METEOR PROCESS Krzysztof Burdzy University of Washington Krzysztof Burdzy METEOR PROCESS
Collaborators and preprints Joint work with Sara Billey, Soumik Pal and Bruce E. Sagan. Math Arxiv: http://arxiv.org/abs/1308.2183 http://arxiv.org/abs/1312.6865 Krzysztof Burdzy METEOR PROCESS
Mass redistribution Krzysztof Burdzy METEOR PROCESS
Mass redistribution (2) space time Krzysztof Burdzy METEOR PROCESS
Related models Chan and Pra� lat (2012) Crane and Lalley (2013) Ferrari and Fontes (1998) Fey-den Boer, Meester, Quant and Redig (2008) Howitt and Warren (2009) Krzysztof Burdzy METEOR PROCESS
Model G - simple connected graph (no loops, no multiple edges) V - vertex set M x t - mass at x ∈ V at time t Assumption: M x 0 ∈ [0 , ∞ ) for all x ∈ V M t = { M x t , x ∈ V } N x t - Poisson process at x ∈ V The Poisson processes are assumed to be independent. The “meteor hit” (mass redistribution event) occurs at a vertex when the corresponding Poisson process jumps. Krzysztof Burdzy METEOR PROCESS
Existence of the process THEOREM If the graph has a bounded degree then the meteor process is well defined for all t ≥ 0. Krzysztof Burdzy METEOR PROCESS
General graph - stationary distribution Example . Suppose that G is a triangle. The following are possible mass process transitions. (1 , 2 , 0) → (0 , 5 / 2 , 1 / 2) Krzysztof Burdzy METEOR PROCESS
General graph - stationary distribution Example . Suppose that G is a triangle. The following are possible mass process transitions. (1 , 2 , 0) → (0 , 5 / 2 , 1 / 2) (1 , π/ 2 , 2 − π/ 2) → (1 + π/ 4 , 0 , 2 − π/ 4) Krzysztof Burdzy METEOR PROCESS
General graph - stationary distribution Example . Suppose that G is a triangle. The following are possible mass process transitions. (1 , 2 , 0) → (0 , 5 / 2 , 1 / 2) (1 , π/ 2 , 2 − π/ 2) → (1 + π/ 4 , 0 , 2 − π/ 4) The state space is stratified. Krzysztof Burdzy METEOR PROCESS
Stationary distribution - existence and uniqueness THEOREM Suppose that the graph is finite. The stationary distribution for the process M t exists and is unique. The process M t converges to the stationary distribution exponentially fast. Proof (sketch). Consider two mass processes M t and � M t on the same graph, with different initial distributions but the same meteor hits. Krzysztof Burdzy METEOR PROCESS
Stationary distribution - existence and uniqueness THEOREM Suppose that the graph is finite. The stationary distribution for the process M t exists and is unique. The process M t converges to the stationary distribution exponentially fast. Proof (sketch). Consider two mass processes M t and � M t on the same graph, with different initial distributions but the same meteor hits. � � t → � � � t − � � M x M x � is non-increasing. x ∈ V t Krzysztof Burdzy METEOR PROCESS
Stationary distribution - existence and uniqueness THEOREM Suppose that the graph is finite. The stationary distribution for the process M t exists and is unique. The process M t converges to the stationary distribution exponentially fast. Proof (sketch). Consider two mass processes M t and � M t on the same graph, with different initial distributions but the same meteor hits. � � t → � � � t − � � M x M x � is non-increasing. x ∈ V t Hairer, Mattingly and Scheutzow (2011) Krzysztof Burdzy METEOR PROCESS
Circular graphs 8 5 7 6 9 4 10 1 2 3 C k - circular graph with k vertices Q k - stationary distribution for M t on C k Krzysztof Burdzy METEOR PROCESS
Circular graphs - moments of mass at a vertex 8 5 7 6 9 4 10 1 2 3 THEOREM E Q k M x 0 = 1 , x ∈ V , k ≥ 1 Krzysztof Burdzy METEOR PROCESS
Circular graphs - moments of mass at a vertex 8 5 7 6 9 4 10 1 2 3 THEOREM E Q k M x 0 = 1 , x ∈ V , k ≥ 1 k →∞ Var Q k M x lim 0 = 1 , x ∈ V Krzysztof Burdzy METEOR PROCESS
Circular graphs - moments of mass at a vertex 8 5 7 6 9 4 10 1 2 3 THEOREM E Q k M x 0 = 1 , x ∈ V , k ≥ 1 k →∞ Var Q k M x lim 0 = 1 , x ∈ V 0 , M x +1 k →∞ Cov Q k ( M x lim ) = − 1 / 2 , x ∈ V 0 Krzysztof Burdzy METEOR PROCESS
Circular graphs - moments of mass at a vertex 8 5 7 6 9 4 10 1 2 3 THEOREM E Q k M x 0 = 1 , x ∈ V , k ≥ 1 k →∞ Var Q k M x lim 0 = 1 , x ∈ V 0 , M x +1 k →∞ Cov Q k ( M x lim ) = − 1 / 2 , x ∈ V 0 0 , M y k →∞ Cov Q k ( M x lim 0 ) = 0 , x �↔ y Krzysztof Burdzy METEOR PROCESS
Circular graphs - correlation and independence 8 5 7 6 9 4 10 1 2 3 0 , M y k →∞ Cov Q k ( M x lim 0 ) = 0 , x �↔ y 0 and M y If x �↔ y then M x 0 do not appear to be asymptotically independent under Q k ’s. 0 ) 2 and M y ( M x 0 seem to be asymptotically correlated under Q k , if x �↔ y . Krzysztof Burdzy METEOR PROCESS
From circular graphs to Z C k - circular graph with k vertices Q k - stationary distribution for M t on C k THEOREM For every fixed n , the distributions of ( M 1 0 , M 2 0 , . . . , M n 0 ) under Q k converge to a limit Q ∞ as k → ∞ . The theorem yields existence of a stationary distribution Q ∞ for the meteor process on Z . Similar results hold for meteor processes on C d k and Z d . Krzysztof Burdzy METEOR PROCESS
Moments of mass at a vertex in Z d THEOREM E Q ∞ M x 0 = 1 , x ∈ V Var Q ∞ M x 0 = 1 , x ∈ V 0 ) = − 1 0 , M y Cov Q ∞ ( M x 2 d , x ↔ y 0 , M y Cov Q ∞ ( M x 0 ) = 0 , x �↔ y Krzysztof Burdzy METEOR PROCESS
Mass fluctuations in intervals of Z THEOREM For every n , � M j E Q ∞ 0 = n , 1 ≤ j ≤ n Krzysztof Burdzy METEOR PROCESS
Mass fluctuations in intervals of Z THEOREM For every n , � M j E Q ∞ 0 = n , 1 ≤ j ≤ n � M j Var Q ∞ 0 = 1 . 1 ≤ j ≤ n Krzysztof Burdzy METEOR PROCESS
Flow across the boundary x x+1 F x t - net flow from x to x + 1 between times 0 and t Krzysztof Burdzy METEOR PROCESS
Flow across the boundary x x+1 F x t - net flow from x to x + 1 between times 0 and t THEOREM Under Q ∞ , for all x ∈ Z and t ≥ 0, Var F x t ≤ 4 . Krzysztof Burdzy METEOR PROCESS
Mass distribution at a vertex of Z A simulation of M x 0 under Q ∞ . 20000 15000 10000 5000 0 1 2 3 4 5 6 Krzysztof Burdzy METEOR PROCESS
Mass distribution at a vertex of Z (2) 20000 15000 10000 5000 0 1 2 3 4 5 6 P Q ∞ ( M x 0 = 0) = 1 / 3 E Q ∞ M x Var Q ∞ M x 0 = 1 , 0 = 1 Is Q ∞ a mixture of a gamma distribution and an atom at 0? No. Krzysztof Burdzy METEOR PROCESS
Mass distribution at a vertex of Z (2) 20000 15000 10000 5000 0 1 2 3 4 5 6 P Q ∞ ( M x 0 = 0) = 1 / 3 E Q ∞ M x Var Q ∞ M x 0 = 1 , 0 = 1 Is Q ∞ a mixture of a gamma distribution and an atom at 0? No. 0 ) n for every n and k . One can find an exact and rigorous value for E Q k ( M x We cannot find asymptotic formulas when k → ∞ . Krzysztof Burdzy METEOR PROCESS
Support of the stationary distribution Assume that | V | = k , and � x ∈ V M x 0 = k . Let S be the simplex consisting of all { S x , x ∈ V } with S x ≥ 0 for all x ∈ V and � x ∈ V S x = k . Krzysztof Burdzy METEOR PROCESS
Support of the stationary distribution Assume that | V | = k , and � x ∈ V M x 0 = k . Let S be the simplex consisting of all { S x , x ∈ V } with S x ≥ 0 for all x ∈ V and � x ∈ V S x = k . Let S ∗ be the set of { S x , x ∈ V } with S x = 0 for at least one x ∈ V . Krzysztof Burdzy METEOR PROCESS
Support of the stationary distribution Assume that | V | = k , and � x ∈ V M x 0 = k . Let S be the simplex consisting of all { S x , x ∈ V } with S x ≥ 0 for all x ∈ V and � x ∈ V S x = k . Let S ∗ be the set of { S x , x ∈ V } with S x = 0 for at least one x ∈ V . THEOREM The (closed) support of the stationary distribution for M t is equal to S ∗ . Krzysztof Burdzy METEOR PROCESS
WIMPs DEFINITION Suppose that M 0 is given and k = � v ∈ V M v 0 . For each j ≥ 1, let { Y j n , n ≥ 0 } be a discrete time symmetric random walk on G with the initial distribution P ( Y j 0 = x ) = M x 0 / k for x ∈ V . We assume that conditional on M 0 , processes { Y j n , n ≥ 0 } , j ≥ 1, are independent. Krzysztof Burdzy METEOR PROCESS
WIMPs DEFINITION Suppose that M 0 is given and k = � v ∈ V M v 0 . For each j ≥ 1, let { Y j n , n ≥ 0 } be a discrete time symmetric random walk on G with the initial distribution P ( Y j 0 = x ) = M x 0 / k for x ∈ V . We assume that conditional on M 0 , processes { Y j n , n ≥ 0 } , j ≥ 1, are independent. Recall Poisson processes N v and assume that they are independent of { Y j n , n ≥ 0 } , j ≥ 1. For every j ≥ 1, we define a continuous time Markov process { Z j t , t ≥ 0 } by requiring that the embedded discrete Markov chain for Z j is Y j and Z j jumps at a time t if and only if N v jumps at time t , where v = Z j t − . Krzysztof Burdzy METEOR PROCESS
WIMPs and convergence rate The rate of convergence to equilibrium for M t cannot be faster than that for a simple random walk. Justification: Consider expected occupation measures. Krzysztof Burdzy METEOR PROCESS
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