NON-UNIQUENESS FOR NON-NEGATIVE SOLUTIONS OF PARABOLIC STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

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1 NON-UNIQUENESS FOR NON-NEGATIVE SOLUTIONS OF PARABOLIC STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS K. BURDZY, C. MUELLER, AND E.A. PERKINS Dedicated to Don Burkholder. Abstract. Pathwise non-uniqueness is established for non-negative solutions of the parabolic stochastic pde X t = 2 X Xp Ẇ ψ, X where Ẇ is a white noise, ψ is smooth, compactly supported and non-trivial, and < p < 1/2. We further show that any solution spends positive time at the function. 1. Introduction Let σ : R R be p-hölder continuous (so σ(x σ(y K x y p, let ψ C 1 c (R (the space of C1 functions on R with compact support, and consider the parabolic stochastic partial differential equation (1.1 X t (t, x = X(t, x σ(x(t, xẇ(t, x ψ. 2 Here Ẇ is a space-time white noise on R R. If σ is Lipschitz continuous, pathwise uniqueness of solutions to (1.1 is classical (see, e.g., [Wal86]. Particular cases of (1.1 for non-lipschitz σ arise in equations modeling populations undergoing migration (leading to the Laplacian and critical reproduction or resampling (leading to the white noise term. For example if σ(x = X and X, we have the equation for the density of onedimensional super-brownian motion with immigration ψ (see Section III.4 of [P1]. If σ(x = X(1 X, ψ = and X [, 1] we get the equation for Date: January 16, Mathematics Subject Classification. Primary 6H15. Secondary 6G6, 6H1, 6H4, 6K35, 6J8. Burdzy s research was supported in part by NSF Grant DMS and by grant N N , MNiSW, Poland. Mueller s work was supported in part by NSF Grant DMS Perkins research was supported in part by an NSERC Discovery Grant. 1

2 2 K. BURDZY, C. MUELLER, AND E.A. PERKINS the density of the stepping stone model on the line [Shi88]. In both cases pathwise uniqueness of solutions remains open while uniqueness in law is obtained by (different duality arguments (see the above references. The duality arguments are highly non-robust and fail, for example if σ(x, X = f(x, XX, which models a critically branching population with branching rate at site x in state X is f(x, X. This is one reason that there is interest in proving pathwise uniqueness in (1.1 under Hölder continuous conditions on σ, corresponding to the classical results of [YW71] for one-dimensional SDE s with Hölder 1/2-continuous diffusion coefficients. In [MP1] pathwise uniqueness for (1.1 is proved if p > 3/4 and in [MMP11] pathwise uniqueness and uniqueness in law are shown to fail in (1.1 when σ(x = X p for 1/2 p < 3/4. Here a non-zero solution to (1.1 is constructed for zero initial conditions and the signed nature of the solution is critical. In the examples cited above the solutions of interest are non-negative and so it is natural to ask whether the results in [MP1] can be improved if there is only one point (say u = where σ(u fails to be Lipschitz, and we are only interested in non-negative solutions. Finding weaker conditions which imply pathwise uniqueness of non-negative solutions in this setting is a topic of ongoing research. In this paper we give counterexamples to pathwise uniqueness of non-negative solutions in the admittedly easier setting where p < 1/2. Even here, however, we will find there are new issues which arise in our infinite dimensional setting. Our methods will also allow us to extend the nonuniqueness result in [MMP11] mentioned above to < p < 1/2. We assume Ẇ is a white noise on the filtered probability space (Ω, F, F t, P, where F t satisfies the usual hypotheses. This means W t (φ is an F t -Brownian motion with variance φ 2 2 for each φ L 2 (R, dx and W t (φ 1 and W t (φ 2 are independent if φ 1, φ 2 φ 1 (xφ 2 (xdx =. A stochastic process X : Ω R R R which is F t previsible Borel measurable will be called a solution to the stochastic heat equation (1.1 with initial condition X : R R if for each φ C c (R, X t, φ = X, φ X s, 2 φ ds σ(x(s, xφ(xw(ds, dx t φ, ψ for all t a.s. (The existence of all the integrals is of course part of the definition. It is convenient to use the space C rap (R of rapidly decreasing continuous functions on R as a state space for our solutions. To describe this space, for f C(R (the continuous functions on R let f λ = sup e λ x f(x, x R

3 NON-UNIQUENESS FOR SPDE 3 and set C rap = {f C(R : f λ < λ > }, C tem = {f C(R : f λ < λ < }. Equip C rap with the complete metric d(f, g = 2 k ( f g k 1, k=1 and C tem is given the complete metric d tem (f, g = 2 k ( f g 1/k 1. k=1 Let C rap be the subspace of non-negative functions in C rap, which is a Polish space. Our primary interest is in the smaller space C rap resulting in stronger non-uniqueness results. A C rap -valued solution to (1.1 is a solution X such that t X(t, is in C(R, C rap, the space of continuous C rap-valued paths for all ω. In general if E is a Polish space we give C(R, E the topology of uniform convergence on compact sets. The following result is proved just as in Theorem 2.5 of [Shi94]. Theorem 1. (Weak Existence of Solutions. Assume ψ and the p-hölder continuous function σ satisfies σ( =. If X C rap, there exists a filtered space (Ω, F, F t, P with a white noise Ẇ and a C rap-valued solution of (1.1. Proof. Our conditions on σ imply the hypothesis on a in Theorem 2.5 of [Shi94], however that reference assumes ψ(x, X satisfies ψ(x, X c X. The proof, however, extends easily to our simpler setting of ψ(x. Here is our main result on non-uniqueness. The proof is given in Section 3. Recall that ψ C 1 c (R. Theorem 2. Consider (1.1 with σ(x = X p for p (, 1/2 and ψ with ψ(xdx >. There is a filtered space (Ω, F, F t, P carrying a white noise Ẇ and two C rap -valued solutions to (1.1 with initial conditions X1 = X 2 = such that P(X1 X 2 >. That is, pathwise uniqueness fails for non-negative solutions to (1.1 for σ, ψ as above. Remarks. 1. The state of affairs in Theorem 2 for ψ = but X non-zero remains unresolved. We expect the solutions to still be pathwise non-unique. The methods used to prove the above theorem do show pathwise uniqueness and uniqueness in law fail if ψ = X = and we drop the non-negativity condition on solutions. Namely, one can construct a non-zero solution to the resulting equation. We will not prove this as stronger results (described above will be shown in [MMP11] using different methods.

4 4 K. BURDZY, C. MUELLER, AND E.A. PERKINS 2. Uniqueness in law holds for non-negative solutions to (1.1 for ψ, σ as above and general initial condition X C rap but now with 1 p 1/2. This may be proved as in [My98] where the case ψ = is treated; for p = 1/2 this is of course the well-known uniqueness of super-brownian motion with immigration ψ. We do not know if uniqueness in law fails for p < 1/2. The presence of a drift will play an important role in the proof of Theorem A key technique in this paper is to consider the total mass M t = X t, 1, and then apply Theorem 4 which, given the Hölder continuity of X(t, x, shows the brackets process [M] to be bounded below by the integral of a power of M. This in turn allows one to apply comparison arguments with one dimensional diffusions. In Section 4 below we prove that in the corresponding stochastic ordinary differential equation, although pathwise uniqueness again fails, uniqueness in law does hold. Of course the SDE is now one-dimensional so on one hand this is not surprising. On the other hand, the manner in which uniqueness in law holds is a bit surprising as the SDE picks out a particular boundary behaviour which has the solution spending positive time at (see Section 4. This leads naturally to the following property for all solutions to the SPDE in Theorem 2. Theorem 3. Assume σ and ψ are as in Theorem 2. Let X be any C rap -valued solution to (1.1 with X =. Then 1(X(s, x xds > for all t > a.s. The proof will be given in Section 5 below. Let b = ψ, 1 >. We note that the above result fails for p = 1/2 since in that case Y t = 4 X t, 1 is a Bessel squared process of parameter 4b satisfying an ordinary sde of the form dy t = 2 Y t db t 4bdt. Such solutions spend zero time at (see for example, the analysis in Section V.48 of [RW]. Finally we state the non-uniqueness result which complements that in [MMP11] in the much easier regime of p < 1/2. The solutions here will be signed. Theorem 4. If < p < 1/2 there is a C rap -valued solution X to X (1.2 t (t, x = X 2 (t, x X(t, x p Ẇ, X(, so that P(X >. In particular uniqueness in law and pathwise uniqueness fail in (1.2.

5 NON-UNIQUENESS FOR SPDE 5 Although the construction in [MMP11] for 1/2 p < 3/4 is more delicate, it is a bit awkward to extend the reasoning to p < 1/2 and so we prefer to present the result here. The proof of Theorem 4 is simpler than that of Theorem 2 in that we can focus on a single process rather than a pair of solutions. The two proofs are similar in that approximate solutions are found by an excursion construction and the key ingredient required for the SPDE setting is Theorem 5 below. Hence we only give a brief sketch of the proof of Theorem 4 at the end of Section A Real Analysis Lemma Theorem 5. If < α, β < 1 and C >, there is a constant K 5 (β, C > such that if f : R R satisfies (2.1 f(x f(y C x y β, then f α dx K 5 ( f dx (αβ1/(β1. Proof. First we use a scaling argument to reduce to the case f(xdx = 1 for which we would have to prove f α K 4. Indeed, if we take b > and let g(x = b β f(bx then g(x g(y = b β f(bx f(by C x y β by the conditions of Theorem 5. Then setting y = bx, we get g(xdx = b β f(bxdx = b (β1 f(ydy = 1 provided ( b = 1 β1 f(ydy. So g satisfies the conditions of Theorem 5 with g = 1, and if we could show that g α (xdx K 4 it would follow, substituting for b, that ( f α (xdx = b αβ1 g α (xdx K 4 αβ1 β1 f(xdx

6 6 K. BURDZY, C. MUELLER, AND E.A. PERKINS as required. Now we concentrate on proving f α K 4 assuming that f = 1 and assuming the Hölder condition (2.1 on f. Let M = sup x f(x, and note the conclusion is obvious if M = so assume it is finite. If M < 1, then since < α < 1, we have f α (xdx f(xdx = 1. On the other hand, if M 1, then the Hölder condition on f implies that f 1 2 on an interval I whose length is bounded below by a constant L > depending only on C, β. So in this case, too, we conclude f α (xdx L 2 α L 2, and Theorem 5 is proved. 3. Proof of Theorem 2 If ψ C 1 c (R, ψ, b = ψdx > and < p < 1/2, we want to construct distinct solutions X, Y to (3.1 X t (t, x = X 2 (t, x (X(t, xp Ẇ(t, x ψ(x, X, X =. Let Cb k denote the space of bounded Ck functions on R with bounded jth order partials for all j k, and set C b = Cb. The standard Brownian semigroup is denoted by (P t, t and p t ( is the Brownian density. Here is an overview of the proof. We will proceed by constructing approximate solutions (X, Y to (3.1 and then let (X, Y be an appropriate weak limit point of (X n, Y n. These approximate solutions will satisfy X Y and Y X, respectively, on alternating excursions away from by M = X, 1 Y, 1. M will equal 2b at the effective start of each excursion. We then calculate an upper bound on the probability that M will hit 1 on a given excursion (see (3.43 below and a lower bound on D = X, 1 Y, 1 hitting an appropriate x (, 1 during each excursion (see (3.58 below. Theorem 5 is used in the proof of the first bound (see (3.36 below. These bounds will then show there is positive probability (independent of of D hitting x before M hits 1. The result follows by taking weak limits as n. The use of Theorem 5 will mean the above upper bound is valid only up to a stopping time Vk which will be large with high probability. This necessitates a padding out of the above excursions after this stopping time, and this technical step unfortunately complicates the construction. Fix > and define (X, Y, D = X Y, the white noise Ẇ and a sequence of stopping times inductively on j as follows. Let T =, U j =

7 NON-UNIQUENESS FOR SPDE 7 Tj, and assume X T2j on [T2j, U 2j ] define = Y T 2j on {T 2j < }. Assuming {T 2j < }, (3.2 Y (t, x, and D (t, = X (t, = 2 T 2j P s ψ( ds C rap. That is, (3.3 X = t 2 X 2ψ for T2j t U 2j. Next let t (YU 2j t, D U2j t in C(R, C rap 2 solve the following SPDE for t U2j, Y (3.4 = t 2 Y ψ (Y p Ẇ, YU 2j, D = t 2 D [(Y D p (Y p ]Ẇ, D U (x = 2 P 2j s ψ(xds. The existence of such a solution on some filtered space carrying a white noise follows as in Theorem 2.5 of [Shi94]. To be careful here one has to construct an appropriate conditional probability given F U 2j and so inductively construct our white noise along with (Y, D. Set Xt = Y t D t for U 2j t T 2j1, where T2j1 = inf{t U2j : Xt, 1 = } (inf =, and also restrict the above definition of (Y, D to [U2j, T 2j1 ]. Therefore on [U2j, T 2j1 ], Y = t 2 Y ψ (Y p Ẇ, YU, 2j X = (3.5 t 2 X ψ (X p Ẇ, XU (x = 2 P 2j s ψ(xds, D = X Y = X Y, X, Y are continuous and C rap -valued. Note that XT = Y 2j1 T =. The precise meaning of the above formulas 2j1 for Y / t and X / t is that equality holds after multiplying by φ Cc and integrating over R and over any time interval in [U2j, T 2j1 ]. Now assume T2j1 < and construct (X, Y and D = X Y = Y X as above but with the roles of X and Y reversed. This means that on [T2j1, U 2j1 ] = [T 2j1, T 2j1 ], (3.6 D (t, = Y (t, = 2 and so (3.7 T 2j1 Y t P s ψ( ds C rap(r and X (t,, = 2 Y 2ψ,

8 8 K. BURDZY, C. MUELLER, AND E.A. PERKINS and on [U 2j1, T 2j2 ], (3.8 X = t 2 X ψ (X p Ẇ, XU, 2j1 X, Y = t 2 Y ψ (Y p Ẇ, YU = 2 P 2j1 s ψ(xds, Y, D = X Y = Y X X, Y are continuous and C rap-valued. Here, as before, we have T 2j2 = inf{t U 2j1 : Y t, 1 = }. Clearly XT = Y 2j2 T on {T 2j2 2j2 < } and T j and so our inductive construction of (X, Y is complete. It is also clear from the construction that if X j (t = X(T j t T and similarly for Y j1 j, then we may assume P((X 2j, Y 2j, T 2j1 T 2j F T 2j (3.9 = P((X, Y, T 1 a.s. on {T 2j < }, and P((Y 2j1, X 2j1, T 2j2 T 2j1 F T 2j1 (3.1 = P((X, Y, T 1 a.s. on {T 2j1 < }. Define J = j=1 [U j 1, T j ], A 1(t, x = ( 1 j 1 [T j,uj ] (sψ(xds, j= and A 2 (t, x = A 1 (t, x. Combine (3.3, (3.5, (3.8 and the fact that on [T2j1, U 2j1 ] we have X t = = 2 X t ψ ψ to see that for a test function φ Cc (R, (3.11 X t, φ = A 1(t, φ X C(R, C rap. ( X s, 2 φ ψ, φ ds 1 J (sφ(xx (s, x p dw(s, x,

9 NON-UNIQUENESS FOR SPDE 9 Similar reasoning gives (3.12 Y t, φ = A 2(t, φ Y C(R, C rap. ( Y s, 2 φ ψ, φ ds 1 J (sφ(xy (s, x p dw(s, x, Since U j = T j, the alternating summation in the definition of A 1 implies that (3.13 sup A i (t, x ψ(x. t It follows from (3.2 and (3.6 (recall that b = ψ(xdx that X (t, x1 J c (t 2 P s ψ(xds 4 1/2 b. Therefore for any T > and φ as above ([ T ] 2 (3.14 E 1 J c (s(x (s, x p φ(xdw(s, x (4 1/2 b 2p T φ 2 2. By identifying the white noise Ẇ with associated Brownian sheet, we may view W as a stochastic process with sample paths in C(R, C tem (R. Using bounds in Section 6 of [Shi94] (see especially the pth moment bounds in the proofs of Theorems 2.2 and 2.5 there it is straightforward to verify that for n, {(X n, Y n, W : n N} is tight in C(R, (C rap 2 C tem. Some of the required bounds are in fact derived in the proof of Lemma 6 below. By (3.13, (3.14 and their analogues for Y, one sees from (3.11 and (3.12 that for any limit point (X, Y, W, X and Y are C rap -valued solutions of (3.1 with respect to the common Ẇ. It remains to show that X and Y are distinct. We know X (t, and Y (t, will be locally Hölder continuous of index 1/4 but it will be convenient to have a slightly stronger statement. We note parenthetically that any other index of Hölder continuity for X (t, and Y (t, would yield the same range for p in Theorem 2, provided that the index were less than 1/2. Let V k = inf{s : x, x R such that X (s, x X (s, x Y (s, x Y (s, x > k x x 1/4 }. We will show in Lemma 6 that lim k sup < 1 P(Vk M = for any M N. Note that on [T2j, U 2j ], Y = and X (t, x X (t, x = 2 p t T 2j (z(ψ(zx ψ(zx dz 2 ψ x x.

10 1 K. BURDZY, C. MUELLER, AND E.A. PERKINS This implies that on the above interval for all real x, x, X (t, x X (t, x Y (t, x Y (t, x 4( ψ ψ x x 1/4, where the inequality holds trivially for x x > 1 since the left side is at most 4 ψ. By symmetry it also holds on [T2j1, U 2j1 ]. We may assume k 4( ψ ψ and so the above implies (3.15 Vk (Uj, T j1 ] { }. j= We fix a value of k which will be chosen sufficiently large below. We will now enlarge our probability space to include a pair of processes ( X t, Ȳ t which will equal ( Xt, 1, Yt, 1 up to time Vk and then switch to a pair of approximate solutions to a convenient SDE. Set p = p2 5 (, 1 2 and K(k = K 5(1/4, k where K 5 is as in Theorem 5. We may assume our (Ω, F, F t, P carries a standard F t -Brownian motion (B s : s Vk, independent of (X, Y, W. To define a law Q on C(R, R 2 [, ], first construct a solution (Ỹ, D, B of (3.16 Ỹ t = b 1( < sds D t = 2b(t 1( < s(ỹ s p K(kdBs, Ỹ, 1( < s[(ỹ s D s p (Ỹ s p ] K(kdB s, D. Such a weak solution may again be found by approximation by solutions of Lipschitz SDE s as in Theorems 2.5 and 2.6 of [Shi94] for the more complicated stochastic pde setting. Set X = Ỹ D and T 1 = inf{t : X t = }, and define (3.17 Q (A = P(( X T, Ỹ 1 T, T 1 1 A.

11 NON-UNIQUENESS FOR SPDE 11 Next we enlarge our space to include ( X, Ȳ so that for finite t T 1 (this time is defined below, (3.18 (3.19 (3.2 (3.21 Ȳ t = Y t V k D t = D t V k, 1, 1 b(t V k 1(s > V k (Ȳ s p K(kdBs, Ȳ, 1(s > V k [( D s Ȳ s p (Ȳ s p ] K(kdB s, D, X t = Ȳ t D t = X t V k, 1 b(t V T 1 = inf{t : X t = }. k 1(s > V k ( X s p K(kdBs, Note that if Vk T 1, then X t = Xt, 1 for t T 1 and so T 1 = T 1. Therefore, T 1 Vk T 1. We conclude that X t V = Ȳ k t V D k t V for k t T 1, thus proving (3.2. To carry out the above construction first build (Ȳ V k t, D V k t, B Vk t B V k by approximation by solutions to SDE s with Lipschitz coefficients as in Theorem 2.5 of [Shi94]. This and a measurable selection argument (see Section 12.2 of [SV] allows us to build the appropriate regular conditional probability Q Y V k,1, D,1 ( V k P(((Ȳ V k t, D V k t, B(Vk t B(Vk, t X, Y, W, where {Q y,d : y, d } is a measurable family of laws on C(R, R 2 R. This then allows us to construct ( X, Ȳ, D as above on an enlargement of our original space which we still denote (Ω, F, F t, P. We also may now prescribe another measurable family of laws {Q y,x : (y, x C(R, R 2 } on C(R, R 2 [, ] such that for each Borel A, w.p. 1, (3.22 Define (3.23 Q X V k,1, Y,1 (A = P(( X V k T, Ȳ 1 T, T 1 1 A X, Y, W. Q 1 (A = P(( X T, Ȳ 1 T, T 1 1 (Q A = E X V k,1, Y,1 (A. V k Next, inductively define ( X t, Ȳ t, t [ T j, T j1 ], and { T j } in a manner reminiscent of that for (X, Y, and consistent with the above construction for j = (set T =. Assume the construction up to T 2j is such that (3.24 X T 2j = Ȳ T 2j = on { T 2j < },

12 12 K. BURDZY, C. MUELLER, AND E.A. PERKINS and (3.25 T 2j = T 2j on {V k > T 2j}. Define (3.26 Xj (t = X ( T j t T j1, and similarly define Ȳj. On { T 2j < V k } set (3.27 On { > T 2j V k } set P(( X 2j, Ȳ2j, T 2j1 T 2j F T 2j σ(x, Y, W = Q X (T 2j V k,1, Y,1 (. (T 2j V k (3.28 P(( X 2j, Ȳ2j, T 2j1 T 2j F T 2j σ(x, Y, W = Q (. These definitions imply T 2j1 = inf{t > T 2j : X t = } and that on our enlarged probability space, conditional on F T 2j and on {Vk > T 2j }, (3.18- (3.2 hold for t [ T 2j, T 2j1 ], while on { > T 2j V k }, for t [ T 2j, T 2j1 ], (3.16, (3.17 and (3.28 give Ȳ t = b 1( T 2j < sds (3.29 D t = 2b( (t T 2j 1( T 2j < s(ȳ s p K(kdB(s, Ȳ, 1( T 2j < s ( (Ȳ s D s p (Ȳ s p K(kdB s, D, X t = 2b ( (t T 2j b 1( T 2j < sds 1( T 2j < s( X s p K(kdB(s. Now assume T 2j1 < and construct ( X t, Ȳ t and D t = X t Ȳ t for t [ T 2j1, T 2j2 ] as above but with the roles of X and Ȳ reversed. This means that on { T 2j1 < V k }, (3.3 P((Ȳ2j1, X 2j1, T 2j2 T 2j1 F T 2j1 σ(x, Y, W = Q Y (T 2j1 V k,1, X,1 (, (T 2j1 V k and on { > T 2j1 V k }, the above conditional probability is again Q. The apparent lack of symmetry in the definitions arises because we have also reversed the roles of X and Y on [ T 2j1, T 2j2 ]. The above definition implies

13 NON-UNIQUENESS FOR SPDE 13 that D t = Ȳ t X t on [ T 2j1, T 2j2 ], T 2j2 = inf{t > T 2j1 : Ȳ t = }, and X ( T 2j2 = Ȳ ( T 2j2 = on { T 2j2 < }. It follows from (3.18, (3.2 (now with t [ T 2j, T 2j1 ] and (3.25 that on {Vk > T 2j1 } we have T 2j1 = T 2j1. Symmetric reasoning shows that on {Vk > T 2j2 }, T 2j2 = T2j2. We have verified (3.24 and (3.25 for j 1. Since T j1 T j (by (3.18,(3.2 and (3.29, T j and our inductive definition is complete. The reasoning above to show T 2j1 = T 2j1 on {Vk > T 2j1 } and the obvious induction also shows that X t V k (3.31 D t V k = X t V k, 1, Ȳ t V k = X t Vk, 1 Yt Vk, 1 = Y t V k, 1, t a.s. The following consequence of the above construction will be important for us: P ( ( X 2j, Ȳ2j, T 2j1 T 2j F T 2j (3.32 and (3.33 = Q 1 ( a.s. on { T 2j < Vk }, P ( (Ȳ2j1, X 2j1, T 2j2 T 2j1 F T 2j1 = Q 1 ( a.s. on { T 2j1 < V k }, P ( ( X 2j, Ȳ2j, T 2j1 T 2j F T 2j = Q ( a.s. on {Vk T 2j}, P ( (Ȳ2j1, X 2j1, T 2j2 T 2j1 F T 2j1 = Q ( a.s. on {V k T 2j1}. Consider, for example, the first equality in (3.32. By (3.27 we have for a Borel set B and A F T 2j, A {V k > T 2j }, (3.34 P ( {( X 2j, Ȳ2j, T 2j1 T 2j B} A ( = E Q X (T 2j V k,1, Y,1 (B1 (T 2j V k A ( ( = E E Q X (T 2j V k,1, Y,1 (B (T 2j V k F T 2j V k 1 A. In the last line we used the fact that Vk > T 2j = T 2j on A to see that A F T 2j Vk. Formula (3.31 shows that our construction of ( X, Ȳ has not increased the information in F T 2j Vk so we may use (3.9. Applying (3.9 and the fact that Vk = T 2j V k θ T2j on {Vk > T 2j }, where (θ t are the shift operators for (X, Y, we conclude from (3.34 that the far left-hand side of

14 14 K. BURDZY, C. MUELLER, AND E.A. PERKINS (3.34 equals ( E Q X V k,1, Y,1 (B 1 V A = E (Q 1 (B1 A, k by (3.23. This gives the first equality in (3.32 and the second inequality holds by a symmetric argument. The proof of (3.33 is easier. Our next goal is to show there is positive probability, independent of, of D hitting some appropriately chosen x (, 1 before X or Ȳ hits 1. By (3.32 and (3.33 the excursions of X Ȳ away from are governed by Q or Q 1, depending on whether or not Vk has occurred. Therefore we need to analyze these two laws. Consider the more complex Q 1 first. Use (3.11, with φ = 1, in (3.2 and the fact that Vk > (by (3.15 to conclude that under Q 1, X t = 2b(t for t and for t T 1, (3.35 X t = b ((t Vk b b = 2b tb N t, 1(s > V k ds 1(s V k X (s, x p dw(s, x 1(s > V k ( X s p K(kdBs where N is a continuous (F t local martingale such that [ N t = 1(s Vk N (sds. X (s, x 2p dx 1(s > Vk ( X ] s K(k ds 2p By the definition of Vk we may apply Theorem 5 with (α, β, C = (2p, 1/4, k and conclude that (3.36 N (s 1(s V k K(k [ X (s, xdx 1(s > V k ( X s 2p K(k = K(k( X s 2p, where (3.31 is used in the last line. Define a random time change τ t by (3.37 t = τt ] ((p/21/(5/4 N (s K(k( X s ds A(τ t, t < A( T 2p 1.

15 NON-UNIQUENESS FOR SPDE 15 The restriction on t ensures we are not dividing by zero in the above integrand because T 1 is the hitting time of by X. Clearly (3.36 implies (3.38 τ (t 1 for t < A( T 1. For t < A( T 1, let (3.39 ˆX(t = X (τ t = 2b bτ(t ˆN(t, where ˆN t = N(τ t is continuous (F τt-local martingale such that ˆN t = τt N (sds = τt N s ds = K(k ˆX(r 2p dr. This follows by using the substitution s = τ r and calculating the differential dτ(r from (3.37. Note also that if ˆTx = inf{t : ˆX(t = x}, then A( T 1 = ˆT. Therefore by (3.39 we may assume there is a Brownian motion ˆB so that (3.4 ˆX(t ˆT = 2b bτ(t ˆT ˆT K(k ˆX(s p d ˆB(s. The scale function for a diffusion defined by a similar formula, but with t ˆT in place of τ(t ˆT, is x } s k (x = exp { 2by1 2p K(k(1 2p dy. That is, s k satisfies (3.41 By Itô s Lemma K(kx 2p s k ( ˆX(t ˆT = s k (2b 2 s k(x bs k(x = on [,, s k ( =. ˆT ˆT bτ (us k( ˆX(u 2p K(k ˆX(u 2 s k( ˆX(ud ˆN(u. s k( ˆX(udu (3.38 and (3.41 show that the integrand in the drift term above is nonpositive, and so s k ( ˆX(t ˆT ˆT 1 is a supermartingale which therefore satisfies This implies that E ( s k ( ˆX(t ˆT ˆT 1 s k (2b. Q 1 ( X t = 1 for some t < T 1 = Q 1 (s k ( ˆX( ˆT 1 ˆT hits s k (1 before ( = lim Q 1 s k ( ˆX(t ˆT ˆT 1 /s k (1 t (3.42 s k (2b/s k (1.

16 16 K. BURDZY, C. MUELLER, AND E.A. PERKINS Under Q add the equations in (3.16 to see that (we write X for X, X t = 2b bt ( X s p K(kdBs, t T 1 = inf{t : X t = }. This is equation (3.4 with t in place of τ t and so the previous calculation applies to again give us (3.42 with Q in place of Q 1. Under either Q i, X t = X t Ȳ t and so we conclude (3.43 Q i ( X t Ȳ t hits 1 for t < T 1 s k(2b/s k (1, i = 1, 2. We next consider the escape probability for D under Q 1. Let x (2b, 1 and T D(, x = inf{t : D t = or x } T 1 Q 1 a.s., the last since D t = X t Ȳ t X t for t T 1 under Q 1. It follows from (3.2, (3.4 and (3.19 that D t = 2b(t for t, and for t T 1 we have, (3.44 D t = 2b (X (s, x p Y (s, x p 1(s Vk dw(s, x 1(s > V k (( X s p (Ȳ s p K(kdB s, which is a non-negative local martingale in t. We have Q 1 ( t < T 1 : D t x ( E D (T D(, x x 1 1(T D(, x <, T 1 < ( = E D (T D(, x T ( 1x 1 E D (T D(, x x 1 1( T ( =. The first term on the right-hand side is the terminal element of a bounded martingale and so ( (3.46 E D (T D(, x T 1 x 1 = 2b/x. It follows from (3.35 that on { T 1 = }, (3.47 {lim sup t X t < } { lim t N t = }, which is a Q 1 -null set by the Dubins-Schwarz theorem which asserts that a continuous martingale is a time-changed Brownian motion. Therefore (3.48 X t hits or 1 for some t T 1, t <, Q 1 a.s., and therefore ( E D (T D(, x x 1 1( T ( = Q 1 ( T 1 = Q1( X t hits 1 for t < T 1 (3.5 s k (2b/s k (1, the last by (3.42.

17 NON-UNIQUENESS FOR SPDE 17 Since lim x s k (x/x = 1, there is an (k > and x = x (k (, 1, such that (3.51 implies s k (2b < 3b and 2b < x s k (1/6. So for and x as above we may use (3.46 and (3.49 in (3.45 and conclude Q 1 ( t [, T 1 : D t x (2b/x (s k (2b/s k (1 > (b/x. Virtually the same proof (it is actually simpler works for Q. Under Q i, D t = X t Ȳ t for t T 1 = inf{t : X t Ȳ t = 1} and so we have proved for x as above, (3.52 Q i ( t [, T 1 ] : X t Ȳ t x b x for i = 1, 2 and <, and (see (3.48 for i = 1 (3.53 X t Ȳ t hits or 1 for t T 1, t finite Q i a.s., i = 1, 2. and Let N 1 = min{j : ( X Ȳ (t T j hits 1 for t < T j1 T j }, N 2 = min{j : X Ȳ (t T j hits x for t < T j1 T j }. Use (3.32, (3.33 and (3.43 to see that P(N 1 > n = E (1(N 1 > n 1P ( X Ȳ (( T n 1 T n doesn t hit 1 F T n 1 ( P(N 1 > n 1 1 s k(2b. s k (1 Therefore, if p 1 = s k(2b s k (1, then (3.54 P(N 1 > n (1 p 1 n1. Similar reasoning using (3.52 in place of (3.43 shows that if p 2 = b x, then for, (3.55 P(N 2 > n (1 p 2 n1. Note that (3.51 shows that (3.56 p 2 p 1 = b s k (1 1 s k (1 2. s k (2b x 3 x

18 18 K. BURDZY, C. MUELLER, AND E.A. PERKINS If n = p 1 1 we get for P(N 2 < N 1 P(N 1 > n P(N 2 > n (1 p 1 n1 (1 p 2 n1 (1 p 1 n1 (1 2p 1 n1 1 2 (e 1 e 2, where the last inequality holds by decreasing (k, if necessary. If t = inf{t : X t Ȳ t 1}, then the above bound implies that for, (3.57 P(sup t t X t Ȳ t x 1 2 (e 1 e 2. Now let Then (3.31 shows that t = inf{t : X t, 1 Yt, 1 1}. if t < V k, then t = t and ( X t, Ȳ t = ( X t, 1, Y t, 1 for all t t, and so by (3.57 for, (3.58 P(sup t t X t, 1 Y t, 1 x 1 2 (e 1 e 2 P(V k t. Now recall we have n so that (X n, Y n, W (X, Y, W weakly on C(R, (C rap 2 C tem, where X and Y are C rap -valued solutions of (3.1. Arguing as in (3.47 and using Dubins-Schwarz, we see that (3.59 lim sup X t, 1 = limsup Y t, 1 = a.s. t t Standard weak convergence arguments now show that {t n } are stochastically bounded. Lemma 6 therefore shows that we may choose a fixed k sufficiently large so that P(V n k t n 1 4 (e 1 e 2 for all n. Using this fixed k throughout we see from (3.58 that for large enough n ( P sup X n t, 1 Yt n, 1 x 1 t t n 4 (e 1 e 2. If t = inf{t : X t, 1 Y t, 1 2} < a.s., by (3.59, then the above implies ( P sup Xt, 1 Y t, 1 x /2 1 t t 4 (e 1 e 2, and so P(X Y 1 4 (e 1 e 2. Lemma 6. For any M N, lim k sup < 1 P(V k M =.

19 NON-UNIQUENESS FOR SPDE 19 Proof. The proof depends on a standard argument in the spirit of Kolmogorov s continuity lemma, so we will omit some details. Fix the time interval [, M]. Define V k (X = inf{s : x, x R such that X (s, x X (s, x > k x x 1/4 }. and Vk (Y likewise. It suffices to prove Lemma 6 for Vk replaced by Vk (X and Vk (Y and so clearly we only need consider Vk (X. Recall that } {Xn is tight in C(R, C rap. So it suffices to choose a constant K > and prove the lemma for X (t, x (Ke x 1/p in place of X. Considering the integral equation for X, and using the fact that ψ Cc 1 (R we see that it is enough to prove Lemma 6 with X replaced by the stochastic convolution N (t, x = p t s (x yϕ (s, ydw(s, y. Here one can use Lemma 6.2 of [Shi94] to handle the drift terms. The term ϕ (s, y is a predictable random field satisfying ϕ (t, x Ke x for all t [, M], x R almost surely. Since our estimates are uniform in, we will omit the superscript on ϕ and N from now on. The constants below may depend on M and K. Now we rely on some standard estimates which are easy to verify. We claim that there exist constants q, K such that for t t δ M and x R, and for δ < 1, δ p 2 s (x ye 2p y dyds K δ 1 2 e q (3.6 x, R R R [p t sδ (x y p t s (x y] 2 e 2p y dyds K δ 1 2 e q x, [p t s (x y δ p t s (x y] 2 e 2 y dyds K δe q x. From these inequalities, it follows in a standard way that for some positive constants q 1, C, C 1, we have P ( N(t δ, x N(t, x λ C exp ( C 1λ 2 (3.61 e q1 x, δ 1 2 P ( N(t, x N(t, x δ λ C exp ( C 1λ 2 e q1 x. δ For example, if we write r ˆM r = [p t s (x y δ p t s (x y]ϕ(s, yw(dy, ds R

20 2 K. BURDZY, C. MUELLER, AND E.A. PERKINS then ˆM r is a continuous martingale and hence a time changed Brownian motion, with time scale r E(r = [p t s (x y δ p t s (x y] 2 ϕ 2 (s, ydyds R r [p t s (x y δ p t s (x y] 2 K 2 e 2 y dyds. Thus, R P ( N(t, x N(t, x δ λ = P( ˆM t λ ( P sup B s λ s E(t and then the reflection principle for Brownian motion and the third inequality in (3.6 (to bound E(t for t M gives the second inequality in (3.61. Now we outline a standard chaining argument, and for simplicity assume that M = 1. Let G n be the grid of points {( } k G n = 2 2n, l 2 n : k 2 2n, l Z. The Borel-Cantelli lemma along with (3.61 now implies that for large enough (random K 1, if n K 1 and p 1, p 2 are neighboring grid points in G n, then (3.62 N(p 1 N(p 2 2 n 4. Now suppose that q i = (t i, x i with x 1 x 2 1, and that each point q i lies in some grid G n. From the above, there is a path from q 1 to q 2 utilizing edges in grids G n, with n n, each edge in the path being a nearest neighbor edge in G n, and with at most 8 edges from a given grid index n. Let n be the least grid index used in this path. We claim that for some constants C > c >, such a path exists with n satisfying c2 2n < t 1 t 2 < C2 2n, c2 n < x 1 x 2 < C2 n. Using the triangle inequality to sum differences of N(t, x over edges of the path, we arrive at a geometric series, and conclude that (3.63 N(q 1 N(q 2 C 1 2 n 4 if n K 1. Although we have only proved the above for grid points, such points are dense in [, T] R, and N(t, x has a continuous version because X(t, x is continuous, and the drift contribution is smooth. Therefore it follows for all points in [, 1] R. We have proved (3.63 for q 1 q 2 C2 K1 where K 1 is stochastically bounded uniformly in. The required result follows.

21 NON-UNIQUENESS FOR SPDE 21 Sketch of Proof of Theorem 4. We carry out an excursion construction of an approximate solution X to (1.2 by starting the ith excursion at ( 1 i ψ, and then run each independent excursion according to a fixed law of a C rap - valued solution to (1.2 with X = ψ, if i is even, and its negative if i is odd, until the total mass hits. At this point a new excursion is started in the same manner. Theorem 5 is used to time change Xt (1 into an approximate solution Y (t = Xτ (1 of Girsanov s equation t (3.64 dy t = Y t p db t, with p < p < 1/2 and dτ (t dt 1. There will be an additional term A (t arising from all the excursion signed initial values up to time t but it will converge to uniformly in t due to the alternating nature of the sum. We now proceed as in the excursion-based construction of non-zero solutions to Girsanov s sde (3.64 to show that one of the excursions of the approximate solutions will hit ±1 before time T with probability close to 1 as T gets large, uniformly in. Let N be the number of excursions of Y until one hits ±1 and let N (T be the number of excursions of Y completed by time T. N is geometric with mean 1 by optional stopping. Let U i ( be the time to completion of the ith excursion of Y. Assuming T 1 N, we have (3.65 P(sup Ys 1 P(N (T N s T P(N (T T 1 P(N > T 1 P(U T 1 ( T (1 T 1 A key step now is to use diffusion theory to show that if Y satisfies (3.64 (pathwise unique until it hits zero then (3.66 P(Y t > for all t T Y = 1 ct 1/(2(1 p as T. If U i (1 is the time of completion of the ith excursion of Y where the excursions now start at ±1, then scaling shows that P(U T 1 ( T = P(U T 1 (1 (2 2p T. (3.66 shows that U T (1/( T 1 2(1 p converges weakly as to a 1 stable subordinator of index α = (2(1 p 1 and so for any η > we may choose T large enough so that for small enough (by (3.65 we have T 1 P(sup Ys 1 P(U T ( T (1 1 s T ( U P T 1(1 ( T 1 T p e T 1 η. 2(1 p The fact that (τ (t 1 allows us to conclude that with probability at least 1 η, uniformly in, the total mass of our approximate solution Xt will hit.

22 22 K. BURDZY, C. MUELLER, AND E.A. PERKINS ±1 for some t T. By taking a weak limit point of the X we obtain the required non-zero solution to ( Pathwise Non-uniqueness and Uniqueness in Law for an SDE. The stochastic differential equation corresponding to (3.1 would be (4.1 X t = X bt (X s p db s, X t t a.s. Here b >, < p < 1/2, B is a standard (F t -Brownian motion on (Ω, F, F t, P and X is F -measurable. A much simpler argument than that used to prove pathwise non-uniqueness in (3.1 allows one to establish pathwise nonuniqueness in (4.1. One only needs to apply the idea behind construction of ( X t, Ȳ t for t V k. In any case the result is undoubtedly known, given the well-known Girsanov examples (see, e.g., Section V.26 in [RW] and so we omit the proof. Theorem 7. There is a filtered probability space (Ω, F, F t, P carrying a standard F t -Brownian motion and two solutions, X 1 and X 2, to (4.1 with X 1 = X 2 = X = such that P(X 1 X 2 >. Weak existence of solutions to (4.1 for a given initial law may be constructed through approximation by Lipschitz coefficients. This is in fact how Shiga [Shi94] constructed solutions to (1.1 and hence is the method used in Theorem 1. As we were not able to verify whether or not uniqueness in law holds in (3.1 it is perhaps interesting that it does hold in (4.1. That is, the law of X is uniquely determined by the law of X. We have not been able to find this result in the literature and since the solutions to (4.1 turn out to have a particular sticky boundary condition at which was not immediately obvious to us, we include the elementary proof here. Theorem 8. Any solution to (4.1 is the diffusion on [, with scale function x { 2b y 1 2p} (4.2 s(x = exp dy 1 2p (with inverse function s 1 on [, s(, speed measure (4.3 m(dx = dx s (s 1 (x 2 s 1 (x 2p b 1 δ (dx on [, s(, and starting with the law of X. In particular if T = inf{t : X t = }, then (4.4 T < implies T and solutions to (4.1 are unique in law. T 1(X s = ds > > a.s.,

23 NON-UNIQUENESS FOR SPDE 23 Proof. The last statement is immediate from the first assertion. To prove X is the diffusion described above, by conditioning on X we may assume X = x is constant. We will show directly that X is the appropriate scale and time change of a reflecting Brownian motion. Note that s is strictly increasing on [, (in fact on the entire real line so that s 1 is well-defined. Note also that { 2b x s 1 2p} (x = exp is of bounded variation and continuous, 1 2p and (4.5 s (x = { s (x2bx 2p if x >, s (x2b x 2p if x <. If L X t (x is the semimartingale local time of X, Meyer s generalized Itô formula (see Section IV.45 of [RW] shows that (4.6 Y t s(x t = Y s (X u Xu p db u b s (X u du 1 L X t (xds (x. 2 Since s is continuous at, 1 L X t 2 (xds (x = 1 (4.7 1(x > L X t 2 (xds (x = 1 1(x > L X t (xs (xdx 2 = b = b So (4.6 and s ( = 1 imply (4.8 Y t = Y s (X u Xu 2p Xu 2p 1(X u > du (by (4.5 s (X u 1(X u > du. s (X u X p u db u b 1(X u = du. Define U = s (X u 2 Xu 2p du and a random time change α : [, U [, by (4.9 α(t s (X u 2 Xu 2p du = t. Clearly α is strictly increasing and is also continuous since X cannot be on any interval. If R(t = Y (α(t for t < U we now show that R is a reflecting Brownian motion on [,, starting at Y, where we extend the definition for t U by appending a conditionally independent reflecting Brownian motion

24 24 K. BURDZY, C. MUELLER, AND E.A. PERKINS starting at the appropriate point. In what follows we may assume t < U as the values of R(t for t U will not be relevant. We have from (4.8 R(t = β t b α(t 1(X u = du β t A t, where β t = t and so β is a Brownian motion starting at Y. A is continuous non-decreasing and supported by {t : X(α(t = } = {t : R(t = }. By uniqueness of the Skorokhod problem (see Section V.6 in [RW] R is a reflecting Brownian motion and A t = L R t (, that is, (4.1 b α(t 1(X u = du = L R t (. Let α 1 : [, [, U denote the inverse function to α. Now differentiate (4.9 to see that (4.11 if X u >, then (α 1 (u = s (X u 2 Xu 2p. We may use (4.1 and (4.11 to conclude that t = = = 1(X u > du α 1 (t 1(X u = du 1(X u > s (X u 2 X 2p d(α 1 (u b 1 L R α 1 (t ( u 1(R(v > s (s 1 (R(v 2 s 1 (R(v 2p dv b 1 L R α 1 (t (, where we have set u = α(v in the last. Therefore if m is as in (4.3, then t = L R (α 1 (t, xdm(x, [, and X(t = s 1 (R(α 1 (t. This identifies X as the diffusion on [, with the given scale function and speed measure. Remarks. (1 One can also argue in the opposite direction. That is, given a diffusion X with speed measure and scale function as above and a given initial law on [,, one can build a Brownian motion B, perhaps on an enlarged probability space, so that X satisfies (4.1, giving us an alternative weak existence proof. (2 One can construct solutions as weak limits of difference equations or equivalently as standard parts of an infinitesimal difference equation. Here one cuts off the martingale part when the solution overshoots into the negative halfline and lets the positive drift with slope b bring it back to R. The smaller the b the longer it takes to become positive, the more time the solution will

25 NON-UNIQUENESS FOR SPDE 25 spend at zero and so the larger the atom of the speed measure at. A short calculation shows that at p = 1/2 the overshoot reduces to t (the time step in the difference equation and so there is no time spent at in the limit. (See Section V.48 of [RW] for the standard analysis. (3 It would appear that (4.1 is not a particular effective tool to study diffusions with drift b on the positive half-line. By just extending the equation to [, we inadvertently pick out a particular case of Feller s possible boundary behaviors at among all diffusions satisfying (4.1 on (,. (This is certainly not a novel observation see the comments in Section V.48 in [RW]. Presumably things can only get worse for the stochastic pde (3.1. In the next section we scratch the surface of this issue and show that all solutions to this stochastic pde spend positive time in the (infinite-dimensional zero state. 5. Proof of Theorem 3 Let X be a solution of (3.1 and define V k = inf{s : x, x such that X(s, x X(s, x > k x x 1/4 }. As in Lemma 6 (but as there is no it is a bit easier, lim k V k = a.s. As in Section 3, we set p = p2 5 and K(k = K 5 (1/4, k. If we define { R X(u,x 2p dx R(u = if X(u, 1 > and u V K(k X(u,1 k, 2p 1 otherwise, then by Theorem 5, R(u 1 for all u. Introduce a random time change τ given by (5.1 τ(t R(u1( X u, 1 > 1( X u, 1 = du = t. Clearly τ is strictly increasing, continuous and well-defined for all t. Differentiate (5.1 to see that τ (tr τ(t 1( X τ(t, 1 > τ (t1( X τ(t, 1 = = 1 for a.a. t (a.a. is with respect to Lebesgue measure, and therefore (5.2 τ (t = R 1 τ(t 1( X τ(t, 1 > 1( X τ(t, 1 = 1 for a.a. t. Now let Y (t = X(τ(t, 1 = bτ(t M(t,

26 26 K. BURDZY, C. MUELLER, AND E.A. PERKINS where M is a continuous local martingale satisfying τ(t M t = X(u, x 2p dxdu = = [ ] X(τ(r, x 2p dx R(τ(r 1 1(Y (r > 1(Y (r = dr K(kY 2p r dr for τ(t V k. We have used (5.2 in the second line. If T k = τ 1 (V k (a stopping time w.r.t the time-changed filtration, we may therefore assume there is a Brownian motion B so that Tk Y (t T k = bτ(t T k K(kY p r db r. If b = K(k 1/(2(p 1 b, then Ŷ (t = K(k1/(2(p 1 Y (t satisfies Ŷ (t T k = b τ(t T k Tk Ŷ p r db r. If s(x = { x exp 2b y 1 2p 1 2p }dy, then an application of Meyer s generalized Itô s formula shows that if Z(t = s(ŷ (t, then for t T k, Z(t = s (ŶrŶ r p db r b s (Ŷrτ (rdr 1 LŶt 2 (xds (x. Here, as before, LŶ is the semimartingale local time of Ŷ. Now argue as in (4.7 to see that for t T k, 1 t LŶt 2 (xds (x = b s (Ŷr1(Ŷr > dr. Therefore if N(t = s (ŶrŶ r p db r and then for t T k, (5.3 A(t = b s (Ŷr(1 τ (r1(ŷr > dr, Z(t = N(t A(t b s (τ (r1(ŷ (r = dr = N(t A(t b 1(Y (r = dr, where we used s ( = 1 and (5.2 in the last line. A is a non-decreasing continuous process by (5.2, N is a continuous local martingale, and N( =

27 NON-UNIQUENESS FOR SPDE 27 A( =. Fix k and assume V k >, and so T k > because T k V k. If { T = inf t : Tk } 1(Y (r = dr >, then by (5.3, Z(t T is a continuous non-negative local supermartingale starting at and so is identically zero. This means Z(r = for r T and so the same holds for Y (r, which by the definition of T and assumption that T k > implies that T = a.s. Since V k a.s. we have shown that w.p. 1 1( X(τ(r, 1 = dr = 1(Y (r = dr > t >. Setting τ(r = u and using (5.2 again (to show τ (r = 1 on { X(τ(r, 1 = } for a.a. r, we see that the above implies The proof is complete. 1( X u, 1 = du > t > a.s. Acknowledgement. It is a pleasure to thank Martin Barlow for a series of helpful conversations on this work. We are grateful to the referee for very helpful suggestions. References [MMP11] C. Mueller, L. Mytnik and E. A. Perkins, Non-uniqueness for parabolic stochastic partial differential equations with Hölder continuous coefficients, in preparation. [My98] L. Mytnik. Weak uniqueness for the heat equation with noise. Ann. Probab. 26 (1998, [MP1] E. A. Perkins and L. Mytnik, Pathwise uniqueness for stochastic heat equations with Hölder continuous coefficients: the white noise case. To appear in Probab. Theory Related Fields (21. [P1] E.A. Perkins. Dawson-Watanabe Superprocesses and Measure-valued Diffusions. In Lectures on Probability Theory and Statistics Ecole d Été de Probabilités de Saint-Flour XXIX Series: Lecture Notes in Mathematics, Vol. 1781, Springer, Berlin, 21. [RW] L.C.G. Rogers and D. Williams, Diffusions, Markov Processes and Martingales V. 2 Itô Calculus, Wiley, 1987, Norwich. [Shi88] T. Shiga. Stepping stone models in population genetics and population dynamics. In S. Albeverio et al., editor, Stochastic Processes in Physics and Engineering, pages D. Reidel, [Shi94] T. Shiga. Two contrasting properties of solutions for one-dimensional stochastic partial differential equations, Can. J. Math. 46 (1994, [SV] D.W. Stroock and S.R.S. Varadhan, Multidimensional Diffusion Processes, Springer-Verlag, 1979, New York. [Wal86] J. B. Walsh. An Introduction to Stochastic Partial Differential Equations. In Lectures on Probability Theory and Statistics Ecole d Été de Probabilités de Saint- Flour XIV Series: Lecture Notes in Mathematics, Vol. 118, Springer, Berlin, 1986.

28 28 K. BURDZY, C. MUELLER, AND E.A. PERKINS [YW71] T. Yamada and S. Watanabe. On the uniqueness of solutions of stochastic differential equations, J. Math. Kyoto U. 11 (1971, Department of Mathematics, U. of Washington, Seattle, USA address: Department of Mathematics, U. of Rochester, Rochester, USA address: Department of Mathematics, UBC, Vancouver, Canada address: