Motion with interaction of a particle system with infinite total - PowerPoint PPT Presentation

Motion with interaction of a particle system with infinite total mass Maksym Tantsiura Institute of Mathematics NAS of Ukraine Supervisor: Andrey Pilipenko Maksym Tantsiura Motion of interacting particle system with infinite total mass

Let (Ω , ℑ , ( ℑ t , t ≥ 0) , P ) be a filtered probability space, w k ( t ) be independent ℑ t -adapted Wiener processes. Suppose that { u k | k ∈ Z } is a nondecreasing numerical sequence such that lim k → + ∞ u k = + ∞ , lim k →−∞ u k = −∞ . Consider the following infinite system of SDEs  dX k ( t ) = a ( X k ( t ) , µ t ) dt + dw k ( t ) , k ∈ Z , t ∈ [0 , T ] ,  µ t = � k ∈ Z δ X k ( t ) , (1) X k (0) = u k , k ∈ Z .  Maksym Tantsiura Motion of interacting particle system with infinite total mass

Space of measures Denote by M the space of all locally finite measures on R with a vague topology τ defined by � � τ ν n → ν ⇔ ∀ f ∈ C c ( R ) : fdν n → fdν, n → ∞ , (2) R R where C c ( R ) is a set of all continuous functions with compact support(see Dawson ’91). Maksym Tantsiura Motion of interacting particle system with infinite total mass

Theorem 1. Suppose that 1. a is a measurable and bounded function: � a � ∞ := sup sup | a ( x, ν ) | < ∞ ; x ∈ R ν ∈ M 2. a finite interaction radius condition is satisfied: ∃ d > 0 ∀ x ∈ R ∀ ν ∈ M : a ( x, ν ) = a ( x, I ( x − d,x + d ) ν ) , where ( I B ν )( A ) = ν ( A ∩ B ) , A, B ∈ B ( R ); 3. there a.s. exists a random sequence { y n | n ∈ Z } such that ∀ n ∈ Z : i : u i ≥ y n min inf t ∈ [0 ,T ] ( u i + w i ( t ) ∧ 0) − sup t ∈ [0 ,T ] ( u i + w i ( t ) ∨ 0) ≥ 2 � a � ∞ T + d. max i : u i <y n Then there exists a unique solution of the equation (1). Maksym Tantsiura Motion of interacting particle system with infinite total mass

Sketch of the proof Veretennikov’s theorem(Veretennikov, ’80) yields that if a function b : R d × [0 , T ] → R d is bounded and measurable then stochastic differential equation � dY ( t ) = b ( t, Y ( t )) dt + dW ( t ) , t ∈ [0 , T ] , (3) Y (0) = Y 0 has a unique strong solution. Here W ( t ) , t ∈ [0 , T ] , is a Wiener process in R d . For every n ∈ N consider a system of equations dX n i ( t ) = a ( X n i ( t ) , µ n  t ) dt + dw i ( t ) , − n ≤ i ≤ n, t ∈ [0 , T ] ,  dX n  i ( t ) = 0 , | i | > n,  (4) µ n t = � k ∈ Z δ X n k ( t ) ,   X n i (0) = u i , i ∈ Z .  Maksym Tantsiura Motion of interacting particle system with infinite total mass

Denote � τ n 1 ,n 2 = inf t ∈ [0 , T ] | ∃ k 1 ∈ ( n 1 , n 2 ] ∃ k 2 / ∈ ( n 1 , n 2 ] | u k 1 + w k 1 ( t ) − u k 2 − w k 2 ( t ) | ∨ | u k 1 − u k 2 − w k 2 ( t ) |∨ � ∨| u k 1 + w k 1 ( t ) − u k 2 | ∨ | u k 1 − u k 2 | ≤ 2 � a � ∞ T + d ∧ T. It can be proved that ∀ t ∈ [0 , τ n 1 ,n 2 ] ∀ n ≥ | n 1 | ∨ | n 2 | : X | n 1 |∨| n 2 | ( t ) = X n i ( t ) . (5) i It follows from condition 3 of the theorem that a.s. ∀ k ∈ Z ∃ n 1 < k ∃ n 2 ≥ k : τ n 1 ,n 2 = T. (6) The unique solution can be obatined as the limit n →∞ X n X k ( t ) = lim k ( t ) , k ∈ Z . (7) Maksym Tantsiura Motion of interacting particle system with infinite total mass

Denote � ∞ 1 exp ( − y 2 / 2 t ) dy, x ∈ R , p w ( t, x ) = P ( sup w ( s ) ≥ x ) = 2 √ (8) 2 πt s ∈ [0 ,t ] x ∨ 0 where w is a Wiener process in R . Theorem 2. Suppose that there exists a deterministic increasing sequence { z n | n ∈ Z } such that: 1. lim n → + ∞ z n = + ∞ , lim n →−∞ z n = −∞ . 2. ∃ ε 1 > 0 ∀ n ∈ Z : � � � 1 − p w ( T, | z n − u i | − � a � ∞ T − d/ 2) > ε 1 i ∈ Z Then there a.s. exists a random sequence { y n | n ∈ Z } such that ∀ n ∈ Z : i : u i ≥ y n min inf t ∈ [0 ,T ] ( u i + w i ( t ) ∧ 0) − sup t ∈ [0 ,T ] ( u i + w i ( t ) ∨ 0) ≥ 2 � a � ∞ T + d. max i : u i <y n Maksym Tantsiura Motion of interacting particle system with infinite total mass

Denote ξ k = max t ∈ [0 ,T ] ( u k + w k ( t )) , (9) η k = min t ∈ [0 ,T ] ( u k + w k ( t )) , (10) and A k = { sup ξ i ≤ z k − d/ 2 − � a � ∞ T, i : u i >z k η i ≥ z k + d/ 2 + � a � ∞ T } . (11) inf i : u i <z k To prove the theorem it is sufficient to verify that P (lim sup A k ) = 1 (12) k → + ∞ and P (lim sup A k ) = 1 . (13) k →−∞ Events A k are dependent, so the second Borel-Cantelli lemma can’t be directly applied. The idea of the proof is to approximate events from some subsequence A k n by independent events A ′ k n . Maksym Tantsiura Motion of interacting particle system with infinite total mass

Theorem 3. Let µ 0 be a Poisson point measure with intensity m. Suppose that µ 0 is independent from { w k , k ∈ Z } and ∃ C m ∀ [ a, b ] ⊂ R : m ([ a, b ]) ≤ C m ( b − a + 1) . If conditions 1 and 2 of Theorem 1 are satisfied then there exist a unique strong solution of the equation (1) for every T > 0 . Maksym Tantsiura Motion of interacting particle system with infinite total mass

For a locally finite measure ν denote ν ([ − n, n ]) Λ( ν ) := lim sup . 2 n n →∞ The value Λ( ν ) is an upper bound for the “average density” of the measure’s ν atoms. For any λ > 0 denote M λ = { ν | Λ( ν ) ≤ λ } . Theorem 4. Suppose that µ 0 = � k ∈ Z δ u k ∈ M λ with λd < 1 and conditions 1 and 2 of Theorem 1 are satisfied. Then there exists a unique strong solution of the equation (1) for any T > 0 . Maksym Tantsiura Motion of interacting particle system with infinite total mass

Sketch of the proof Denote A T = � a � ∞ T + d/ 2 , f T ( x ) = p w ( T, | x | − A T ) ∧ 1 / 2 , � S T ( x ) = f T ( x − u i ) , i ∈ Z It can be proved that for some small T 0 > 0 � n − n S T 0 ( x ) dx lim sup < 1 / 2 . (14) 2 n n →∞ Note that ln(1 − y ) ≥ − 2 y, y ∈ [0 , 1 / 2] . Hence if S T 0 ( x ) < 1 / 2 , then � � ln(1 − p w ( T 0 , | u i − x |− A T 0 )) ≥ − 2 p w ( T 0 , | u i − x |− A T 0 ) = − 2 S T 0 ( x ) ≥ − 1 . i ∈ Z i ∈ Z Lemma. Let { X k ( t ) | t ∈ [0 , T ] , k ∈ Z } be a solution of the equation (1), Λ( µ 0 ) < + ∞ . Then P ( ∀ t ∈ [0 , T ] : Λ( µ 0 ) = Λ( µ t )) = 1 . Maksym Tantsiura Motion of interacting particle system with infinite total mass