An exit measure construction of the total local time of super-Brownian motion

We use a renormalization of the total mass of the exit measure from the complement of a small ball centered at x ∈ R for d ≤ 3 to give a new construction of the total local time L of super-Brownian motion at x.


Introduction and main results
The local time of super-Brownian motion (SBM) has been well studied by many authors, e.g., Adler and Lewin [1], Barlow, Evans and Perkins [2], Krone [9], Sugitani [14], etc. It may be formally defined as the density function of the occupation measure of super-Brownian motion. Let M F = M F (R d ) be the space of finite measures on (R d , B(R d )) equipped with the topology of weak convergence of measures. A super-Brownian motion X = (X t , t ≥ 0) starting at µ ∈ M F is a continuous M F -valued strong Markov process defined on some filtered probability space (Ω, F, F t , P ) with X 0 = µ a.s. Write µ(φ) = φ(x)µ(dx) for any measure µ. It is well known that super-Brownian motion is the solution to the following martingale problem (see [13], II.5): For any φ ∈ C 2 b (R d ), where (M t (φ)) t≥0 is a continuous F t -martingale such that M 0 (φ) = 0 and the quadratic variation of M (φ) is For any 0 ≤ t ≤ ∞, the occupation measure of super-Brownian motion X up to time t is the random measure defined by In dimensions d ≤ 3, the occupation measure I t has a density, L x t , called the local time of X, which satisfies Moreover, Theorems 2 and 3 of Sugitani [14] imply that (t, denotes the closed support of a measure µ. The extinction time of X is a.s. finite (see, e.g., Chp II.5 in [13]) and so we set L x = L x ∞ to be the (total) local time of X. We define the range, R, of X to be R = Supp(I ∞ ). Now consider SBM under the canonical measure N x0 , which is a σ-finite measure on (1.4) has the law, P X0 , of a super-Brownian motion X starting from X 0 . We refer the readers to Theorem II.7.3(c) of [13] for more details. The global continuity of the total local time L x under N x0 is given in [6] (see, e.g., Theorem 1.2 of the same reference). By (1.4) we may decompose the total local time L x under P X0 as (1.5) Intuitively the total local time L x measures the amount of mass distributed by super-Brownian motion on the singleton x. This mechanism is pretty similar to the exit measure from the complement of a small ball centered at x. To define the exit measure in an appropriate way, we first recall Le Gall's Brownian snake.
Let W = ∪ s≥0 C([0, s], R d ) be equipped with the natural metric (see, e.g., Chp. IV.1 of Le Gall [11]). For any w ∈ W, we write ζ(w) = s if w ∈ C([0, s], R d ). We call ζ(w) the lifetime of w. The Brownian snake W = (W t , t ≥ 0) is a W-valued continuous strong Markov process. Let ζ t = ζ(W t ) and useŴ (t) = W t (ζ t ) to denote the tip of the snake at time t. Recall the canonical measure N x of super-Brownian motion from above. By slightly abusing the notation, we let N x denote the excursion measure of the snake, on C([0, ∞), W), starting from the trivial path at x ∈ R d with zero lifetime. Then we may use the Brownian snake W to construct a measure-valued process X(W ) = (X t (W ), t ≥ 0) under N x such that the law of X(W ) under N x is equal to that of a super-Brownian motion under the canonical measure N x , thus justifying our abusive notation. We use X t (W ) to denote the super-Brownian motion associated with the snake W instead of the integral with respect to X t . This should be clear if one recalls that W is not a function on R d but the snake. The construction of the super-Brownian motion X(W ) by the snake W is not important for our discussion here, and so we refer the interested readers to Theorem IV.4 of [11] for more information. If Ξ = j∈J δ Wj is a Poisson point process on W with intensity N X0 (dW ) = N x (dW )X 0 (dx), then it follows from (1.4) that ECP 26 (2021), paper 40. has the law, P X0 , of a super-Brownian motion X starting from X 0 . It also follows from (1.5) that the total local time L x under P X0 may be decomposed as Now we turn to the exit measure. The exit measure from an open set G, under P X0 or N X0 , is a random finite measure supported on ∂G and is denoted by X G (see Chp. V of [11] for the construction of the exit measure). Intuitively X G represents the mass started at X 0 which is stopped at the instant it leaves G. We note [11] also suffices as a reference for the properties of For any K 1 , K 2 non-empty, set In what follows we will only be considering exit measures X G for G = G x0 ε with x 0 ∈ R d and ε > 0 as above. Under N x we have the range R of super-Brownian motion X = X(W ), defined by R = S(I ∞ ) with I ∞ as in (1.2), may also be written as (see, e.g., equation (8) in the proof of Theorem IV.7(iii) of [11]) For any x ∈ G, under N x we may use the definition of exit measure in Chp. V of [11] to get (see also (2

.3) of [8])
X G is a finite random measure supported on ∂G ∩ R a.e. (1.10) The extension of (1.10) to N X0 is immediate as N X0 (dW ) = N x (dW )X 0 (dx). It also works under P X0 as we may, equivalently, set (see, e.g., (2.23) of [12]) (1.11) where Ξ is a Poisson point process on W with intensity N X0 .
Throughout the rest of the paper, we will always work with this càdlàg version. For any ε > 0, set (1.12) The following result gives a new construction of the total local time L x in terms of the local asymptotic behavior of the exit measures at x. This result is also useful in the construction of a boundary local time measure whose support is the topological boundary of the range of super-Brownian motion in d = 2 and d = 3 (see [7]).
Exit measure construction of the local time Theorem 1.1. Let d = 2 or d = 3 and X 0 ∈ M F (R d ). For any x ∈ S(X 0 ) c , we have X G x ε (1)ψ 0 (ε) converges in measure to L x under N X0 or P X0 as ε ↓ 0, (1.13) where ψ 0 is as in (1.12). Moreover, in d = 3 the convergence holds N X0 -a.e. or P X0 -a.s.
, 0 ≤ r < r 0 } with r 0 = d(x, S(X 0 ))/2 is indeed a martingale (see the proof of the above theorem in Section 3). This allows us to use martingale convergence to conclude a.s. convergence in d = 3. In d = 2, we already know from Proposition 6.2(b) of [8] that the family {X G x r 0 −r (1), 0 ≤ r < r 0 } is a martingale, and so one can check that A will be a submartingale in d = 2. Whether or not a.s. convergence holds in d = 2 remains unresolved.

The special Markov property
We will state the special Markov property for the Brownian snake from [10] that plays an essential role in our proof. We first deal with N X0 . Recall that we are working with exit measures X where s → W η G s is continuous (see p. 401 of [10]). Intuitively one may think of E G as the σ-field generated by the excursions of W inside G. Write the open set {u : In this way, we have W i are the excursions of W outside G for each i ∈ I. Proposition 2.3 of [10] implies that X G is E G -measurable and Corollary 2.8 of the same reference gives the following special Markov property: Conditional on E G , the point measure i∈I δ W i is a Poisson point measure with intensity N X G .

(2.2)
Here N X G (dW ) = N x (dW )X G (dx) is a (random) intensity measure on the space of the snake, i.e. C([0, ∞), W). Consider G = G x ε1 and D = G x ε2 with ε 1 > ε 2 > 0. We can define the exit measure X D (W i ) for each W i following the construction of exit measure in Chapter V.1 of [11]. As in (2.6) of [8], one may conclude If U is an open subset of S(X 0 ) c , then L U , the restriction of the total local time L x to U , is in C(U, R) which is the set of continuous functions on U . Here are some consequences of (2.2) that are already proved in Proposition 2.2(a) of [8].
The σ-finiteness of N X0 is not an issue here as we may define the above conditional expectation by, e.g., using Radon-Nikodym derivative.
We will need a version of the above under P X0 as well, which follows immediately from Proposition 2.3 of [8].

Construction of the total local time by exit measure
In this section we will give the proof of Theorem 1.
where V λ is the unique solution to Here δ 0 is the Dirac delta function and the above differential equation is interpreted in a distributional sense. One can check that V λ is radially symmetric and we may write V λ (|x|) for V λ (x). Recall ψ 0 from (1.12). It is known that (see, e.g., p. 187 of [4]) V λ is smooth in R d \{0}, and near the origin, Lemma 8 of [3] gives that Proof of Theorem 1.1. The outline for the proof is as follows: First we get some L 2 convergence, associated with X G x ε and L x , using the Laplace transforms. Then we show that this implies the convergence in measure. When d = 3, we prove there is an a.s. limit by the martingale arguments. It is then immediate that L x , as the limit of convergence in measure, is in fact the a.s. limit, thus completing the proof.
We first consider the N X0 case. Fix any x ∈ S(X 0 ) c and let δ := d(x, S(X 0 )) > 0. For any λ > 0 and 0 < ε < δ/2, we have In the second equality we have used the fact that the exit measure X G x ε is supported on ∂G x ε by (1.10) and then apply the radial symmetry of V λ to get V λ (x − y) = V λ (|x − y|) = V λ (ε) for any y ∈ ∂G x ε . The above still holds true if we replace λ with 2λ in (3.5). Use the above in (3.4) to arrive at We first deal with I 1 .