STOCHASTIC DIFFERENTIAL EQUATIONS WITH INTERACTION AND THE LAW OF ITERATED LOGARITHM

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1 Theory of Stochastic Processes Vol. 18 (34, no. 2, 212, pp M. P. LAGUNOVA STOCHASTIC DIFFEENTIAL EQUATIONS WITH INTEACTION AND THE LAW OF ITEATED LOGAITHM We consider a one-dimensional stochastic differential equation with interaction with no drift part. For single trajectories, we obtain the result similar to the law of iterated logarithm for a Wiener process. 1. Introduction The main object of this paper is the one-dimensional stochastic differential equation (SDE with interaction with no drift part dx(u, t = σ(x(u, t, µ t dw(t (1 x(u, = u, u µ t = µ x(, t 1, t. Such equations and stochastic flows driven by them were introduced by A.A. Dorogovtsev [1]. It occurs that the properties of the solutions to such equations differ from those of the solutions to usual SDEs [5]. So it is of interest to compare their asymptotic behavior with the behavior of the diffusion processes. We want to obtain some kind of the law of iterated logarithm for trajectories of (1. For the stochastic differential equations without interaction, such results are known (see, e.g., [2], [3]. The difficulty of our case is that the behavior of a single trajectory x(u, t, t depends on all other trajectories through the measure-valued process µ t, t. 2. Main results Let M 1 be the space of all probability measures on having the first moment with the Wasserstein metric [1]. Consider Eq. (1 with σ : M 1 being global Lipschitz, and µ M 1. Under such conditions, there exists a unique strong solution to (1 such that x is a flow of homeomorphisms ([1]. The following result shows that, for the equation without drift (1, the trajectories of different particles remain not far from one another at infinity. Lemma 1. For all u, v, there exists (x(u, t x(v, t a.e. Proof. Fix u > v., by the random time change [4], x(u, t x(v, t = u v + t (σ(x(u, s, µ s σ(x(v, s, µ s dw(s = ( t = u v + w u,v (σ(x(u, s, µ s σ(x(v, s, µ s 2 ds 21 Mathematics Subject Classification. Primary 6H1; Secondary 6G17. Key words and phrases. Law of iterated logarithm, stochastic flow, stochastic differential equation with interaction, measure-valued process. 54

2 STOCHASTIC DIFFEENTIAL EQUATIONS WITH INTEACTION for some Wiener process w u,v. Since x(u, t x(v, t > for all t and t is a continuous random process, (σ(x(u, s, µ s σ(x(v, s, µ s 2 ds (σ(x(u, s, µ s σ(x(v, s, µ s 2 ds τ u,v < +. Here, τ u,v is the time of the first strike of (u v by w u,v. So with probability 1, ( (x(u, t x(v, t = w u,v (σ(x(u, s, µ s σ(x(v, s, µ s 2 ds. The proof is completed. Of course, the difference x(u, t x(v, t is a martingale, which does not change the sign. So, it has a it at infinity by the general theory [4]. But we present a direct proof, since we need the precise formula for the characteristics of x(u, t x(v, t. The corollary of this lemma is that the law of iterated logarithm holds (or does not hold for all u simultaneously. Indeed, since x(u, x(v, is bounded, ( x(u, t x(v, t = a.e. So, we can obtain the law of iterated logarithm only for some fixed u, for example, u =. Denote z(u, t = x(u, t x(, t. We need the following result. Lemma 2. Let β > be a positive number. sup t z(u, t (2 u 1 + u 1+β < a.e. Proof. We will consider only u >, the proof for u < being similar. Consider some sequence u n +. We have ( t z(u n, t = u n + w un, σ 2 (x(u, s, µ s ds. Hence, P sup z(u n, t > u 1+β n P sup w un,(s > u 1+β n u n = t s τ un, Put u n = n 2/β. we have n=1 P sup t = u n u 1+β n = u β n. z(u n, t > u 1+β n < +, and, by the Borel Cantelli lemma with probability 1, there exists N N such that sup z(u n, t < u 1+β n for all n N. t Since x is a homeomorphism, for u > u N, u [u n, u n+1 ], sup t so (2 holds. The proof is completed. z(u, t < sup z(u n+1, t < 2 2 β +2 u 1+β, t

3 56 M. P. LAGUNOVA Consider now Eq. (1 with σ(u, µ = b(u vµ(dv for some real function b. Eq. (1 takes the form dx(u, t = (3 b(x(u, t x(v, tµ (dvdw(t x(u, = u. Now, the interaction of particles in a random environment depends on distances between them. Equation (3 can be considered as a stochastic analog of the Kuramoto model for interacting oscillators [6]. Theorem 1. Let the coefficients of (3 satisfy the following conditions: 1 b is global Lipschitz with some constant L, 2 sup ( ;] b < + ; inf [; b >, 3 µ M 1+α for some α >. P x(, t D = P where D = u : u [inf [;+ b; sup ( ;] b]. Proof. For every u, we denote ζ u = z(u, t. x(, t D = 1, For every u, this it exists almost everywhere. Now, let S be a (countable set of all rational numbers and all atoms of measure µ., with probability 1, Let z(u, t ζ u, t for all u S. ξ u = ζ u, u S S v u ζ v, u / S (the it exists, since ζ v is non-decreasing. ξ u is also non-decreasing, and it is continuous except for a countable number of jumps. Hence, and µ u : ξ u ζ u =, µ u : z(u, t ξ u, t = 1. With regard for the proof of Lemma 1, we have ( 2ds (4 (b(z(u, s z(v, s b( z(v, sµ (dv < a.e. Applying Lemma 2 with β = α, we have, for all s, b(z(u, s z(v, s b( z(v, s L z(u, s LC(ω(1 + u 1+α. Thus, by the Lebesgue theorem, (b(z(u, s z(v, s b( z(v, sµ (dv (5 (b(ξ u ξ v b( ξ v µ (dv, s a.e. From (4 and (5, we have, with probability 1, b(ξ u ξ v µ (dv = b( ξ v µ (dv, u.

4 STOCHASTIC DIFFEENTIAL EQUATIONS WITH INTEACTION Now let u n +, n. Applying the Fatou lemma to the functions b(ξ un ξ v 1I v<un (which are bounded below by a constant by condition 2 of the theorem, we have (b(ξ un ξ v 1I v<unµ (dv n un b(ξ un ξ v µ (dv = n n b(ξ un ξ v µ (dv = u n + (we used Lemma 2 and the condition µ M 1+α. Now, we have and, thus, (6 Analogously, (7 b( ξ v µ (dv b(ξ un ξ v 1I v<un inf b(z, n z b( ξ v µ (dv b( ξ v µ (dv inf b. [;+ sup b. ( ;] b( ξ v µ (dv Now returning to the process x, we have, for some Wiener process w, t x(, t = b( z(v, sµ (dvdw(s = Denote If T (t, t, Else, Analogously, ( t ( = w T (t = t ( 2ds b( z(v, sµ (dv. b( z(v, sµ (dv 2ds. T (t ( 2 ( 2. = b( z(v, tµ (dv = b( ξ v µ (dv t x(, t = = w (T (t T (t = 2T (t lln T (t t b( ξ v µ (dv. x(, t w (T ( = = = x(, t = Together with (6 and (7, this proves the theorem. b( ξ v µ (dv. b( ξ v µ (dv.

5 58 M. P. LAGUNOVA Corollary. Under the conditions of the theorem, if b is such that then P u(b(u b( (, u, x(, t = b( = P x(, t = b( = 1. Example. Let µ = 1 2 δ δ π 2, b(u = 1 + cos u + sin u1i u. dx(, t = ( 1 2 b( b(x(, t x(π 2, tdw(t Now dx( π 2, t = (1 2 b( b(x(π, t x(, tdw(t. 2 dz( π 2, t = 1 2 sin z(π 2, tdw(t. The it ξ π = z(π 2 2, t takes only values and π. Due to the symmetry, So, we have P (ξ π = = P (ξ π = π = , P b( ξ v µ (dv = 2 = P b( ξ v µ (dv = 1 P x(, t = 2 = P x(, t = 1 = 1 2. = 1 2. eferences 1. A. A. Dorogovtsev, Measure-valued Markov processes and stochastic flows on abstract spaces, Stoch. ep. 76 (24, no.5, G. L. Kulinič, On the law of iterated logarithm for one-dimensional diffusion processes, Teor. Veroyat. Primen. 29 (1984, L. Caramellino, Strassen s law of the iterated logarithm for diffusion processes for small time, Stoch. Process. Their Appl. 74 (1998, O. Kallenberg, Foundations of Modern Probability, Springer, berlin, M. P. Karlikova, On a weak solution of an equation for an evolutionary flow with interaction, Ukr. Math. J. 57 (25, no. 7, S. H. Strogatz, From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Bifurcations, patterns and symmetry, Phys. D 143 (2, no. 1-4, 1-2. Institute of Mathematics of the NAS of Ukraine address: marie-la@mail.ru