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Mathematical Model of Coalescing Diffusion Particles System With Coalescing particles system Coalescing particles system with variable weights Markov property and martingale problem Stationary case Heavy diusion particles system with drift Mathematical model of coalescing diusion particles system with variable weights Vitalii Konarovskyi Yuriy Fedkovych Chernivtsi National University Jena, 2013 [email protected] 1 / 47 Coalescing particles system Coalescing particles system with variable weights Markov property and martingale problem Stationary case Heavy diusion particles system with drift Main object of investigation System of diusion particles on the real line that 1 start from some set of points with masses; 2 move independently up to the moment of the meeting; 3 coalesce; 4 have their mass adding after sticking; 5 have their diusion changed correspondingly to the changing of the mass ( 2 1 ). σ = m 2 / 47 Coalescing particles system Coalescing particles system with variable weights Markov property and martingale problem Stationary case Heavy diusion particles system with drift Interaction particles systems. Singular interaction Aratia ow Arratia R. A. '79 3 / 47 Coalescing particles system Coalescing particles system with variable weights Markov property and martingale problem Stationary case Heavy diusion particles system with drift Interaction particles systems. Singular interaction Arratia ow, mathematical description fx(u; t); t ≥ 0; u 2 Rg such that 1 x(u; ·) is a Brownian motion; 2 x(u; 0) = u, u 2 R; 3 x(u; t) ≤ x(v; t), u < v, t ≥ 0; 4 hx(u; ·); x(v; ·)it = 0; t < τu;v; where τu;v = infft : x(u; t) = x(v; t)g. 4 / 47 Coalescing particles system Coalescing particles system with variable weights Markov property and martingale problem Stationary case Heavy diusion particles system with drift Total time of runs of the particles Let a set of dierent points be dense in [0,U]. fuk; k 2 Ng τ1 = 1; 8 k−1 9 < Y = τk = inf 1; t : (x(uk; t) − x(uj; t)) = 0 ; k ≥ 2: : j=1 ; 5 / 47 Coalescing particles system Coalescing particles system with variable weights Markov property and martingale problem Stationary case Heavy diusion particles system with drift Total time of runs of the particles Theorem 1 1 The sum P is nite a.s. and does not depend on the set . τn fuk; k 2 Ng n=1 Theorem 2 For all t the set fx(u; t); u 2 [0;U]g has a nite number of dierent points. Dorogovtsev A. A. '04 6 / 47 Coalescing particles system Coalescing particles system with variable weights Markov property and martingale problem Stationary case Heavy diusion particles system with drift Special stochastic integral with respect to Arratia ow Theorem 3 Let a : R ! R be a measurable bounded function then the series 1 τn X Z a(x(un; s))dx(un; s) n=1 0 is convergent in L2 and their sum does not depend on the set fuk; k 2 Ng. Denote τ(u) U 1 τn Z Z X Z a(x(u; s))dx(u; s) = a(x(un; s))dx(un; s): 0 0 n=1 0 Theorem 4 U τ(u) The random processes R R a(x(u; s))dx(u; s);U ≥ 0, is a σ(x(u; ·); u 2 [0;U])-martingale. 0 0 Dorogovtsev A. A. '06 7 / 47 Coalescing particles system Coalescing particles system with variable weights Markov property and martingale problem Stationary case Heavy diusion particles system with drift Girsanov theorem for diusion processes with coalescence Let a : R ! R be a bounded Lipschitz continuity function Diusion ow with coalescing fy(u; t); t ≥ 0; u 2 Rg such that · R 1 M(u; ·) = y(u; ·) − a(y(u; s))ds is a Brownian motion, for all u 2 R; 0 2 y(u; 0) = u, u 2 R; 3 y(u; t) ≤ y(v; t), u < v, t ≥ 0; 4 hM(u; ·); M(v; ·)it = 0; t < σu;v; where σu;v = infft : y(u; t) = y(v; t)g. 8 / 47 Coalescing particles system Coalescing particles system with variable weights Markov property and martingale problem Stationary case Heavy diusion particles system with drift Girsanov theorem for diusion processes with coalescence Theorem 5 The distribution of y is absolutely continuous with respect to the distribution of x in the space D([0;U]; C([0; 1])) with the density 8 U τ(u) U τ(u) 9 <>Z Z 1 Z Z => p(x) = exp a(x(u; s))dx(u; s) − a(x(u; s))ds : > 2 > :0 0 0 0 ; Dorogovtsev A. A. '07 9 / 47 Coalescing particles system Coalescing particles system with variable weights Markov property and martingale problem Stationary case Heavy diusion particles system with drift Asymptotic behaviour Law of the iterated logarithm for a Wiener process jx(u; t) − uj lim p = 1 a.s. t!0 2t ln ln t−1 Asymptotic in the sup-norm jx(u; t) − uj lim sup p = 1 a.s. t!0 u2[0;1] t ln t−1 Shamov A. '10 Asymptotic of cluster size Let ν(t) = λfu : x(u; t) = x(0; t)g; t ≥ 0. Then a.s. ν(t) ν(t) lim p ≥ 1; lim p ≤ 1: t!0 2t ln ln t−1 t!0 2 t ln ln t−1 Dorogovtsev A. A., Vovchanskii M. B. '13 10 / 47 Coalescing particles system Coalescing particles system with variable weights Markov property and martingale problem Stationary case Heavy diusion particles system with drift Brownian web Brownian web System of brownian particles which 1 start from all time-space point in R × R; 2 move independently up to the moment of the meeting; 3 coalesce; Fontes L. R. G., Isopi M., Newman C. M., Ravishankar K. '04 11 / 47 Coalescing particles system Coalescing particles system with variable weights Markov property and martingale problem Stationary case Heavy diusion particles system with drift Interaction particles systems. Singular interaction Coalescing system of non-Brownian particles System of particles in a metric space such that 1 every particle is described by the Markov process; 2 every two particles move independently up to the moment of the meeting; 3 particles coalesce. Le Jan Y. '04, Evens S. N. '12 12 / 47 Coalescing particles system Coalescing particles system with variable weights Markov property and martingale problem Stationary case Heavy diusion particles system with drift Interaction particles systems. Singular interaction Main property such systems All subsystem of such system may be described as a separate system 13 / 47 Coalescing particles system Coalescing particles system with variable weights Markov property and martingale problem Stationary case Heavy diusion particles system with drift Interaction particles systems. Systems concerning particles masses - Coalescing Brownian particles which have some masses and these masses vary by the some law. The mass does not inuence motion of the particles Dawson D. A. '01, '04. - Stochastic dierential equation with interaction ( R dx(u; t) = a(x(u; t); µt; t)dt + b(x(u; t); µt; t; q)W (dt; dq) R −1 x(u; 0) = u; µt = µ0 ◦ x(·; t) : Dorogovtsev A. A. '07. - The cases of nite and innite numbers of particles with sticking that have mass and speed and their motion obey the laws of mass conservation and inertion Sinai Ya. G. '96. 14 / 47 Coalescing particles system Coalescing particles system with variable weights Markov property and martingale problem Stationary case Heavy diusion particles system with drift Coalescing particles system with variable weights xk(0) = k; mk(0) = 1; k 2 Z 15 / 47 Coalescing particles system Coalescing particles system with variable weights Markov property and martingale problem Stationary case Heavy diusion particles system with drift Construction of nite system fwk(t); t ≥ 0; k = −n; : : : ; ng is a system of Wiener processes n n x (·) = Fn(w (·)); n where w (·) = (−n + w−n(·); : : : ; n + wn(·)). 16 / 47 Coalescing particles system Coalescing particles system with variable weights Markov property and martingale problem Stationary case Heavy diusion particles system with drift Properties of nite system Theorem 6 The process xn(t); t ≥ 0 satises the following conditions ◦) n is a continuous square integrable martingale with respect to the 1 xk (·) ltration n n Ft = σ(xi (s); s ≤ t; i = −n; : : : ; n); ◦) n , ; 2 xk (0) = k k = −n; : : : ; n ◦) n n , , ; 3 xk (t) ≤ xl (t) k < l t ≥ 0 t ◦) n R 1 , , 4 hxk (·)it = mn(s) ds t ≥ 0 0 k where n n n ; mk (t) = jfi : 9s ≤ t xi (s) = xk (s)gj ◦) n n n , 5 hxk (·); xl (·)it = 0; t < τk;l where n n . τk;l = infft : xk (t) = xl (t)g Moreover, if a process y(·) satises the conditions 1◦)5◦) then the processes y(·) and xn(·) have the same distributions in the space (C([0; 1)))2n+1. 17 / 47 Coalescing particles system Coalescing particles system with variable weights Markov property and martingale problem Stationary case Heavy diusion particles system with drift Some property of a system of Wiener processes Lemma 7 Let fwk(t); t ≥ 0; k = 0; 1;:::g be a system of independent Wiener processes. Dene T T ξk = max fk + wk(t)g ; ηk = min fk + wk(t)g : t2[0;T ] t2[0;T ] Then for every T > 0 and δ 2 (0; 1) T T P lim max ξk < n + δ; ηn+1 > n + δ = 1: n!1 k=1;:::;n 18 / 47 Coalescing particles system Coalescing particles system with variable weights Markov property and martingale problem Stationary case Heavy diusion particles system with drift Stabilization fwk(t); t ≥ 0; k = 0; 1;:::g is a xed system of Wiener processes. n n x (·) = Fn(w (·)), n N for all P 9N 8n ≥ N 8t 2 [0;T ] xk (t) = xk (t) = 1; k 2 Z The sequence n converges almost surely in C .
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