In a recent work, Fleischmann and Mueller (2004) showed the existence of a
super-Brownian motion in R^d, d=2,3, with extra birth at the origin. Their
construction made use of an analytical approach based on the fundamental
solution of the heat equation with a one point potential worked out by
Albeverio et al. (1995). The present note addresses two properties of this
measure-valued process in the three-dimensional case, namely the scaling of the
process and the large scale behavior of its mean

Fleischmann, Klaus

Mueller, Carl

Vogt, Pascal

English

arXiv

Louisiana State University

The large scale behavior of super-Brownian motion in three dimensions with a single point source

In a recent work, Fleischmann and Mueller (2004) showed the existence of a super-Brownian motion in R^d, d=2,3, with extra  birth at the origin. Their construction made use of an analytical approach based on the fundamental solution of the heat equation with a one point potential worked out by Albeverio et al. (1995). The present note addresses two properties of this measure-valued process in the three-dimensional case, namely the scaling of the process and the large scale behavior of its mean

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Weierstraÿ-Institutfür Angewandte Analysis und Stohastikim Forshungsverbund Berlin e.V.Preprint ISSN 0946  8633On the large sale behaviorof superBrownian motion in three dimensionswith a single point soureKlaus Fleishmann1, Carl Mueller2, Pasal Vogt3submitted: July 21, 20061 Weierstrass Institute forApplied Analysis and StohastisMohrenstr. 3910117 BerlinGermanyE-Mail: eishmwias-berlin.de2 Department od MathematisUniversity of RohesterRohester, NY 14627USAE-Mail: mlrmath.rohester.edu3 IFB AGNeumarkt-Galerie50667 KölnGermanyE-Mail: Pasal.VogtifbAG.omNo. 1154Berlin 2006W I A S2000 Mathematis Subjet Classiation. 60J80, 60K35.Key words and phrases. Super-Brownian motion with singular mass reation, expeted mass,Shrödinger equation with one-point-potential .Edited byWeierstraÿ-Institut für Angewandte Analysis und Stohastik (WIAS)Mohrenstraÿe 3910117 BerlinGermanyFax: + 49 30 2044975E-Mail: preprintwias-berlin.deWorld Wide Web: http://www.wias-berlin.de/SINGLE POINT SOURCE SBM 1Abstract. In a recent work, Fleischmann and Mueller (2004) showed the existence of asuper-Brownian motion in Rd, d = 2, 3, with extra birth at the origin. Their constructionmade use of an analytical approach based on the fundamental solution of the heat equationwith a one point potential worked out by Albeverio et al. (1995). The present noteaddresses two properties of this measure-valued process in the three-dimensional case,namely the scaling of the process and the large scale behavior of its mean.1. IntroductionA super-Brownian motion in R with a single point source δ0 was constructed in Engländer & Fleischmann[EF00]. It was shown that its expected mass grows exponentially in time, and is in the mass-rescaled limitdistributed in space as x 7→ e−|x|. In Engländer & Turaev [ET02] it is even proved that the randommeasures themselves grow in law exponentially as time increases, and are otherwise in the mass-rescaledlimit spatially situated with the same shape except an overall random factor. The probabilistic effectbehind the non-trivial existence of the model is the fact that a Brownian particle in R hits the origin withcertainty and that it has there a non-degenerate local time, serving as an additional birth rate for therandom creation of mass.In higher dimensions, a Brownian particle fails to hit the origin, and a local time would degenerate.Nevertheless, Fleischmann & Mueller [FM04] succeeded in constructing a super-Brownian motion in Rd,d = 2, 3, with a single point source. They heavily used well-known analytical facts from mathematicalphysics concerning Laplace operators with one-point-potentials. Heuristically, some additional rescalingenters the regularization of the delta function (serving as single point source). Properties of this newsuper-Brownian motion are not known so far. The purpose of the present note is to get some progress bystudying its scaling and the large scale behavior of its expectation in the three-dimensional case.1.1. The heat equation with one-point-potential. The Schrödinger equation with a one-point-poten-tial is studied in quantum theory to describe singular electromagnetic effects on quantum particles, see e.g.the monograph Albeverio et al. [AGHKH88, Part I]. By analytic continuation, solutions to the Schrödingerequation can be (at least formally) obtained via solutions of the heat equation.Formally, the heat equation with a one-point-potential is given by(1) ∂tu = ∆u + δ(α)0 u =: ∆(α)u,where ∂t denotes the derivative with respect to time, ∆ is the d–dimensional Laplacian, and u : (0,∞) ×Ṙd → R+ is a time-space field, where Ṙd := Rd \ {0} with the Euclidean metric is locally compact. If wedenote by Bε(y) an open ball around y ∈ Rd of radius ε > 0, then having in mind that ε−d1Bε(0) ≈ δ0 ,the operator ∆(α) := ∆ + δ(α)0 is heuristically the limit as ε ↓ 0 of the operator(2) ∆(α)ε := ∆ + h(d, α, ε) ε−d1Bε(0),where h(d, α, ε) is some additional rescaling factor which depends on a parameter α at least. Restrictingto d = 3, the function h can be chosen as(3) h(3, α, ε) := π24 ε − 8π2αε2, α ∈ R, ε > 0,(cf. [AGHKH88, (H.74)]).Physically, α in the case α < 0 is related to the scattering length slα := −(4πα)−1 of the free Laplaceoperator ∆ with respect to the interaction Laplacian ∆(α). Roughly speaking, the scattering length de-scribes the average distance a free particle manages to go before any interaction takes place. So, if α ↓ −∞the scattering length slα ↓ 0 becomes smaller and we expect more interaction. For α ≥ 0 there is no properphysical interpretation of slα as the point spectrum of ∆(α) is empty (see [AGHKH88, Theorem I.1.4]).The fundamental solution pα to the equation(4) ∂tu = ∆(α)u on (0,∞) × Ṙd, d = 2, 3,which provides the basis for the analytical construction of the superprocess in [FM04], have been computedin Albeverio et al. [ABD95]. In d = 3 (the two-dimensional case is analytically more delicate, which is thereason we restrict to d = 3), the one-point-interaction heat kernel pα is given by(5) pαt (x, y) = pt(x, y) +2t|x||y|pt(|x| + |y|)−8παt|x||y|∫ ∞0du e−4παu pt(u + |x| + |y|),2 KLAUS FLEISCHMANN, CARL MUELLER, AND PASCAL VOGTt > 0, x, y 6= 0, where p is the usual free heat kernel defined by,(6) pt(x, y) := (4πt)−d/2 exp(− |y − x|2/4t),and with a slight abuse of notation,(7) pt(r) := (4πt)−d/2 exp(−r2/4t), t > 0, r ≥ 0.Also recall the scaling of the free heat kernel, i.e. for all k, t > 0 and x, y ∈ Rd,(8) pt(x, y) = kd/2pkt(k1/2x, k1/2y).Note, that the last term in (5) is always finite and disappears for α = 0. Moreover, α 7→ pα is pointwisecontinuous and decreasing, and we have the (pointwise) convergences pα ↑ +∞ as α ↓ −∞ (i.e. thefundamental solution explodes which can be interpreted as immediate interaction), whereas pα ↓ p asα ↑ +∞ leads the free case (i.e. the interaction disappears).Rigorously, the family {∆(α) : α ∈ R} of operators are defined as all self-adjoint extensions on theHilbert space L2(Ṙd, dx) of the Laplacian ∆ acting on C∞com(Ṙd), the space of unboundedly differentiablefunctions on Ṙd = Rd \ {0} with compact support (see e.g. [AGHKH88, Chapters I.1 and I.5]). Hence,although the pα differ from the free heat kernel p, they solve the heat equation(9) ∂tpαt (x, y) = ∆pαt (x, y) on (0,∞) × Ṙd,with the Laplacian ∆ acting either on the variable x or y. In particular, (t, x, y) 7→ pαt (x, y) is jointlycontinuous on (0,∞) × Ṙd × Ṙd. Let us denote by Sα the semigroup associated with the kernel pα, i.e.(10) Sαt ϕ(x) :=∫Ṙddy ϕ(y) pαt (x, y).Note that Sα is not a contraction semigroup and so there is no stochastic process generated by this flow.The following Lemma shows that the kernel pα has a similar scaling behavior as the free heat kernel p.Lemma 1 (Scaling of the pα). We have, for all k, t > 0, x, y ∈ Ṙ3, and α ∈ R,(11) pαt (x, y) = k3/2 pk−1/2αkt (k1/2x, k1/2y).Proof. That follows immediately from the definition (5) of the pα and the scaling (8) of the free heat kernelp. 1.2. The flow associated with the one-point-interaction heat kernel. This section is devoted tointroduce a space of functions Φ on which the flow Sα acts as a strongly continuous linear semigroup (see[FM04, Section 2] for details). Let φ denote the weight and reference function(12) φ(x) := |x|−1, x ∈ Ṙ3 = R3 \ {0}.For fixed ̺ ∈ (1, 2), let H = H̺ denote the space of measurable functions ϕ on Ṙ3 for which(13) ‖ϕ‖H :=(∫Ṙ3dxφ(x) |ϕ(x)|̺)1/̺< ∞.Then(H, ‖ · ‖H)is a Banach space, where as usual we do not distinguish between equivalence classes andtheir representatives. Now, let Φ = Φ̺ denote the set of all continuous functions ϕ : Ṙ3 → R such thatϕ ∈ H and(14) 0 ≤ ϕ ≤ C φ for some constant C = Cϕ > 0.We endow Φ with the topology inherited from H. Note that the set C+com = C+com(Ṙ3) of all non-negative,continuous functions on Ṙ3 with compact support is contained in Φ. We remark that ϕ ∈ Φ might have asingularity at x = 0 of order |x|−ξ with 0 < ξ < 1 . The linear semigroup Sα introduced in (10) is stronglycontinuous on the cone Φ = Φ̺, cf. Corollary 2.12 in [FM04].SINGLE POINT SOURCE SBM 31.3. Super-Brownian motion with a single point source. Denote by M = M(Ṙ3) the set of all(Radon) measures µ on Ṙ3 such that 〈µ, ϕ〉 < ∞ for all ϕ ∈ Φ. Recalling that C+com ⊂ Φ, endow M withthe vague topology.Fix a constant η > 0 (branching rate). Suppose 0 < β < 1 (the finite variance branching case β = 1 hasbeen excluded in [FM04] for d = 3 for technical reasons). Then for each α ∈ R, there is a non-degenerateM–valued (time-homogeneous) Markov process Xα such that for (deterministic) starting measures µ ∈M and for ϕ ∈ Φ,(15) − log Eµ exp〈Xαt ,−ϕ〉 =〈µ, v(t, ·)〉, t > 0,where{v(t, x) : t ≥ 0, x ∈ Ṙ3}is the unique non-negative solution of the integral equation related to theΦ-valued evolution equation(16){∂tv = ∆(α)v − η v1+β on (0,∞),v(0+, · ) = ϕ(see [FM04, Theorem 4.4]). That is,(17) v(t, x) =∫Ṙ3dy pαt (x, y)ϕ(y) − η∫ t0ds∫Ṙ3dy pαt−s(x, y) v1+β(s, y),t > 0, x ∈ Ṙ3. Clearly, the first moments of Xα are determined by the Sα flow to be(18) Eµ〈Xαt , ϕ〉 = 〈µ, Sαt ϕ〉,for all starting measures µ ∈ M, t ≥ 0, and ϕ ∈ Φ.2. Large scale behavior2.1. A scaling property.Proposition 2 (A scaling property). Let t, k > 0, µ ∈ M, and α, λk ∈ R. Then(19){k−1/βXλkαkt (k1/2 · )∣∣ Xλkα0 = k1/βµ(k−1/2 · )}L={Xk1/2λkαt∣∣Xk1/2λkα0 = µ}.Of course, the cases λk = k−1/2 or even α = 0 are particularly nice.Proof. For ϕ ∈ Φ fixed,(20)〈k−1/βXλkαkt (k1/2 dy), ϕ〉=〈Xλkαkt , k−1/βϕ(k−1/2 · )〉,hence, by (15) and (17),− log Ek1/βµ(k−1/2 · ) exp〈k−1/βXλkαkt (k1/2 dy), ϕ〉= − log Ek1/βµ(k−1/2 · ) exp〈Xλkαkt , k−1/βϕ(k−1/2 · )〉=〈k1/βµ(k−1/2 · ), v(kt, · )〉=〈µ, k1/βv(kt, k1/2 · )〉(21)where{v(t′, x′) : t′ ≥ 0, x′ ∈ Ṙ3}is the non-negative solution of the integral equation related to thefunction-valued evolution equation(22){∂tv = ∆(λkα)v − η v1+β on (0,∞),v(0+, · ) = k−1/βϕ(k−1/2 · ).More precisely,k1/βv(kt, k1/2x) = k1/β∫Ṙ3dy pλkαkt (k1/2x, y) k−1/βϕ(k−1/2y)(23)− k1/βη∫ kt0ds∫Ṙ3dy pλkαkt−s(k1/2x, y) v1+β(s, y).By a change of variable,k1/βv(kt, k1/2x) =∫Ṙ3dy k3/2pλkαkt (k1/2x, k1/2y)ϕ(y)(24)− k1/βη∫ t0ds k∫Ṙ3dy k3/2pλkαkt−ks(k1/2x, k1/2y) v1+β(ks, k1/2y).4 KLAUS FLEISCHMANN, CARL MUELLER, AND PASCAL VOGTHence, by Lemma 1,k1/βv(kt, k1/2x) =∫Ṙ3dy pk1/2λkαt (x, y)ϕ(y) − k1/βη∫ t0ds k∫Ṙ3dy pk1/2λkαt−s (x, y) v1+β(ks, k1/2y).Since 1/β + 1 − (1/β)(1 + β) = 0 we see that k1/βv(kt, k1/2x) =: wk(t, x) satisfies the equation(25) wk(t′, x′) =∫Ṙ3dy pk1/2λkαt′ (x′, y)ϕ(y) − η∫ t′0ds∫Ṙ3dy pk1/2λkαt′−s (x′, y)w1+βk (s, y),t′ > 0, x′ ∈ Ṙ3. By uniqueness of solutions of the log-Laplace equation (17) and by (15), claim (19)follows. 2.2. Expectation of the scaled Xα. Before we can state the result, we have to introduce some notation.The limiting measure will be expressed by means of the kernel(26) ϑαt (x, y) :=2t|x| |y|pt(|y|)−8παt|x| |y|∫ ∞0du e−4παu pt(u + |y|),for α ∈ R, t > 0, and x, y ∈ Ṙ3. Note that the integral is always finite, hence for α = 0 the second termdisappears. Moreover, the kernel ϑα is always non-negative. This holds trivially whenever α < 0, and tosee this for α > 0, use the estimate(27) pt(u + |y|)≤ pt(|y|).We extend the definition of ϑα by setting(28) ϑαt (x, y) :≡{0, if α = +∞,+∞, if α = −∞.The so defined kernels ϑα turn out to be pointwise continuous in α ∈ [−∞, +∞] (which follows from thearguments of the proof of Theorem 3 below).Theorem 3 (Large scale behavior of the mean). For t > 0, α, λk ∈ R, and all starting measuresXα0 = µ ∈ M satisfying 〈µ, φ〉 < ∞, we have the convergence in M,(29) limk↑∞k−1/2 Eµ[Xλkαkt (k1/2 dy)]=〈µ, ϑα∗t ( · , y)〉dy,provided that α∗ := limk↑∞ k1/2λkα ∈ [−∞, +∞].Proof. Fix ϕ ∈ C+com(Ṙ3). Using formula (18) for the first moment of Xα and substitution, we obtaink−1/2 Eµ〈Xλkαkt , ϕ(k−1/2 · )〉= k−1/2∫Ṙ3µ(dx)∫Ṙ3dy pλkαkt (x, y)ϕ(k−1/2y)= k−1/2∫Ṙ3µ(dx)∫Ṙ3dy k3/2 pλkαkt (x, k1/2y)ϕ(y).(30)By Lemma 1 this is equal to(31) k−1/2∫Ṙ3µ(dx)∫Ṙ3dy pk1/2λkαt (k−1/2x, y)ϕ(y).Inserting according to definition (5) of pα, we get three terms, we will deal with separately.1◦ (First term). The first term equals,(32) k−1/2∫Ṙ3µ(dx)∫Ṙ3dy pt(k−1/2x, y)ϕ(y).This double integral is finite and vanishes as k ↑ ∞. To see this, let us restrict the outer integral first to|x| > K where we specify K ≥ 1 later. We call this restricted integral IK . We use ϕ ≤ Cφ (since ϕ ∈ Φ)and, with S denoting the free heat flow,(33) Stφ ≤ Cφ, t ≥ 0,with changed constant C (see [FM04, Lemma 2.1]) to arrive at(34) IK ≤ Ck−1/2∫|k−1/2x|>Kµ(dx)φ(k−1/2x) = C∫|x|>k1/2Kµ(dx)φ(x)SINGLE POINT SOURCE SBM 5which can be made arbitrarily small uniformly in k by choosing K sufficiently large (by our assumption onµ). It remains to deal with the case |x| ≤ K for fixed K. We split the internal integral in (32) as follows.First, if |k−1/2x − y| ≥ |y|/2, then(35) pt(k−1/2x, y) ≤ pt(|y|/2),which leads to the bound(36) k−1/2∫|x|≤Kµ(dx)∫Ṙ3dy pt(|y|/2)ϕ(y) −→ 0 as k ↑ ∞,the µ(dx)-integral is finite as 〈µ, φ〉 < ∞. On the other hand, if |k−1/2x−y| < |y|/2, then −k−1/2|x|+ |y| <|y|/2 which implies |y| < 2k−1/2|x| ≤ 2K. Hence as pt(k−1/2x, y) ≤ Ct−3/2 and ϕ ≤ Cφ, we get the upperestimate(37) Ct k−1/2∫|x|≤Kµ(dx)∫|y|< 2Kdy φ(y) −→ 0 as k ↑ ∞(the dy-integral is finite, since we are in dimension three).2◦ (Second term). The second term reads(38)∫Ṙ3µ(dx)∫Ṙ3dy2t|x| |y|pt(k−1/2|x| + |y|)ϕ(y) =: IIk .Observe that,(39) pt(k−1/2|x| + |y|)↑ pt(|y|)as k ↑ ∞.We can apply the monotone convergence theorem to obtain the limit,(40) limk↑∞IIk =∫Ṙ3µ(dx)2t|x|∫Ṙ3dy1|y|pt(y)ϕ(y),where finiteness follows from 〈µ, φ〉 < ∞ and ϕ ≤ Cφ. Hence, this summand gives the first part of thekernel ϑα∗.3◦ (Third term). It remains to insert the scaled third term from (5) into (31) which reads as(41) k−1/2∫Ṙ3µ(dx)∫Ṙ3dy−8πtk1/2λkαk−1/2|x| |y|∫ ∞0du e−4πk1/2λkαu pt(u + k−1/2|x| + |y|)ϕ(y).We distinguish several cases: If λkα = 0 for all sufficiently large k, then the third term disappears and weare done. From now on assume λkα 6= 0 for all k. Substituting u 7→ 4πk1/2|λkα|u into (41) yields,(42)∫Ṙ3µ(dx)∫Ṙ3dy−2t sign(λkα)|x| |y|∫ ∞0du e−sign(ρkα) u pt( u4πk1/2|λkα|+ k−1/2|x| + |y|)ϕ(y).Now let k1/2|λkα| → ∞. We may consider a monotone subsequence of k1/2λkα. Clearly,(43) pt( u4πk1/2|λkα|+ k−1/2|x| + |y|)↑ pt(|y|)as k ↑ ∞,and by monotone convergence the expression (42) converges along the subsequence to(44) −∫Ṙ3µ(dx)2t sign(α∗)|x|∫ ∞0du e−sign(α∗)u∫Ṙ3dy1|y|pt(y)ϕ(y),which is independent of the choice of the subsequence. Note that(45) sign(α∗)∫ ∞0du e−sign(α∗) u ={1 if sign(α∗) = 1,+∞ if sign(α∗) = −1.In the first case the second and the third limiting terms cancel.Next, let k1/2λkα → 0. Note, that(46) pt( u4πk1/2|λkα|+ k−1/2|x| + |y|)≤ pt( u4πk1/2|λkα|)e−|y|2/4t .In this case the double integral (42) is in absolute value bounded by(47)∫Ṙ3µ(dx)∫Ṙ3dy2t|x| |y|e−|y|2/4t ϕ(y)∫ ∞0du eu pt( u4π k1/2|λkα|),6 KLAUS FLEISCHMANN, CARL MUELLER, AND PASCAL VOGTwhich tends to 0 as k ↑ ∞ as the µ(dx) and dy-integrals are finite and in the du-integral the pt-termcompensates the eu.It remains to deal with the case k1/2λkα → α∗ ∈ Ṙ1. Note, that we only have to justify to changethe limit and integration in (42), as substituting u 7→(4π|α∗|)−1u leads to the desired expression. Tojustify the interchange, we estimate as in (46). The resulting µ(dx) and dy-integrals are independent of kand finite, whereas to dominate in the second integral we use k1/2|λkα| ≤ |α∗|+ 1 for all sufficiently largek. Remark 4 (Large scale total mass). Taking λk ≡ 1 and choosing formally ϕ = 1 as test function in(29) yields(48) limk↑∞k−1/2 Eµ 〈Xαkt, 1〉 =0 if α > 0,2t 〈µ, φ〉∫R3dy1|y|pt(y) if α = 0,∞ if α < 0.A rigorous argument can be given along the lines of the previous proof. 32.3. Discussion and open problems. Let us comment on the three cases α∗ = +∞, α∗ ∈ R, andα∗ = −∞ in Theorem 3. In the first case, the limiting mass disappears, more precisely, the scaled expressionEµ[Xλkαkt (k1/2 dy)]is of order o(k1/2). Roughly speaking, if α∗ = +∞, then there are no interactions in thescaling limit (free case). In the second case, α∗ ∈ R, the former expectation is about k1/2〈µ, ϑα∗t ( · , y)〉dy.Note that these measures are decreasing in α∗. Finally, if α∗ = −∞, we have immediate interaction in thelarge scale limit leading to the explosion of the expected mass.Clearly, to describe only the large scale behavior of the expected processes is unsatisfactory. It is desirableto get something similar for the processes themselves. Recall that in the one-dimensional case the largetime behavior of the process itself is known from [ET02]. However, we stress the fact, that the process inthree dimensions is expected to have quite different features. For instance, if α = 0, then according toRemark 4 the total mass grows with a power order, whereas in one dimension the growth is exponential.Moreover, in the three-dimensional case one needs additionally to contract the normalized measures to geta limit. For the measures themselves, scaled as in Theorem 3, there might be extinction in law despiteconvergence of their expectations.Another open problem is the large scale behavior of EXα in the two-dimensional case, in which thefundamental solutions pα from [ABD95] are analytically more delicate, see e.g. [FM04, formula (2.30)]. Inparticular, a scaling property as in Lemma 1 is not available.Acknowledgement. We thank Hagen Neidhardt of the WIAS for a helpful discussion on semigroups.References[ABD95] S. Albeverio, Z. Brzeźniak, and L. Dabrowski. Fundamental solution of the heat and Schrödinger equations withpoint interaction. J. Funct. Anal., 130:220–254, 1995.[AGHKH88] S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden. Solvable Models in Quantum Mechanics. Springer-Verlag, New York, 1988.[EF00] J. Engländer and K. Fleischmann. Extinction properties of super-Brownian motions with additional spatiallydependent mass production. Stoch. Proc. Appl., 88(1):37–58, 2000.[ET02] J. Engländer and D. Turaev. A scaling limit theorem for a class of superdiffusions. Ann. Probab., 30(2):683–722,2002.[FM04] K. Fleischmann and C. Mueller. Super-Brownian motion with extra birth at one point. SIAM J. Math. Analysis,36(3):740–772, 2004.Weierstraß Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, D-10117 Berlin, GermanyE-mail address: fleischm@wias-berlin.deDepartment of Mathematics, University of Rochester, Rochester, NY 14627, USAE-mail address: cmlr@math.rochester.eduifb AG, Neumarkt-Galerie, D-50667 Köln, GermanyE-mail address: Pascal.Vogt@ifbAG.com