A Rough Super-Brownian Motion

matapvr 000/6fl:rough_superbrownian_mo file: 2020/08/06 ver. imsart-aop arXiv:1905.05825v3 [math.PR] 17 Sep 2020 narno niomn BWE.Sc rcs saparticula a is process Such on (BRWRE). process environment random a in srsrce to restricted is h oe ...... Motion . . . . Super-Brownian . Rough . . . . . the . of . . . . . Properties . . . . Motion . Super-Brownian . . . Rough Hamiltonian . 5 The Anderson . . . & . PAM Continuous . . . . and 4 Discrete ...... Model 3 . . . The . . . . . 2 Notations . . . . 1 . . . . . Acknowledgements Work the of Structure Probability of Annals the to Submitted okflosteoiia etn n tde h eairu behavior the studies and setting original the sup of follows study work [ the (see to motion rise super-Brownian gave so-called and the al. et Dawson by results early [ survival as potential atcea rate at particle a nrdcin...... 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Anderson lic ok h dimension the work, ran- c s is ess 1 ...... ntevleo a of value the on s 1 ie it to birth gives , dmwalk ndom ...... 25 14 12 1 6 4 4 3 2 increments ∆x 1/n, temporal increments ∆t 1/n2. The particular nature of our problem requires us to couple≃ the diffusive scaling with≃ the scaling of the environment: This is done via an “averaging parameter” ̺ d/2, while the noise is assumed to scale to space white noise (i.e. ξn(x) nd/2). ≥ The diffusive scaling≃ of spatial branching processes in a random environment has already been studied, for example by Mytnik [Myt96]. As opposed to the current setting, the environment in Mytnik’s work is white also in time. This has the advantage that the model is amenable to probabilistic martingale arguments, which are not available in the static noise case that we investigate here. Therefore, we replace some of the probabilistic tools with arguments of a more analytic flavor. Nonetheless, at a purely formal level our limiting process is very similar to the one obtained by Mytnik: See for example the SPDE representation (2) below. Moreover, our approach is reminiscent of the conditional duality appearing in later works by Cri¸san [Cri04], Mytnik and Xiong [MX07]. Notwithstanding these resemblances, we shall see later that some statistical properties of the two processes differ substantially. At the heart of our study of the BRWRE lies the following observation. If u(t,x) indi- cates the numbers of particles in position x at time t, then the conditional expectation given the realization of the random environment, w(t,x)= E[u(t,x) ξ], solves a linear PDE with stochastic coefficients (SPDE), which is a discrete version of the| parabolic Anderson model (PAM): (1) ∂ w(t,x)=∆w(t,x)+ ξ(x)w(t,x), (t,x) R Rd, w(0,x)= w (x). t ∈ >0 × 0 The PAM has been studied both in the discrete and in the continuous setting (see [Kön16] for an overview). In the latter case (ξ is space white noise) the SPDE is not solvable via Itô integration theory, which highlights once more the difference between the current setting and the work by Mytnik. In particular, in dimension d = 2, 3 the study of the continuous PAM requires special analytical and stochastic techniques in the spirit of rough paths, such as the theory of regularity structures [Hai14] or of paracontrolled distributions [GIP15]. In dimension d = 1 classical analytical techniques are sufficient. In dimension d 4 no solution is expected to exist, because the equation is no longer locally subcritical in≥ the sense of Hairer [Hai14]. The dependence of the subcriticality condition on the dimension is explained by the fact that white noise loses regularity as the dimension increases. Moreover, in dimension d = 2, 3 certain functionals of the white noise need to be tamed with a technique called renormalization, with which we remove diverging singularities. In this work, we restrict to dimensions d = 1, 2 as this simplifies several calculations. At the level of the 2-dimensional BRWRE, the renormalization has the effect of slightly tilting the centered potential by considering instead an effective potential: ξn(x)= ξn(x) c , c log(n). e − n n ≃ So if we take the average over the environment, the system is slightly out of criticality, in the biological sense, namely births are less likely than deaths. This asymmetry is counterintuitive at first. Yet, as we will discuss later, the random environment has a strongly benign effect on the process, since it generates extremely favorable regions. These favorable regions are not seen upon averaging,and they have to be compensatedfor by subtracting the renormalization. The special character of the noise and the analytic tools just highlighted will allow us, in a nutshell, to fix one realization of the environment - outside a null set - and derive the following scaling limits. For “averaging parameter” ̺ > d/2 a law of large numbers holds: The process converges to the continuous PAM. Instead, for ̺ = d/2 one captures fluctuations from the branching mechanism. The limiting process can be characterized via duality or a martingale problem (see Theorem 2.12) and we callit rough super-Brownian motion (rSBM).

imsart-aop ver. 2020/08/06 file: rough_superbrownian_motion.tex date: September 18, 2020 ROUGH SUPER-BM 3

In dimension d = 1, following the analogous results for SBM by [KS88, Rei89], the rSBM admits a density which in turn solves the SPDE: (2) ∂ µ(t,x)=∆µ(t,x)+ξ(x)µ(t,x)+ 2νµ(t,x)ξ˜(t,x), (t,x) R R, t ∈ ≥0 × with µ(0,x)= δ0(x), where ξ˜ is space-timep white noise that is independent of the space white noise ξ, and where ν = EΦ+. The solution is weak both in the probabilistic and in the analytic sense (see Theorem 2.18 for a precise statement). This means that the last product represents a stochastic integral in the sense of Walsh [Wal86] and the space-time noise is constructed from the solution. Moreover, the product ξ µ is defined only upon testing with functions in the random domain of the Anderson Hamiltonian· H =∆+ξ, a random operator that was introduced by Fukushima-Nakao [FN77] in d = 1 and by Allez-Chouk [AC15] in d = 2, see also [GUZ20, Lab19] for d = 3. One of the main motivations for this work was the aim to understand the SPDE (2) in d = 1 and the corresponding martingale problem in d = 2. For ξ˜= 0, equation (2) is just the PAM which we can only solvewith pathwise methods, while for ξ = 0 we obtain the classicalSBM, for which the existence of pathwise solutions is a long standing open problem and for which only probabilistic martingale techniques exist. (See however [CT19] for some progress on finite-dimensional rough path differential equations with square root nonlinearities). Here we combine these two approaches via a mild formulation of the martingale problem based on the Anderson Hamiltonian. A similar point of view was recently taken by Corwin-Tsai [CT18], and to a certain extent also in [GUZ20]. Coming back to the rSBM, we conclude this work with a proof of persistence of the process in dimension d = 1, 2. More precisely we even show that with positive probability we have µ(t, U) for all open sets U Rd. This is opposed to what happens for the classical SBM, where→∞ persistence holds only in⊂ dimension d 3, whereas in dimensions d = 1, 2 the process dies out: See [Eth00, Section 2.7] and the references≥ therein. Even more striking is the difference between our process and the SBM in random, white in time, environment: Un- der the assumption of a heavy-tailed spatial correlation function Mytnik and Xiong [MX07] prove extinction in finite time in any dimension. Note also that in [Eth00, MX07] the process is started in the Lebesgue measure, whereas here we prove persistence if the initial value is a Dirac mass. Intuitively, this phenomenon can be explained by the presence of “very favorable regions” in the random environment.

Structure of the Work. In Assumption 2.1 we introduce the probabilistic requirements on the random environment. These assumptions allow us to fix a null set outside of which certain analytical conditions are satisfied, see Lemma 2.4 for details. We then introduce the model, (a rigorous construction of the random Markov process is postponed to Section A of the Appendix). We also state the main results in Section 2, namely the law of large numbers (Theorem 2.9), the convergence to the rSBM (Theorem 2.12), the representation as an SPDE in dimension d = 1 (Theorem 2.18) and the persistence of the process (Theorem 2.20). We then proceed to the proofs. In Section 3 we study the discrete and continuous PAM. We recall the results from [MP19] and adapt them to the current setting. We then prove the convergence in distribution of the BRWRE in Section 4. First, we show tightness by using a mild martingale problem (see Remark 4.1) which fits well with our analytical tools. We then show the duality of the process to the SPDE (7) and use it to deduce the uniqueness of the limit points of the BRWRE. In Section 5 we derive some properties of the rough super-Brownian motion: We show that in d = 1 it is the weak solution to an SPDE, where the key point is that the random measure admits a density w.r.t. the Lebesgue measure, as proven in Lemma 5.1. We also show that the process survives with positive probability, which we do by relating it to the rSBM on a finite box with Dirichlet boundary conditions and by applying the spectral theory for the Anderson Hamiltonian on that box. For this we rely on [CvZ19] and [Ros20]. imsart-aop ver. 2020/08/06 file: rough_superbrownian_motion.tex date: September 18, 2020 4

Acknowledgements. The authors would like to thank Adrian Martini for thoroughly reading this work and pointing out some mistakes and improvements. The main part of the work was done while N.P. was employed at Max-Planck-Institut für Mathematik in den Naturwissenschaften Leipzig & Humboldt-Universität zu Berlin, and while T.R. was employed at Humboldt-Universität zu Berlin. N.P. gratefully acknowledges ﬁnancial support by the DFG via the Heisenberg program. T.R. gratefully acknowledges ﬁnancial support by the DFG / FAPESP: this paper was developed within the scope of the IRTG 1740 / TRP 2015/50122-0.

N N N Zd 1. Notations. We define = 1, 2,... , 0 = 0 and ι = √ 1. We write n for 1 Zd N { } ∪ { } Zd −Rd the lattice n , for n , and since it is convenient we also set ∞ = . Let us recall the basic constructions from∈ [MP19], where paracontrolled distributions on lattices were developed. Define the Fourier transforms for k,x Rd ∈ F −2πιhx,ki F −1 2πιhx,ki Rd (f)(k)= dxf(x)e , Rd (f)(x)= dkf(k)e Rd Rd Z Z as well as for x Zd , k Td (with Td = (R/(nZ))d the n-dilatation of the torus Td): ∈ n ∈ n n 1 F (f)(k)= f(x)e−2πιhx,ki, k Td , n nd ∈ n x∈Zd Xn F −1 2πιhx,ki Zd n (f)(x)= dkf(k)e , x n. Td ∈ Z n σ S S ′ Consider ω(x)= x for some σ (0, 1). We then define ω and ω as in [MP19, | | S ∈ S ′ Definition 2.8]. Roughly speaking ω is a subset of the usual Schwartz functions, and ω consists of so-called ultradistributions, with more permissive growth conditions at infinity. Let ̺(ω) be the space of admissible weights as in [MP19, Definition 2.7]. For our purposes it suffices to know that for any a R≥0, l R, the functions p(a) and e(l) belong to ̺(ω), where ∈ ∈

σ p(a)(x)=(1+ x )−a, e(l)(x)= e−l|x| . | | Moreover, we ﬁx functions ̺, χ in Sω supported in an annulus and a ball respectively, such −j that for ̺−1 = χ and ̺j( )= ̺(2 ), j N0, the sequence ̺j j≥−1 forms a dyadic partition · · ∈ { } d of the unity. We also assume that supp(χ), supp(̺) ( 1/2, 1/2) and write jn N for the d⊂ − Zd R∈ smallest index such that supp(̺j) n[ 1/2, 1/2] . For j < jn and ϕ: n we deﬁne the Littlewood-Paley blocks 6⊆ − →

n −1 n −1 ∆ ϕ = F ̺jFn(ϕ) , ∆ ϕ = F (1 ̺j)Fn(ϕ) . j n jn n − −1≤j

REMARK 1.1. In this setting we can decompose the (for n = a priori ill-posed) product of two distributions as ϕ ψ = ϕ 4 ψ+ϕ ψ+ψ 4 ϕ, with: ∞ · n n n n ϕ 4 ψ = ∆

1≤i≤jn |i−j|≤1 X −1≤Xi,j≤jn n n where ∆

Now we consider time-dependent functions. Fix a time horizon T > 0 and assume we are given an increasing family of normed spaces X = (X(t))t∈[0,T ] with decreasing norms (X(t) X(0) is allowed). Usually we will use this to deal with time-dependent ≡ C α Zd R weights and take X(t) = ( n, e(l + t)) for some α, l . We then write CX for the space of continuous functions ϕ: [0, T ] X(T ) endowed∈ with the supremum norm → ϕ CX = supt∈[0,T ] ϕ(t) X(t) . For α (0, 1) we sometimes quantify the time regularity k k α k k ∈ via C X = f CX : f α < , where { ∈ k kC X ∞} f(t) f(s) X(t) f α = f + sup k − k . k kC X k kCX t s α 0≤s

N Zd LEMMA 1.2. The estimates below hold uniformly over n (recall that ∞ = Rd). Consider z, z , z , z ̺(ω) and α, β R. We ﬁnd that: ∈ ∪ {∞} 1 2 3 ∈ ∈ ϕ 4 ψ C α(Zd ;z z ) . ϕ Lp (Zd ;z ) ψ C α(Zd ;z ), k k p n 1 2 k k n 1 k k n 2

ϕ 4 ψ α+β d . ϕ β d ψ C α Zd , if β< 0, Cp (Z ;z z ) Cp (Z ;z ) ( n;z2) k k n 1 2 k k n 1 k k

α+β β α Zd if ϕ ψ C (Zd ;z z ) . ϕ C (Zd ;z ) ψ C ( ;z2) α+β> 0. k k p n 1 2 k k p n 1 k k n Similar bounds hold if we estimate ψ in a Cp Besov space and ϕ in C = C∞. And for γ [0, 1), ε [0, 2γ] [0, α), 0 <α< 2 and δ> 0 we can bound: ∈ ∈ ∩ γ,α (3) ϕ γ−ε/2,α−ε d . ϕ L Zd . Lp (Z ;z) p ( n;z) k k n k k Moreover, for the operator C (ϕ,ψ,ζ) = (ϕ 4 ψ) ζ ϕ(ψ ζ) we have: 1 − β+γ α Zd β Zd γ Zd C1(ϕ,ψ,ζ) C (Zd ;z z z ) . ϕ C ( ;z1) ψ C ( ;z2) ζ C ( ;z3), k k p n 1 2 3 k k p n k k n k k n if β+γ< 0, α+β+γ> 0. imsart-aop ver. 2020/08/06 file: rough_superbrownian_motion.tex date: September 18, 2020 6

PROOF. The ﬁrst three estimates are shown in [MP19, Lemma 4.2] and the fourth estimate comes from [MP19, Lemma 3.11]. In that lemma the case ε = 2γ < α is not included, but it follows by the same arguments (since [GP17, Lemma A.1] still applies in that case). The last estimate is provided by [MP19, Lemma 4.4].

For two functions ψ, ϕ: Rd R we deﬁne ψ, ϕ = dx ψ(x)ϕ(x) and ψ ϕ(x)= Rd → Zd hR i ∫ 1 ∗ ψ(x ), ϕ( ) for x , whereas if ψ, ϕ: n we write ψ, ϕ n = d x∈Zd ψ(x)ϕ(x) h −· · i ∈ → h i n n and ψ ϕ(x)= ψ(x ), ϕ( ) for x Zd . n n n P Finally,∗ for a metrich −· space E· wei denote∈ with D([0, T ]; E) and D([0, + ); E) the Skorohod space equipped with the Skorohod topology (cf. [EK86, Section 3.5]).∞ We will also write M (Rd) for the space of positive ﬁnite measures on Rd with the weak topology, which is a Polish space (cf. [DMS93, Section 3]).

2. The Model. We consider a branching random walk in a random environment (BR- Zd N WRE). This is a process on the lattice n, for n and d = 1, 2, and we are interested in the limit n . The evolution of this process depends∈ on the environment it lives in. Therefore, we ﬁrst→∞ discuss the environment before introducing the Markov process. n A deterministic environment is a sequence ξ n∈N of potentials on the lattice, i.e. func- n Zd R { } p,n F p,n Pp,n tions ξ : n . A random environment is a sequenceof probability spaces (Ω , , ) → n n p,n d together with a sequence ξ N of measurable maps ξ :Ω Z R. { p }n∈ p × n → n ASSUMPTION 2.1 (Random Environment). We assume that for every N, Zd n ξp (x) x∈ n is a set of i.i.d. random variables on a probability space (Ωp,n, F p,n, Pp,n)∈which{ satisfy:} −d/2 n (4) n ξp (x)=Φ in distribution, for a random variable Φ with ﬁnite moments of every order such that E[Φ] = 0, E[Φ2] = 1.

n Rd REMARK 2.2. It follows that ξp converges in distribution to a white noise ξp on , in the sense that ξn,f ξ (f) for all f C (Rd). h p in → p ∈ c To separate the randomness coming from the potential from that of the branching random n walks it will be convenient to freeze the realization of ξp and to consider it as a deterministic environment. But we cannot expect to obtain reasonable scaling limits for all deterministic environments. Therefore, we need to identify properties that hold for typical realizations of random potentials satisfying Assumption 2.1. The reader only interested in random environments may skip the following assumption and use it as a black box, since by Lemma 2.4 below it is satisﬁed under Assumption 2.1.

ASSUMPTION 2.3 (Deterministic environment). Let ξn be a deterministic environment n n n n F −1 F n and let X be the solution to the equation ∆ X = χ(D)ξ = n (χ nξ ) in the sense explained in [MP19, Section 5.1], where− χ is a smooth function equal to 1 outside of ( 1/4, 1/4)d and equal to zero on ( 1/8, 1/8)d. Consider a regularity parameter − − α (1, 3 ) in d = 1, α ( 2 , 1) in d = 2. ∈ 2 ∈ 3 We assume that the following holds: (i) There exists ξ C α−2(Rd,p(a)) such that for all a> 0: ∈ a>0 n n n α−2 d sup ξ TC α−2(Zd ,p(a)) < + and E ξ ξ in C (R ,p(a)). n k k n ∞ → imsart-aop ver. 2020/08/06 file: rough_superbrownian_motion.tex date: September 18, 2020 ROUGH SUPER-BM 7

(ii) For any a, ε > 0 we can bound: −d/2 n −d/2 n sup n ξ+ C −ε(Zd ,p(a)) + sup n ξ C −ε(Zd ,p(a)) < + n k k n n k | |k n ∞ as well as for any b > d/2:

−d/2 n sup n ξ+ L2(Zd ,p(b)) < + . n k k n ∞ Moreover, there exists ν 0 such that the following convergences hold: ≥ E nn−d/2ξn ν, E nn−d/2 ξn 2ν + → | |→ in C −ε(Rd,p(a)). −d/2 (iii) If d = 2 there exists a sequence cn R with n cn 0 and there exist X C α(Rd,p(a)) and X ξ {C 2}⊂α−2(Rd,p(a)) which→ satisfy for all a> 0: ∈ a>0 ⋄ ∈ a>0 n n n T sup X C α(Zd ,p(a)) +T sup (X ξ ) cn C 2α−2(Zd ,p(a)) < + n k k n n k − k n ∞ and E nXn X in C α(Rd,p(a)) and E n (Xn ξn) c X ξ in C 2α−2(Rd,p(a)). → − n → ⋄ S ′ Rd We say that ξ ω( ) is a deterministic environment satisfying Assumption 2.3 if there ∈ n exists a sequence ξ n∈N such that the conditions of Assumption 2.3 hold. The next result{ establishes} the connection between the probabilistic and the analytical conditions. To formulate it we need the following sequence of diverging renormalization constants: χ(k) (5) κn = dk n log(n), T2 l (k) ∼ Z n with ln being the Fourier multiplier associated to the discrete Laplacian ∆n and χ as in Assumption 2.3.

¯n LEMMA 2.4. Given a random environment ξp n∈N satisfying Assumption 2.1, there p F p Pp { } n exists a probability space (Ω , , ) supporting random variables ξp n∈N such that ¯n n n p { } ξp = ξp in distribution and such that ξp (ω , ) n∈N is a deterministic environment satisfying Assumption 2.3 for all ωp Ωp. Moreover{ · } the sequence c in Assumption 2.3 can be ∈ n chosen equal to κn (see Equation (5)) outside of a null set. Similarly, ν is strictly positive and deterministic outside of a null set and equals the expectation E[Φ+].

PROOF. The existence of such a probability space is provided by the Skorohod representation theorem. Indeed it is a consequence of Assumption 2.1 that all the convergences hold in the sense of distributions: The convergences in (i) and (iii) follow from Lemma B.2 if d = 1 and from [MP19, Lemmata 5.3 and 5.5] if d = 2 (where it is also shown that we can choose cn = κn). The convergence in (ii) for ν = E[Φ+] is shown in Lemma B.1. After changing the probability space the Skorohod representation theorem guarantees almost sure convergence, so setting ξn,ξ,cn,ν = 0 on a null set we ﬁnd the result for every ωp. (There is a small subtlety in the application of the Skorohod representation theorem because C γ(Rd,p(a)) is not separable, but we can restrict our attention to the closure of smooth compactly supported functions in C γ(Rd,p(a)), which is a closed separable subspace).

n NOTATION 2.5. A sequence of random variables ξp n∈N defined on a common probability space (Ωp, F p, Pp) which almost surely satisfies{ Assumption} 2.3 is called a controlled random environment. By Lemma 2.4, for any random environment satisfying Assumption 2.1 imsart-aop ver. 2020/08/06 file: rough_superbrownian_motion.tex date: September 18, 2020 8 we can find a controlled random environment with the same distribution. For a given controlled random environment we introduce the effective potential: ξn (ωp,x)= ξn(ωp,x) c (ωp)1 . p,e p − n {d=2} Given a controlled random environment we define H ωp as the random Anderson Hamilto- nian and its domain DH ωp (see Lemma 3.5). If the environment is deterministic we drop all indices p.

We pass to the description of the particle system. This will be a (random) Markov process Zd N n Zd N on the space E = 0 0 of compactly supported functions η : n 0, whose construc- x7→y → tion is discussed in Appendix A. We deﬁne η (z)= η(z)+(1{y}(z) 1{x}(z))1{η(x)≥1} x± − and η (z) = (η(z) 1{x}(z))+. Moreover, Cb(E) is the Banach space of continuous and ± Zd bounded functions on E endowed with the discrete topology. For F Cb(E),x n we write: ∈ ∈ ∆nF (η)= n2 (F (ηx7→y) F (η)), d±F (η)= F (ηx±) F (η). x − x − y∼x X DEFINITION 2.6. Fix an “averaging parameter” ̺ 0 and a controlled random envi- n Pn p D ≥ ronment ξp . Let be the measureon Ω ([0, + ); E) deﬁned as the “semidirect product p × ∞ p measure” (cf. (26)) Pp ⋉Pω ,n, where for ωp Ωp the measure Pω ,n on D([0, + ); E) is n∈ p n p ̺ ∞ the law under which the canonical process up (ω , ) started in up (ω , 0) = n 1{0}(x) is the Markov process with generator · ⌊ ⌋

p p L n,ω : D(L n,ω ) C (E), → b where L n,ωp (F )(η) is deﬁned by:

(6) η ∆nF (η) + (ξn ) (ωp,x)d+F (η) + (ξn ) (ωp,x)d−F (η) x · x p,e + x p,e − x x∈Zd Xn h i n,ωp and the domain D(L ) consists of all F Cb(E) such that the right-hand side of (6) lies n n∈ in Cb(E). To up we associate the process µp with the pairing µn(ωp,t)(ϕ) := n̺ −1un(ωp,t,x)ϕ(x) p ⌊ ⌋ p x∈Zd Xn Zd R n D M Rd for any function ϕ: n . Hence µp is a stochastic processwith values in ([0, + ); ( )), with the law induced by→Pn. ∞

REMARK 2.7. Although not explicitly stated, it is part of the deﬁnition that ωp Pωp,n(A) is measurable for Borel sets A B(D([0, + ); E)). 7→ ∈ ∞ Since all particles evolve independently, we expect that for ̺ the law of large numbers applies. This is why we refer to ̺ as an averaging parameter.→∞

NOTATION 2.8. In the terminology of stochastic processes in random media, we refer to Pωp,n n n n Pn as the quenched law of the process up (or µp ) given the noise ξp . We also call the total law. As before, if the process is deterministic we drop the index p everywhere.

We can now state the main convergence results of this work. We will ﬁrst prove quenched versions and the total versions are then easy corollaries. We start with a law of large numbers. imsart-aop ver. 2020/08/06 file: rough_superbrownian_motion.tex date: September 18, 2020 ROUGH SUPER-BM 9

THEOREM 2.9. Let ξ be a deterministic environment satisfying Assumption 2.3 and let ̺ > d/2. Let w be the solution of PAM (1) with initial condition w(0,x)= δ0(x), as constructed in Proposition 3.1 (cf. Remark 3.2). The measure-valued process µn from Deﬁnition 2.6 converges to w in probability in the space D([0, + ); M (Rd)) as n + . ∞ → ∞ PROOF. The proof can be found in Section 4.1.

If the averaging parameter takes the critical value ̺ = d/2, we see random ﬂuctuations in the limit and we end up with the rough super-Brownian motion (rSBM). As in the case of the classical SBM, the limiting process can be characterized via duality with the following equation: κ (7) ∂ ϕ = H ϕ ϕ2, ϕ(0) = ϕ , t − 2 0 for ϕ C∞(Rd), ϕ 0, where we recall that H is the Anderson Hamiltonian. With some 0 ∈ c 0 ≥ abuse of notation (since the equation is not linear) we write Utϕ0 = ϕ(t).

DEFINITION 2.10. Let ξ be a deterministic environment satisfying Assumption 2.3, let κ> 0 and let µ be a process with values in the space C([0, + ); M (Rd)), such that µ(0) = ∞ δ0. Write F = Ft t∈[0,+∞) for the completed and right-continuous filtration generated by µ. We call µ a rough{ } super-Brownian motion (rSBM) with parameter κ if it satisfies one of the three properties below: (i) For any t 0 and ϕ C∞(Rd), ϕ 0 and for U ϕ the solution to Equation (7) with ≥ 0 ∈ c 0 ≥ · 0 initial condition ϕ0, the process ϕ N 0 (s)= e−hµ(s),Ut−sϕ0i, s [0,t], t ∈ is a bounded continuous F martingale. −∞ Rd C ζ Rd (ii) For any t 0 and ϕ0 Cc ( ) and f C([0,t]; ( , e(l))) for some ζ > 0 and l< t, and≥ for ϕ solving∈ ∈ − t ∂ ϕ + H ϕ = f, s [0,t], ϕ (t)= ϕ , s t t ∈ t 0 it holds that s s M ϕ0,f (s) := µ(s), ϕ (s) µ(0), ϕ (0) dr µ(r),f(r) , 7→ t h t i−h t i− h i Z0 defined for s [0,t], is a continuous square-integrable F martingale with quadratic variation ∈ − s M ϕ0,f = κ dr µ(r), (ϕ )2(r) . h t is h t i Z0 (iii) For any ϕ DH the process: ∈ t Lϕ(t)= µ(t), ϕ µ(0), ϕ dr µ(r), H ϕ , t [0, + ), h i−h i− h i ∈ ∞ Z0 is a continuous F martingale, square-integrable on [0, T ] for all T > 0, with quadratic variation − t Lϕ = κ dr µ(r), ϕ2 . h it h i Z0 Each of the three properties above characterizes the process uniquely: imsart-aop ver. 2020/08/06 file: rough_superbrownian_motion.tex date: September 18, 2020 10

LEMMA 2.11. The three conditions of Deﬁnition 2.10 are equivalent. Moreover, if µ is a rSBM with parameter κ, then its law is unique.

PROOF. The proof can be found at the end of Section 4.1.

n THEOREM 2.12. Let ξ n∈N be a deterministic environment satisfying Assumption 2.3 { } n and let ̺ = d/2. Then the sequence µ n∈N converges to the rSBM µ with parameter κ = 2ν in distribution in D([0, + ); M (R{d)).} ∞ PROOF. The proof can be found at the end of Section 4.1.

REMARK 2.13. Lemma 2.11 gives the uniqueness of the rSBM for all parameters κ> 0, but Theorem 2.12 only shows the existence conditionally on the existence of an environment 1 which satisﬁes Assumption 2.3, which leads to the constraint ν (0, 2 ] because we should 2∈ think of ν = E[Φ+] for a centered random variable Φ with E[Φ ] = 1. But we establish the existence of the rSBM for general κ> 0 in Section 4.2.

REMARK 2.14. We restrict our attention to the Dirac delta initial condition for simplicity, but most of our arguments extend to initial conditions µ M (Rd) that satisfy µ, e(l) < for all l< 0. In this case only the construction of the∈ initial value sequence h n i ∞ µ (0) n∈N is more technical, because we need to come up with an approximation in terms {of integer} valued point measures (which we need as initial condition for the particle system). This can be achieved by discretizing the initial measure on a coarser grid.

The previous results describe the scaling behavior of the BRWRE conditionally on the environment, and we now pass to the unconditional statements. To a given random environment n ξp satisfying Assumption 2.1 (not necessarily a controlled random environment) we associate S ′ Rd n −d n a sequence of random variables in ω( ) by deﬁning ξp (f)= n x ξp (x)f(x). The se- Pn Pp,n ⋉Pωp,n S ′ Rd D M Rd quence of measures = on ω( ) ([0, + ); ( )) is then such that Pp,n n Pωp,n × ∞ P n n is the law of ξp and is the quenched law of the branching process µp given ξp (cf. Appendix A).

n p COROLLARY 2.15. The sequence of measures P converges weakly to P = Pp ⋉Pω on S ′ Rd D M Rd Pp S ′ Rd ω( ) ([0, + ); ( )), where is the law of the space white noise on ω( ), ωp × ∞ and P is the quenched law of µp given ξp which is described by Theorem 2.9 if ̺ > d/2 or by Theorem 2.12 if ̺ = d/2.

S ′ Rd D M Rd PROOF. Consider a function F on ω( ) ([0, + ); ( )) which is continuous E × n n ∞ E and bounded. We need the convergence limn F (ξp ,µ ) F (ξp,µ) . Up to changing → n the probability space (which does not affect the law) we may assume that ξp is a controlled random environment. We condition on the noise, rewriting the left-hand side as

E n n Eωp,n n p n Pp p F (ξp ,µ ) = F (ξp (ω ),µ ) (d ω ). Z Under the additional property of being a controlled random environment and for ﬁxed ωp Ωp, the conditional law Pωp,n on the space D([0, + ); M (Rd)) converges weakly to the∈ measure Pωp given by Theorem 2.9 respectively Theorem∞ 2.12, according to the value of ̺. We can thus deduce the result by dominated convergence.

For ̺ > d/2 the process of Corollary 2.15 is simply the continuous parabolic Anderson model. For ̺ = d/2 it is a new process. imsart-aop ver. 2020/08/06 file: rough_superbrownian_motion.tex date: September 18, 2020 ROUGH SUPER-BM 11

DEFINITION 2.16. For ̺ = d/2 we call the process µ of Corollary 2.15 an SBM in static random environment (of parameter κ> 0).

In dimension d = 1 we characterize the process µ as the solution to the SPDE (2). First, we rigorously deﬁne solutions to such an equation.

DEFINITION 2.17. Let d = 1, κ> 0, and π M (R). A weak solution to ∈ ωp (8) ∂tµp(t,x)= H µp(t,x)+ κµp(t,x)ξ˜(t,x), µp(0) = π, q is a couple formed by a probability space (Ω, F , P) and a random process µ :Ω C([0, + ); M (R)) p → ∞ such that Ω=Ωp Ω¯ and P is of the form Pp ⋉Pωp with (Ωp, Pp) supporting a space × white noise ξp and (Ω, P) supporting an independent space-time white noise ξ˜, such that the following properties are fulfilled for almost all ωp Ωp: ∈ F ωp ¯ Pωp • There exists a filtration t t∈[0,T ] on the space (Ω, ) which satisfies the usual con- { p } p 2 ditions and such that µp(ω , ) is adapted and almost surely lies in L ([0, T ]; L (R, e(l))) · p for all p< 2 and l R. Moreover, under Pω the process ξ˜(ωp, ) is a space-time white noise adapted to the∈ same filtration. ·

• The random process µ satisﬁes for all ϕ DH ωp and for all t 0: p ∈ ≥ t p p ωp dxµp(ω ,t,x)ϕ(x)= ds dxµp(ω ,s,x)(H ϕ)(x) R R Z Z0 Z t p p + ξ˜(ω , ds, dx) κµp(ω ,s,x)ϕ(x)+ ϕ(x)π(dx), 0 R R Z Z q Z with the last integral understood in the sense of Walsh [Wal86].

THEOREM 2.18. For π = δ0 and any κ> 0 there exists a weak solution µp to the SPDE (8) in the sense of Deﬁnition 2.17. Thelaw of µp as a random process on C([0, + ); M (R)) is unique and it corresponds to an SBM in static random environment of parameter∞ κ.

PROOF. The proof can be found at the end of Section 5.1.

As a last result, we show that the rSBM is persistent in dimension d = 1, 2.

DEFINITION 2.19. We say that a random process µ C([0, + ); M (Rd)) is super- ∈ ∞ R∞d exponentially persistent if for any nonzero positive function ϕ Cc ( ) and for all λ> 0 it holds that: ∈ P lim e−tλ µ(t), ϕ = > 0. t→∞ h i ∞ THEOREM 2.20. Let µp be an SBM in static random environment. Then for almost all ωp Ωp the process µ (ωp, ) is super-exponentially persistent. ∈ p · The result follows from Corollary 5.6 and the preceding discussion.

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3. Discrete and Continuous PAM & Anderson Hamiltonian. Here we review the solution theory for the PAM (1) in the discrete and continuous setting and the interplay between the two. Recall that the regularity parameter α from Assumption 2.3 satisﬁes: (9) α (1, 3 ) in d = 1, α ( 2 , 1) in d = 2. ∈ 2 ∈ 3 We recall some results from [MP19] regarding the solution of the PAM on the whole space (see also [HL15]), and regarding the convergence of lattice models to the PAM. We take an ζ C Rd M γ0 C α0 Rd initial condition w0 p ( , e(l)) and a forcing f p ( , e(l)), and we consider the equation ∈ ∈

(10) ∂tw =∆w + ξw + f, w(0) = w0 and its discrete counterpart n n n n n n n (11) ∂tw = (∆ + ξe )w + f , w (0) = w0 . To motivate the constraints on the parameters appearing in the proposition below, let us first formally discuss the solution theory in d = 1. Under Assumption 2.3 it follows from the Schauder estimates in [MP19, Lemma 3.10] that the best regularity we can expect at α∧(ζ+2)∧(α0 +2) a fixed time is w(t) Cp (R, e(k)) for some k R. In fact we lose a bit of regularity, so let ϑ <∈ α be “large enough” (we will see soon∈ what we need from ϑ) and C ϑ R assume that ζ +2 ϑ and α0 +2 ϑ. Then we expect w(t) p ( , e(k)), and the Schauder estimates suggest≥ the blow-up γ ≥= max (ϑ + ε ζ) /2, γ∈ for some ε> 0, which has to { − + 0} be in [0, 1) to be locally integrable, so in particular γ0 [0, 1). If ϑ + α 2 > 0 (which is possible because in d = 1 we have 2α 2 > 0), then∈ the product w(t)ξ−is well defined C α−2 R − and in p ( , e(k)p(a)), so we can set up a Picard iteration. The loss of control in the weight (going from e(k) to e(k)p(a)) is handled by introducing time-dependent weights so C ϑ Rd that w(t) p ( , e(l + t)). In the setting of singular SPDEs this idea was introduced by Hairer-Labbé∈ [HL15], and it induces a small loss of regularity which explains why we only obtain regularity ϑ < α for the solution and the additional +ε/2 in the blow-up γ. In two dimensions the white noise is less regular, we no longer have 2α 2 > 0, and we need paracontrolled analysis to solve the equation. The solution lives in a− space of paracontrolled distributions, and now we take ϑ> 0 such that ϑ + 2α 2 > 0. We now need − additional regularity requirements for the initial condition w0 and for the forcing f. More t precisely, we need to be able to multiply (Ptw0)ξ and 0 Pt−sf(s)ds ξ, and therefore we require now also ζ+2+(α 2) > 0 and α0+2+(α 2) > 0, i.e. ζ, α0 > α. We do not provide the− details of the construction− andR refer to [MP19− ] instead, where the two-dimensional case is worked out (the one-dimensional case follows from similar, but much easier arguments).

PROPOSITION 3.1. Consider α as in (9), any T > 0, p [1, + ], l R, γ [0, 1) and ∈ ∞ ∈ 0 ∈ ϑ,ζ,α0 satisfying: (2 α, α), d = 1, (12) ϑ − ζ> (ϑ 2) ( α), α0 > (ϑ 2) ( α), ∈ (2 2α, α), d = 2, − ∨ − − ∨ − ( − n ζ d n γ α d and let w Cp (Z , e(l)) and f M 0 C 0 (Z , e(l)) be such that 0 ∈ n ∈ p n E nwn w , in C ζ (Rd, e(l)), E nf n f in M γ0 C α0 (Rd, e(l)). 0 → 0 p → p

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Then under Assumption 2.3 there exist unique (paracontrolled) solutions wn, w to Equation n (11) and (10). Moreover, for all γ> (ϑ ζ)+/2 γ0 and for all ˆl l+T , the sequence w is L γ,ϑ Zd ˆ − ∨ ≥ uniformly bounded in p ( n, e(l)):

n n n α (13) sup w γ,ϑ d . sup w ζ d + sup f M γ0 C 0 Zd , Lp (Z ,e(ˆl)) 0 Cp (Z ,e(l)) p ( n,e(l)) n k k n n k k n n k k where the proportionality constant depends on the time horizon T and the norms of the objects in Assumption 2.3. Moreover E nwn w in L γ,ϑ(Rd, e(ˆl)). → p REMARK 3.2. We consider the case p< to start the equation in the Dirac measure −d d ∞ δ0. Indeed, δ0 lies in C (R , e(l)) for any l R. This means that ζ = d, and in d = 1 we can choose ϑ small enough such that (12) holds.∈ But in d = 2 this is not− sufﬁcient, so we use d(1−p)/p d instead that δ0 Cp (R , e(l)) for p [1, ] and any l R, so that for p [1, 2) the conditions in (12∈) are satisﬁed. ∈ ∞ ∈ ∈

NOTATION 3.3. We write t t t T nwn + ds T n f n, t T w + ds T f 7→ t 0 t−s s 7→ t 0 t−s s Z0 Z0 for the solution to Equation (11) and (10), respectively.

Proposition 3.1 provides us with the tools to make sense of Property (ii) in the definition of the rSBM, Definition 2.10. To make sense of the last Property (iii), we need to construct the Anderson Hamiltonian. In finite volume this was done in [FN77, AC15, GUZ20, Lab19], respectively, but the construction in infinite volume is more complicated, for example because the spectrum of H is unbounded from above and thus resolvent methods fail. Hairer- −1 Labbé [HL18] suggest a construction based on spectral calculus, setting H = t log Tt, but this gives insufficient information about the domain. Therefore, we use an ad-hoc approach which is sufficient for our purpose. We define the operator in terms of the solution map (Tt)t≥0 to the parabolic equation. Strictly speaking, (Tt)t≥0 does not define a semigroup, since due to the presence of the time-dependent weights it does not act on a fixed Banach space. But we simply ignore that and are still able to use standard arguments for semigroups on Banach spaces to identify a dense subset of the domain (compare the discussion below to [EK86, Proposition 1.1.5]). However, in that way we do not learn anything about the spectrum of H . In finite volume, (Tt)t≥0 is a strongly continuous semigroup of compact operators and we can simply define H as its infinitesimal generator. It seems that this would be equivalent to the construction of [AC15] through the resolvent equation. We first discuss the case d = 1. Then ξ C α−2(R,p(a)) for all a> 0 by assumption, 3 H ∈ C ϑ R where α (1, 2 ). In particular, u = (∆+ξ)u is well defined for all u ( , e(l)) with ϑ>∈2 α and l R, and H u C α−2(R, e(l)p(a)). Our aim is to identify∈ a subset of C ϑ(R, e(−l)) on which∈ H u is even∈ a continuous function. We can do this by defining for t> 0 t Atu = Tsu ds. Z0 Then A u C ϑ(R, e(l+t)), and by definition t ∈ t t H A u = H T u ds = ∂ T u ds = T u u C ϑ(R, e(l + t)). t s s s t − ∈ Z0 Z0 imsart-aop ver. 2020/08/06 file: rough_superbrownian_motion.tex date: September 18, 2020 14

Moreover, the following convergence holds in C ϑ(R, e(l+t+ε)) for all ε> 0: t+1/n 1/n lim n(T1/n id)Atu = lim n Tsu ds Tsu ds = H Atu. n→∞ − n→∞ − Zt Z0 Therefore, we define ϑ DH = A u : u C (R, e(l)), l R,t [0, T ] . { t ∈ ∈ ∈ } ϑ R ϑ R Since for u C ( , e(l)) the map (t Ttu)t∈[0,ε] is continuous in the space C ( , e(l+ε)) ∈ ϑ 7→ m m wecanfindforall u C (R, e(l)) a sequence u m∈N DH such that u u C ϑ(R,e(l+ε)) ∈ m { −1 } ⊂ k − k → 0 for all ε> 0. Indeed, it suffices to set u = m Am−1 u. The same construction also works for H n instead of H . In the two-dimensional case (∆+ξ)u would be well defined whenever u C β(R2, e(l)) 2 ∈ with β > 2 α for α ( 3 , 1). But in this space it seems impossible to find a domain that is mapped to− continuous∈ functions. And also (∆+ξ)u is not the right object to look at, we have to take the renormalization into account and should think of H =∆+ξ .So we first need an appropriate notion of paracontrolled distributions u for which can define−∞ H u as a distribution. As in Proposition 3.1 we let ϑ (2 2α, α). ∈ − DEFINITION 3.4. Consider X = ( ∆)−1χ(D)ξ and X ξ defined as in Assumption 2.3. We say that u (resp. un) is paracontrolled− if u C ϑ(R2, e(l))⋄ for some l R, and ∈ ∈ u♯ = u u 4 X C α+ϑ(R2, e(l)). − ∈ Then set H u =∆u + ξ 4 u + u 4 ξ + u♯ ξ + C (u, X, ξ)+ u(X ξ), 1 ⋄ where C1 is defined in Lemma 1.2. The same lemma also shows that H u is a well defined distribution in C α−2(R2, e(l)p(a)).

The operator Tt leaves the space of paracontrolled distributions invariant, and therefore the same arguments as in d = 1 give us a domain DH such that for all paracontrolled u m m there exists a sequence u N DH with u u C ϑ R2 0 for all ε> 0. For { }m∈ ⊂ k − k ( ,e(l+ε)) → general u C ϑ(R2, e(l)) and ε> 0 we can ﬁnd a paracontrolled v C ϑ(R2, e(l)) with ∈ ∈ u v C ϑ R2 < ε, because T u is paracontrolled for all t> 0 and converges to u in k − k ( ,e(l+ε)) t C ϑ(R2, e(l+ε)) as t 0. Thus, we have established the following result: → LEMMA 3.5. Under Assumption 2.3 let ϑ be as in Proposition 3.1. There exists a domain D C ϑ Rd H C ϑ Rd H l∈R ( , e(l)) such that u = limn n(T1/n id)u in ( , e(l+ε)) for all ⊂ ϑ d − ϑ d u DH C (R , e(l)) and ε> 0 and such that for all u C (R , e(l)) there is a sequence ∈m S∩ m ∈ u m∈N DH with u u C ϑ(R2,e(l+ε)) 0 for all ε> 0. The same is true for the { } ⊂ H n k −Rd k Zd→ discrete operator (with replaced by n). 4. The Rough Super-Brownian Motion.

4.1. Scaling Limit of Branching Random Walks in Random Environment. In this section n we consider a deterministic environment, that is a sequence ξ n∈N satisfying Assumption 2.3, to which we associatethe Markov process µn as in Deﬁnition{ }2.6: Our aim is to prove that the sequence µn converges weakly, with a limit depending on the value of ̺. First, we prove tightness for the sequence µn in D([0, T ]; M (Rd)) for ̺ d/2. Then, we prove uniqueness in law of the limit points and thus deduce the weak convergenc≥ e of the sequence. Recall that d d for µ M (R ) and ϕ Cb(R ) we use both the notation µ, ϕ and µ(ϕ) for the integration of ϕ against∈ the measure∈ µ. h i imsart-aop ver. 2020/08/06 file: rough_superbrownian_motion.tex date: September 18, 2020 ROUGH SUPER-BM 15

REMARK 4.1. Fix t> 0. For any ϕ L∞(Zd ; e(l)), for some l R: ∈ n ∈ (14) [0,t] s M n,ϕ(s)= µn(T n ϕ) T nϕ(0) ∋ 7→ t s t−s − t is a centered martingale on [0,t] with predictable quadratic variation s M n,ϕ = µn n−̺ nT n ϕ 2 + n−̺ ξn (T n ϕ)2 dr. h t is r |∇ t−r | | e | t−r Z0 SKETCHOFPROOF. Consider a time-dependent function ψ. We use Dynkin’s formula t n n and an approximation argument applied to the function (s,µ) Fψ(s,µ )= µ (ψ(s)): By t 7→ truncating Fψ and discretizing time and then passing to the limit, we obtain for suitable ψ that s µn(ψ(s)) µn(ψ(0)) µn(∂ ψ(r)+H nψ(r)) dr s − 0 − r r Z0 is a martingale with the correct quadratic variation. Now it suffices to note that for r [0,t] : ∂ T n ϕ = H nT n ϕ. ∈ r t−r − t−r For the remainder of this section we assume that ̺ d/2. To prove the tightness of the measure-valued process we use the following auxiliary≥ result, which gives the tightness of n the real-valued processes t µt (ϕ) n∈N. The main difficulty in the{ 7→ proof lies} in handling the irregularity of the spatial environment. n n For this reason we consider first the martingale [0,t] s µs (Tt−sϕ) (cf. (14)) instead n ∋ 7→ of the more natural process s µs (ϕ). We then exploit the martingale to prove tightness for µn(ϕ). Here we cannot apply7→ the classical Kolmogorov continuity test, since we are considering a pure jump process. Instead we will use a slight variation, due to Chentsov [Che56] and conveniently exposed in [EK86, Theorem 3.8.8].

∞ d n LEMMA 4.2. For any l R and ϕ C (R , e(l)) the processes t µ (t)(ϕ) n∈N form a tight sequence in D([0∈, + ); R).∈ { 7→ } ∞ PROOF. It is sufficient to prove that for arbitrary T > 0 the given sequence is tight in D([0, T ]; R). Hence fix T > 0 and consider 0 <ϑ< 1 as in Proposition 3.1. In the following computation k R may change from line to line, but it is uniformly bounded for l R and T > 0 varying in∈ a bounded set. ∈ Step 1. Here the aim is to establish a second moment bound for the increment of the F n n process. Let ( t )t≥0 be the filtration generated by µ . We will prove that the following conditional expectation can be estimated uniformly over 0 t t+h T : ≤ ≤ ≤ σ σ (15) E µn (ϕ) µn(ϕ) 2 F n . hϑ µn(ek|x| )+ µn(ek|x| ) 2 , | t+h − t | | t t | t | In fact, via the martingales defined in (14), one canh start by observing that: i E µn (ϕ) µn(ϕ) 2 F n | t+h − t | | t = E M n,ϕ(t+h) M n,ϕ(t)+µn(T nϕ ϕ) 2 F n | t+h − t+h t h − | | t t+h . E µn n−̺ nT n ϕ 2 + n−̺ ξn (T n ϕ)2 dr F n ϕ,T r |∇ t+h−r | | e | t+h−r t Zt σ + hϑ µn(ek|x| ) 2, | t |

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n L ϑ Zd where the last term appears since h Th ϕ ( n, e(k)). The ﬁrst term on the right hand side can be bounded for any ε> 0 by:7→ ∈ t+h µn T n n−̺ nT n ϕ 2 + n−̺ ξn (T n ϕ)2 dr t r−t |∇ t+h−r | | e | t+h−r (16) Zt t+h σ σ . µn ek|x| + (r t)−2εek|x| dr. t − Zt Here we have used Lemma D.1 to ensure that Zd is smooth on the lattice together with the a- ϕ n priori bound (13) of Proposition 3.1 and with Lemmata| D.2 and D.3, which show respectively a gain of regularity via the factor n−̺ and a loss of regularity via the discrete derivative n, to obtain: ∇ −̺ n n 2 lim sup n Tr ϕ C ϑ˜(Zd ,e(2(l+r))) = 0, n→∞ r∈[0,T ] k |∇ | k n for 0 < ϑ<ϑ˜ 1+̺/2 (we can choose ϑ sufﬁciently large so that the latter quantity is positive). Since ϑ>− 0 and the term is positive, one has by comparison: n −̺ n n 2 k|x|σ −̺ n n 2 Tr−t n Tt+h−rϕ . e n Tt+h−rϕ C ϑ˜(Zd ,e(2(l+T ))). |∇ | k |∇ | k n −̺ n Moreover, according to Assumption 2.3 for ̺ d/2 the term n ξe is bounded in C −ε Zd ≥ | | ( n,p(a)) whenever ε> 0. It then follows from the uniform bounds (13) from Propo- sition 3.1 and by applying (3) from Lemma 1.2, together with similar arguments to the ones just presented, that: n −̺ n n 2 sup sup s T (n ξ (T ϕ) ) M 2εC ε Zd s e r ( n,e(k)) n∈N r∈[0,T ]k 7→ | | k n −̺ n n 2 ϑ+ε . sup sup s T (n ξ (T ϕ) ) +ε,ϑ s e r L 2 (Zd ,e(k)) n∈N r∈[0,T ] k 7→ | | k n −̺ n n 2 . sup sup n ξ (T ϕ) C −ε Zd < . e r ( n,e(k)) n∈N r∈[0,T ] k | | k ∞ This completes the explanation of (16). So overall, integrating over r we can bound the conditional expectation by:

σ σ σ σ h1−2εµn(ek|x| )+ hϑ µn(ek|x| ) 2 hϑ µn(ek|x| )+ µn(ek|x| ) 2 , t | t | ≤ t | t | assuming 1 2ε ϑ. This completes the proof of (15h ). i Step 2. Now− we≥ are ready to apply Chentsov’s criterion [EK86, Theorem 3.8.8]. We have to multiply two increments of µn(ϕ) on [t h, h] and on [t,t+h] and show that for some κ> 0: − (17) E ( µn (ϕ) µn(ϕ) 1)2( µn(ϕ) µn (ϕ) 1)2 . h1+κ | t+h − t |∧ | t − t−h |∧ We use (15) to bound: E ( µn (ϕ) µn(ϕ) 1)2( µn(ϕ) µn (ϕ) 1)2 | t+h − t |∧ | t − t−h |∧ E µn (ϕ) µn(ϕ) 2 µn(ϕ) µn (ϕ) ≤ | t+h − t | | t − t−h | σ σ . hϑE µn(ek|x| )+ µn(ek|x| ) 2 µn(ϕ) µn (ϕ) t | t | | t − t−h | h σ i . hϑE 1+ µn(ek|x| ) 2 µn(ϕ) µn (ϕ) . | t | | t − t−h | h i imsart-aop ver. 2020/08/06 file: rough_superbrownian_motion.tex date: September 18, 2020 ROUGH SUPER-BM 17

n k|x|σ 4 By the Cauchy-Schwarzinequality together with (15) andthe momentboundfor µt (e ) from Lemma C.1 one obtains: | |

σ E (1 + µn(ek|x| ) 2) µn(ϕ) µn (ϕ) | t | | t − t−h | σ 1/2 1/2 . 1+ E µn(ek|x| ) 4 E µn(ϕ) µn (ϕ) 2 . hϑ/2. | t | | t − t−h | Combining all the estimates one ﬁnds:

n n 2 n n 2 3 ϑ E ( µ (ϕ) µ (ϕ) 1) ( µ (ϕ) µ (ϕ) 1) . h 2 . | t+h − t |∧ | t − t−h |∧ 2 Since ϑ> 3 , this proves Equation (17) for some κ> 0. In particular, we can apply [EK86, Theorem 3.8.8] with β = 4, which in turn implies that the tightness criterion of Theo- rem 3.8.6 (b) of the same book is satisﬁed. This concludes the proof of tightness for n t µ (t)(ϕ) N. { 7→ }n∈ Consequently, we ﬁnd tightness of the process µn in the space of measures.

n d COROLLARY 4.3. The processes t µ (t) N form a tight sequencein D([0, ); M (R )). { 7→ }n∈ ∞ PROOF. We apply Jakubowski’s criterion [DMS93, Theorem 3.6.4]. We ﬁrst need to verify the compact containment condition. For that purpose note that for all R> 0 the set M Rd 2 M Rd 2 2 KR = µ ( ) µ( ) R is compact in ( ). Here µ( )= Rd x dµ(x). { ∈ | | · | ≤ } n 2 | · | | | Since the sequence of processes µ ( ) N are tight by Lemma 4.2, we ﬁnd for all n∈ R T,ε> 0 an R(ε) such that { | · | }

supP sup µn(t)( 2) R(ε) ε, | · | ≥ ≤ n t∈[0,T ] ∞ Rd as required. Second we note that Cc ( ) is closed under addition and the maps µ d n 7→ µ(ϕ) ϕ∈C∞ (Rd) separate points in M (R ). Since Lemma 4.2 shows that t µ (t)(ϕ) is { } c 7→ tight for any ϕ C∞(Rd), we can conclude. ∈ c Next we show that any limit point is a solution to a martingale problem.

n LEMMA 4.4. Any limit point of the sequence t µ (t) n∈N is supported in the space of continuous function C([0, + ); M (Rd)), and{ it satisﬁes7→ Property} (ii) of Deﬁnition 2.10 with κ = 0 if ̺ > d/2, and κ =∞ 2ν if ̺ = d/2.

PROOF. First, we address the continuity of an arbitrary limit point µ. Since M (Td) is endowed with the weak topology, it is sufﬁcient to prove the continuity of t µ(t), ϕ for all ϕ C (Rd). In view of Corollary 4.3, up to a subsequence: 7→ h i ∈ b µn, ϕ µ, ϕ in D([0, ); R). h i → h i ∞ Then by [EK86, Theorem 3.10.2] in order to obtain the continuity of the limit point it is sufﬁcient to observe that the maximal jump size is vanishing in n: n n −̺ sup µt , ϕ µt−, ϕ . n ϕ L∞ . t≥0 |h i − h i| k k

ϕ0,f Next, we study the limiting martingale problem. First we will prove that the process Mt from Deﬁnition 2.10 is a martingale. Then we will compute its quadratic variation. imsart-aop ver. 2020/08/06 file: rough_superbrownian_motion.tex date: September 18, 2020 18

Step 1. We ﬁx a limit point µ and study the required martingale property. For f, ϕ0 as n ζ0 d required, observe that Zd is uniformly bounded in C Z for any and ϕ0 = ϕ0 n ( n; e(l)) ζ0 > 0 n | ζ d R, and similarly Zd is uniformly bounded in C Z , with an applica- l f = f n C([0,t]; ( n)) ∈ | n tion of Lemma D.1. Hence by Proposition 3.1 the solutions ϕt to the discrete equations n H n n n n n ∂sϕt + ϕt = f , ϕt (t)= ϕ0 ϑ d converge in L (R , e(l)) to ϕt, up to choosing a possibly larger l. At the discrete level we ﬁnd, analogously to (14), that s M ϕ0,f,n(s) := µn(s), ϕn(s) µn(0), ϕn(0) + dr µn(r),f n(r) , t h t i − h t i h i Z0 for s [0,t] is a centered square-integrable martingale. Moreover this martingale is bounded in L2∈uniformly over n, since the second moment can be bounded via the initial value and the predictable quadratic variation by t E M ϕ0,f,n 2(s) . dr T n n−̺ nϕn(r) 2+n−̺ ξn (ϕn(r))2 | t | r |∇ t | | | t Z0 h i ϕ0,f F and the latter quantity is uniformly bounded in n. To conclude that Mt is an martingale ϕ0,f,n −ϕ0,f note that by assumption Mt converges to the continuous process Mt . From [EK86, Theorem 3.7.8] we obtain that for 0 s r t and for bounded and continuous Φ: D([0,s]; M ) R ≤ ≤ ≤ → 0 E[Φ(µ )(M ϕ0,f (r) M ϕ ,f (s))] |[0,s] t − t 0 n ϕ0,f,n ϕ ,f,n = lim E[Φ(µ [0,s])(M (r) M (s))] = 0 n | t − t

ϕ0,f by the martingale property. From here we easily deduce the martingale property of Mt . ϕ0,f Step 2. We show that Mt has the correct quadratic variation, which should be given as the limit of s M ϕ0,f,n = drµn(r) n−̺ nϕn(r) 2 + n−̺ ξn (ϕn(r))2 . h t is |∇ t | | | t Z0 We only treat the case ̺ = d/2, the case ̺ > d/2 is similar but easier becausethen we can use d/2−̺ −̺ n Lemma D.2 to gain some regularity from the factor n , so that n ξ C ε Zd 0 ( n,p(a)) for some ε> 0 and for all a> 0. k | |k → n First we assume, leaving the proof for later, that for any sequence ψ n∈N with n { } lim ψ C −ε Rd = 0 for some a> 0 and all ε> 0: n k k ( ,p(a)) s 2 (18) E sup drµn(r) ψn (ϕn(r))2 0. · t −→ s≤t Z0 By Assumption 2.3 we can apply this to ψn = n−̺ ξn 2ν, and deduce that along a subsequence we have the following weak convergence in|D([0|−,t]; R): · 2 2 M ϕ0,f,n M ϕ0,f,n M ϕ0,f drµ(r) 2ν(ϕ )2(r) . t · − h t i· −→ t · − t Z0 Note also that the limit lies in C([0,t]; R). If the martingales on the left-hand side are uniformly bounded in L2 we can deduce as before that the limit is a continuous L2 martingale, and conclude that − s M ϕ0,f = drµ(r) 2ν(ϕ )2(r) . h t is t Z0 imsart-aop ver. 2020/08/06 file: rough_superbrownian_motion.tex date: September 18, 2020 ROUGH SUPER-BM 19

As for the uniform bound in L2, note that it follows from Lemma C.1 that

E ϕ0,f,n 4 sup sup Mt (s) < + . n 0≤s≤t | | ∞ For the quadratic variation term we estimate: s 2 E M ϕ0,f,n 2 s dr E µn(r) n−̺ nϕn(r) 2 + n−̺ ξn (ϕn(r))2 , |h t is| ≤ |∇ t | | | t Z0 which can be bounded via the second estimate of Lemma C.1. Step 3. Thus, we are left with the convergence in (18). By introducing the martingale from Equation (14) we find that E µn(r) ψn(ϕn(r))2 2 | t | r 2 . T nψn(ϕn(r))2 2(0) + dq T n n−̺ n T n [ψn(ϕn(r))2] (19) | r t | q ∇ r−q t Z0 h + n−̺ ξn (T n [ψn(ϕn(r))2])2 (0). | | r−q t We start with the first term. By Proposition 3.1 we know that for all ε>i0 and 0 <ϑ< 1 satisfying ϑ + 3ε< 1 and for l> 0 sufficiently large: n n n 2 ϑ+ε r T [ψ (ϕ (r)) ] +ε,ϑ r t L 2 Zd k 7→ k ( n;e(3l)) (20) n n 2 . ψ C −ε(Zd ;p(a)) ϕ L ϑ(Zd ;e(l)) k k n k k n n . ψ C −ε(Zd ;p(a)). k k n Together with Equation (3) from Lemma 1.2 and (20), we thus bound: n n n 2 2 −4ε n n n 2 2 Tr ψ (ϕt (r)) (0) . r r Tr ψ (ϕt (r)) L 2ε,ε(Zd ;e(l)) | | k 7→ | k n −4ε n 2 . r ψ C −ε(Zd ;p(a)). k k n Now we can treat the first term in the integral in (19). We can choose 0 <ϑ< 1 and ε> 0 with ϑ + 3ε< 1 such that 0 < ϑ˜ = ϑ 1+ d/4. We then apply Lemmata D.2 and D.3, which − − d guarantee us respectively a regularity gain from the factor n 4 and a regularity loss from the derivative n, to obtain: ∇ −d/4 n n n n 2 2 n n n 2 2 n Tr−q[ψ (ϕt (r)) ] C ϑ˜(Zd ;e(6l)) . Tr−q[ψ (ϕt (r)) C ϑ(Zd ;e(3l)) k| ∇ k n k k n −(ϑ+3ε) n 2 . (r q) ψ C −ε(Zd ;p(a)), − k k n where the last step follows similarly to (20). Overall we thus obtain the estimate: r dq T n n−̺ n T n [ψn(ϕn(r))2] 2)(0) q ∇ r−q t Z0 r n 2 −(ϑ+3ε) n 2 . ψ C −ε(Zd ;p(a)) dq (r q) . ψ C −ε(Zd ;p(a)). k k n − k k n Z0 Following the same steps, one can treat the second term in the integral in (19). We now use the same parameter ε both for the regularity of n−̺ ξn and of ψn, in view of Assumption 2.3, and choose ϑ, ε as above with the additional constraint| | ϑ + 5ε< 1. Then we can argue as follows: −̺ n n n n 2 2 −(ϑ+3ε) n 2 n ξ (Tq [ψ (ϕt (r)) ]) C −ε(Zd ;e(2l)p(a)) . q ψ C −ε(Zd ;p(a)) k | | k n k k n imsart-aop ver. 2020/08/06 file: rough_superbrownian_motion.tex date: September 18, 2020 20 and hence: r dq T n(n−̺ ξn (T n[ψn(ϕn(r))2])2)(0) q | | q t Z0 r n 2 −(ϑ+3ε) −2ε . ψ C −ε(Zd ;p(a)) dq (r q) q k k n − Z0 n 2 . ψ C −ε(Zd ;p(a)), k k n where in the last step we used that ϑ + 5ε< 1. This concludes the proof.

Our ﬁrst main result, the law of large numbers, is now an easy consequence.

PROOFOF THEOREM 2.9. Recall that now we assume ̺ > d/2. In view of Corollary 4.3 we can assume that along a subsequence µnk µ in distribution in D([0, + ); M (Rd)). It ⇒ ∞∞ Rd thus suffices to prove that µ = w. The previous lemma shows that for ϕ Cc ( ) the process s µ(s)(T ϕ) T ϕ(0) is a continuous square-integrable martingale∈ with vanishing 7→ t−s − t quadratic variation. Hence, it is constantly zero and µ(t)(ϕ)= Ttϕ(0) = (Ttδ0)(ϕ) almost surely for each fixed t 0. Note that T δ is well-defined, as explained in Remark 3.2. Since ≥ · 0 µ is continuous, the identity holds almost surely for all t> 0. The identity µ(t)= Ttδ0 then ∞ Rd follows by choosing a countable separating set of smooth functions in Cc ( ).

Now we pass to the case ̺ = d/2. To deduce the weak convergence of the sequence µn we have to prove that the distribution of the limit points is unique. For that purpose we ﬁrst introduce a duality principle for the Laplace transform of our measure-valued process, for which we have to study Equation (7). We will consider mild solutions, i.e. ϕ solves (7) if and only if κ t ϕ(t)= T ϕ ds T (ϕ(s)2). t 0 − 2 t−s Z0 We shall denote the solution by ϕ(t)= Utϕ0, which is justiﬁed by the following existence and uniqueness result:

∞ d PROPOSITION 4.5. Let T,κ> 0, l0 < T and ϕ0 C (R , e(l0)) with ϕ0 0. For − ∈ ϑ d≥ l = l0 + T and ϑ as in Proposition 3.1 there is a unique mild solution ϕ L (R , e(l)) to Equation (7): ∈ κ ∂ ϕ = H ϕ ϕ2, ϕ(0) = ϕ . t − 2 0

We write Utϕ0 := ϕ(t) and we have the following bounds:

Ck{Tt ϕ0}t∈[0,T ]k ∞ Rd 0 U ϕ T ϕ , U ϕ L ϑ Rd . e CL ( ,e(l)) . ≤ t 0 ≤ t 0 k{ t 0}t∈[0,T ]k ( ,e(l)) PROOF. We deﬁne the map I (ψ)= ϕ, where ϕ is the solution to κ ∂ ϕ = H ψ ϕ, ϕ(0) = ϕ . t − 2 0 ϑ Rd If l0 < T , then (Ttϕ0)t∈[0,T ] L ( , e(l)) for l = l0 + T , and thus a slight adaptation of the arguments− for Proposition 3.1∈ shows that I satisﬁes

ϑ d ϑ d Ckψk ∞ Rd I : L (R , e(l)) L (R , e(l)), I (ψ) L ϑ Rd . e CL ( ,e(l)) → k k ( ,e(l))

imsart-aop ver. 2020/08/06 file: rough_superbrownian_motion.tex date: September 18, 2020 ROUGH SUPER-BM 21 for some C> 0. Moreover, for positive ψ this map satisﬁes the bound 0 I (ψ)(t) T ϕ , ≤ ≤ t 0 so in particular we can bound I (ψ) CL∞ (Rd,e(l)) Ttϕ0 t∈[0,T ] CL∞(Rd,e(l)). Now, de- 0 k k m ≤ k{m−1 } k m ﬁne ϕ (t,x)= Ttϕ0(x) and then iteratively ϕ = I (ϕ ) for m 1. This means that ϕ solves the equation: ≥ κ ∂ ϕm = H ϕ ϕm−1ϕm. t − 2 Hence our a priori bounds guarantee that

m Ck{Tt ϕ0}t∈[0,T ]k ∞ Rd sup ϕ L ϑ(Rd,e(l)) . e CL ( ,e(l)) . m k k By compact embedding of L ϑ(Rd, e(l)) L ζ (Rd, e(l′)) for ζ<ϑ, l′ < l we obtain convergence of a subsequence in the latter space.⊂ The regularity ensures that the limit point is indeed a solution to Equation (7). The uniqueness of such a ﬁxed-point follows from the fact that the difference z = ϕ ψ of two solutions ϕ and ψ solves the well posed linear equation: H κ − ∂tz = + 2 (ϕ+ψ) z with z(0) = 0, and thus z = 0.

We proceed by proving some implications between Properties (i) (iii) of Deﬁni- tion 2.10. −

LEMMA 4.6. In Deﬁnition 2.10 the following implications hold between the three properties: (ii) (i), (ii) (iii). ⇒ ⇔

PROOF. (ii) (i): Consider U·ϕ0 as in point (i) of Definition 2.10, which is well defined in view of Proposition⇒ 4.5. An application of Itô’s formula and Property (ii) of Definition 2.10 with ϕ (s)= U ϕ , guarantee that for any F C2(R), and for f(r)= κ (U ϕ )2: t t−s 0 ∈ 2 t−r 0 t F ( µ(t), ϕ )= F ( µ(s),U ϕ )+ dr F ′( µ(r),U ϕ ) µ(r),f(r) h 0i h t−s 0i h t−r 0i h i Zs 1 t t + F ′′( µ(r),U ϕ )d M ϕ0,f + F ′( µ(r),U ϕ )dM ϕ0,f (r), 2 h t−r 0i h t ir h t−r 0i t Zs Zs ϕ0,f 2 where d Mt r = µ(r), κ(Ut−rϕ0) dr = µ(r), 2f(r) dr. We apply this for F (x)= e−x, so thath F ′′i= hF ′ and the two Lebesguei h integrals cancel.i Since F ′ is bounded for positive x the stochastic− integral is a true martingale and we deduce property (i). D n n n N (ii) (iii): Let ϕ H and t> 0 and let 0= t0 t1 . . . tn = t, n , be a sequence⇒ of partitions of∈ [0,t] with max ∆n := max≤ ≤(tn ≤ tn) 0. Then∈ k≤n−1 k k≤n−1 k+1− k → µ(t), ϕ µ(0), ϕ h i−h i n−1 n n n = µ(t ), ϕ µ(t ), T∆n ϕ + µ(t ), T∆n ϕ ϕ h k+1 i−h k k i h k k − i k=0 X n−1 T n ϕ ϕ ϕ,0 n ϕ,0 n n n ∆k = Mtn (tk+1) Mtn (tk ) +∆k µ(tk ), − . k+1 − k+1 h ∆n i k=0 k X We start by studying the second term on the right hand side: n−1 T n ϕ ϕ n n ∆k ∆k µ(tk ), n− h ∆k i Xk=0 imsart-aop ver. 2020/08/06 file: rough_superbrownian_motion.tex date: September 18, 2020 22

n−1 T∆n ϕ ϕ n n k H n n H = ∆k µ(tk ), n− ϕ +∆k µ(tk ), ϕ h ∆k − i h i Xk=0 h i n−1 =: R + ∆n µ(tn), H ϕ . n k h k i Xk=0 By continuity of µ the second term on the right hand side converges almost surely to the Riemann integral t µ(r), H ϕ dr. Moreover, from the characterization (ii) we get 0 h i E[µ(s)(ψ)] = µ(0), Tsψ and h R i s 2 2 2 E[µ(s)(H ϕ) ] . µ(0), (Ts(H ϕ)) + dr µ(0), Tr (Ts−rH ϕ) , h i 0 h i Z h i which is uniformly bounded in s [0,t]. So the sequence is uniformly integrable and converges also in L1 and not just almost∈ surely. Moreover, n−1 n n −1 E[ Rn ] . ∆ µ0, Ttn ( (∆ ) (T∆n ϕ ϕ) H ϕ ) , | | k k | k k − − | k=0 X n −1 and since Lemma 3.5 implies that maxk≤n−1(∆ ) (T∆n ϕ ϕ) converges to H ϕ in k k − C ϑ(Rd, e(l)) for some l R and ϑ> 0 (so in particular uniformly), it follows from Proposi- ∈ tion 3.1 and the assumption µ0, e(l) < for all l R that E[ Rn ] 0. Thus, we showed that h i ∞ ∈ | | → t Lϕ = µ(t), ϕ µ(0), ϕ µ(r), H ϕ dr t h i − h i− h i Z0 n−1 ϕ,0 n ϕ,0 n = lim Mtn (tk+1) Mtn (tk ) , n→∞ k+1 − k+1 k=0 X and the convergence is in L1. By taking partitions that contain s [0,t) and using the mar- ϕ,0 ϕ ϕ ϕ ∈ tingale property of Mr we get E[L (t) Fs]= L (s), i.e. L is a martingale. By the same arguments that we used to show the uniform| integrability above, Lϕ(t) is square integrable for all t> 0. To derive the quadratic variation we use again a sequence of partitions contain- ing s [0,t) and obtain ∈ E Lϕ(t)2 Lϕ(s)2 F = E (Lϕ(t) Lϕ(s))2 F − s − s E ϕ,0 n ϕ,0 n 2 F = lim Mtn (tk+1) Mtn (tk ) s n→∞ k+1 − k+1 k:tn >s Xk+1 n tk+1 2 E n F = lim κ dr µ(r), (Ttk+1−rϕ) s n→∞ n h i k:tn >s tk k+1 h Z i X t 2 = E κ dr µ(r), ϕ Fs . s h i h Z i Since the process κ · dr µ(r), ϕ2 is increasing and predictable, it must be equal to Lϕ . 0 h i h i (iii) (ii): Let t 0, ϕ0 DH , and let f : [0,t] DH be a piecewise constant function ⇒ R≥ ∈ → (in time; it might seem more natural to take f continuous, but since we did not equip DH with a topology this has no clear meaning). We write ϕ for the solution to the backward equation

(∂s+H )ϕ = f, ϕ(t)= ϕ0, imsart-aop ver. 2020/08/06 file: rough_superbrownian_motion.tex date: September 18, 2020 ROUGH SUPER-BM 23

t D which is given by ϕ(s)= Tt−sϕ0 + s Tr−sf(r)dr. Note that by assumption ϕ(r) H for n n n N ∈ all r t. For 0 s t, let 0= t0 t1 . . . tn = s, n , be a sequence of partitions of ≤ ≤ ≤ n ≤R ≤ n ≤ n ∈ [0,s] with maxk≤n−1 ∆k := maxk≤n−1(tk+1 tk ) 0. Similarly to the computation in the step “(ii) (iii)” we can decompose: − → ⇒ µ(s), ϕ(s) µ(0), ϕ(0) = h i−h i n−1 tn n n k+1 ϕ(tk+1) n ϕ(tk+1) n = L (tk+1) L (tk )+ dr µ(r),f(r) + Rn, − n h i tk Xk=0 Z with

n−1 n tk+1 n n n −1 n H n Rn = dr µ(r), ϕ(tk+1) µ(tk ), (∆k ) (T∆k id)ϕ(tk+1) n h i−h − i tk Xk=0 Z n + µ(t ), Tr−tn f(r) µ(r),f(r) . h k k i−h i 1 By similar arguments as in the step (ii) (iii) we see that Rn converges to zero in L , and ⇒ s therefore s µ(s), ϕ(s) µ(0), ϕ(0) 0 dr µ(r),f(r) is a martingale. Square integrability and7→ the h right formi−h of the quadratici− variationh are showni again by similar arguments as before. R D ϕ0,f By density of H it follows that Mt is a martingale on [0,t] with the required quadratic ∞ Rd C ζ Rd variation for any ϕ0 Cc ( ) and f C([0,t]; ( )) for ζ > 0. This concludes the proof. ∈ ∈

Characterization (i) of Deﬁnition 2.10 enables us to deducethe uniquenessin law and then to conclude the proof of the equivalence of the different characterizations in Deﬁnition 2.10.

PROOFOF LEMMA 2.11. First, we claim that uniqueness in law follows from Prop- ∞ Rd erty (i) of Definition 2.10. Indeed, we have for 0 s t and ϕ Cc ( ), ϕ 0 that −hµ(t),ϕi −hµ(s),U ϕi ≤ ≤ ∈ ≥ E e Fs = e t−s . For s = 0 we can use the Laplace transform and the lin- earity of ϕ µ(t), ϕ to deduce that the law of ( µ(t), ϕ1 ,..., µ(t), ϕn ) is uniquely 7→ h i h i ∞h Rd i determined by (i) whenever ϕ1,...,ϕn are positive functions in Cc ( ). By a monotone class argument (cf. [DMS93, Lemma 3.2.5]) the law of µ(t) is unique. We then see induc- tively that the finite-dimensional distributions of µ = µ(t) t≥0 are unique, and thus that the law of µ is unique. { } It remains to show the implication (i) (ii) to conclude the proof of the equivalence of the characterizations in Definition 2.10⇒. But we showed in Lemma 4.4 that there exists a process satisfying (ii), and in Lemma 4.6 we showed that then it must also satisfy (i). And since we just saw that there is uniqueness in law for processes satisfying (i) and since Property (ii) only depends on the law and it holds for one process satisfying (i), it must hold for all processes satisfying (i). (Strictly speaking Lemma 4.4 only gives the existence for κ = 2ν (0, 1], but see Section 4.2 below for general κ.) ∈ n Now the convergence of the sequence µ N is an easy consequence: { }n∈ PROOFOF THEOREM 2.12. This follows from the characterization of the limit points from Lemma 4.4 together with the uniqueness result from Lemma 2.11.

imsart-aop ver. 2020/08/06 file: rough_superbrownian_motion.tex date: September 18, 2020 24

4.2. Mixing with a classical Superprocess. In Section 4.1 we constructed the rSBM of 1 parameter κ = 2ν, for ν deﬁned via Assumption 2.1 which leads to the restriction ν (0, 2 ]. This section is devoted to constructing the rSBM for arbitrary κ> 0.Wedosobymeansofan∈ interpolation between the rSBM and a Dawson-Watanabe superprocess (cf. [Eth00, Chapter k 1]). Let Ψ be the generating function of a discrete ﬁnite positive measure Ψ(s)= k≥0 pks n E and ξp a controlled random environment associated to a parameter ν = [Φ+]. We consider the quenched generator: P

p L n,ω (F )(η)= η ∆nF (η) + (ξn ) (ωp,x)d1 F (η) ψ x · p,e + x x∈Zd Xn n p −1 ̺ (k−1) + (ξp,e)−(ω ,x)dx F (η)+ n pkdx F (η) Xk≥0 with the notation dkF (η) = F (ηx;k) F (η), where for k 1 we write ηx;k(y) = x − ≥ − (η(y)+k1{x}(y))+.

ASSUMPTION 4.7 (On the Moment generating function). We assume that Ψ′(1) = 1 (critical branching, i.e. the expected number of offsprings in one branching/killing event is 1) and we write σ2 =Ψ′′(1) for the variance of the offspring distribution.

Now we introduce the associated process. The construction of the process u¯n is analogous to the case without Ψ, which is treated in Appendix A.

DEFINITION 4.8. Let ̺ d/2 and let Ψ be a moment generating function satisfying ≥ n the previous assumptions. Consider a controlled random environment ξp associated to a 1 Pn Pp ⋉Pn,ωp p D parameter ν (0, 2 ]. Let = be the measure on Ω ([0, + ); E) such that for fixed ωp ∈Ωp, under the measure Pn,ωp the canonical process× on D([0∞, + ); E) is the ∈ n p n ̺ ∞ L ωp,n Markov process u¯p (ω , ) started in u¯p (0) = n 1{0}(x) associated to the generator Ψ n · ⌊ ⌋ defined as above. To u¯p we associate the measure valued process µ¯n(ωp,t), ϕ = u¯n(ωp,t,x)ϕ(x) n̺ −1 h p i p ⌊ ⌋ x∈Zd Xn Zd R n p D M Rd for any bounded ϕ: n . With this definition µ¯p takes values in Ω ([0, T ]; ( )) with the law induced by→Pn. ×

∞ Zd R REMARK 4.9. As in Remark 4.1 we see that for ϕ L ( n, e(l)) with l the process M¯ n,ϕ(s) :=µ ¯n(s)(T n ϕ) T nϕ(0) is a martingale with∈ predictable quadratic∈ variation: t t−s − t s M¯ n,ϕ = dr µ¯n(r) n−̺ nT n ϕ 2 + (n−̺ ξn +σ2)(T n ϕ)2 . h t is |∇ t−r | | e | t−r Z0 In viewof this Remark, we can follow the discussionof Section 4.1 to deduce the following result (cf. Corollary 2.15).

PROPOSITION 4.10. The sequence of measures Pn as in Deﬁnition 4.8 converge weakly as measures on Ωp D([0, T ]; M (Rd)) to the measure Pp Pωp associated to a rSBM of × 2 × parameter κ = 1 d 2ν+σ , in the sense of Theorem 2.12 and Corollary 2.15. In short, we {̺= 2 } write µn µ . p → p In particular the rSBM is also the scaling limit of critical branching random walks whose branching rates are perturbed by small random potentials. imsart-aop ver. 2020/08/06 file: rough_superbrownian_motion.tex date: September 18, 2020 ROUGH SUPER-BM 25

5. Properties of the Rough Super-Brownian Motion.

5.1. Scaling Limit as SPDE in d=1. In this section we characterize the rSBM in dimension d = 1 as the solution to the SPDE (8) in the sense of Deﬁnition 2.17. For that purpose we ﬁrst show that the random measure µp admits a density with respect to the Lebesgue measure.

LEMMA 5.1. Let µ be a one-dimensional rSBM of parameter ν. For any β< 1/2, p [1, 2/(β+1)) and l R, we have: ∈ ∈ E p µ p β R < . k kL ([0,T ];B2,2( ,e(l))) ∞ ∞ R PROOF. Let t > 0 and ϕ Cc ( ). By Point (ii) of Deﬁnition 2.10 the process ϕ ∈ Mt (s)= µ(s), Tt−sϕ µ(0), Ttϕ , s [0,t], is a continuous square-integrable martin- h i − h ϕ i s∈ 2 gale with quadratic variation Mt s = 0 µ(r), (Tt−rϕ) . Using the moment estimates of Lemma C.1, which by Fatou’sh lemmai alsoh hold for the limiti µ of the µn , this martingale property extends to ϕ C ϑ(R, e(k)) forR arbitrary k R and ϑ> 0. In{ particular,} for such ϕ we get ∈ ∈ t E[ µ(t), ϕ 2] . T ((T ϕ)2)(0) dr + (T ϕ)2(0). h i r t−r t Z0 E 2 2jβ E 2 −2l|x|σ Now note that µ(t) β = j 2 [ µ(t), Kj (x ) ]e dx, so we apply k kB2,2 (e(l)) h − · i this estimate with ϕ = K ( x): j ·− P R t (21) E[ µ(t), K (x ) 2] . T ((T K (x ))2)(0)dr+(T K (x ))2(0). h j −· i r t−r j −· t j −· Z0 jα −k|x|σ We start by proving that C α R . for any . Indeed, using that Kj (x ) 1 ( ,e(k)) 2 e k> 0 k − · k (i−j)d i−j Ki is an even function and writing K˜i−j = 2 K0(2 ) K0 if i, j 0 and appropriately adapted if i = 1 or j = 1, we have: · ∗ ≥ − − −k|y|σ ∆i(Kj(x ))e(k) L1 (R) = 1{|i−j|≤1} Ki Kj(x y) e dy k − · k Rd | ∗ − | Z ˜ −k|x−2−j y|σ = 1{|i−j|≤1} Ki−j(y) e dy R | | Z ˜ k|2−j y|σ −k|x|σ −k|x|σ . 1{|i−j|≤1} Ki−j(y) e dy . 1{|i−j|≤1}e , R | | Z −2k|y|σ −jσ σ where in the last step we used that K˜i−j(y) . e and 2 2 < 2. Now, for ζ< 0 satisfying the assumptions| | of Proposition 3.1 and≤ for p [1, ] and sufﬁ- ciently small ε> 0: ∈ ∞

TsKj(x ) C ε(R,e(k+s)) . TsKj(x ) 1− 1 +ε p C p R k − · k k − · k 1 ( ,e(k+s)) jζ (ζ−1+ 1 −2ε)/2 −k|x|σ . 2 s p e . To control the ﬁrst term on the right hand side of (21), we apply this with p = 2 and obtain for t [0, T ] and ζ> 1/2 ∈ − t T ((T K (x ))2)(0) dr r t−r j −· Z0 imsart-aop ver. 2020/08/06 file: rough_superbrownian_motion.tex date: September 18, 2020 26

t 2 . Tr((Tt−rKj(x )) ) C ε (R,e(2k+T )) dr k −· k ∞ Z0 t 2 . Tr((Tt−rKj(x )) ) C 1+ε(R,e(2k+T ))dr k −· k 1 Z0 t − 1+2ε 2 . r 2 (Tt−rKj(x )) C ε(R,e(2k)) dr k −· k 1 Z0 t − 1+2ε 2 . r 2 Tt−rKj(x ) C ε(R,e(k))dr k −· k 2 Z0 t ζ− 1 −2ε − 1+2ε jζ 2 −k|x|σ 2 . r 2 (2 (t r) 2 e ) dr − Z0 2jζ −2k|x|σ 1− 1+2ε +ζ− 1 −2ε 2jζ −2k|x|σ ζ−3ε 2 e t 2 2 = 2 e t , ≃ t −α −β 1−α−β where we used that 0 r (t r) dr t for α,β < 1. The second term on the right hand side of (21) is bounded− by ≃ R 2 2 (TtKj(x )) (0) . (TtKj(x )) C ε (R,e(2k+2T )) − · k − · k ∞ 2 2jζ ζ−1−2ε −2k|x|σ . TtKj(x ) C ε (R,e(k+T )) . 2 t e . k − · k ∞ Note that this estimate is much worse than the first one (because t [0, T ] is bounded above). We plug both those estimates into (21) and set ζ = β ε and∈ k> l to obtain 2 −β−1−3ε − − − E µ(t) β . t for β< 1/2 and for l R. So finally for p [1, 2) k kB2,2 (e(l)) ∈ ∈ T T p p (−β−1−3ε) p E E 2 µ p β R = µ(t) β dt . t dt, k kL ([0,T ];B2,2( ,e(l))) k kB2,2 (e(l)) Z0 Z0 p and now it suffices to note that there exists ε> 0 with ( β 1 3ε) 2 > 1 if and only if p< 2/(β + 1). − − − −

COROLLARY 5.2. In the setting of Proposition 5.1 we have almost surely √µ L2([0, T ]; L2(R, e(l))) for all T > 0 and l R. ∈ ∈ PROOFOF THEOREM 2.18. We follow the approach of Konno and Shiga [KS88]. Ap- plying Corollary 2.15 for κ (0, 1/2] or Proposition 4.10 for κ> 1/2, we obtain an SBM in ∈ p p ωp static random environment µp, which is a process on (Ω D([0, T ]; M (R)), F , P ⋉P ), with F being the product sigma algebra. Enlarging the probability× space, we can moreover assume that the process is defined on (Ωp Ω¯, F p F¯, Pp ⋉ P¯ωp ) such that the probability space (Ω¯, F¯, P¯) supports a space-time white× noise⊗ξ¯ which is independent of ξ. More precisely, we are given a map ξ :Ωp Ω S ′(Rd [0, T ]) which has the law of space-time white noise and does not depend on×Ωp→, i.e. ξ(ωp,×ω)= ξ(ω). For ωp Ωp let F ωp be the usual augmentation of the (random) filtration ∈ { t }t∈[0,T ] generated by µ(ωp, ) and ξ¯. For almost all ωp Ωp the collection of martingales t ϕ p · ∈ 7→ L (ω ,t) for t [0, T ], ϕ DH ωp defines a (random) worthy orthogonal martingale measure M(ωp, dt,∈dx) in the∈ sense of [Wal86], with quadratic variation Q(A B [s,t]) = t R × × s µ(r)(A B)dr for all Borel sets A, B (first we define Q(ϕ ψ [s,t]) = t ∩ ⊂ × × µ(r), ϕψ dr for ϕ, ψ DH ωp , then we use Lemma 5.1 with p = 1 and β (0, 1/2) Rs toh extend thei quadratic variation∈ and the martingales to indicator functions of Borel∈ sets). We R imsart-aop ver. 2020/08/06 file: rough_superbrownian_motion.tex date: September 18, 2020 ROUGH SUPER-BM 27 can thus build a space-time white noise ξ˜ by defining for ϕ L2([0, T ] R): ∈ × p ˜ p M(ω , ds, dx)ϕ(s,x) ξ(ω , ds, dx)ϕ(s,x) := 1{µ(ωp ,s,x)>0} R R µ(ωp,s,x) Z[0,T ]× Z[0,T ]× ¯ p + ξ(ds, dx)ϕ(s,x)1{µ(ωp ,s,x)=0}. R Z[0,T ]× By taking conditional expectations with respect to ξp we see that ξ˜ and ξp are independent, and by definition the SBM in static random environment solves the SPDE (8). Conversely, it is straightforward to see that any solution to the SPDE is a SBM in static random environment of parameter ν = κ/2. Uniqueness in law of the latter then implies uniqueness in law of the solution to the SPDE.

5.2. Persistence. In this section we study the persistence of the SBM in static random environment µp and we prove Theorem 2.20, i.e. that µp is super-exponentially persistent. For the proof we rely on the related work [Ros20] which constructs, for integer L> 0, a L killed SBM in static random environment µp , in which particles are killed once they leave d L L the box ( L/2, L/2) . The processes µp are coupled with µp so that almost surely µp µp for all L.− In particular, the following result holds. ≤

n LEMMA 5.3. Let µ¯ be an rSBM associated to a random environment ξ N satisfying p { p }n∈ Assumption 2.1. Thereexists a probability spaceof the form (Ωp D([0, + ); M (Rd)), F p, Pp ⋉ Pωp × ∞p ) supporting a rSBM µp such that µp = µp in distribution. Moreover Ω supports a spatial white noise ξ and there exists a null set N Ωp such that: p 0 ⊆ c N 1. For all ω N0 and L 2 the random Anderson Hamiltonian associated to ξp with ∈ ∈ d ωp Dirichlet boundary conditions on ( L/2, L/2) , H , on the domain DH ωp is well de- d,L d,L − ϑ d fined (cf. [CvZ19]). Moreover, DH ωp C (( L/2, L/2) ) for any ϑ< 2 d/2. Finally d,L ⊆ − − p p H ω the operator has discrete spectrum. If λ(ω ,L) 0 is the largest eigenvalue of d,L, then ≥ L L d the associated eigenfunction eλ(ωp,L) satisfies eλ(ωp,L)(x) > 0 for all x ( 2 , 2 ) . L ∈ −d 2. There exist random variables µ N with values in D([0, ); M (R )) satisfying { p }L∈2 ∞ µL(ωp,t) µL+2(ωp,t) µ (ωp,t) and µL(0) = δ . Moreover, for all ω N c p ≤ p ≤···≤ p p 0 ∈ 0 and ϕ DH ωp : ∈ d,L t ϕ p L p ωp K (ω ,t)= µ (t), ϕ µ (ω , 0), ϕ dr µ(r), Hd ϕ , t 0 L h p i−h p i− h ,L i ≥ Z0 is a continuous centered martingale (w.r.t. the filtration generated by µL(ωp, )) with p · quadratic variation Kϕ = 2ν t dr µ(r), ϕ2 . h L it 0 h i PROOF. For the first point see [CvZ19R ] and [Ros20, Lemma 2.4]. The second statement is proved in [Ros20, Corollary 3.9]

d Analogously to the previous section we denote with t Tt the semigroup associated to H ωp p 7→ d,L for some ﬁxed L,ω which will be clear from the context. Now we shall prove that ∞ Rd p p given a nonzero positive ϕ Cc ( ) and λ> 0, for almost all ω there exists L = L(ω ) with ∈ Pωp −tλ L p (22) lim e µp (ω , t, ), ϕ = > 0. t→∞ h · i ∞ This implies Theorem 2.20. imsart-aop ver. 2020/08/06 file: rough_superbrownian_motion.tex date: September 18, 2020 28

L The reason for working with µp is that the spectrum of the Anderson Hamiltonian on ( L/2, L/2)d is discrete, and its largest eigenvalue almost surely becomes bigger than λ for L− . Given this information, (22) follows from a simple martingale convergence argument,→∞ see Corollary 5.6 below.

REMARK 5.4. For simplicity we only treat the case of (killed) rSBM with parameter ν (0, 1/2]. For ν> 1/2 we need to use the constructions of Section 4.2, after which we can follow∈ the same arguments to show persistence.

p H ωp Let us write λ(ω ,L) for the largest eigenvalue of the Anderson Hamiltonian d,L with Dirichlet boundary conditions on ( L/2, L/2)d. − LEMMA 5.5. There exist c , c > 0 such that for almost all ωp Ωp: 1 2 ∈ (i) In d = 1 (by [Che14, Lemmata 2.3 and 4.1]): λ(ωp,L) lim = c1. L→+∞ log(L)2/3 (ii) In d = 2 (by [CvZ19, Theorem 10.1]): λ(ωp,L) lim = c2. L→+∞ log(L)

COROLLARY 5.6. Let d 2 and λ> 0 and let µp be an SBM in static random en- ≤ N L vironment, coupled for all L 2 to a killed SBM in static random environment µp on L L d L ∈ p p [ 2 , 2 ] with µp µp (as described in Lemma 5.3). For almost all ω Ω a there exists an − p ≤ p L p ∈ L0(ω ) > 0 such that for all L L0(ω ) the killed SBM µp (ω , ) satisfies (22). In particular, for almost all ωp Ωp the process≥ µ (ωp, ) is super-exponentially· persistent. ∈ p · p p p PROOF. In view of Lemma 5.5, for almost all ω Ω we can choose L0(ω ) such that the ∈p p largest eigenvalue of the Anderson Hamiltonian λ(ω ,L) is bigger than λ for all L L0(ω ). Now we fix ωp such that the above holds true and thus drop the index p (i.e.: we≥ will use p a purely deterministic argument). We also fix some L L0(ω ) and write λ1 instead of p ≥ λ(ω ,L) for the largest eigenvalue. Finally, let e1 be the strictly positive eigenfunction with e1 L2((− L , L )d) = 1 associated to λ1. By Lemma 5.3 we find for 0 s 0, and therefore ∈ − d d T ((e−λ1re )2)(0) e e−λ1rT (e−λ1re )(0) r 1 ≤ k 1k∞ r 1 = e e−λ1re (0) . e−λ1r. k 1k∞ 1 It follows that E(t) converges almost surely and in L2 to a random variable E( ) 0 as t , and since E[E( )] = E(0) = e (0) > 0 we know that E( ) is strictly positive∞ ≥ with →∞ ∞ 1 ∞ imsart-aop ver. 2020/08/06 file: rough_superbrownian_motion.tex date: September 18, 2020 ROUGH SUPER-BM 29 positive probability. For ϕ 0 nonzero with support in [ L/2, L/2]d we show in Lemma 5.7 that: ≥ −

p (23) e−λ1t µL(t), ϕ e , ϕ E( ), as t , in L2(Pω ) h i → h 1 i ∞ →∞ so that we get from the strict positivity of e1 and from the fact that λ1 > λ P lim e−λt µL(t), ϕ = P(E( ) > 0) > 0. t→∞ h i ∞ ≥ ∞ This completes the proof.

ϑ LEMMA 5.7. In the setting of Corollary 5.6, let ϕ Cd and let ψ = ϕ e , ϕ e . Then ∈ − h 1 i 1 p Eω −λ1t L p 2 (24) lim e µp (ω ,t), ψ = 0. t→∞ | h i| h i PROOF. As before we omit the subscript p from the notation, as well as the dependence on the realization ωp of the noise. Using the martingale µL(s), T d ψ , we get h t−s i t (25) E µL(t), ψ 2 . T d(ψ) 2(0) + dr T d (T d ψ)2 (0). |h i| | t | r t−r Z0 h i Let λ2 < λ1 be the second eigenvalue of the Anderson Hamiltonian (the strict inequality is a consequence of the Krein-Rutman theorem, cf. [Ros20, Lemma 2.4]). The main idea is to leverage that:

d λ2t T ψ 2 e ψ 2 , k t kL ≤ k kL since ψ is orthogonal to the first eigenfunction. The only subtlety is that of course the value of a function in 0 is not controlled by its L2 norm. To go from L2 to a space of continuous functions, we use that for all ϑ as in Equation (12) and sufficiently close to 1: d d d T f C ϑ . T f ϑ− d . T f C ϑ 1 d 2/3 2 2/3 d,2 k k k kCd k k d d . T f ϑ− d . T f C ϑ . f L2 , 1/3 C 2 1/3 2 k k d,2 k k k k in view of the regularizing properties of the semigroup T d (which hold with the same parameters as in Proposition 3.1, cf. [Ros20, Theorem 2.3], see also the same article for the definition of Besov spaces with Dirichlet boundary conditions, for all current purposes identical to the classical spaces) and by Besov embedding theorems. Let us consider the second term in (25), for t 2. With the previous estimates, we bound it as follows: ≥ 1 t d d 2 d d 2 dr Tr (Tt−rψ) (0) + dr Tr (Tt−rψ) (0) Z Z 0 1 1 t d 2 d d 2 . dr T ψ C ϑ + dr T (T ψ) L2 k t−r k d k r−1 t−r k Z0 Z1 1 t d 2 λ1(r−1) d 2 . dr Tt−r−1ψ L2 + dr e (Tt−rψ) L2 k k 1 k k Z0 Z imsart-aop ver. 2020/08/06 file: rough_superbrownian_motion.tex date: September 18, 2020 30

1 t−1 t

d 2 λ1(r−1) d 2 λ1(r−1) 2 . dr T ψ 2 + dr e T ψ 2 + dr e ψ C ϑ k t−r−1 kL k t−r−1 kL k k d Z0 Z1 t−Z1 t . dr e2λ2(t−r)+λ1 r . e2λ2 t(1 + e(λ1 −2λ2)t + t) . (e2λ2 t + eλ1t)(1 + t), Z0 where we used that for any λ R one can bound t eλs ds 1 (1 + eλt + t). Plugging this ∈ 0 ≤ |λ| estimate into (25), we obtain R E e−λ1t µL(t), ψ 2 . e−2λ1te2λ2(t−1) + e−2λ1t e2λ2t + eλ1t (1 + t) | h i| h i . e−λ1t + e−2(λ1−λ2)t(1 +t). This proves (24).

REMARK 5.8. The connection of extinction or persistence of a branching particle system to the largest eigenvalue of the associated Hamiltonian is reminiscent of conditions appearing in the theory of multi-type Galton-Watson processes: See for example [Har02, Section 2.7]. The martingale argument in our proof can be traced back at least to Everett and Ulam, as explained in [Har51, Theorem 7b].

APPENDIX A: CONSTRUCTION OF THE MARKOV PROCESS This section is dedicated to a rigorous construction of the BRWRE. For simplicity and NZd without loss of generality we will work with n = 1. Since the space 0 is harder to deal with NZd Zd and we do not need it, we considerthe countable subspace E = 0 0 of functions η : d ′ → N0 with η(x) = 0, except for ﬁnitely many x Z . We endow E with the distance d(η, η )= ′ ∈ x∈Zd η(x) η (x) , under which E is a discrete and hence locally compactseparable metric space. Recall| − the notations| from Section 2. Below we will construct “semidirect product measures”P of the form Pp ⋉Pωp on Ωp D([0, + ); R), by which we mean that there exists a Markov kernel κ such that for A F×p,B B∞(D([0, + ); R)): ⊂ ⊂ ∞ p (26) Pp ⋉Pω (A B)= κ(ωp,B)dPp(ωp) × ZA p p p LEMMA A.1. Assume that for any ω Ω the potential ξp(ω ) is uniformly bounded ∈ p and consider π E. There exists a unique probability measure Pπ on Ω=Ω D([0, + ); E) ∈ P Pp ⋉Pω×p Pω∞p endowed with the product sigma algebra, such that π is of the form π , with π being the unique measure on D([0, + ); E) under which the canonical process u is a Markov jump process with u(0) = π whose∞ generator is given by L ωp : D(L ωp ) C (E), with → b p L ω (F )(η)

= η ∆ F (η) + (ξ ) (ωp,x)d+F (η) + (ξ ) (ωp,x)d−F (η) , x · x p + x p − x Zd xX∈ h i where the domain D(L ωp ) is the set of functions F C (E) such that the right-hand side ∈ b lies in Cb(E).

imsart-aop ver. 2020/08/06 file: rough_superbrownian_motion.tex date: September 18, 2020 ROUGH SUPER-BM 31

PROOF. The construction for fixed ωp Ωp is classical. Indeed, the generator has the ∈ p form of [EK86, (4.2.1)], with λ(η)= x∈Zd ηx(2d+ ξp (ω ,x)), and we only need to rule | 1 | out explosions by verifying that almost surely N =+ , where Y is the associated P k∈ λ(Yk ) ∞ discrete time Markov chain. This is the case, since ξ is bounded and thus P p 1 1 1 & =+ N λ(Yk) N x Yk(x) ≥ N c+k ∞ Xk∈ Xk∈ Xk∈ P L ωp with c = x π(x). It follows via classical calculations that is the generator associated to the process u. This allows us to define for fixed ωp the law κ(ωp, ) of our process on P · D([0, + ); E). To construct the measure Pπ we have to show that κ is a Markov kernel, which amounts∞ to proving measurability in the ωp coordinate. But κ depends continuously on ξp, which we can verify by coupling the processes for ξp and ξ˜p through a construction based on Poisson jumps at rate K> ξ , ξ˜ and then rejecting the jumps if an independent k pk∞ k pk∞ uniform [0, K] variable is not in [0, ξp(x) ] respectively in [0, ξ˜p(x) ]. Since ξp is measurable in ωp, also κ is measurable in ωp. | | | |

Next, we extend the construction to potentials of sub-polynomial growth:

p ∞ Zd p p LEMMA A.2. Let ξp(ω ) a>0 L ( ,p(a)) for all ω Ω and consider π E. ∈ P Pp ⋉Pωp ∈ p D ∈ There exists a unique probability measure π = π on Ω=Ω ([0, + ); E) en- T Pωp × D ∞ dowed with the product sigma algebra, where π is the unique measure on ([0, + ); E) under which the canonical process u is a Markov jump process with u(0) = π and∞ with generator L ωp and D(L ωp ) deﬁned as in the previous lemma.

PROOF. Let us fix ωp Ωp. Consider the Markov jump processes uk started in π with L ωp,k ∈ k generator associated to ξp (x) = (ξp(x) k) ( k) whose existence follows from k ∧ ∨ − the previous result. The sequence u k∈N is tight (this follows as in Lemma 4.2 and Corol- lary 4.3, keeping n fixed but letting{ k}vary) and converges weakly to a Markov process u. Indeed, for k, R N let τ k be the first time with supp(uk(τ k )) Q(R), where Q(R) is the ∈ R R 6⊂ square of radius R around the origin, and let τR be the corresponding exit time for u. Then we get for all k> max ξ (x) , for all T > 0, and all F C (D([0, T ]; E)): x∈Q(R) | p | ∈ b ωp k ωp E [F ((u (t)) )1 k ]= E [F ((u(t)) )1 ], π t∈[0,T ] {τR ≤T } π t∈[0,T ] {τR ≤T } where we used that the exit time τR is continuous because E is a discrete space. Moreover, k from the tightness of u k∈N it follows that for all ε> 0 and T > 0 there exists R N with P k { } ∈ supk (τR T ) < ε. This proves the uniqueness in law and that u is the limit (rather than ≤ k subsequential limit) of u k∈N. Similarly we get the Markov property of u from the Markov k { } property of the u k∈N and from the convergence of the transition functions. It remains to{ verify} that L ωp is the generator of u. But for large enough R we have Pωp 2 + π (τR h)= O(h ) as h 0 , because on the event τR h at least two transitions must have≤ happened (recall→ that π is compactly supported).{ We≤ can} thus compute for any F C (E): ∈ b Eωp Eωp k 2 π F (u(h)) = π F (u (h)) + O(h ). The result on the generator then follows from the previous lemma. As before, we now have constructed a collection of probability measures κ(ωp, ) as the limit of the Markov ker- nels κk(ωp, ). Since measurability is preserved when passing· to the limit, this concludes the proof. · imsart-aop ver. 2020/08/06 file: rough_superbrownian_motion.tex date: September 18, 2020 32

APPENDIX B: SOME ESTIMATES FOR THE RANDOM NOISE In this section we prove parts of Lemma 2.4, i.e. that a random environment satisfying Assumption 2.1 gives rise to a deterministic environment satisfying Assumption 2.3.

LEMMA B.1. Let a, ε, q > 0 and b > d/2. Under Assumption 2.1 we have

E −d/2 n q E −d/2 n 2 sup n (ξp )+ C −ε(Zd ,p(a)) + n (ξp )+ L2(Zd ,p(b)) < + , k k n k k n ∞ n and the same holds if we replace (ξn) with ξn . Furthermore, for ν = E[Φ ], the following p + | p | + convergences hold true in distribution in C −ε(Rd,p(a)): E nn−d/2(ξn) ν, E nn−d/2 ξn 2ν. p + −→ | p | −→ n n PROOF. We prove the result only for (ξp )+, since then we can treat (ξp )− by con- n sidering ξp ( Φ is still a centered random variable). Now note that we can rewrite E −d/2− n −q [ n (ξp )+ Lq (Zd ,p(a))] as k k n −dE −d/2 n q q E q −aq n [ n (ξp )+ (x)] p(a)(x) . [ Φ ] (1 + y ) dy, | | | | | | Rd | | x∈Zd Xn Z which is ﬁnite whenever aq > d. From here the uniform bound on the expectations follows by Besov embedding. Convergence to ν is then a consequenceof the spatial independenceof the noise ξn, since it E E n n −d is easy to see that (ξp )+ ν, ϕ = O(n ) for all ϕ with compactly supported Fourier transform. h − i The following result is a simpler variant of [MP19, Lemma 5.5] for the case d = 1, hence we omit the proof.

LEMMA B.2. Fix ξn satisfying Assumption 2.1, d = 1, a, q > 0 and α< 2 d/2. We have: − E n q E n n sup ξp C α−2(Zd ,p(a)) < + , ξp ξp, n k k n ∞ → α−2 d where ξp is a white noise on R and the convergence holds in distribution in C (R ,p(a)).

APPENDIX C: MOMENT ESTIMATES n Here we derive uniform bounds for the moments of the processes µ n∈N. As a conven- tion, in the following we will write E and P for the expectation and the{ probability} under the distribution of un. For different initial conditions η E we will write E , P . ∈ η η n LEMMA C.1. Fix q,T > 0. For all n N, consider the process µ (t) t≥0 as in Def- n d ∈ n n { }2 d inition 2.6. Consider then Z R with , Zd with C R for ϕ : n ϕ 0 ϕ = ϕ n ϕ ( , e(l)) some l R. Then → ≥ | ∈ ∈ sup sup E µn(t)(ϕn) q < + . n t∈[0,T ] | | ∞ R n If for all ε> 0 there exists an l such that supn ϕ C −ε(Rd,e(l)) < + , we can bound for all γ (0, 1): ∈ k k ∞ ∈ sup sup tγE µn(t)(ϕn) q < + . n t∈[0,T ] | | ∞ imsart-aop ver. 2020/08/06 file: rough_superbrownian_motion.tex date: September 18, 2020 ROUGH SUPER-BM 33

PROOF. We prove the second estimate, since the ﬁrst estimate is similar but easier n (Lemma D.1 below controls ϕ C ϑ Zd for all ϑ< 2 in that case). Also, we assume with- ( n,e(l)) out loss of generality that q k2. Ask usual,we use the conventionof freely increasing the value ≥ E n n n n of l in the exponential weight. Let us start by recalling that µ (t)(ϕ ) = Tt ϕ (0). More- over, via the assumption on the regularity, Proposition 3.1 and Equation (3) from Lemma 1.2 guarantees that for any γ (0, 1) there exists a δ = δ(γ,q) >0 such that ∈ n n sup t Tt ϕ L γ/q,δ (Zd ,e(l)) < + . n k 7→ k n ∞ By the triangle inequality it thus sufﬁces to prove that for any γ> 0: γE n n n n q sup sup t µ (t)(ϕ ) Tt ϕ (0) < + . n t∈[0,T ] | − | ∞ Note that we can interpret the particle system un as the superposition of n̺ independent n n ⌊ ⌋ n particle systems, each started with one particle in zero; we write u = u + + u ̺ . To 1 · · · ⌊n ⌋ lighten the notation we assume that n̺ N. We then apply Rosenthal’s inequality, [Pet95, ∈ Theorem 2.9] (recall that q 2) and obtain (with (f, g)= x∈Zd f(x)g(x)): ≥ n n̺ P q E µn(t)(ϕn) T nϕn(0) q = E n−̺(un(t), ϕn) n−̺T nϕn(0) | − t | k − t k=1 X n̺

. n−̺q E (un(t), ϕn) T nϕn(0) q | k − t | k=1 X n̺ q 2 + n−̺q E (un(t), ϕn) T nϕn(0) 2 | k − t | k=1 X . n−̺(q−1)E (un(t), ϕn) q + n−̺E (un(t), ϕn) 2 q/2 | 1 | | 1 | ̺q q − 2 −γ n n + n t t Tt ϕ L γ/q,δ (Zd ,e(l)) k 7→ k n for the same δ> 0 and l R as above. The two scaled expectations are of the same form, in the second term we simply∈ have q = 2. To control them, we deﬁne for p N the map ∈ p,n ̺(1−p) n n p m n (t,x)= n E (u (t), ϕ ) . ϕ 1{x} | 1 | p,n As a consequence of Kolmogorov’s backward equation each mϕn solves the discrete PDE (see also Equation (2.4) in [ABMY00]): p−1 p,n n p,n −̺ n p i,n p−i,n ∂ m n (t,x)= H m n (t,x)+ n (ξ ) (x) m n (t,x)m n (t,x), t ϕ ϕ e + i ϕ ϕ Xi=1 p,n ̺(1−p) n p with initial condition mϕn (0,x)= n ϕ (x) . We claim that this equation has a unique p,n | | (paracontrolled in d = 2) solution mϕn , such that for all γ> 0 there exists δ = δ(γ,p) > 0 n,p with sup m n L γ,δ Zd < . Once this is shown, the proof is complete. We proceed n ϕ ( n,e(l)) k k ∞ n,1 n n by induction over p. For p = 1 we simply have mϕn (t,x)= Tt ϕ (x). For p 2 we use that ̺(1−p) n p ≥ by Lemma D.2 we have n ϕ (x) C κ Zd 0 for some κ> 0 and we assume ( n,e(l)) that the induction hypothesisk holds| for all| kp′ 0, we may assume also that κ > γ. We choose then γ′ < γ such that for some δ(γ′,p) > 0: p−1 i,n p−i,n sup mϕn mϕn < + . n M γ′ C δ(γ′ ,p) Zd ∞ i=1 ( n,e(l)) X imsart-aop ver. 2020/08/06 file: rough_superbrownian_mo tion.tex date: September 18, 2020 34

−̺ n Since by Assumption 2.3 n (ξ ) C −ε Zd is uniformly bounded in n for all ε,a> 0, e + ( n,p(a)) the above bound is sufﬁcientk to controlk the product: p−1 −̺ n i,n p−i,n sup n (ξe )+ mϕn mϕn < + . n M γ′ C −ε Zd ∞ i=1 ( n,e(l)) X n,p Now the claimed bound for mϕn follows from an application of Proposition 3.1. For non- integer q we simply use interpolation between the bounds for p

α d α d LEMMA D.1. Let ϕ C (R ) for α R>0 N. Then ϕ Zd C (Z ) and ∈ ∈ \ | n ∈ n sup ϕ Zd C α Zd . ϕ C α Rd . n ( n) ( ) n∈N k | k k k n β d For the extension of ϕ Zd we have E (ϕ Zd ) ϕ in C (R ) for all β < α. | n | n → n n n PROOF. Letuscall Zd We have to estimate ∞ Zd , and for that purpose ϕ = ϕ n . ∆j ϕ L ( ) | k k n n n we consider the cases j < jn and j = jn separately. In the ﬁrst case we have ∆j ϕ (x)= Zd d Kj ϕ(x)=∆jϕ(x) for x n because, as supp(̺j) n( 1/2, 1/2) , the discrete and the continuous∗ convolutions coincide.∈ Therefore: ⊂ − n jα ∆ ϕ L∞(Zd ) ∆jϕ L∞(Rd) 2 ϕ C α . k j k n ≤ k k ≤ k k For j = j we have ̺n ( ) = 1 χ(2−jn ), where χ S is one of the two functions n jn ω generating the dyadic partition· of− unity, a· symmetric∈ smooth function such that χ = 1 in a ball around the origin. By construction we have ̺n (x) 1 for x near the bound- jn d −j ≡ d ary of n( 1/2, 1/2) , and therefore supp(χ(2 n )) n( 1/2, 1/2) . Let us deﬁne ψn = −1 −−jn −1 −jn · ⊂ − F χ(2 )= F d χ(2 ). Then n · R · n−dψ (x)= F ψ (0) = χ(2−jn 0) = 1, n n n · x∈Zd Xn and for every monomial M of strictly positive degree we have, since ψn is an even function,

−d −1 −jn n ψ (x)M(x) = (ψ M)(0) = F d (χ(2 )FRd M)(0) = M(0) = 0, n n ∗ R · x∈Zd Xn Zd where we used that the Fourier transform of a polynomial is supported in 0. Thus for x n n n ∈ we get ∆ ϕ (x)= ϕ(x) (ψn n ϕ)(x), that is: jn − ∗ 1 ϕ(x) (ψ ϕ)(x)= ψ ϕ( ) ϕ(x) ∂kϕ(x)( x)k (x), − n ∗n − n ∗n · − − k! ·− 1≤|Xk|≤⌊α⌋ with the usual multi-index notation and where as above we could replace the discrete convo- d α d lution n with a convolution on R . Moreover, since ϕ C (R ) and α> 0 is not an integer, we can∗ estimate ∈

1 k ⊗k α ϕ( ) ∂ ϕ(x)( x) . y ϕ C α(Rd), · − k! ·− ∞ Rd | | k k 0≤|k|≤⌊α⌋ L ( ) X and from here the estimate for the convolution holds by a scaling argument. The convergence then follows by interpolation. imsart-aop ver. 2020/08/06 file: rough_superbrownian_motion.tex date: September 18, 2020 ROUGH SUPER-BM 35

Zd −κ The following result shows that multiplying a function on n by n for some κ> 0 gains regularity and gives convergence to zero under a uniform bound for the norm.

LEMMA D.2. Consider z ̺(ω) and p [1, ], α R and a sequence of functions f n C α(Zd , z) with uniformly∈ bounded norm:∈ ∞ ∈ ∈ p n n sup f C α(Zd ,z) < + . n k k p n ∞ −κ n C α+κ Zd Then for any κ> 0 the sequence n f is bounded in p ( n, z): −κ n n α+κ d . C α Zd sup n f Cp (Z ,z) sup f ( ,z) n k k n n k k p n

−κ n n β d and n E f converges to zero in Cp (R , z) for any β < α + κ.

PROOF. By deﬁnition, we only encounter Littlewood-Paley blocks up to an order jn log (n). Hence 2j(α+κ−ε)n−κ . 2jαn−ε for j j and ε 0, from where the claim follows.≃ 2 ≤ n ≥

C α Zd Rd Now we study the action of discrete gradients. We write p ( n, z; ) for the space of Zd Rd C α Zd maps ϕ: n such that each component lies in p ( n, z) with the naturally induced norm. →

LEMMA D.3 ([MP19], Lemma 3.4). The discrete gradient ( nϕ) (x)= n(ϕ(x+ ei ) ϕ(x)) ∇ i n − for i = 1,...,d (with e the standard basis in Rd) and the discrete Laplacian ∆nϕ(x)= { i}i n2 d (ϕ(x+ ei ) 2ϕ(x)+ϕ(x ei )) satisfy: i=1 n − − n n n P ϕ C α−1(Zd ,z;Rd) . ϕ C α(Zd ,z), ∆ ϕ C α−2(Zd ,z) . ϕ C α(Zd ,z), k∇ k p n k k p n k k p n k k p n for all α R and p [1, ], where both estimates hold uniformly in n N. ∈ ∈ ∞ ∈ PROOF. For ∆n this is shown in [MP19, Lemma 3.4]. The argument for the gradient n is essentially the same but slightly easier. ∇

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