Entrance laws for coalescing and annihilating Brownian motions

The system is often studied starting from a specific entrance law, informally starting with a particle at every x ∈ R (for example this is the case with the Arratia flow). The intuition is that the instantaneous reactions bring the system instantaneously into M 0 . However a complete description of entrance laws requires care. See [4] where a complete classification of the entrance laws for θ = 1 is found using duality with the continuum voter model. However, the Pfaffian formulae for the intensities of the one dimensional marginals clarify the picture and lead to a simple description of the complete set of entrance laws, which is the purpose of this note. The usefulness of this Pfaffian property is illustrated in [1] where Fredholm Pfaffians are used to study empty intervals and exit measures.

We recall the Pfaffian structure of X t , yielding explicit Lebesgue intensities ρ t (x 1 , . . . , x n ) for X t . These were studied first for θ = 0 for the Arratia flow, and the analogous flow for θ = 1, in [5], and then more generally in [2] and [3] for mixed systems, together with associated systems where certain branching or immigration mechanisms are present. Started from a deterministic initial condition µ ∈ M 0 the point process X t is, at any fixed time t > 0, a Pfaffian point process. This means that its intensities are given in terms of a Pfaffian

where the kernel

, where V 2 = {(x, y) : x < y}, is constructed from the initial condition µ as we now recall. Firstly K t is in ‘derived form’, that it is derived from a scalar kernel K t : V 2 → R via the relation

for t > 0 and x < y,

and

is the unique bounded solution to the heat equation

satisfying the initial condition K t → K 0 in distribution as t ↓ 0 on V 2 , where

The proof of this Pfaffian structure is based on the Markov duality formula E[(-θ) Xt(x 1 ,x 2 )+Xt(x 3 ,x 4 )+…+Xt(x 2n-1 ,x 2n ) ] = pf(K t (x i , x j ) : i, j ≤ 2n).

(

For coalescing-annihilating random walks this is Lemma 7 of [2]. Expression (5) is the corresponding continuous limit obtained by following the arguments of Section 3 of that paper.

Here, and throughout, we write µ(a, b) (and µ(a, b] e.t.c.) as shorthand for µ((a, b)). We also use the convention that 0 k = I(k = 0), so that for instance when θ = 0 the expression (-θ) X 0 (x,y) becomes the indicator I(X 0 (x, y) = 0) that there are no particles inside (x, y).

Write (T t ) for the Markov semigroup of (X t ) acting on bounded measurable F : M 0 → R.

Recall that an entrance law for (X t ) is a family of laws (Q t : t > 0) on M 0 so that

Clearly, entrance laws form a convex set and our aim is classify the extreme points of this set.

Notation. We write (K f t ) for the solution to the heat equation (3) with initial condition

for the family of laws on M 0 , when it exists, where Q f t is the law of the point process with intensities ρ

(n) t

given via (1) for the kernels K f t arising as in (2) from the scalar kernel K f t .

As explained above, starting from a deterministic condition, the law of (X t : t > 0) is given by Q f with f (x, y) = (-θ) X 0 (x,y) . The aim of this note is to show that all entrance laws are mixtures of Q f for suitable functions f : V 2 → R.

Theorem 1 The extreme elements of the set of entrance laws for

) is given by

and for θ ∈ [0, 1)

The duality function used to analyse mixed systems is, for θ ∈ [0, 1] and µ ∈ M 0 ,

Here is the key underlying lemma, whose proof is delayed until the end of this note.

The weak- * closure in L ∞ (V 2 ), as dual to L 1 (V 2 ), of the set

of finite spin functions is Cθ , where C1 = C 1 and for θ ∈ [0, 1) it is the set

where S = S i ∪ S c is the disjoint decomposition of a closed set S into its isolated points S i and its cluster points S c .

Remarks. 1. The superset Cθ ⊇ C θ , when θ ∈ [0, 1), will label entrance laws (via the map f → Q f ); however only the set C θ will label extremal entrance laws.

All the functions here lie in the unit ball

} and the weak- * topology is metrizable on this ball (since

Proof of Theorem. We first check that Q f , for f ∈ Cθ , do form entrance laws. We follow the steps from [5] where an entrance law for the cases θ ∈ {0, 1} when f ≡ 0 was constructed (called there a ‘maximal’ entrance law and informally corresponding to starting a particle at every point in R as for the Arratia flow). Fix f ∈ Cθ . By Lemma 2 we can choose a sequence (µ n ) so that s µn → f weak- * . Let (X (n) t ) be the corresponding particle system with the initial condition µ n . At a fixed t > 0, the corresponding scalar kernels

where

t , together with their derivatives

t , converge bounded pointwise to K f t and its associated derivatives. This convergence of the kernels K (n) t implies that the associated Pfaffian point processes X (n) t converge in law to a limiting point process X t with law Q f t (see Lemma 10 in [2]). Moreover the Markov duality formula extends to hold for the limit, that is when

when t > 0, x 1 < x 2 < . . . < x 2n

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