ON THE REGULARITY OF MANY-PARTICLE DYNAMICAL SYSTEMS PERTURBED BY WHITE NOISE

We consider a system of finite number of particles that are moving in R under mutual interaction. It is assumed that the particles are subjected to some additional random forces which cause diffusion motion of the particles. The latter is described by a system of stochastic differential equations of the first order for noninertia particles and the second order for inertial particles. The coefficient of the system are unbounded because the interaction force tends to infinity if the distance between two particles tends to zero. The system is called regular if no particle can hit the other. We investigate conditions of regularity. This article is dedicated to the memory of Roland L. Dobrushin.

It is impossible to survey all aspects of the theory of randomly perturbed dynamical systems.I would like to mention the article of N.M.Krylov and N.N.Bogolubov [6] and the work of I.I.Gikhman [1], where the notion of stochastic differential equation was introduced to prove a well- known statement of Krylov and Bogolubov.Stochastic differential equations became the main tool for the investigation of randomly perturbed dynamical systems.A variety of the results of such kind are collected in the book of R. Khasminskii [5].I also use the results from the mono- graphs by I.I.Gikhman and A.V. Skorokhod [2] and A.V. Skorokhod [9].
In monograph [8], the system of randomly interacting particles was considered under the assumption that random perturbations are generated by some Poisson random measures.The coefficients of corresponding stochastic equations are supposed to be smooth.
428 ANATOLI V. SKOROKHOD At that time, I discussed some of the results of this monograph with Roland Dobrushin and he asked me whether all the particles of the system could stick together.Trying to answer his question, I accomplished this article.
The description of nonperturbed several particle mechanical systems can be found in the book of Watson [10].
We consider the system of N particles that are moving in the space /i d and mutually interacting.It is supposed that the interaction force of two particles depends only on the distance between them; this force is directed from one particle to another (in the case of attraction) or it has the opposite direction (in the case of repulsion).Besides, the particles are subjected to some random forces which cause additional diffusion motion of the particles.
Let xl(t),...,xN(t be the positions of the particles at time t.We assume that xl(t),...,xN(t are stochastic processes satisfying some system of stochastic differential equations. The form of the equations depends on some additional assumptions Noninertia particles In this case, forces and random perturbations are acting directly on particles; the velocities of particles are not defined.The system of the stochastic differential equations is of the form N dxk(t) N-1E Fikdt + rdwk(t), k 1,...,N, (1) ,=1 where Fik is the force that is caused by ith particle at the kth one, a > 0 is a parameter, and {wi(t), 1,..., N} is a system of independent Wiener processes in Rd.

Inertia particles
In this case, particles have velocities.We denote the velocity of the ith particle by 5ci(t).
In the case (AF) particles are attracting and in the case (RF) they are repulsing.
Because of conditions (AF) and (RF) and formula (3) the coefficients of equations ( 1) and ( 2) are unbounded and undefined if xk(t xi(t for some k,i, t. We can consider the solutions of equations ( 1) and ( 2) only on the interval [0,), where is a stopping time for which A(t)-1-Ii#klxi( t)-xk(t)l >0 for t< and A(-)-0 if <cx.
On the Regularity of Many-Particle Dynamical Systems 429 The existence and uniqueness of the solution for any initial condition such that A(0)> 0 is the consequence of general theorems on local existence and uniqueness of the solution for stochastic differential equations (see, for example [2], p. 506).
A system is called regular if P{ < c} -0 for any initial condition.
to the investigation of the conditions under which the system is regular.
This article is devoted 2. Noninertia Particles Two-particle systems We consider equation ( 1) for N 2 with Fik given the formula (3).following equation for xl(t -x2(t z(t): where w(t) Wl(t)-w2(t) is also a Wiener process distributed as wl(t).Set x(t) (z(t),z(t)).X(t) is a real-valued stochastic process that satisfies the relation We can write the (4) (We use Ito's formula.)Set 2r2d-p (here d is the dimension of the phase space Rd).Let l{s < } z(s) l(z(s),dw(s)) + / l{s > }1 z l(z, dw(s)), 0 where is a stopping for which z(t) > 0 if t < and z(-) 0 if < oe, z C= Rd, z =/: 0 is a fixed vector.(t) is a Wiener process and X(t) satisfies the stochastic differential equation: We consider the solution of this equation with a positive initial value X(0), (-sup{t: X(t) > 0).
Theorem 1: a) P{-+ oc}-1 for any initial condition if and only if tion LO(x)-0 and the function q(x)-f H(u) f H-l(v)dvdt satisfies the equation Lt(x)- 1.

X It
Now the proof of the theorem is the consequence of the results of W. Feller [4] (see also [9], p. 42).
2) If F(u) 0 and d 3 then the system is regular because 3) Let F(u) cu-, > 0, c > 0, then the system is regular.
4) Let F(u) clog I c ,c > 0, then the system is regular if < -1 and the system is irregular c d 1 2a2 f 2 > y- The system of many noninertia partiel We use expression (3) for Fik and set Fii-O.
It is easy to see that for Zl,... z N E R d we have that (z zj, (Fik Fjk) Note that the process is a martingale.characteristic" ,3 0 (8) After some calculations we can obtain the following expression for its square (, )t 8cr2N / R(s)ds. -(s) E ( x i ( s ) -x j ( s ) , d w i ( s ) -d w j ( s ) ) .
The system is regular if -2 > --1 and the system is irregular if 2 2 b) Let F satisfy condition (RF).Then the system is regular.
Proof: We will use the following statement: if the system is irregular for N N O then it is irregular for N > N 0. To prove this statement we only have to note that for the system of N O + 1 particles we can choose the initial value ZN0 + 1(0) so far from z0(0),..., XNo(O that the influence of ZNo + l(t) on the Zo(t),...,ZNo(t is negligibly small.So the system is irregular for all N if it is irregular for N-2.But there exists the possibility that the system is regular for N 2, 3,..., N O and it is irregular for N N O + 1.In this case where P{ < i,j<No+l > 0).
In part a) of the theorem we consider first F'(t)- As it follows from Theorem 1, the solution of this equation is irregular if and the solution is regular if It is easy to see that if then 012r2 Assume that F(t) satisfies condition a) with c (0, oo).stochastic differential equation" 1 dY(t) cdt + V/8er2Ny-(t)dW (t) and let y sup{t: Y(s) > 0 for s (0, t)}, c 2c + 2d.cr2)N(N 1).
Denote by Y(t) the solution of (11 1 It follows from condition a) that fl(t)-R2(t)7(t), where 7(t)is a bounded measurable adapted function.It follows from relation (9) that Since measures corresponding to the stochastic processes (t) and (t)+ f (82N) 27(s)ds are 0 equivalent due to Girsanov's theorem (see [3]), the measures corresponding to Y(t) and R(t) are equivalent if Y(0)= R(0).Consequently, P{ < cx} > 0 if P{y < } > 0 and P{ < c} 0 if P{y < cx} 0. So, part a) is proved for c G (0,).We also have to consider the case c 0 and the case c + x.
For this, we need the following statement.
Lemma 1: Let F and F* satisfy the conditions of part a) of the theorem.We denote by S and * the corresponding systems of stochastic differential equations.Let F'(u)u <_ F*'(u)u.
Then, if S* is irregular then S is also irregular; hence if S is regular, then S* is regular.
The proof of the statement can be rendered in the same way as the comparison theorem (see [7], p. 124).
To prove statement b) we note that Lemma 1 is true if F and F* satisfy condition But if F 0 then the equation for R is of the form: dR(t) 2. cr2N(N 1)dr + V/8r2Nv/R(t)dv (t). (13) 1 The function H(u) for this equation is H(u)-u (N-1)and f H(u)du-+ c if N > 2 for any d= 1,2 0 So the solution of equation ( 13) is regular.Therefore, it is regular for any F satisfying condi- tion (RF).
Then z(t) is a solution of the equation z( t + d(t) F'( z(t) ).l z(t where w(t)is the same as in equation (4).
We consider the conditions under which there exist z(O) and (0) such that z(to) 0 for some o > O.In this case we will say that the system is irregular.If z(t) =/: 0 for any initial condition and > 0 then we say that the system is regular.
for any continuous real-valued martingale It(t) with the square characteristic (It, It)t" To prove the statement we note that for the Wiener process w(t) we have t>01+ Itl c< --1.
The system is regular if c) limuou2F(u) O, d > 2.
where Remark: Let F satisfy condition (RF).Then,   0Using Lemma 3 we can prove that for any t >O,