Pathwise uniqueness for a SPDE with H\"older continuous coefficient driven by \alpha-stable noise

In this paper we study the pathwise uniqueness of solution to the following stochastic partial differential equation (SPDE) with H\"older continuous coefficient: \begin{eqnarray*} \frac{\partial X_t(x)}{\partial t}=\frac{1}{2} \Delta X_t(x) +G(X_t(x))+H(X_{t-}(x)) \dot{L}_t(x),~~~ t>0, ~x\in\mathbb{R}, \end{eqnarray*} where $\dot{L}$ denotes an $\alpha$-stable white noise on $\mathbb{R}_+\times \mathbb{R}$ without negative jumps, $G$ satisfies the Lipschitz condition and $H$ is nondecreasing and $\beta$-H\"older continuous for $1<\alpha<2$ and $0<\beta<1$. For $G\equiv0$ and $H(x)=x^\beta$, in Mytnik (2002) a weak solution to the above SPDE was constructedand the pathwise uniqueness of the solution was left as an open problem. In this paper we give an affirmative answer to this problem for certain values of $\alpha$ and $\beta$. In particular, for $\alpha\beta=1$, where the solution to the above equation is the density of a super-Brownian motion with $\alpha$-stable branching (see also Mytnik (2002)), our result leads to its pathwise uniqueness for $1<\alpha<\sqrt{5}-1$. The local H\"older continuity of the solution is also obtained in this paper for fixed time $t>0$.


Introduction
It was proved by Konno and Shiga (1988) as well as Reimers (1989) that one-dimensional binary branching super-Brownian motion has a jointly continuous density that is a random field {X t (x) : t > 0, x ∈ R} satisfying the following continuous-type stochastic partial different equation (SPDE): for SPDE (1.1) remains open even though it has been studied by many authors; see [14,15,21,6]. The main difficulty comes from the unbounded drift coefficient and the non-Lipschitz diffusion coefficient. Mytnik (2002) considered the following jump-type SPDE and constructed a weak solution: whereL is a one sided α-stable white noise on R + × R without negative jumps, 1 < α < 2 and 0 < β < 1. Put p := αβ < 2. For the case p = 1, the solution is the density of a super-Brownian motion with α-stable branching. The weak uniqueness of the solution to (1.2) also follows from a martingale problem and the pathwise uniqueness is still unknown. Mytink and Perkins (2003) showed that the density has a continuous version at any fixed time. Recently, Fleischmann et al. (2010) showed that this continuous version is locally Hölder continuous with index η c := 2/α − 1, and Fleischmann et al. (2011) showed that it is also Hölder continuous with indexη c := (3/α − 1) ∧ 1 at any given spatial point. For p = 1, the weak uniqueness of solution to SPDE (1.2) and the regularities of solution X t (·) at fixed time t are unknown.
Throughout this paper, we always assume that 1 < α < 2 and 0 < β < 1. Our goal is to establish the pathwise uniqueness of solution to (1.2) under certain conditions of α and β. In particular, for p = 1 we show that the pathwise uniqueness holds for 1 < α < 4 − 2 √ 2. To prove the pathwise uniqueness we need the local Hölder continuity of the solution at fixed time t > 0 (Fleischmann et al. (2010) only proved this result for the case p = 1).
To continue with our introduction we present some notation. Let B(R) be the set of Borel functions on R. Let B(R) denote the Banach space of bounded Borel functions on R furnished with the supremum norm · . We use C(R) to denote the subset of B(R) of bounded continuous functions. For any integer n ≥ 1 let C n (R) be the subset of C(R) of functions with bounded continuous derivatives up to the nth order. Let C n c (R) be the subset of C n (R) with compact support. We use the superscript "+" to denote the subsets of positive elements of the function spaces, e.g., B(R) + . For f, g ∈ B(R) write f, g = R f (x)g(x)dx if it exists. Let M (R) be the space of finite Borel measures on R endowed with the weak convergence topology. For µ ∈ M (R) and f ∈ B(R) we also write µ(f ) = f dµ.
Equation (1.2) is a formal SPDE that is understood in the following sense: For any f ∈ S (R), the (Schwartz) space of rapidly decreasing infinitely differentiable functions on R, where L(ds, dx) is a one sided α-stable white noise on R + × R without negative jumps. Let if there is a pair (X, L) defined on the same filtered probability space (Ω, F , F t , P), which satisfies the usual conditions, so that (i) L is an α-stable white noise on R + × R without negative jumps.
Our main result, Theorem 4.1, is that the pathwise uniqueness holds for solution to equation (1.3) under certain conditions of α and β. To prove the pathwise uniqueness we need the local Hölder continuity of the solution at fixed time t > 0. For p = 1, the proof for the local Hölder continuity of X t (x) is based on the following equation (Fleischmann et al. (2010)): where M (ds, dz, dx) denotes a random measure on (0, ∞) 2 ×R with compensatorM (ds, dz, dx) = dsm(dz)X s (x)dx. Equation (1.6) was established for super-Brownian motion. But for p = 1 the solution to (1.3) is not a density of super-Brownian motion and we can not obtain the equivalent of equation (1.6). So, inspired by Li (2006, 2012), we transform (1.3) into the following SPDE (see Proposition 2.1): where f ∈ S (R) andÑ 0 (ds, dz, du, dv) is a compensated Poisson measure on (0, ∞) 2 ×R×(0, ∞) with intensity dsm(dz)dudv. By modifying the proof of [3, Theroem 1.2(a)] and using (1.7), we can obtain the local Hölder continuity (in the spatial variable) with index η c := 2/α − 1 for the continuous version of the solution to (1.3). Notice that η c ↑ 1 as α ↓ 1, which is quite different from that of continuous-type SPDE whose local Hölder index is typically smaller than 1 2 . This fact is key to proving the pathwise uniqueness.
We outline our approach here. By an infinite-dimensional version of the Yamada-Watanabe argument for ordinary stochastic differential equations (see Mytnik et al. (2006)), the problem of showing the pathwise uniqueness is reduced to showing the analogue of the local time term is zero (see proof of Theorem 4.1 and Lemma 4.6). That is to show that goes to zero as m, n → ∞. HereŨ s is the difference of continuous versions of the two weak solutions to (1.3),Ṽ s denotes the difference of continuous versions of these two solutions with power β, ψ n and Ψ are two positive test functions, Φ m is a nonnegative function and τ k is a stopping time.
Inspired by an argument of Mytnik and Perkins (2011), for fixed s, m and x, denote by x s,m (x) ∈ [−1, 1] a value satisfying The key to the proof of Lemma 4.6 is to split the local time term I m,n,k 4,1 (t) into two terms, where one term is bounded from above by and the other term is bounded from above by where M is a countable dense subset of [0, T ].
Observe that for fixed s, the continuous versionX s of the weak solution to (1.3) satisfies sup |x|≤K,|y|∨|v|≤1 So, it is natural to apply the Hölder continuity of x →X s (x) to find a collection of suitable stopping times (τ k ) so that lim k→∞ τ k = ∞ almost surely, and the first term I m,n,k 4,1,1 (t) can be bounded by 4m(na n ) −1 (2/m) 2ηβ which tends to zero as m and n jointly go to infinity in a certain way (if η > 1 2 ). It is hard to show that the supremum or integral with respect to s ∈ (0, T ] on the right side of (1.8) is finite. To this end the time τ k is chosen so that a Riemann type "integral" of the right hand side of (1.8) over s ∈ [0, τ k ] is finite. Concerning the second term is bounded away from zero. Now we can show that the integrand of I m,n 4,1,2 (t) is bounded from above by a deterministic function which also converges to zero as m, n → ∞ under certain conditions of α and β.
The paper is organized as follows. In Section 2 we first present some properties of the weak solution to equation (1.3). The local Hölder continuity for the continuous version of the solution is established in Section 3. In Section 4 we prove the main result of pathwise uniqueness of solutions to (1.3). In Section 5, the proofs of Proposition 2.3 and Lemma 2.4 are presented.
Notation: Throughout this paper, we adopt the conventions for any y ≥ x ≥ 0. Let C denote a positive constant whose value might change from line to line. We write C ε or C ′ ε if the constant depends on another value ε ≥ 0. We sometimes write R + for [0, ∞). Let λ denote the Lebesgue measure on R and (P t ) t≥0 denote the transition semigroup of a one-dimensional Brownian motion. For t > 0 and x ∈ R write p t (x) := (2πt) − 1 2 exp{−x 2 /(2t)}. We always use N 0 (ds, dz, du, dv) to denote the Poisson random measure corresponding to the compensated Poisson measureÑ 0 (ds, dz, du, dv).

Weak solution
In this section we establish some properties of the weak solution for (1.3), which will be used in next two sections.
Proof. (i) Suppose that (X, L) is a weak solution of (1.3). Then by the argument in Introduction, and we use the convention that 0 · ∞ = 0. Then for all B ∈ B(0, ∞) and a ≤ b ∈ R, Then by [7, p.93], on an extension of the probability space, there exists a Poisson random measure N 0 (ds, dz, du, dv) on (0, ∞) 2 × R × (0, ∞) with intensity dsm(dz)dudv so that LetÑ 0 (ds, dz, du, dv) = N 0 (ds, dz, du, dv) − dsm(dz)dudv. Then by (1.4) it is easy to see that for each f ∈ S (R), (ii) The proof is essentially the same as that of [10,Theorem 9.32]. Suppose that {X t : t > 0, x ∈ R} satisfies (1.7). Define the random measure N (ds, dz, du) on (0, ∞) 3 by It is easy to see that N (ds, dz, du) has a predictable compensator Then N (ds, dz, du) is a Poisson random measure with intensity dsm(dz)du (see [8,Theorems II.1.8 and II.4.8]). Define the α-stable white noise L by We then have for each f ∈ S (R). (X, L) is thus a weak solution to (1.3). 2 We need the following assumption on the weak solution: Assumption 2.2 Suppose that (X, L) is a weak solution to (1.3). For p = αβ > 1, there exists a constant q > 3p 3−α so that In the rest of this section, we always assume that (X, L) is a weak solution to (1.3) satisfying Assumption 2.2 with deterministic initial value X 0 ∈ M (R). Then it follows from Proposition 2.1, {X t (x) : t > 0, x ∈ R} satisfies (1.7). Proposition 2.3 For any t > 0 and f ∈ B(R) we have Moreover, P-a.s., λ-a.e. x. (2. 2) The proof is given in Appendix.
Lemma 2.4 Let 0 <p < α and T > 0 be fixed. Then for each 0 < t ≤ T there is a set K t ⊂ R of Lebesgue measure zero so that The proof is also given in Appendix.

t). Then by Lemma 2.4 and dominated convergence, both
tend to zero as t → 0.
Similarly, both tend to zero as t → 0. The proof is completed. 2

Hölder continuities
In this section we establish the local Hölder continuity of the weak solution to (1.3) at fixed time t > 0. Throughout this section we always assume that (X, L) is a weak solution to (1.3) satisfying Assumption 2.2 with deterministic initial value X 0 ∈ M (R). For n ≥ 1 and 0 ≤ k ≤ 2 n , put Theorem 3.1 For fixed t > 0, with probability one X t has a continuous versionX t . Moreover, for each η < η c = 2 α − 1, the continuous versionX t is locally Hölder continuous of order η, i.e. for any compact set K ⊂ R, Moreover, for each T > 0 and subsequence {n ′ : n ′ ≥ 1} of {n : n ≥ 1}, we have lim inf For any x ∈ R and s > 0, let Proof. Let r, δ and δ 1 satisfy the conditions in Lemma 2.5 and rδ > 1. By (2.9) and Corollary 1.2(ii) of [20], for each 0 < ε < r − 1/δ and T > 0, almost surely by the Markov inequality. By (2.5) we have Then lim sup Therefore, Theorem 3.1 holds with η c replaced by ε. Now via the same argument as in Proposition 2.9 of [3] one can finish the proof. 2 Lemma 3.3 For fixed t > 0, with probability one X t has a continuous versionX t . Moreover, for any compact subset K of R and δ ∈ (1, α), we have Proof. The first assertion follows immediately from Corollary 1.2(i) of [20] and Lemma 2.5. It follows from Lemma 2.4 that Then one can complete the proof of (3.1) by (3.2), Lemma 2.5 and Corollary 1.2(iii) of [20]. 2 . Similar to [3, Lemma 2.12] we can prove the following lemma.
Then we have Proof. We assume that K ⊂ [0, 1] for simplicity. Observe that It is elementary to check that Then (3.3) follows from the Markov inequality. Using (2.5) one gets For t ≥ 0 and ψ ∈ B(R) define the discontinuous martingales For i = 1, 2 let ∆M i s (y) denote the jumps of M i (ds, dy). Write |K| := sup x∈K |x|. Similar to [3, Lemma 2.14] one can show the following result.
Lemma 3.5 Let ε > 0 and γ ∈ (0, α −1 ). Then for each ε > 0 and n ≥ 1, Proof. Since the proofs are similar, we only present the first one. Let c > 0. For n ≥ 1 and Then by the Markov inequality for all n ≥ 1 and 1 ≤ k ≤ 2 n , By Lemma 2.4, which implies The desired result then follows.

2
Let L be the space of measurable functions ψ on R + × R so that Similar to [3, Lemma 2.15], we can prove the next result.
Lemma 3.6 Given ψ ∈ L with ψ ≥ 0, there exists a spectrally positive α-stable process The proof is similar to that of [3, Lemma 2.15].
Proof of Theorem 3.1. By Lemma 3.2 we only consider the case α < 3/2. The proof is a modification of that of [3, Theroem 1.2(a)] which proceeds as follows. Let X 0 ∈ M (R) be fixed. We assume that T = 1 in this proof. Recall that λ = 1 α − γ and Also recall that n k = k 2 n for n ≥ 1 and 0 ≤ k ≤ 2 n . Then where ψ + n (s, u) and ψ − n (s, u) are, respectively, the positive part and the negative part of One can see that both ψ + n (s, u) and ψ − n (s, u) satisfy the assumptions of Lemma 3.6, and there is a stable process L 1 and L 2 so that By Lemmas 3.4 and 3.5, Note that on A ε n the jumps of M 1 s (x) do not exceed Then the jumps of are bounded by Similarly, one can see that the jumps of are bounded by Combining with (3.9) we conclude that the jumps of on A ε n are bounded by By an abuse of notation we write L T ± n for L 1 Then Observe that Applying Lemma 3.4 with δ = α and r = 1 one gets Combining this with (3.10) we have Using (3.14) of [3], and [3, Taking γ :=η c−η 4 , we have which together with (3.7) gives By [5, Lemma III.5.1], it is easy to see that for all n ≥ 1 and δ > 0, whereZ n,ε n k (x) is a continuous version of Z n,ε n k (x), It is easy to check that Observe that for each m, n ≥ 1 and 1 This implies η > 0. It follows from (3.12) that for each m ≥ 0, Then by Fatou's lemma and (3.6) for each subsequence {n ′ } of {n}, First letting r → ∞ and then letting ε → 0 we immediately have lim inf almost surely by (3.13). It follows from (2.5) that Then lim sup This completes the proof. 2

Pathwise uniqueness
In this section we prove the pathwise uniqueness for (1.3). Recall that η c = 2 α − 1. If (X, L) and (Y, L), with X 0 = Y 0 ∈ M (R), are two weak solutions to equation (1. 3) defined on the same filtered probability space (Ω, F , F t , P) satisfying Assumption 2.2, then with probability one, for each t > 0 we have Throughout this section we always assume that the assumptions of Theorem 4.1 hold. The proof of Theorem 4.1 adopts the arguments from [14,15]. By considering a conditional probability, we may assume that the initial states X 0 and Y 0 are deterministic. For n ≥ 1 define a n := exp{−n(n + 1)/2}.

Suppose that T > 0 and that Ψ is a nonnegative and compactly supported infinitely differentiable function on
By (4.3) and a stochastic Fubini's theorem, it is easy to see that f (s)ds.
For fixed K > 0 and η ∈ (0, η c ) define a stopping time σ k by By Theorem 3.1 one sees that lim k→∞ σ k = ∞ almost surely.
In the rest of this section we always write τ k := min{γ k , σ k }.  We first present the main proof.
Proof of Theorem 4.1. By the continuity of x →Ũ t (x), for each x ∈ R and t > 0, Note that φ ′ n ≤ 1. Then for all x m → x as m → ∞, we have that which converges to zero as m, n → ∞. Now by (4.6) and Fatou's lemma Together with (4.4) and Lemmas 4.3-4.5 we have Letting k → ∞ in the above inequality we have This gives (34) of [14]. By the same argument in the proof of Theorem 1.6 of [14], one has for every t > 0. It follows that X t , f = Y t , f almost surely for every t > 0 and f ∈ S (R). By the right-continuities of t → X t , f and t → Y t , f we have P{ X t , f = Y t , f for all t > 0} = 1 for every f ∈ S (R). Considering a suitable sequence {f 1 , f 2 , · · · } ⊂ S (R) we can conclude (4.2). 2 We now present the proofs of Lemmas 4.3-4.5.
Proof of Lemma 4.3. By the same argument as Lemma 2.2(b) of [14], By the continuity of x →Ũ s (x), it is easy to see that Observe that Note that

Now by (2.4) and dominated convergence one finishes the proof. 2
Proof of Lemma 4.4. For t ≥ 0 and m, n ≥ 1 let and I m,n 3,2 (t) := I m,n 3 (t) − I m,n 3,1 (t). By the Burkholder-Davis-Gundy inequality, forᾱ ∈ (α, α β ∧ 2) and T > 0, For the last inequality we used the fact that for all x i ∈ R and n ≥ 1. Since |H n (y, z)| ≤ |z| for all y, z ∈ R and Ψ is bounded, by Hölder inequality and Lemma 2.4, Similarly, It follows that for T > 0 E sup Then by [16, p.38], t → I m,n 3 (t ∧ τ ) is a martingale, which implies (4.5). 2 To prove Lemma 4.5, we only need to show the following two lemmas. Then Lemma 4.7 For m, n ≥ 1 and t ≥ 0 let Then I m,n 4,2 (t) → 0 as m, n → ∞.
Proof of Lemma 4.7. Observe that D n (y, z) ≤ 2|z| for all n ≥ 1 and y, z ∈ R, and This inequality together with (4.8) and dominated convergence leads to lim m,n→∞ tends to zero as m, n → ∞. IfŨ s (x) > 0, then there is a strictly positive constant c 2 so that U s (x − u m ) ≥ c 2 for all u ∈ [−1, 1] and for m large enough. Thus, which implies Ũ s , Φ m x + zmhṼ s (x − y m )Φ(y) ≥ c 2 > 0 for all z, h ≥ 0 and for m large enough. Since supp(ψ n ) ⊂ (a n , a n−1 ), by (3.3) of [11], we have for m, n large enough. The proof for the caseŨ s (x) < 0 is similar. 2 For m, n, k ≥ 1 and t > 0 define Before proving Lemma 4.6 we need to show two more lemmas. almost surely.
Together with (4.10), one can see that lim n∞ E M m,n,k (x, y, z, h, t n ) − M m,n,k (x, y, z, h, t) = 0, x, y ∈ R\{K t ∪K t }.
which completes the proof. 2 We are now ready to show Lemma 4.6.
Step 1. We first estimate I m,n,k 4,1,1 (t). SinceX s andỸ s are the continuous versions of X s and Y s for fixed s > 0, almost surely. Observe that One can also obtain the same estimation forỸ . Then by (4.14) we havē almost surely. We then have Step 2. We then estimate I m,n 4,1,2 (t). Observe that In fact, if 1]. Then by the continuity of u →Ṽ s (u) and the mean value theorem, which implies (4.16). It follows from (4.16) that Then by (4.17) and (4.18), Finally, Combining with (4.13) and (4.15), we finish the proof. 2

Appendix
Before proving Proposition 2.3, we state two lemmas. For λ > 0 let where τ k is a stopping time τ k defined by Proof. We consider a partition ∆ n : It follows that where I i (s) := 1 (t i−1 ,t i ] (s). By dominated convergence we have Observe that q > p for the case p > 1. By Hölder inequality, for all measurable function tends to zero as as ∆ n → 0. Then by dominated convergence and (5.5) it is easy to see that as ∆ n → 0, and Now it is obvious that (5.1) follows from (5.3)-(5.4) and (5.6)-(5.7). 2 Proof. By Lemma 5.1 for all k, λ ≥ 1 we have By Fatou's lemma, By monotone convergence, Then we can finish the proof easily by dominated convergence. 2 Now we are ready to present proof of Proposition 2.3.
(ii) Now we consider the case 1 < p < α in three steps.

By (2.2) again, we have
Then by (5.17) one sees that (2.3) holds with T replaced by T 0 .
Step 3. Similar to Step 1, for T ∈ [0, T 0 ], This together this with (5.18) shows that (2.3) holds with T replaced by 2T 0 . This completes the proof.