine, Kyiv, Ukraine address: glinkate@gmail.com URL:

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