Joint H\"older continuity of local time for a class of interacting branching measure valued diffusions

Using a Tanaka representation of the local time for a class of superprocesses with dependent spatial motion, as well as sharp estimates from the theory of uniformly parabolic partial differential equations, the joint H\"older continuity in time and space of said local times is obtained in two and three dimensional Euclidean space.


Introduction
Given a stochastic process {µ t : t ≥ 0} built on a probability space (Ω, F , P) and valued in the space M(R d ) of all positive Radon measures on R d , a Borel measurable process which maps any (t, x) ∈ [0, ∞) × R d to Λ x t ∈ [0, ∞) is called the local time of {µ t } if, for any continuous function with compact support φ ∈ C c (R d ), there holds P-almost surely at each time t ≥ 0 (with finite integrals on both sides) (1.1) Here and henceforth we write φ, µ = R d φ(x)µ(dx), for any µ ∈ M(R d ) and any µintegrable φ. Equation (1.1) ensures the unicity of the local time Λ x t dx as a Radon measure, when it exists, P-almost surely for every fixed t ≥ 0. If in addition Λ x t is integrable with respect to Lebesgue measure λ 0 on R d , for each fixed t, then it is the Radon-Nikodym derivative (in space) with respect to λ 0 of the occupation time process t 0 µ s ds, itself a time-averaged measure valued process with (hopefully) more regular paths than µ t itself.
For example, Super-Brownian motion {µ t } has a density f t = dµ t /dλ 0 when d = 1 (Konno and Shiga [15]) but not in the cases d ≥ 2 (see Dawson and Hochberg [3] and Perkins [22] [23]). When d = 1 the choice Λ x t = t 0 f s (x)ds is immediate but when d ≥ 2 the fractal dimension of support for µ t renders the proof of existence of Λ x t a lot more involved. The sharpest estimates for the closed support of Super-Brownian motion are found in Dawson and Perkins [5] when d ≥ 3 and Le Gall and Perkins [18] when d = 2.
The existence and the joint space-time continuity of paths for the local time of Super-Brownian motion, when d ≤ 3, were first obtained by Iscoe [14] and Sugitani [27]. Their results were variously sharpened and generalized, first by Adler and Lewin [1] for super stable processes and Krone [16] for superdiffusions; then by many others, most notably Ethier and Krone [9] for some related Fleming-Viot processes with diffusive mutations; López-Mimbela and Villa [21], who streamlined and unified the various definitions of the local time and clarified their interrelations in the above cases; Li and Xiong [19], who offered an alternative (trajectorial) definition of the local time when the superprocess is degenerate, that is, a purely atomic measure valued process, and proved its joint Hölder continuity, as well as scaling limit theorems, using a representation in terms of stochastic integrals with respect to the excursions of an underlying Poisson random measure.
More recently, Dawson et al. [6] proved the existence of the local time for the (larger still) class of superprocesses with dependent spatial motion (SDSM) first introduced in Wang [29], [30]. This class, which contains Super-Brownian motion and some superdiffusions as special cases, describes the movement of a cloud of infinitely many particles, each subject to critical branching and diffusing according to second order partial differential operators with spatially dependent coefficients, among a random environment prescribed by a Brownian sheet. Following Hochberg [12] in his description of Super-Brownian motion, the resulting M(R d )-valued process can been colourfully described as the high density limit for an instantaneous smearing of the diffusive trajectories performed by approximating clusters of finitely many newborn particles, further subjected to the motion of an ambiant random medium.
Our main result (Theorem 2.3) is the joint Hölder continuity for the local time Λ x t of SDSM, in time and space (t, x), when d ≤ 3. It is formulated next, in Section 2, together with all the notation and preliminary statements required for this purpose. The proof of our main result is in Section 3 and uses: a duality argument for the evaluation of the moments of {µ t }, inspired by Sugitani [27], Krone [16] and Ethier and Krone [9]; sharp inequalities due to Aronson [2] and Ladyzenskaja et al. [17] for the fundamental solutions of those uniformly parabolic partial differential equations associated with the Markov semigroup for SDSM; estimates for the Green function of this semigroup; and a Tanaka formula from Dawson et al. [6], which involves stochastic integrals with singular integrands. Some incidental proofs are relegated to Section 4.
2 Conditions, estimates and main results

Basic notation
For any topologically complete and separable metric space S (hereafter, a Polish space), B(S) denotes its Borel σ-field, B(S) the Banach space of real valued bounded Borel measurable functions on S with the supremum norm · ∞ and C(S) the space of real valued continuous functions on S. Subscripts b or c on a space of functions refers to its subspace of bounded or compactly supported functions, respectively, as in C b (S) and C c (S). As usual C([0, ∞), S) denotes the Polish space of continuous trajectories into S with the topology of uniform convergence on compact time sets. S m denotes the m-fold product of S. When S = R d , subscript 0 indicates those functions vanishing at ∞ while superscript k ≥ 1 means continuous derivatives up to and including order k (possibly infinite). We shall make use mostly of C 0 (R d ), those bounded continuous functions vanishing at ∞, its The main set of functions of interest here is defined for any real number a ≥ 0 with I a (x) = (1 + |x| 2 ) (−a/2) and We will also need , the space of bounded continuous functions with all derivatives bounded, up to and including order 1 in the time variable up to time t and order 2 in the md space variables, including mixed derivatives of that order. When no ambiguity is present we also write the partial derivatives (of functions and distributions) in abridged form ∂ ∂x q and so on.
Given any positive Radon measure µ ∈ M(R d ) and any p ∈ [1, ∞), we write L p (µ) for the Banach space of real valued Borel measurable functions on R d , with finite norm Let M(R d ) be the space of all positive Radon measures on R d and M 0 (R d ), its subspace of finite positive Radon measures. For any real number a ≥ 0, define the main set of measures of interest here as The topology τ a of M a (R d ) is defined in the following way: is a Polish space (see Iscoe [13] and Konno and Shiga [15]). For instance, the Lebesgue measure λ 0 on R d belongs to M a (R d ) for any a > d. Furthermore, both dx = λ 0 (dx) are used indifferently when calculating Lebesgue integrals.
The short form µ m = µ ⊗ . . . ⊗ µ denotes the m-fold product measure of µ ∈ M a (R d ) by itself and we write I a,m for the product I a,m (x) = I a (x 1 ) · . . . · I a (x m ), keeping in mind that I −1 a,m f (x) = I −1 a,m (x) · f (x) means the product, not the composition of functions.

Sufficient conditions
The following basic assumptions are valid throughout this paper.
, P) be a filtered probability space with a right continuous filtration {F t } t≥0 , satisfying the usual hypotheses and upon which all our processes are built, notably a Brownian sheet W on R d and a countable family Recall that a Brownian sheet on R d is an R 1 -valued random set function W on the Borel σ-field B(R d ×R + ) defined on (Ω, F , {F t } t≥0 , P) such that both of the following statements hold: for every A ∈ B(R d ) having finite Lebesgue measure λ 0 (A), the process W (A×[0, t]) is a square-integrable {F t }-martingale; and for every pair A i ∈ B(R d ×R + ), i = 1, 2, having finite Lebesgue measure with A 1 ∩ A 2 = ∅, the random variables W (A 1 ) and W (A 2 ) are independent, Gaussian random variables with mean zero, respective variance λ 0 (A i ) and W (A 1 ∪ A 2 ) = W (A 1 ) + W (A 2 ) holds P-almost surely (see Walsh [28] or Perkins [24]).
The mathematical description of the diffusive motion of SDSM requires the following second order partial differential operators: , where, for p, q = 1, . . . , d, the local (or individual) diffusion coefficient is and the global (or common) interactive diffusion coefficient is , for every i, j = 1, · · · , d. For every m ≥ 1, the dm × dm diffusion matrices (Γ ij pq ) 1≤i,j≤m;1≤p,q≤d of real valued functions defined by (2.3) are strictly positive definite everywhere on R d , that is, there exists two positive constants λ * and Λ * such that for any ξ = (ξ (1) , · · · , ξ (m) ) ∈ (R d ) m we have Hypothesis (2) ensures that operator G m is not only uniformly elliptic, but also that it generates a Feller semigroup Ethier and Kurtz [10,Chapter 8]). It can be written as when t > 0, for every f ∈ C 0 ((R d ) m ), with a transition probability density q m t (x, y) > 0 which is jointly continuous in (t, x, y) ∈ (0, ∞) × (R d ) m × (R d ) m everywhere and such that P m t f (x), as a function of (t, x), belongs to ∪ t≥0 C 1,2 b ([0, t] × (R d ) m ), for every choice of f ∈ C 0 ((R d ) m ) (see Stroock and Varadhan [26,Chapter 3]).
The rest of the assumptions collectively constrain the family of initial measures. When the initial state of SDSM is a finite measure, the total mass process of SDSM is equivalent to a one-dimensional continuous state critical branching process and extinction occurs almost surely (see Wang [29], [30]). Unbounded measures as initial states give rise to alternative phenomena, provided that the potential explosion of mass near the initial time t = 0 is curtailed in order to allow the local time of SDSM to exist at all (see Dawson et al. [6]). This explosion can occur in either situation, as exemplified next.
Example 2.1. Let ϕ t is the transition density of a Brownian motion particle on R d ϕ s (y) : When the initial measure is δ, the Dirac measure which puts mass 1 at the origin 0 ∈ R d , we get, for all t > 0 and d ≥ 1, sup 0<s≤t ϕ s (y − ·), δ < ∞ if and only if y = 0.
Under less restrictive conditions than Hypothesis (2), Aronson [2, Theorem 10] proved that, for the transition probability density q 1 from (2.7), there exist four positive constants a * , b, c and A * such that holds for any x, y ∈ R d and s > 0. Example (2.1) thus remains valid in the uniformly elliptic case as well: an initial measure with an atom makes the existence of the local time for SDSM questionable in general, hence the need for an additional restriction.
Hypothesis 3. For all t > 0, the initial measure µ 0 ∈ ∪ a≥0 M a (R d ) verifies Remark: Any measure µ 0 which is absolutely continuous with respect to λ 0 and has a Radon-Nikodym derivative which is either bounded or finitely λ 0 -integrable, satisfies Hypothesis (3), notably the measures I a (x)dx for all choices of a ≥ 0. Combined with Aronson's inequalities (2.9), the requirement of finiteness in (2.10) is equivalent to its special case Υ t (ϕ, µ 0 ) < ∞ obtained by setting q 1 s (·, y) = ϕ s (y − ·) from (2.8). The proof of joint continuity of the local time most likely requires such a uniform bound, since our efforts at localizing this to a neighbourhood of the origin, succeed at the cost of a strengthening of the observation that, for every w ∈ R d , the translate µ 0 (· − w) of any measure µ 0 ∈ M a (R d ) is also in M a (R d ). This is immediately seen though the inequality I a (x + w) ≤ 2 (a/2) I −1 a (w)I a (x), valid for every x, w ∈ R d .
This is true for any finite measure µ 0 ∈ M 0 (R d ) ⊂ M a (R d ) and every a ≥ 0, hence any Radon measure with compact support and any measure with a finitely λ 0 -integrable Radon-Nikodym derivative with respect to λ 0 ; as well as any measure with a bounded Radon-Nykodym derivative with respect to λ 0 , in this last case provided a > d.
Finally, observe that when Hypotheses (2) and (4) are combined, Hypothesis (3) is equivalent to the weaker requirement of the existence of an ǫ > 0 such that holds. The proof of this last statement is in Subsection 4.1.

Sharp estimates
We need some important properties of fundamental solutions to uniformly parabolic partial differential equations, specifically here, the Kolmogorov backward equation Lu = 0, recalling that operator L = G 1 − ∂ t is uniformly parabolic whenever G 1 is uniformly elliptic.
Lemma 2.1. Under Hypothesis (2) and for each choice of T > 0 and d ≥ 1, there are positive constants a 1 and a 2 such that, for all nonnegative integers r and s verifying 0 ≤ 2r + s ≤ 2, There also exists a unique fundamental solution Γ(x, t; ξ, τ ) to Lu = 0, started at position ξ ∈ R d at initial time τ ∈ [0, T ) and evaluated at new position x ∈ R d after spending time t − τ > 0 to reach t ∈ (τ, T ]. Moreover, there exist two constants c > 0 and c 0 > 0, such that, for all nonnegative integers r, s 1 , s 2 , . . . , s d verifying 0 ≤ l = 2r +s 1 +s 2 +. . .+s d ≤ 2 and writing for the identity, there holds, for every choice of α ∈ (0, 1),

SPDE and dual process
Let us now describe the SDSM class itself. Under Hypotheses (1) and (2), Dawson et al. [6] characterize SDSM by way of a wellposed martingale problem, the unique solution of which is the law of a measure valued diffusion (Markov process with continuous paths) {µ t : t ≥ 0} which satisfies and are continuous square-integrable {F t }-martingales, mutually orthogonal for every choice of φ ∈ K a (R d ) and driven respectively by a Brownian sheet W and a square-integrable martingale measure M with quadratic variation Here the filtration of choice is For any a ≥ 0 and any initial value µ 0 ∈ M a (R d ), this unique law on the Borel subsets of C([0, ∞), M a (R d )) will henceforth be denoted by P µ 0 and the corresponding expectation by E µ 0 . Parameter γ > 0 is related to the branching rate of the particle system and σ 2 > 0 is the variance of the limiting offspring distribution.
Almost sure statements, including those referring to (1.1), will henceforth be meant to hold P µ 0 -almost surely.
The properties of law P µ 0 are determined by way of a function valued dual process for SDSM due to Dawson et al. [6], a version of the original first built by Dawson and Hochberg [3] and later generalized by Dawson and Kurtz [4] as well as others thereafter.
Let {J t : t ≥ 0} be a decreasing càdlàg Markov jump process on the nonnegative integers {0, 1, 2, . . .}, started at J 0 = m and decreasing by 1 at a time, with Poisson waiting times of intensity γσ 2 l(l − 1)/2 when the process has reached value l ≥ 2. The process is frozen in place when it reaches value 1 and never moves if it is started at either m = 0 or 1. Write {τ k : 0 ≤ k ≤ J 0 −1} for the sequence of jump times of {J t : t ≥ 0} with τ 0 = 0 and τ J 0 = ∞. At each such jump time a randomly chosen projection is effected on the function valued process of interest, as follows. Let {S k : 1 ≤ k ≤ J 0 } be a sequence of random operators which are conditionally independent given {J t : t ≥ 0} and satisfy for any m ≥ 2 and y = ( The process Y is a well-defined B-valued strong Markov process for any starting point Y 0 ∈ B, since all linear operators Φ m ij and P m t are contractions on bounded functions. (When m = 0 we simply write B((R d ) 0 ) = R d and Y t = P 0 t acts as the identity mapping on constant functions.) Clearly, {(J t , Y t ) : t ≥ 0} is also a strong Markov process.
We show next that Y t ∈ B still holds for all t > 0 when Y 0 ∈ B, under some mild conditions. The following extends several of the results from Dawson et al. [6]. For the proof of the new features, see Subsection 4.2. (1) and (2) are satisfied. For any a ≥ 0, m ≥ 1,

Theorem 2.2. Assume that Hypotheses
with both sides finite in every case. Moreover, for any p ≥ 1, every f ∈ L p (µ m 0 ) also belongs P µ 0 -almost surely to L p (µ m t ) as well as and therefore also both hold P µ 0 -almost surely, whenever the respective right hand side is finite.
Equation (2.17) is called the duality identity between the law of SDSM and that of its dual process (J, Y ).

Tanaka representation and main result
For the single particle transition density q 1 t (0, x) (from 0) exhibited in (2.7) for the semigroup P 1 t associated with generator G 1 from (2.2), its Laplace transform (in the time variable) is given by for any λ > 0. Formally Q 0 is known as Green's function for density q 1 t and exhibits a potential singularity at x = 0. By (2.12), for all x ∈ R d {0} we can also write for any i ∈ {1, 2, · · · , d}, with the derivative taken in the distributional sense. When d = 1, under Hypothesis (2), there holds I −1 a |Q λ | p ∞ < ∞ for any a > 0 and p ≥ 1 -see Inequality (4.5) in Subsection 4.3 -so the bounds (2.18) and (2.19) are applicable in that case (though it is not the one of interest here). Unfortunately, when d ≥ 2, even in the special case of the gaussian kernel ϕ, there holds Q λ ∞ = ∞. Nevertheless, the · µ 0 ,p -norms, with p ∈ [1, ∞) instead, yield preliminary bounds.
at t = 0 and so does Q λ ∈ L p (µ 0 ) ∩ L p (λ 0 ), in each of the following three cases: (i) when p = 1, for all d ≥ 1; (ii) when p = 2 and d ≤ 3; (iii) for all p ≥ 1 when d = 1 or 2. Furthermore, there holds (iv) in all dimensions d ≥ 1 and for any i ∈ {1, 2, · · · , d}, In the special case of Lebesgue measure µ 0 = λ 0 ∈ M a (R d ) (when a > d), the pointwise part of each of statements (i), (ii) and (iv) pertaining to t = 0 was obtained in [6], where ∂ x Q λ ∈ L 2 (λ 0 ) is also proved to hold when d = 1 and to fail when d ≥ 2. This pointwise finiteness suffices to prove the existence of the local time under Hypotheses (1), (2) and (3), as their proof does not require the use of Hypothesis (4); but the uniform extensions presented here are useful in our proof of its joint Hölder continuity. The proof of Proposition 2.1 is found in Subsection 4.3. Our main result can now be stated. (1) and (2), with d = 1, 2 or 3, select any a ≥ 0 and µ 0 ∈ M a (R d ) satisfying Hypothesis (3), with ambiant Brownian sheet W and martingale measure M. For every choice of (t, x) ∈ [0, ∞) × R d ,

Theorem 2.3. Under Hypotheses
is the local time for SDSM {µ t }. It satisfies both (1.1) P µ 0 -almost surely for every choice of φ ∈ C c (R d ) and square-integrability for every (t,

23)
Moreover, under the additional Hypothesis (4), there exists a version of Λ x t which is Hölder jointly continuous in (t, x) ∈ [0, ∞) × R d . Explicitely, for every exponent α ∈ (0, 1) in space, as well as for every choice of n > 1, there is a constant c = c(T, d, h, q 1 , n, α, λ) > 0 such that, for every x, z ∈ R d and 0 < s < t ≤ T there holds This last inequality suffices for an application of Kolmogorov's continuity criterion (such as Theorem I.2.1 of Revuz and Yor [25]), which states the existence of a modificatioñ Λ x s of Λ x s such that holds for every γ ∈ [0, 1) and β ∈ [0, 1/2). Equation (2.22) is from Dawson et al. [6] and is called the Tanaka formula for SDSM. It was proved there under a slightly weaker version of the current Hypothesis (3). Its right hand side is a Schwartz distribution and (1.1) holds in the distributional sense. The value of the local time is independent of λ > 0 but it varies with d. What is new here of course is the last assertion, regarding joint Hölder continuity. Its proof occupies the next section, where we also explore the impact of Hypothesis (4).

Joint continuity of the local time
The proof of Theorem 2.3 proceeds next, using some of the sharp inequalities previously mentionned, by comparing the higher moments of both the local time for SDSM {µ t } and its space-time differences, against those associated with a special function controlling the growth of the two martingales emanating from the branching and random environment mechanisms within the Tanaka formula (2.22). The argument is in the spirit of a similar result by Ethier and Krone [9]) for the Fleming-Viot process, for which they use Super-Brownian motion as the benchmark.
Denote byQ λ the Laplace transform in the special case of Super-Brownian motion, where h ≡ 0 and c is the identity matrix: The two are linked by way of the following technical result.
Lemma 3.1. Fix λ > 0 and d ≥ 1. The following hold for all x ∈ R d : Proof: Elementary inequality (4.5) applied toQ λ (x) ≤G λ/2 (x) sup s≥0 e −λs/2 s 1/2 yields the first statement. For every u > 0 and v > 0, there holds with z = uv/(u + v) and w = u + v, as seen by expanding the right side. The Jacobian J(w, z; u, v) = |v − u|/(u + v) = 1 − 4z/w of this change of variable is non null except on the line v = u. Keeping in mind that, for all w > 0 and z > 0, each dimension amongst d = 1, 2 or 3 satisfies w −d/2 ≤ w −3/2 + w −1/2 and z 1−d/2 ≤ z −1/2 + z 1/2 everywhere, there is a constant K = K(d, λ) > 0 such that after splitting the second integral, taking advantage of the monotonicity of the various functions, using the change of variable s = 1 − 4z/w with dw = 4zds/(1 − s) 2 and integrating by parts. Finally note that inequality (4.5) in the Appendix yields ∞ 0 e −4λz z 1/2 ϕ z (x)dz ≤G 2λ (x) · sup z≥0 ze −2λz ≤G 2λ (x) · max(1, 1/2λ) and thatG λ (x) is decreasing in λ at every x ∈ R d . Remark: Under Hypothesis (2), Aronson's inequalities (2.9) ensure that both statements remain valid when the specificQ λ (x) is replaced by the general Q λ (x), provided the constants in front ofG, as well as the scale parameters inside it, are adjusted accordingly. The same is true when bounding either |∂ x i Q λ (x)| or |∂ x iQ λ (x)| above by a scaled multiple ofG, using inequality (2.12) instead. The key inequality in Lemma (3.1) is of course the second one. It controls the size and fluctuations of the square ofQ λ (x), which drives the quadratic variation process for the branching martingale, by way of the more volatile size and fluctuations ofG λ (x), which drives the quadratic variation process for the random environment martingale.
Proof: We first prove (3.5). For any Borel measurable, real valued function f on some (R d ) n with n ≥ 1, define mapping T n X f (Y ) = f (X − Y ) and observe the following three properties: T n−1 X n−1 • Φ n ij = Φ n ij • T n Xn holds everywhere with X n = (x, . . . , x) ∈ (R d ) n ; while T n Xn does not necessarily commute with P n u in general, it does so when the kernel q 1 is symmetric, in particular when q 1 u (·, y) = ϕ u (y − ·), so we can write T n Xn •P n u =P n u • T n Xn withP n the heat semigroup on (R d ) n ; and finally, writing the dual as Y u = Y u (f ) to indicate the dependency on Y 0 = f andỸ u =Ỹ u (f ) for its special case in (2.16) involving P n cu instead of P n u , that is, the dual to a rescaled Super-Brownian motion, upper bound (2.9) applied to q n instead of q 1 implies Using bounds (2.18) and (2.19) from Theorem 2.2 with f = I a,n and constant C to differentiate from c already in use here, we get and hence (3.5) follows from Hypothesis (4). The proof of (3.6) ensues similarly from Hypothesis (3) via holds before the first jump (for any ǫ > 0) by the Markov property so that, in the event {τ 1 > u} where the first jump does not occur prior to time u, there holds for any b > 0 which is finite uniformly in both x ∈ R d and u ∈ (0, t] by Hypothesis (3). Next, at or immediately after the first jump time τ 1 of the function valued dual, when some randomly selected coordinate j is replaced by some other i, but before the second jump, hence on {τ 1 ≤ u < τ 2 } and for any ǫ > 0, we get with min(τ 1 + s i , τ 1 + s j )/2 ≤ z ij ≤ max(τ 1 + s i , τ 1 + s j )/2 using (3.3). Trajectory by trajectory, the function valued dual is therefore bounded everywhere on ] Ju for the selected pair (i, j) with J u = n − 1 and therefore, once again uniformly, with n 2 such pairs, there holds On {τ 2 ≤ u < τ 3 }, either the second jump involves two new coordinates and finiteness ensues for the same reason, with two factors (s i +s j )(s k +s ℓ ) inside (3.9) and (3.10) instead of just one; or coordinate i is involved in both jumps, to or from a new site k, for the second jump, yielding a different factor of the form (τ 2 + s k + τ 2 − τ 1 + z ij ) ≥ s k + min(s i , s j )/2. The finiteness of triple integral the finiteness of which finishes the argument for the validity of (3.6).
That (3.5) and (3.6) together imply (3.7) is a consequence of inequalities relegated to Subsection 4.3 and already used in the proof of Proposition 2.1, where starting with (4.7) we used the supremum Υ from (2.10) in order to handle all at once the powers p ≥ 1 required in the proof of Proposition 2.1. Going back to (4.7) with p = 1 yields a sharper version of (4.9), which nevertheless remains valid when substitutingG for |∂ x i Q λ (x)| and setting values a 1 = 1, a 2 = 1/2, η = 2a 2 p/a = 1/a and η * = η/2a 2 = η = 1/a. We obtain instead for all x ∈ R d and using any a > 0 in the base case where µ 0 ∈ M 0 (R d ).
is a square-integrable F t -martingale, for every λ > 0 and p ∈ {1, 2, . . . , d}, with quadratic variation given by Proof: The statement regarding (3.11) is from Dawson et al. [6] so we focus on the joint Hölder continuity. Given any suitably measurable real valued functions H ≥ 0 and F ≥ 0, there holds, for any z ∈ R d , n ≥ 1 and 0 < s < t < ∞, by successively expanding the square, applying Fubini's theorem and using Hölder's inequality with conjugate exponents n and n/(n − 1), treating the case n = 1 separately. From (3.12) we get upper bounds for both the time lag 3.13) and the space lag for the quadratic variation process after Hypothesis (2) first takes care of the finiteness of constant Combining (2.12) and (2.13), for any α ∈ (0, 1), the constants c = c(α) > 0 and c 0 = c 0 (α) > 0 from (2.13) can be respectively increased and decreased, so that both for every x, z ∈ R d , with G λ * from (3.2) rewritten as ∞ 0 e −λs s −(d+1)/2 exp {−c 0 |x| 2 s }ds after putting u = s/2c 0 and λ * = 2c 0 λ to standardize the exponential kernel and get rid of c 0 , by incorporating it into both λ * and c. This transforms (3.13) and (3.14) into with a new c = c(T, d, h, q 1 , n, α) > 0 and , yields the existence of a universal constant C n > 0 for each n ≥ 1 such that, for every x, z ∈ R d and every 0 ≤ s ≤ t ≤ T , there holds this last inequality using (3.15), (3.16) and Lemma 3.2, incorporating its bounds and C n into the new c = c(T, d, h, q 1 , n, α, λ, µ 0 ) > 0 which does not depend on x, z, s or t.
Theorem 3.4. Assume that Hypotheses (1), (2), (3) and (4) are satisfied for some a ≥ 0 and µ 0 ∈ M a (R d ). For either d = 1, 2 or 3, the random field is a square-integrable F t -martingale, for every λ > 0, with quadratic variation given by and withQ λ * from (3.1). The rest of the argument follows the proof of Theorem 3.3.
Proof: [Proof of Theorem (2. 3)] All we need to show here is that every term on the right hand side of (2.22) satisfies individually the Hölder-type upper bound (2.24) for their respective norm, as defined by the left hand side. The last two terms do, as a direct consequence of Theorems 3.3 and 3.4. The other three are less difficult and handled similarly.
Using (3.12), with F (x, z, w) = |Q λ (z − w) − Q λ (x − w)| and H(y, w) = I B (y) for any set B with λ 0 (B) = 1, the third term (ignoring the constant λ) has its spatial increments bounded, by way of the Cauchy-Schwarz inequality and then (3.12), to get and the expectation E µ 0 of the spatial difference taken on both sides is treated identically to (3.20) by setting s = 0. The time difference is treated in the same fashion, using F (x, z, w) = |Q λ (x − w)| and (3.19) instead.
The first two terms of (2.22) follow by way of (2.12), (2.13) and (3.8), just as in the proof of Theorem 3.4.
and therefore, by Fubini's theorem, any measure µ 0 ∈ M a (R d ) satisfies hence sup ǫ≤s≤t ϕ s (y − ·), µ 0 < ∞ holds λ 0 -almost everywhere in y ∈ R d (and hence on a dense subset of R d ). Since the mapping y → ϕ s (y − x) is concave for each fixed x ∈ R d and s > 0, it sends any convex combination y = αw ]. This means that the dense set is also a convex set. Therefore the dense set is all of R d and we get, over any ball B ∈ B(R d ), using the dyadic grid in order to trap this ball inside the convex hull of finitely many points from the dense subset, sup y∈B sup ǫ≤s≤t ϕ s (y − ·), µ 0 < ∞ for any µ 0 ∈ ∪ a≥0 M a (R d ) and under no further assumptions. The need for Hypothesis (4) is illustrated next. Because of lim |x|→∞ ϕ t (x)I −1 a (x) = 0 for any a, the radius w = w(d, a, t) > 0 of the smallest ball centered at 0 outside of which ϕ t ≤ I a holds, is such that, for all x ∈ R d , we have where the indicator function of set N ∈ B(R) is 1 N (x) = 1 if x ∈ N and 0 elsewhere. Hence there is a constant c = c(d, a, t) > 0 such that ϕ t (x) ≤ cI a (x) holds everywhere and we get under Hypothesis (4), using (4.2). This is the required bound ensuring that the two statements are equivalent.
The size of the integrand near 0 affects the finiteness of this integral significantly so we partition [0, ∞) p into blocks close to and away from the origin. For any subset of indices A n ⊂ {1, 2, . . . , p} of cardinal n ≤ p, with A 0 = ∅ and complement A c n = {1, 2, . . . , p}\A n , write A n for the subset of those points (s 1 , s 2 , . . . , s p ) ∈ [0, ∞) p such that s i ≤ η holds for all i ∈ A n and B n for the one such that s i > η for all i ∈ A c n , with η ≥ 0 arbitrary but fixed. The symmetry of mappings S , Σ S and γ(S) in their coordinates makes the integrals to follow, over these sets, dependent only on the cardinality of the sets involved and not on their actual selection, so this ambiguous notation is not problematic and its simplicity fits our purpose. Indeed we can write, for any permutation invariant function of the form F (S) = p i=1 G(s i ) = i∈An G(s i ) j∈A c n G(s j ) for some G (which is positive in all our applications so let G map [0, ∞) into itself here), where p n denotes the usual binomial coefficient, A * n = [0, η] n denotes the projection of A n onto its constrained coordinates and similarly for B * n = (η, ∞) p−n . (The integrals over A * 0 = ∅ and B * p = ∅ are set equal to 1, here and in the rest of the proof.) The next step consists in selecting a specific value of η, as small as possible, in order to control the tail asymptotics of the multiple integral (4.3) decomposed along the lines of (4.4), that is, its value over the largest sets B * n possible, for every n ∈ {1, 2, . . . , p}, thus enabling sharper bounds on as small a neighbourhood of the origin as possible, as this is where the singularity makes integrability most difficult, in the worst case over the set A * p . The value η = 2a 2 p/a turns out to be the smallest one for which the following calculations carry through.
So we bound the term in B * n first. For every v ∈ [0, ∞), γ > 0 and β ≥ 0, there holds There ensues successively, since on B * n we have min {i=1,2,...,p−n} s i > η which implies γ(S p−n ) < a 2 (p − n)/η = a(p − n)/2p ≤ β = a/2 when 0 ≤ n < p and η = 2a 2 p/a, that there holds, for any a > 0 and any integers d ≥ 1 and p ≥ 1, (a+d)/2 Γ(p − n + a/2)λ p−n+a/2 < ∞, (4.6) keeping in mind that for this calculation we write S = S p−n , dS = dS p−n = p−n i=1 ds i , S = S p−n = p−n i=1 s i , Σ S = Σ S p−n = p−n i=1 s i and γ(S) = γ(S p−n ) = a 2 p−n i=1 s −1 i . Here we write the usual Gamma function Γ(t) = ∞ 0 v t−1 e −v dv. Let us denote the last line of (4.6) by K(a 2 , d, p, n, a, λ), emphasizing that this bound is uniform in x in spite of the compensator term I −1 a (x) up front which is required due to the generality of starting measure µ 0 . Define constant C p = a p 1 max 0≤n<p p n K(a 2 , d, p, n, a, λ), eliminate e −λΣ S ≤ 1 and combine (4.3), (4.4) and (4.6) to get, for any time u ≥ 0, Note the absence of mollifier I a (x) ≤ 1 when n = p, making this last term more difficult to control. We also have, for every u ≥ 0 and n ≤ p, where Υ η * (ϕ, µ u ) is from (2.10) and we used both 1/2γ(S n ) ≤ η/2a 2 n ≤ η * := η/2a 2 on the set A * n and the inequality between the geometric and the harmonic means. The double integrals in (4.7) become, again for every u ≥ 0 and n ≤ p, We have just proved that for every measure µ 0 ∈ M a (R d ) satisfying Hypothesis (3), there holds |Q λ |, µ 0 < ∞ in all dimensions d ≥ 1, |Q λ | 2 , µ 0 < ∞ when d ≤ 3 and |Q λ | p , µ 0 < ∞ for all p ≥ 1 when d = 1 or 2. Notice that Hypothesis (4) is not needed for these statements.