Strong solutions of non-colliding particle systems

We study systems of stochastic differential equations describing positions x_1,x_2,...,x_p of p ordered particles, with inter-particles repulsions of the form H_{ij}(x_i,x_j)/(x_i-x_j). We show the existence of strong and pathwise unique non-colliding solutions of the system with a colliding initial point x_1(0)\leq ...\leq x_p(0) in the whole generality, under natural assumptions on the coefficients of the equations.


Introduction
Consider the following system of SDEs describing positions of p ordered particles evolving in R. Here (B i ) i=1,...,p denotes a collection of one-dimensional independent Brownian motions. Throughout the whole paper we assume that the coefficients of the equations are continuous and that the functions H ij are non-negative and symmetric in the sense (2.1). The SDEs systems (1.1) contain the following ones where G(x, y) = g 2 (x)h 2 (y) + g 2 (y)h 2 (x), β > 0 and g, h, b : R → R. Let S p denote the space of symmetric p × p real matrices and H p the space of Hermitian p × p matrices. It was shown in [11] that for the starting point having no collisions and for β = 1, this system describes the eigenvalue processes of the S p -valued process X t satisfying the following matrix valued stochastic differential equation dX t = g(X t )dW t h(X t ) + h(X t )dW T t g(X t ) + b(X t )dt, where the functions g, h, b act spectrally on S p and W t is a Brownian matrix of dimension p × p. When β = 2, the system (1.2) is satisfied by the eigenvalues of the H p -valued process Y t which is a solution of whereW t is a complex Brownian matrix of dimension p × p. In the last case, for some special choices of g, h and b, the systems (1.2) contain the canonical Dyson Brownian Motion (g = 1 2 , h = 1, b = 0) and the eigenvalue processes of the complex Wishart (Laguerre) processes (g = √ x, h = 1, b = const > p − 1). Recall that the Dyson Brownian Motion is obtained as p independent Brownian particles conditioned not to collide (see [8,10]) and the Laguerre eigenvalue process as p independent Squared Bessel particles conditioned not to collide (see [17]). The general case β ∈ R + in (1.2) corresponds to the β-versions of the processes described by (1.2) with β = 1 and is important in modern statistical physics (see for example [9]). On the other hand, Dyson Brownian Motions are a special case of Brownian particle systems with an interacting potential (see [21]).
Thus the systems (1.1) contain Dyson Brownian Motions, Squared Bessel particle systems, Jacobi particle systems, their β-versions, non-colliding Brownian and Squared Bessel particles, potential-interacting Brownian particles and other particle systems crucial in mathematical physics and physical statistics (see [15,16]). Note that the singularities (x i − x j ) −1 make the SDEs system (1.1) difficult to solve, especially when the starting point has a collision, i.e.
In this paper we prove the existence of strong and pathwise unique non-colliding solutions of (1.1), with a degenerate colliding initial point x(0), in the whole generality, under natural assumptions on the coefficients of the equations (1.1), formulated and discussed in details in Section 2. The Theorem 1 ensures, that even if starting from the most degenerate collision state the particles x i will diffract instantly and never more collide. This statement is proved in the strong, trajectorial solution sense. In this way, we answer in a very general setting a question raised by Rogers and Shi ([21,(5i)]) in the context of potential-interacting Brownian particles: is the finite particle process well defined by its SDEs system? As observed by Grabiner in [10], starting the process from a collision point makes impossible the usual conditioning procedure and the existence of strong solutions of corresponding SDEs is highly unclear even in the case of processes conditioned not to collide.
In some particular cases (Dyson Brownian Motions, some Squared Bessel particle systems) these difficulties have been overcome and the existence of strong solutions of (1.1) has been established by Cépa and Lépingle in [3,4,18], using the technique of Multivalued SDEs (MSDEs). The MSDEs theory was used in [7] and [23] in order to show the strong existence of solutions of radial Dunkl and Heckman-Opdam SDEs with more general singularities.
However, the approach via MSDEs can not be applied to the equations of the general form given in (1.1) and consequently, the existence of the strong solutions has been an open question in many important examples.
Note that some existence results were proved in last years without use of MSDEs (see [14,5,6]) but under the hypothesis of starting without collision (Chybiryakov's proof of [5,Prop. 6.8,p.170] does not work for a collision starting point).
Our approach is based on the classical Itô calculus, applied to elementary symmetric polynomials in p variables X = (x 1 , . . . , x p ) y n = e n (X) = i 1 <...<in as well as to symmetric polynomials of squares of differences between particles The main advantage of the semimartingales y n and V n is that singularities disappear in their stochastic descriptions. Moreover, the processes V n control the collisions between particles.
In the next Section 2 we present and explain technical assumptions of the main result of the paper, Theorem 1, formulated at the end of Section 2. In Sections 3 and 4 we develop the stochastic analysis of symmetric polynomial stochastic processes y n and V n . These sections provide the main ingredients of the proof of Theorem 1. In Section 5 we show that the system (1.1) has a weak continuous solution. Next, we prove the pathwise uniqueness of solutions of the system (1.1) and we conclude with a proof of Theorem 1. The last Section 6 contains applications to important classes of particle systems.

Assumptions and Main Result
As it was mentioned in the Introduction, our general assumptions on the coefficients of the equations are • the functions σ i , b i , H ij are continuous for every i, j = 1, . . . , p and i = j; • the functions H ij are non-negative and the following symmetry condition holds H ij (x, y) = H ji (y, x), x, y ∈ R. (2.1) describes the repulsive force with which the j-th particle located in x j acts on the i-th particle located in x i . The symmetry assumptions on H ij mean that if j > i, i.e. x j > x i , then the upper particle x j pushes the lower particle x i down with the same force as the lower one pushes the upper one up. Next three conditions are adaptations of standard regularity assumptions occurring in the theory of SDEs (without singularities), which usually guarantee uniqueness and non-explosion of solutions. Note that we remain in the context of one-dimensional Yamada-Watanabe theorem, where the coefficients in the martingale part are allowed to be 1/2-Hölder continuous and the drift part coefficients are Lipschitz continuous.
(C1) There exists a function ρ : Moreover, the functions b i are Lipschitz continuous or non-increasing (C2) There exists c > 0 such that The conditions (C2) are standard conditions on the growth of the coefficients of SDE which give finiteness of the solutions for every t > 0, however the sublinear growth of b i can be replaced by non-positivity of b i (x)x for large x.
The last group of conditions (A1)-(A5) relates mainly to the singular part of the equations. Condition (A1) will be crucial for the proof of the pathwise uniqueness of solutions in Section 5. Conditions (A2)-(A5) are introduced in order to ensure non-collisions of the particles, which is a crucial property of a solution of (1.1) to show the existence of its strong solution.
(A1) For every i = j and w < x < y < z we have describes the force with which the particles x j and x i repel each other, the condition (2.2) means that the force decreases as the particles move away from each other in such a way that the first particle goes down and the other goes up.
(A2) There exists c ≥ 0 such that for every i = j we have This assumption ensures that the repulsive forces between the particles are sufficiently large relatively to the martingale part to prevent collisions of the particles caused by σ i (x i )dB i and σ j (x j )dB j . Moreover, this assumption is optimal in many important examples such as β versions of Dyson's Brownian motion model (see Section 6 for more details), i.e. for σ = 1, b = 0 and H ij (x, y) = β/2 the condition (A2) holds if and only if β ≥ 1, which is a necessary and sufficient condition for the system to have no collisions. (A3) There exists c ≥ 0 such that for every x < y < z and i < j < k This condition is used to ensure that the repulsive forces between particles do not cause collisions.
Since H is non-negative, two particles are pushed off from each other, but if we add another particle, which is above the previous ones, then two additional forces appear which push the first two particles down. The condition (A3) implies that the additional forces do not cause a collision between two original particles. It can be easily seen that the conditions (A2) and (A3) do not ensure that the particles become immediately distinct if we start from a collision point (consider the example of generalized squared Bessel particles with integer order α ∈ {0, 1, . . . , p − 2} starting from zero). Thus, if x i (0) = x j (0) = x for some i < j, we will distinguish two situations. When σ 2 i (x) + σ 2 j (x) > 0 or H ij (x, x) > 0, then the process X will instantly leave the initial collision point thanks to the martingale part or the repulsive forces, respectively (see Proposition 7). We call such a phenomenon a "diffraction" of particles. Consequently, in the study of the particle process we must pay special attention to starting from a collision in an element of the sets We will call elements of the sets G kl "degenerate points", and if we will say that the particle process is starting from a "multiple degenerate" point. The next condition (2.4) guarantees that there is a force coming from the whole drift at x such that at most one particle can stay at the point x, so the multiple degeneracy will disappear (Proposition 4).
(A4) The sets G kl consist of isolated points and for every x ∈ G kl we have for every y 1 , . . . , y p−2 ∈ R.
We use the convention that multiplying by the indicator 1 R\{x} (y j ) = 0 always gives 0, i.e. the whole j-th term of the second sum in (2.4) disappears when y j = x. Finally, we consider the following monotonicity property of the drift coefficients b i in (1.1) This condition comes up naturally because if b i (x) > b j (x) then the particle x i could catch up with the particle x j thanks to the bigger drift force.
In the case, when the coefficients of the equations do not depend on i and j, i.e. σ i (x) = σ(x), b i (x) = b(x) and H ij (x, y) = H(x, y), simple sufficient conditions for (A1)-(A5) are discussed in more detail in Section 6.
Remark 1. If we know that the particle system (x 1 , . . . , x p ) lives on some subset I = [a, b] ⊂ R, then we can restrict all the conditions to x, y ∈ I.
Before formulating Theorem 1, the main result of the paper, recall that, accordingly to [20, IX(1.2)], a pair (X, B) is a solution of the system (1.1) if all the integrals appearing in its integral form are meaningful. In particular, the integrals of the drift parts of (1.1) will be understood, if needed, as improper Riemann integrals. Theorem 1. Consider the system (1.1) with an initial condition If the conditions (C1), (C2) and (A1)-(A5) hold, then there exists a unique strong non-exploding solution [X(t)] t≥0 of (1.1) such that the first collision time Remark 2. If we drop the condition (C2) in Theorem 1, then there exists a unique strong solution, possibly admitting explosions, such that the first collision time T is not shorter than the lifetime of the solution.
Remark 3. It is enough to assume (A2)-(A5) to show that there exists a solution of (1.1) having no collisions after the start. The additional conditions (C1) and (A1) ensure the pathwise uniqueness of the solutions and consequently the existence of a unique strong solution.

Stochastic description of the basic symmetric polynomials
We denote the elementary symmetric polynomials in p variables X = (x 1 , . . . , x p ) and of degree n = 1, 2, . . . , p by e n (X) = We use the following notational conventions: e 0 (X) = 1 and e −1 (X) = 0. We will also consider incomplete polynomials. For any fixed collection i.e. it is the sum of all products of length n which do not contain any of the specified variables x j 1 , x j 2 , . . . , x j k . There is obviously no one-to-one correspondence between (x 1 , . . . , x p ) and (e 1 , . . . e p ) since changing the order of the arguments does not affect the values of their symmetric polynomials. But if we restrict the arguments to the open set Thus e is a diffeomorphism between C + and e(C + ), which is an open subset of R p . Let us denote by By the continuity of ordered roots of a polynomial as functions of its coefficients (see for example [19]), f extends to a continuous function 3.1. Symmetric polynomials of particles. In the following proposition we determine the SDEs system for the symmetric polynomials in (x 1 , . . . , x p ) verifying the system (1.1) whenever there are no collisions between particles.
In the proof, as well as in some other proofs in this paper, we use the property . Then the symmetric polynomials e n (X), n = 1, . . . , p, are continuous semimartingales described until the first collision time of (x i ) i=1,...,p by the system of SDEs where {U n ; n = 1 . . . , p} is a family of one-dimensional Brownian motions such that Proof. By an application of Itô formula and the fact that e n (X) = x i e x i n−1 (X) + e x i n (X) for any i we get that for every t smaller than the first collision time of ( Thus there exist Brownian motions U n , n = 1, . . . , p, such that and (3.2) holds. Moreover, by the symmetry property (C2) of H ij (x, y) and the fact that for any j = i This ends the proof.
Note the following remarkable property of the stochastic differential equations describing the polynomial processes e n (X), n = 1, . . . , p: the singularities (x i − x j ) −1 appearing in (1.1) are no longer present in (3.1). Now, using the map f : e(C + ) → C + we get rid of x i 's in the system (3.1). We will shorten the notation e x i n to e i n . We denote by y elements of e(C + ).
Proposition 2. Define the following functions on e(C + ): (i) The functions a n , q n and s n,m , n, m = 1, . . . , p, are continuous on e(C + ).
(ii) Let X = (x 1 , . . . , x p ) be a solution of (1.1) on C + . The symmetric polynomial processes y n (t) = e n (X(t)), t ≥ 0, n = 1, . . . , p, satisfy the system of SDEs dy n = a n (y 1 , . . . , y p )dU n + q n (y 1 , . . . , y p )dt, n = 1, . . . , p, where {U n ; n = 1 . . . , p} is a family of one-dimensional Brownian motions satisfying a n dU n , a m dU m = s n,m dt, n = m, n, m = 1, . . . , p. Definition 1. Let (y 1 , . . . , y p ) be a solution of (3.3)-(3.4) with y(0) ∈ e(C + ). We define stochastic process Λ = (λ 1 , . . . , λ p ), where Thus, from now on, whenever we write λ i (t) we mean the process λ i (t) = f i (y(t)), defined from a solution y(t) of (3.3)-(3.4), using the inverse symmetric polynomial map f . Obviously we have y n = e n (λ 1 , . . . , λ p ) = e n (Λ), n = 1, . . . , p and whenever y(0) ∈ e(C + ), i.e. λ i (0) = λ j (0) for every i = j, then Λ = (λ 1 , . . . , λ p ) is a solution of (1.1) up to the first collision time. It is thus natural to interpret Λ = (λ 1 , . . . , λ p ) as a system of particles related to the solution y of (3.  Proof. Let y = (y 1 , . . . , y p ) be a solution of (3.3). We define Applying Itô formula to (3.3) we get Using (3.4) one sees easily that It follows that there exists a Brownian motion W t such that The rest of the proof is similar to the proof of the classical theorem on non-explosion of solutions of a SDE, see Theorem 2.4 in [13]. For the convenience of the reader we provide the proof. Set τ n = inf{t > 0 : R t ≥ n}. Using the fact that the expectation of the martingale part vanishes and (C2) we get Continuity of the paths and the Lebesgue dominated convergence theorem imply that the function t → ER t∧τn is continuous. By the integral version of the Gronwall's lemma and taking n → ∞ we obtain that R t is finite almost surely for every t ≥ 0. Thus, all the processes λ 2 i (t) ≤ R t are finite for every t and consequently every y n (t) = e n (Λ(t)) is finite almost surely. It implies that the explosion time of the solution y(t) is infinite almost surely.
3.3. Instant exit from a multiple degenerate point. Starting from this section, we study the behavior of the particles λ i associated to a solution y of (3.3).
In the next proposition we use the condition (A4) to show that if there are at least two particles λ k , λ l starting from the same degenerate point, i.e. a point x belonging to a set G kl , then immediately all particles, except perhaps one, are pushed off that point, i.e. there might be at most one particle which stays at x. Observe that Proposition 4 does not imply the instant diffraction of the particles, i.e. it could possibly happen that λ k (t) = λ l (t) on some time interval (0, ǫ) with positive probability. This problem together with the non-degenerate case, i.e. starting from a collision in x / ∈ G kl , will be considered later in Proposition 7.
Proposition 4. Let y = (y 1 , . . . , y p ) be a solution of (3.3) and assume that (A4) holds. If λ k (0) = λ l (0) = x ∈ G kl for some k < l then Proof. Without loss of generality we can assume that λ k−1 (0) < λ k (0) and λ l (0) < λ l+1 (0), i.e. there are exactly l − k + 1 particles starting from x, and we denote by S = {k, k + 1, . . . , l − 1, l} the set of indices of these particles. Moreover, we can and do assume that x = 0. Additionally, we will denote by γ(t) = p i=1 1 {λ i (t)=x} the number of particles staying at time t in x. Let γ = γ(0) = l − k + 1 and note that we have γ ≥ 2. Suppose by contradiction that the first time, when at least one of the particles moves from 0, i.e. the random variable is greater than zero with positive probability. Using continuity of the paths we can see that there existsτ 0 (positive with positive probability) such that i.e. we have γ particles remaining equal to zero on [0,τ 0 ) and p − γ nonzero particles on this time interval. Since each product forming part of the sum defining e n (Λ), where n = p − γ + 1 contains at least one of the zero particle, we have e n (Λ)(t) = 0 for t ∈ [0,τ 0 ). In particular its drift part is zero on [0,τ 0 ).
Seeing that for j / ∈ S we have e λ i ,λ j n−1 (Λ) = 0, we can write Note also that for every i = k, i, k ∈ S we have e λ i n−1 (Λ) = e λ k n−1 (Λ) and this common quantity is equal to the product of p − γ non zero particles. Consequently, we obtain that for t <τ 0 the drift part of y n = e n (Λ) is from one side identically zero, but from the other side it is equal to Since the assumption (A4) ensures that the above-given sum over S is nonzero, it implies that for every i ∈ S we have e λ i n−1 (Λ) = 0 on (0,τ 0 ), which is a contradiction with the fact that e λ i n−1 (Λ) is equal to the product of particles which are nonzero on [0,τ 0 ). Now we reason by induction on γ = γ(0) ≥ 2 in order to prove the statement of the Proposition. Note that if γ = 2, we have τ = τ 0 . The above-given arguments show that in this situation, for every t > 0 we have P 2 (τ > t) = 0.

Polynomials of squares of differences between particles and collision times
4.1. Symmetric polynomials of squares of differences between particles. For any 1 ≤ i, j ≤ p we put a ij = (λ i − λ j ) 2 and define the family of the processes where n = 1, 2, . . . , N = p(p − 1)/2. The process V n is a sum of all products of the length n of the squared differences between particles. In particular is the squared Vandermonde determinant and Note that these processes and their zeros control the collisions between particles. For Since V n as a function of (λ 1 , . . . , λ p ) is a symmetric polynomial, it can by expressed as a polynomial of y 1 = e 1 (Λ), . . . , y p = e p (Λ). Processes y 1 , . . . , y p are defined as a solution of the system (3.3)-(3.4), so by Itô formula, V n are semimartingales. This is a reason why we consider the squares of differences between particles instead of studying the differences themselves. We begin with exploring the semimartingale structure of V n , n = 1, . . . , N in the case when V N (0) > 0, i.e. the system starts from a point having no collisions. Then the semimartingale Λ = (λ 1 , . . . , λ p ) is a solution of the system (1.1) up to the first collision time. Thus we can apply the Itô formula to (1.1) which significantly simplifies calculations. Since Consequently, the martingale part of dV n is described by Finally, the drift part of V n is (using the notation i = j = k when no equality between any two of the indices i, j, k holds) Observe that using the relation e so it is well-defined even if any collision occurs. Moreover, the second part is non-negative and the first one vanishes when V n = 0.
Similarly as for SDEs for symmetric polynomials y i = e i (X), the stochastic differential equations describing V n (Λ) do not contain singularities (x i − x j ) −1 .
We finish by observing that the polynomials V n fulfill equations (4.1) and (4.2) also if V N (0) = 0, with λ i = f i (Y ), i = 1, . . . , p. This follows from the fact that the polynomials V n are smooth polynomial functions of the semimartingales y 1 , . . . , y p , satisfying the SDEs system (3.3). Thus, by the unicity of the martingale and drift part of a semimartingale, the Itô formula applied to (3.3) gives the equations (4.1) and (4.2). But computing derivatives of the functions V n = V n (f (Y )) required in the Itô formula does not depend on the initial condition V N (0). Such argument allows us to avoid looking for explicit relations between the polynomials V n and e n . A similar argument was used in [2].
We resume the results of this subsection in the following proposition.

4.2.
Collision time when starting from a regular state. In this section we consider the first collision time defined in terms of the semimartingale V N by with standard convention that inf ∅ = ∞.
We begin with a generalization of Theorem 5 from [11]. We show that under certain conditions on coefficients of the equation the particles never collide when the starting point does not have any collisions. Proof. The proof is similar to the proof of Theorem 5 from [11]. Defining U t = − 1 2 ln V N (t) on [0, T ) and applying the Itô formula we obtain the martingale part of U equal to and the following representation of the finite-variation part Note that conditions (A2) and (A3) together with the assumptions on b i (x) ensure that there exists c ≥ 0 such that drift[U ] t ≤ ct, which implies finiteness of the finite-variation part of U whenever t is bounded. Applying McKean argument we obtain the result.  Proof. First we assume that if λ i (0) = λ j (0) = x ∈ R for some i = j, then σ 2 i (x) + σ 2 j (x) + H ij (x, x) > 0, i.e. Λ(0) belongs to the set Since the functions σ 2 i (x) and H ij (x, x) are continuous, the set E is open. The continuity of the paths implies that there exists a positive stopping time τ such that on the interval [0, τ ) the system stays in E, i.e. if λ i (0) = λ j (0), then and if λ i (0) = λ j (0) for some i = j, then Now inductively we show that if τ n > 0 with positive probability, then the probability that τ n−1 > 0 is also positive. Note that if V n (t) = 0 on [0, τ n ), then its finite-variation part vanishes on [0, τ n ∧ τ ). Since then (λ i − λ j )e Now let us fix i = j. If λ i (t) = λ j (t) at t ∈ [0, τ n ∧ τ ) then by (4.4) we have λ i (0) = λ j (0) and consequently by (4.5) one of the functions σ 2 i (x), H ij (x, x) is positive at λ i (t). Then, the above-given equality implies that e a ij n−1 (A) = 0 at t. If λ i (t) = λ j (t), since V n (t) = 0 and Finally, if we assume that τ N > 0 with positive probability, then the first part of the proof implies that τ 1 > 0 with positive probability, i.e. if a pair of particles remains glued for some positive time, then all of the particles are glued for some time. But for n = 1 the formula given in (4.6) reads as for every t ∈ [0, τ 1 ∧ τ ) and it is a contradiction with (4.5). Note that if there exists i = j such that λ i (0) = λ j (0) (i.e. V n (0) > 0 for some n ≥ 1) then this finite induction can be stopped at level n, because we obtain then a contradiction with the continuity of the paths. Now we consider the remaining case Λ(0) / ∈ E. It means that the process Λ starts from a multiple degenerate point. By condition (A4) and Proposition 4 we obtain that the system immediately visits the set E. Thus, the standard argument based on the Markov property and the continuity of the paths together with the above-given proof for the case when Λ(0) ∈ E give Proof. If V N (0) > 0 then it is just the result given in Proposition 6. If V N (0) = 0, then for every t > 0, by continuity of the paths and Proposition 7, we have Consequently, by the Markov property where the last equality follows from Proposition 6. If we now define for every s ≥ 0 then obviously T 0 = T is the first collision time and {T 2/n = ∞}, P 0 (T = ∞) = lim n→∞ P 0 (T 2/n = ∞), but once again by the Markov property, Proposition 6 and the fact that V N (t) > 0 a.s. for every t > 0, we have The proof is complete.
Proof. We consider symmetric polynomials V n (t) of squares of differences between particles x 1 (t), . . . , x p (t). All we proved on non-collisions of the process Λ in Theorem 2 and the preceding propositions was based on the Itô calculus applied to the system (1.1), so it applies to any solution X of (1.1).

Existence and uniqueness of a non-colliding solution
Thus, for every t > s > 0, using the smoothness of f and Itô formula we have where B i are one dimensional independent Brownian motions. Here we have used the bijectivity of Itô formula, i.e. if we apply the Itô formula for the smooth and invertible function h and semimartingale X and then for h −1 and h(X) we arrive at the original semimartingale representation for X. By continuity, λ i (s) tends to λ i (0) whenever s goes to 0. Moreover, s 0 σ i (λ i (u))dB i (u) is a continuous martingale starting from 0 and it converges to 0 when s → 0 almost surely. Thus the drift integral in (5.1) converges almost surely when s → 0. It means that for t smaller than the explosion time we have where the last integral is understood as an improper integral whenever x / ∈ C + . Obviously, if (C2) holds, then the explosion time and consequently the first collision time are infinite. This ends the proof.
Remark 4. Note that the assumptions (A2)-(A5) were used only to ensure that V N (t) > 0 for every t > 0. So even if (A2)-(A5) do not hold, but we can show that V N (t) > 0 for every t > 0, then we can construct a solution of (1.1) in the way described above.
In the next Theorem we use the conditions (C1) and (A1) to show the pathwise uniqueness of the solutions of (1.1).
Proof. Let (X, B) and (X, B) be two solutions of (1.1) having common starting point X(0) = X(0) ∈ C + , where B is a Brownian motion in R p . Corollary 1 implies that the particles X = (x 1 , . . . , x p ) do not collide after the start and the same is true forX = (x 1 , . . . ,x p ).
The condition (C1) together with Lemma 3.3 from [20], p. 389, implies that the local time of Z i = x i −x i at 0 is zero. Note that we can apply the Tanaka formula to the process Z i (if ever the drift integral in the SDE for Z i is an improper integral in 0, we write the Tanaka formula for |Z i | on [s, t] and consider s → 0. The local time at 0 of the process (Z i (u)) s≤u≤t converges a.s. to the local time at 0 of the process (Z i (u)) 0≤u≤t ). Thus, The Lipschitz condition imposed on b i (x) implies that there exists c ≥ 0 such that Moreover, the assumptions (A1) and (2.1) on the functions H ij ensure that the first term is non-positive. Indeed, we can write it in the following form Note that by the symmetry property (2.1) we have F ij + F ji = 0. If sgn(x i −x i ) = sgn(x j −x j ), it follows that the term indexed by i, j in (5.2) vanishes. If the signs of the differences between particles with and without tilde are different, then the term indexed by i, j in (5.2) equals 2sgn (x i −x i ) F ij and (A1) implies that Consequently, we have obtained that and the Gronwall Lemma ends the proof.
Note that the systems (6.1) contain as a special case the systems (1.2) related to eigenvalues of matrix stochastic processes. The conditions (A1)-(A5) may be simplified to (A2') For all x, y or, otherwise, for every y 1 , . . . , Corollary 2. Suppose that σ is at least 1/2-Hölder and b is Lipschitz (see condition (C1)) and that the condition (C2) of non-explosion holds. If the conditions (A1')-(A4') are verified, then the system (6.1) with X(0) ∈ C + has a strong pathwise unique solution, with no collisions and no explosions for t > 0.

6.2.
Interacting Brownian particles. In this subsection we consider the following interacting Brownian particle systems including and essentially bigger than Dyson Brownian particle systems and the systems considered by Cépa, Lépingle (see [3]): 2) Corollary 3. Let the functions σ i be at least 1/2-Hölder and b i be Lipschitz (i.e. they verify condition (C1)), with b i (x) ≤ b j (x) if i < j, and let γ > 0. We suppose that the conditions hold for all x ∈ R. Then the system (6.2) with x(0) ∈ C + has a strong pathwise unique solution, with no collisions and no explosions for t > 0.
Proof. We apply the Theorem 1 with constant positive H = γ. The conditions (A1), (A3) and (A4) are satisfied (observe that the sets G kl from (A4) are empty.) The rest of the assumptions of Theorem 1 hold thanks to the assumptions of the Corollary.
6.4. Non-colliding Squared Bessel particles and related processes. In this section we consider the processes satisfying the following system of SDEs When σ i (x) = 2 √ x, i = 1, . . . , p, these processes are called β-Wishart processes and contain for β = 2 the non-colliding Squared Bessel particle systems studied in [16]. For applications of these classes of particle systems, see [11] and [16]. The β-Wishart processes were studied in [7].
Corollary 5. Let α ≥ p − 1 and β ≥ 1. Suppose that the functions σ i are defined on R and verify the condition (C1) and the estimate Then the system (6.5) with an initial condition 0 ≤ x 1 (0) ≤ x 2 (0) ≤ . . . ≤ x p (0) has a unique strong solution for t ∈ [0, ∞). Moreover, the process x 1 verifies x 1 (t) ≥ 0 and there are no collisions between the processes x i (t) for t > 0.
Note that the Corollary 5 strengthens Corollary 6 of [11]. When σ i (x) = 2 √ x and α > p, Corollary 5 was proved in [18] by the methods of MSDEs, see also [7]. The proof of the last corollary applies to more general SDEs systems of the form and we obtain the following corollary strengthening Corollary 4 of [11].
The generalized Squared Bessel particle systems of the form (6.7) for any α ∈ R will be studied in a forthcoming paper [12]. On the other hand, defining y i = √ x i where the processes x i are solutions of the system (6.5) with σ i (x) = 2 √ x, we obtain Bessel particle systems and the results of Corollary 5 can be transfered to those systems, cf. [18].
In the special case σ i = 1, b i = 0 they arise as radial Heckman-Opdam processes and were studied in [23].
Corollary 8. Let the functions σ i be at least 1/2-Hölder and b i be Lipschitz (i.e. they verify condition (C1)), with b i (x) ≤ b j (x) if i < j, and let γ > 0. We suppose that the conditions b i (x)x ≤ c(1 + |x| 2 ), σ 2 i (x) ≤ 2γ, i = 1, . . . , p hold for all x ∈ R. Then the system (6.9) with x(0) ∈ C + has a strong pathwise unique solution, with no collisions and no explosions for t > 0.
Note that the same proof works in much greater generality and gives the following Corollary.
Finally observe that the results and techniques of this section and Section 6.3 may be applied to systems (6.4), e.g. for φ(x) = ln | sinh x| and φ ′ (x) = coth x.