arXiv:1411.4907v1 [math.PR] 18 Nov 2014 h aeOnti-hebc rcs,wsoiial ie otepr the to given originally equation: was ferential process, Ornstein-Uhlenbeck Ornstein-Uhlenbeck name The catalytic The 2 ex give also versions annealed The spaces. infinit fields. Sobolev result random catalytic of the Gaussian in of class and motion notion a catalysts super-Brownian the measure-valued spaces of involve is limits esta processes was paper fluctuation It These as this process. arise of affine can dimensional topic processes infinite Co main of such finit examples The general processes give the which affine of [K-R]). Other characterization a 03]). and (cf. [Du modelling financial al O in case et arise dimensional Duffie finite the (e.g. In o processes study [Ito-1]). extensive (e.g. been has Itˆo processes there K. Uhlenbeck of work the with Beginning Introduction 1 medium. random a me in moment processes annealed processes, and quenched Ornstein-Uhlenbeck catalytic processes, concepts: Key h aayi rsenUlnekPoeswt Superproce with Process Ornstein-Uhlenbeck catalytic The h dnicto fbscpoete fteqece n an and quenched the of resu properties main basic The joint of identification the superprocess. the of a transform nee by functional-Laplace we given characteristic which catalyst motion with super-Brownian set process unified of a properties provides basic that proce some Ornstein-Uhlenbeck processes exa and affine for superprocesses, of processes, catalyst, class random the measure-valued to a this with processes U) h anojcieo hswr st td aua ls fc of class natural a study to is work this of objective main The tcatcprildffrnileutos uepoess mea superprocesses, equations, differential partial Stochastic oadA asn([email protected]) Dawson A. Donald nvria e aaopn(Mxc ) Mexico ( Papaloapan del Universidad altnUiest Cnd ) (Canada University Carleton unMne Perez-Abarca Juan-Manuel ([email protected]) coe ,2018 1, October Catalyst Abstract 1 aayi - rcs n aaytpoesand process catalyst and process O-U catalytic sswt uepoesctls.W hnreview then We catalyst. superprocess with sses eldvrin fteeprocesses. these of versions nealed igi hc ove Ornstein-Uhlenbeck view to which in ting pe ue-rwinmto.W relate We motion. super-Brownian mple, n nrdc h Ornstein-Uhlenbeck the introduce and d t r h ffiecaatrzto fthe of characterization affine the are lts n aayi - rcse aea state as have processes O-U catalytic ing srs hrceitcLpaefunctional, Laplace characteristic asures, h ls fifiiedmninlOrnstein- dimensional infinite of class the f Upoessbln otefml faffine of family the to belong processes -U iesoa ffiepoessi known is processes affine dimensional e lse n[A 2 htctltcO-U catalytic that 12] [PAD in blished -nesl-osadHso processes Heston and x-Ingersoll-Ross cs ecie ytesohsi dif- stochastic the by described ocess ue-rwinctltcmedium. catalytic super-Brownian a tltcOnti-hebc (O- Ornstein-Uhlenbeck atalytic mlso nnt iesoa non- dimensional infinite of amples iesoa - processes O-U dimensional e uevle rcse,affine processes, sure-valued ss dX = θ(a X )dt + σdB t − t t where θ > 0,a R,σ > 0 are parameters, Xt R and Bt is Brownian motion. The corresponding infinite dimensional analogue∈ has developed into what∈ is now known as the generalized Ornstein-Uhlenbeck process :
dXt = AXtdt + dWt.
Here Xt takes values in some Hilbert space H; A (H) and Wt t≥0 is a Hilbert-space-valued Wiener 2 ∈ L { } ∂ Wt process. One important case is the cylindrical Wiener process whose distributional derivative: ∂t∂x is space- time white noise.
In this paper, we will consider the Generalized Ornstein-Uhlenbeck (OU) process in catalytic media, that is, a process that satisfies a stochastic evolution equation of the form:
dX(x, t)= AX(x, t)+ dWµ(x, t) (2.1) where A and X are as before, but this time, µ = µt t≥0 is a measure-valued function of time and Wµ is a Wiener process based on µ(t), i.e. W defines a random{ } set function such that for sets A : ∈E (i) W (A [0,t]) is a random variable with law (0, t µ (A))ds. µ × N 0 s (ii) if A B = φ then Wµ(A [0,t]) and Wµ(B [0R,t]) are independent and Wµ((A B) [0,t]) = W (A∩ [0,t]) + W (B [0×,t]). × ∪ × µ × µ × As it will be seen later, µt will play the role of the catalyst. For the rest of the discussion, we will assume that µt is a measure-valued Markov process, for example, the super-Brownian motion (SBM).
As a simple example, consider the case of a randomly moving atom µt = δBt where Bt is a Brownian motion in Rd starting at the origin. In the case of a random catalyst there are two processes to consider. The first is the solution of the perturbed heat equation conditioned on a given realization of the catalyst process - this is called the quenched case. The second is the process with probability law obtained by averaging the laws of the perturbed heat equation with respect to the law of the catalytic process - this is called the annealed case. We will now determine the behavior of the annealed process and show that it also depends on the dimension as expressed in the following:
Rd Theorem 2.1. Let X(t, x) be the solution of (2.1) where µt = δBt with Bt a Brownian motion in and
1 with A = 2 ∆. Then X is given by: t
X(t, x)= p(t s,x,y)WδB(s) (dy, ds) (2.2) Rd − Z0 Z then, the annealed variance of X(t, x) is given by:
1/4 d =1, x =0 E [Var X(t, x)] = < d =1, x =0 ∞ 6 d 2 ∞ ≥ Proof. The second moments are computed as follows:
t 1 x B(s) 2 EX2(t, x)= E exp k − k ds. 2π(t s) − (t s) Z0 − −
2 In the case d = 1, x = 0 the expectation in the last integral can be computed using the Laplace transform
2 MX of the χ1 distribution as: t 1 B2(s) E exp ds 2π(t s) − t s Z0 − − with: B2(s) 1 s B2(s) E exp = E exp − t s 2π −t s s − − 1 s 1 t s 1/2 = M = − 2π X −t s 2π t + s − A trigonometric substitution shows:
1 t 1 1 EX2(t, 0) = ds = . 2π (t2 s2)1/2 4 Z0 − For x = 0 and d = 1, using a spatial shift we have 6 t 1 (B(s) x)2 EX2(t, x)= E exp − ds 2π(t s) − (t s) Z0 − − t − 2 2 1 − (y x) − y = e t−s e 2s dyds 2π(t s)(2πs)1/2 Z0 − Z t − 2 1 − (y x) e t−s dyds ≤ 2π(t s)(2πs)1/2 Z0 − Z t 1 C ds < . ≤ (s(t s))1/2 ∞ Z0 −
2 When d 2 and the perturbation is δ as before, then kB(s) k is distributed as χ2 , and its Laplace ≥ B(t) s d transform is: 1 d/2 M (t)= x 1 2t − So: t ds EX2(t, 0) = (t2 s2)d/2 Z0 − Which is infinite when d 2 at x = 0. A modified calculation show this is true everywhere. ≥
Remark 2.1. Note that the annealed process is not Gaussian since a similar calculation can show that E(X4(t, x)) = 3(E(X2(t, x)))2. 6
3 Affine Processes and Semigroups
In recent years affine processes have raised a lot of interest, due to their rich mathematical structure, as well as to their wide range of applications in branching processes, Ornstein-Uhlenbeck processes and mathematical finance.
In a general affine processes are the class of stochastic processes for which, the logarithm of the charac- teristic function of its transition semigroup has the form x, ψ(t,u) + φ(t,u). h i Important finite dimensional examples of such processes are the following SDE’s:
3 Ornstein-Uhlenbeck process: z(t) satisfies the Langevin type equation • dz(t) = (b βz(t))dt + √2σdB(t) − This is also known as a Vasieck model for interest rates in mathematical finance. Continuous state branching immigration process: y(t) 0 satisfies • ≥ dy(t) = (b βz(t))dt + σ 2y(t)dB(t) − with branching rate σ2, linear decay rate β and immigrationp rate b. This is also called the Cox- Ingersoll-Ross (CIR) model in mathematical finance.
The Heston Model (HM): Also of interest in mathematical finance, it assumes that St the price of • an asset is given by the stochastic differential equation: 1 dSt = µStdt + VtStdBt
where, in turn, Vt the instantaneous volatility, is a CIRp process determined by: dV = κ(θ V )dt + σ V (ρdB1 + 1 ρ2dB2) t − t t t − t 1 2 and dBt , dBt are Brownian motions with correlationp ρ, θ is thep long-run mean, κ the rate of reversion and σ the variance. A continuous affine diffusion process in R2: r(t)= a y(t)+ a z(t)+ b where • 1 2 dy(t) = (b β y(t))dt 1 − 11 + σ11 2y(t)dB1(t)+ σ12 2y(t)dB2(t) dz(t) = (b β y(t) β z(t))dt 2 p− 21 − 22 p + σ21 2y(t)dB1(t)+ σ22 2y(t)dB2(t)+ √2adB0(t) p p where B0,B1,B2 are independent Brownian motions. It can be seen that the general affine semigroup can be constructed as the convolution of a homogeneous semigroup ( one in which φ = 0) with a skew convolution semigroup which corresponds to the constant term φ(t,u).
Definition 3.1. A transition semigroup (Q(t)t≥0) with state space D is called a homogeneous affine semigroup (HA-semigroup) if for each t 0 there exists a continuous complex-valued function ψ(t, ) := ≥ · (ψ (t, ), ψ (t, )) on U = C (iR) with C = a + ib : a R ,b R such that: 1 · 2 · − × − { ∈ − ∈ } exp u, ξ Q(t, x, dξ) = exp x, ψ(t,u) , x D,u U. (3.3) {h i} {h i} ∈ ∈ ZD The HA-semigroup (Q(t)t≥0) given above is regular, that is, it is stochastically continuous and the derivative ψ′(0,u) exists for all u U and is continuous at u = 0 (see [K-R]). t ∈
Definition 3.2. A transition semigroup (P (t))t≥0 on D is called a (general) affine semigroup with the
HA-semigroup (Q(t))t≥0 if its characteristic function has the representation
exp u, ξ P (t, x, dξ) = exp x, ψ(t,u) + φ(t,u) , x D,u U (3.4) {h i} {h i } ∈ ∈ ZD where ψ(t, ) is given in the above definition and φ(t, ) is a continuous function on U satisfying φ(t, 0) = 0. · · Further properties of the affine processes can be found in ( [Du 03], [DaLi 06] and [K-R] ). We will see that the class of catalytic Ornstein-Uhlenbeck processes we introduce in the next section forms a new class of infinite dimensional affine processes.
4 4 A brief review on super-processes and their properties
Given a measure µ on Rd and f (Rd), denote by µ,f := f dµ, let M (Rd) be the set of finite ∈ B h i Rd F measures on Rd and let C (Rd) denote the continuous and positive functions, we also define: 0 + R C (Rd)= f C(Rd): f(x) x p < ,p> 0 p ∈ k · | | k∞ ∞ M (Rd)= µ M(Rd):(1+ x p)−1 dµ(x) is a finite measure p ∈ | |
Definition 4.1. The (α, d, β)-superprocess Zt is the measure-valued process, whose Laplace functional is given by: E [exp( ψ,Z )] = exp[ U ψ,µ ] µ M (Rd), ψ C (Rd) µ − h ti − h t i ∈ p ∈ p + where µ = Z0, and Ut is the nonlinear continuous semigroup given by the the mild solution of the evolution equation:
1+β u˙(t) = ∆α u(t) u(t) , 0 < α 2, 0 <β 1 − ≤ ≤ (4.5) u(0) = ψ, ψ D(∆ ) . ∈ α + here ∆α is the generator of the α-symmetric stable process, given by:
∆ u(x)= A(d, α) u(x+y)−u(x) dy α Rd |y|d+2α A(d, α)= π2α−d/2Γ((R d 2α)/2)/Γ(α) − Then, u(t) satisfies the non-linear integral equation: t u(t) = U ψ U (u1+β(s))ds. t − t−s Z0 This class of measure-valued processes was first introduced in [Wat 68], an up-to-date exposition of these processes is given in [Li 11]. It can be verified that the process with the above Laplace functional is a finite d measure-valued Markov process with sample paths in D(R+,Mp(R )). The special case α = 2, β = 1 is d called super-Brownian motion (SBM) and this has sample paths in C(R+,Mp(R )) d Consider the following differential operator on Mp(R ): 1 δ2F δF LF (µ)= µ(dx) 2 + µ(dx)∆ (x) (4.6) 2 Rd δµ(x) Rd δµ Z Z Here the differentiation of F is defined by δF = lim(F (µ + ǫδx) F (µ))/ǫ δµ(x) ǫ↓0 − where δx denotes the Dirac measure at x. The domain (L) of L will be chosen a class containing such D d functions F (µ) = f( µ, φ1 ,..., µ, φ1 ) with smooth functions φ1,...,φn defined on R having compact support and a boundedh smoothi functionh i f on Rd. As usual µ, φ := φ(x)µ(dx). h i R The super-Brownian motion is also characterized as the unique solution to the martingale problem given by (L, (L)). The process is defined on the probability space (Ω, , P , Z ) and D F µ { t}t≥0 P (Z(0) = µ)=1, P (Z C([0, ),M (Rd))) = 1. µ µ ∈ ∞ p
An important property of super-Brownian motion is the compact support property discovered by Iscoe [Is 88], that is, the closed support S(Zt) is compact if S(Z0) is compact.
5 5 Catalytic OU with the (α, d, β)-superprocess as catalyst
5.1 Formulation of the process The main object of this section is the catalytic OU process given by the solution of 1 dX(t, x)= ∆X(t, x)dt + W (dt, dx), X (t, x) 0 (5.7) 2 Zt 0 ≡
where Zt is the (α, d, β)-superprocess, in the following discussion, we will assume that the catalyst and the white noise are independent processes. In order to study this process we first note that conditioned on the process Zt , the process X is Gaussian. We next determine the second moment structure of this Gaussian process.{ }
Proposition 5.1. The variance of the process X(t) given by (5.7), is:
t 2 2 EX (t, x)= p (t s,x,u)Zs(du)ds Rd − Z0Z and its covariance by:
t EX(t, x)X(t,y)= p(t s,x,u)p(t s,y,u)Zs(du)ds Rd − − Z0Z Proof. In order to compute the covariance of X(t); recall that the solution of (5.7) is given by the stochastic convolution: t
X(t, x)= p(t s,x,u)WZs (ds, du) Rd − Z0Z From which, the covariance is computed as:
t t E 2 E X (t, x)= p(t r, x, u)p(t s, x, w) [WZr (dr, du)WZs (ds, dw)] Rd Rd − − Z0Z Z0Z The covariance measure is defined by:
. E CovWZ (dr, ds; du, dw) = [WZr (dr, du)WZs (ds, dw)]
= δs(r) dr δu(w) Zr(dw)
The second equality is due to the fact that WZr is a white noise perturbation and has the property of independent increments in time and space.
5.2 Affine structure of the catalytic OU-process
We will compute now the characteristic functional of the annealed process E[exp(i φ, Xt )], where by defi- nition: h i . d φ, Xt = X(t, x)φ(x) dx, φ C(R ) h i Rd ∈ Z which is well-defined since Xt is a random field and also the characteristic functional-Laplace functional of the joint process X(t),Z(t) . For this, we need the following definitions and properties, for further details see [Is 86]. { }
6 Definition 5.1. Given the (α, d, β)-superprocess Zt, we define the weighted occupation time process Yt by
t < ψ,Y >= < ψ,Z > ds ψ C (Rd). t s ∈ p Z0
Remark 5.1. The definition coincides with the intuitive interpretation of Yt as the measure-valued process satisfying: t Y (B)= Z (B)ds, for B (Rd) t s ∈B Z0 Theorem 5.2. It can be shown ([Is 86]) that, given µ M (Rd) and φ, ψ C (Rd) , and p < d + α then ∈ p ∈ p + the joint process [Zt, Yt] has the following Laplace functional:
E [exp( < ψ,Z > < φ, Y >)] = exp[ ], t 0, µ − t − t − t ≥
φ where Ut is the strongly continuous semigroup associated with the evolution equation:
1+β u˙(t) = ∆αu(t) u(t) + φ − (5.8) u(0) = ψ
A similar expression will be derived when φ is a function of time, but before, we need the following:
Definition 5.2. A deterministic non-autonomous Cauchy problem, is given by:
u˙(t)= A(t)u(t)+ f(t) 0 s t (NACP) ≤ ≤ u(s)= x where A(t) is a linear operator which depends on t.
Similar to the autonomous case, the solution is given in terms of a two parameter family of operators U(t,s), which is called the propagator or the evolution system of the problem (NACP), with the following properties:
U(t,t)= I,U(t, r)U(r, s)= U(t,s). • (t,s) U(t,s) is strongly continuous for 0 s t T . • → ≤ ≤ ≤ The solution of (NACP) is given by: • t u(t)= U(t,s)x + U(t, r)f(r)dr (5.9) Zs in the autonomous case the propagator is equivalent to the semigroup U(t,s)= T . • t−s
Theorem 5.3. Let µ M (Rd), and Φ: R1 C (Rd) be right continuous and piecewise continuous such ∈ p + → p + that for each t> 0 there is a k> 0 such that Φ(s) k (1 + x p)−1 for all s [0,t]. Then ≤ · | | ∈ t E exp Ψ,Z Φ(s),Z ds = exp U Φ Ψ,µ µ − h ti− h si − t,t0 Z0 7 Φ 1+β where U is the non-linear propagator generated by the operator Au(t) = ∆αu(t) u (t)+Φ(t), that is, t,t0 − Φ u(t)= Ut,t0 Ψ satisfies the evolution equation:
1+β u˙(s) = ∆αu(s) u(s) + Φ(s), t0 s t − ≤ ≤ (5.10) u(t )=Ψ 0. 0 ≥
Φ Proof. The existence of Ut,t0 follows from the fact that Φ is a Lipschitz perturbation of the maximal monotone non-linear operator ∆ ( ) u1+β( ) (refer to [Br] pp. 27). α · − · α Note that the solution u(t) depends continuously on Φ, to see this, denote by Vt the subgroup generated by ∆α, then the solution u(t) can be written as:
t u(t)= V (Ψ) + V α (Φ(s) u1+β(s)) ds t t−s − Z0 from which the continuity follows.
So we will assume first that Φ is a step function defined on the partition of [0,t] given by 0 = s0 Taking a Riemann sum approximation: t E exp Ψ,Z Φ(s),Z ds µ − h ti− h si Z0 N−1 E t t = lim µ exp φk,Z k t φN +Ψ,Zt . N→∞ "− N N − N #! kX=1 D E Denote by U Φ and U ΦN the non-linear semigroups generated by ∆ u(t) u1+β(t)+Φ(t) and ∆ u(t) t t α − α − 1+β u (t)+ΦN (t) respectively. Conditioning and using the Markov property of Zt we can calculate: t E exp Ψ,Z Φ(s),Z ds µ − h ti− h si Z0 N sk = lim Eµ exp Ψ,Zt φk,Zs ds N→∞ "− h i− sk−1 h i #! Xk=1 Z N−1 sk sN = lim Eµ Eµ exp Ψ,Zt φk,Zs ds φN−1,Zs ds N→∞ "− h i− sk−h1 i − sN−h1 i # Xk=1 Z Z σ(Z ), 0 s s )) | s ≤ ≤ N−1 8 N−1 sk t = lim Eµ exp φk,Zs ds Eµ exp( Ψ,Zt φN ,Zs ds) N→∞ − sk−h1 i ! " − h i− sN−h1 i # Xk=1 Z Z σ(Z ), 0 s s ) | s ≤ ≤ N−1 N−1 sk t E E = lim µ exp φk,Zs ds Zs exp( Ψ,Zt φN ,Zs ds) N→∞ N−1 − skh−1 i ! " − h i− sNh−1 i #! Xk=1 Z Z N−1 sk E ΦN = lim µ exp φk,Zs ds exp( Us −s − Ψ,ZsN−1 ) N→∞ N N 1 − skh−1 i ! − ! Xk=1 Z D E N−1 sk E ΦN = lim µ exp Us −s − Ψ,Zt φk,Zs ds N→∞ N N 1 "− − sk−1 h i #! D E Xk=1 Z ΦN where Us−sN−1 Ψ is the mild solution of the equation: u˙(s) = ∆ u(s) u(s)1+β + φ , s s s α − N−1 N−1 ≤ ≤ N u(sN−1)=Ψ. Performing this step N times, one obtains: t E exp Ψ,Z Φ(s),Z ds µ − h ti− h si Z0 E ΦN ΦN ΦN = lim µ exp Us −s Us − −s − Us −s − Ψ,Zs0 N→∞ − 1 0 ··· N 1 N 2 N N 1 h D Ei E ΦN = lim µ exp Ut Ψ,Z0 N→∞ − h D Ei ΦN ΦN = lim exp U0 Ut Ψ,µ N→∞ − h D Ei = exp U ΦΨ,µ − t ΦN I where in the fourth line U0 = . The last equality follows because the solution depends continuously on Φ as noted above and the result follows by taking the limit as N . → ∞ Now let Xt be the solution of: 1 dX(t, x)= ∆X(t, x)dt + W (dt, dx) X(0, x) 0 (5.11) 2 Zt ≡ with values in the space of Schwartz distributions on Rd. The previous result will allow us to compute the characteristic-Laplace functional E[exp(i φ, X λ, Z )] of the pair [X ,Z ]. h ti − h ti t t Theorem 5.4. The characteristic-Laplace functional of the joint process [Xt,Zt] is given by : E [exp(i φ, X λ, Z )] = exp( u(t, φ, λ),µ ) (5.12) µ h ti − h ti − h i 9 where u(t, φ, λ) is the solution of the equation: ∂u(s, x) 1+β 2 = ∆αu(s, x) u (s, x)+ Gφ(t s, x), 0 s t ∂s − − ≤ ≤ (5.13) u(0, x)= λ with Gφ defined as: Gφ(t,s,z)= p(t s,x,z)φ(x)dx Rd − Z Proof. Denote by , the standard inner product of L2 and recalling the following property of the Gaussian h· ·i processes: E[exp(i φ, X )] = exp( Var φ, X ) h ti − h ti We get 2 [ φ, Xt ]= X(t, x)X(t,y)φ(x)φ(y) dx dy h i Rd Rd Z Z Hence: 2 Var[ φ, Xt ] = E[ φ, Xt ]= φ(x)E[X(t, x)X(t,y)]φ(y) dx dy h i h i Rd Rd Z Z = φ(x)Γt(x, y)φ(y) dx dy Rd Rd Z Z where by definition: t Γt(x, y)= E[X(t, x)X(t,y)] = p(t s,x,z)p(t s,y,z) Zs(dz) ds Rd − − Z0Z So: t Var[ φ, Xt Zt] = φ(x)p(t s,x,z)p(t s,y,z)φ(y) Zs(dz) dx dy ds h i | 0 Rd Rd Rd − − Z tZ Z Z = gφ(t s,z) Zs(dz) ds Rd − Z0Z In the last line: . gφ(t s,z) = p(t s,x,z)p(t s,y,z)φ(x)φ(y) dx dy − Rd Rd − − Z Z 2 = p(t s,x,z)φ(x) dx Rd − Z Note that the function: . Gφ(t s,z) = p(t s,x,z)φ(x) dx − Rd − Z considered as a function of s, satisfies the backward heat equation: ∂ 1 G (t s,z)+ ∆G (t s,z)=0, 0 s t ∂s φ − 2 φ − ≤ ≤ 10 with final condition: Gφ(t)= φ(x) So: t E[exp(i φ, X )] = exp < G2 (t s,z),Z (dz) > ds h ti − φ − s Z0 2 where Gφ is given by the above expression. With this, assuming Z0 = µ we have the following expression for the Laplacian of the joint process [Xt,Zt]: E [exp(i φ, X λ, Z )] µ h ti − h ti = E [E [exp(i φ, X λ, Z ) σ(Z ), 0 s t]] µ µ h ti − h ti | s ≤ ≤ = E [exp( λ, Z )E [exp(i φ, X ) σ(Z ), 0 s t]] (measurability) µ − h ti µ h ti | s ≤ ≤ t E 2 = µ exp Gφ(t s,z)Zs(dz) ds λ, Zt − Rd − − h i Z0 Z t = E exp G2 (t s,z),Z ds λ 1,Z (by definition) µ − φ − s − h ti Z0 = exp( u(t),µ ) ( by Theorem 5.3) − h i and the result follows after renaming the variables. Remark 5.2. We can generalize this to include a second term 1 dX(t, x)= ∆X(t, x)dt + W (dt, dx)+ b(t, x)W (dt, dx), X(0, x)= χ(x) (5.14) 2 Zt 2 where Zt is the (α, d, β)-superprocess, and W2(dt, dx) is space-time white noise. In this case the characteristic- Laplace functional has the form E [exp(i φ, X λ, Z )] (5.15) µ,χ h ti − h ti = exp u(t, φ, λ),µ q(t, x , x )φ(x )φ(x )dx dx + i p(t,x,y)χ(x)φ(y)dxdy) . − h i− 1 2 1 2 1 2 i Z Z Z Z This is an infinite dimensional analogue of the general affine property defined in (3.4). Rd 1 Remark 5.3. If in Theorem 5.4, is replaced by a finite set E and 2 ∆ and ∆α are replaced by the gen- erators of Markov chains on E, then the analogous characterization of the characteristic-Laplace functional remains true and describes a class of finite dimensional (multivariate) affine processes. 6 Some properties of the quenched and annealed catalytic O-U processes In this section we formulate some basic properties of the catalytic O-U process in the case in which the catalyst is a super-Brownian motion, that is α = 2 and β = 1. For processes in a random catalytic medium 11 Zs :0 s t a distinction has to be made between the quenched result above, which gives the process {conditioned≤ ≤ on}Z and the annealed case in which the process is a compound stochastic process. In the quenched case the process is Gaussian. The corresponding annealed case leads to a non-Gaussian process. These two cases will of course require different formulations. For the quenched case, properties of Xt are obtained for a.e. realization. On the other hand for the annealed case we obtain results on the annealed laws P ∗(X( ) A)= P Z (X( ) A)P C (dZ), · ∈ · ∈ µ ZC([0,t],C([0,1])) Z where P (X( ) A) denotes the probability law for the super-Brownian motion Xt in the catalyst Z( ) C([0, · ),∈ C([0, 1])) . { } { · ∈ ∞ } 6.1 The quenched catalytic OU process In order to exhibit the role of the dimension of the underling space we now formulate and prove a result for the quenched catalytic O-U process on the set [0, 1]d Rd. ⊂ Theorem 6.1. In dimension d 1 and with Z M (Rd), consider the initial value problem: ≥ 0 ∈ F dX(t, x)= AX(t, x)dt + W (dx, dt) t 0, x [0, 1]d Zt ≥ ∈ X(0, x)=0 x ∂[0, 1]d ∈ where A = 1 ∆ on [0, 1]dwith Dirichlet boundary conditions. Then for almost every realization of Z , X(t) 2 { t} 1 has β-H¨older continuous paths in H−n for any n>d/2 and β < 2 , where H−n is the space of distributions on [0, 1] defined below in subsection 7.1. 6.2 The annealed O-U process with super-Brownian catalyst (α =2, β =1): State space and sample path continuity Now, we apply the techniques developed above to determine some basic properties of the annealed O-U process X( ) including identification of the state space, sample path continuity and distribution properties of the random· field defined by X(t) for t> 0. Theorem 6.2. Let X(t) be the solution of the stochastic equation: dX(t, x) = ∆X(t, x)dt + WZ (dt, dx) t 0, x R (CE) t ≥ ∈ X(0, x)=0 x R ∈ (i) For t> 0, X(t) is a zero-mean non-Gaussian leptokurtic random field. (ii) if Z0 = δ0(x), then the annealed process X(t) satisfies E X(t) 4 < k kL2 ∞ and has continuous paths in L2(R) 12 7 Proofs In this section we give the proofs of the results formulated in Section 6. 7.1 Proof of Theorem 6.1. d Proof. We first introduce the dual Hilbert spaces Hn,H−n. let ∆ denote the Laplacian on [0, 1] with Dirichlet boundary conditions. Then ∆ has a CONS of smooth eigenfunctions φ with eigenvalues λ { n} { n} −p which satisfy (1 + λj ) < if p>d/2, (see [Kr 99]). j ∞ P N Let E0 be the set of f of the form f(x)= cjφj (x), where the cj are constants. For each integer n, positive j=1 or negative, define the space H = f HP : f < where the norm is given by : n { ∈ 0 k kn ∞} f 2 = (1 + λ )nc2. k kn j j j X The H−n is defined to be the dual Hilbert space corresponding to Hn. The solution of (6.1) is given by: t −λk (t−s) X(t, x)= e φk(x)φk(y)WZs (dy, ds) 0 [0,1]d Z Z kX≥1 Let t −λk (t−s) Ak(t)= e φk(y)WZs (dy, ds) d Z0 Z[0,1] Then X(t, x)= φk(x)Ak(t) kX≥1 we will show that X(t) has continuous paths on the space Hn, which is isomorphic to the set of formal eigenfunction series ∞ f = akφk kX=1 for which f = a2 (1 + λ )n < k kn k k ∞ X E 2 t Fix some T > 0, we first find a bound for sup Ak(t) , let Vk(t)= 0 [0,1]d φk(x)W (Zs(dx), ds), integrat- t≤T ing by parts in the stochastic integral, we obtain: R R t t A (t)= e−λk(t−s)dV (s)= V λ e−λk(t−s)V (s) ds k k k − k k Z0 Z0 Thus: t sup A (t) sup V (t) (1 + λ eλk (t−s)ds) 2 sup V (t) | k |≤ | k | k ≤ | k | t≤T t≤T Z0 t≤T 13 Hence: E sup A2 (t) 4E sup V 2(t) k ≤ k t≤T t≤T 16E V 2(T ) (Doob’s inequality) ≤ k T 2 d = 16 φk(x)Zs(dx)ds 16TZT ([0, 1] )= CT. d ≤ Z[0,1] Z0 Therefore: E 2 −n −n sup Ak(t)(1 + λk) CT (1 + λk) . (7.16) t≤T ≤ kX≥1 kX≥1 Using now: (1 + λ )−p < if p>d/2 k ∞ kX≥1 then ( 7.16) is finite if n>d/2 and clearly: E[ X(t) 2 ]= [A2 (t)](1 + λ )−n < (7.17) k k−n k k ∞ kX≥1 and hence X(t) H a.s.. Moreover, if s> 0, ∈ −n [ X(t + s) X(t) 2 ]= E[(A (t + s) A (t))2](1 + λ )−n Cs. k − k−n k − k k ≤ X Then since conditioned on Z, X(t) is Gaussian, (7.17) together with [DpZ 92], Proposition 3.15, implies that 1 there is a β-H¨older continuous version with β < 2 . 7.2 Second moment measures of SBM In this section we evaluate the second moment measures of SBM, E (Z1(dx)Z2(dy)), which are needed to 4 compute E[ Zt ] in the the annealed catalytic OU process. k kL2 To accomplish this, given any two random measures Z1 and Z2, then it is easy to verify that: E( Z , A ), E( Z , A ), A (Rd) h 1 i h 2 i ∈B E( Z , A Z ,B ), A,B (Rd) h 1 i h 2 i ∈B are well defined measures which will be called the first and second moment measures. Similarly, given the measure-valued process Zt t≥0 one can define the n-th moment measure of n given random variables denoted by: { } (dx ,...,dx )= E (Z (dx )Z (dx ) Z (dx )) . Mt1...tn 1 n t1 1 t2 2 ··· tn n The main result of this section, is the following: 1 Proposition 7.1. The super-Brownian motion Zt in R has the following first and second moment measures: (i) if Z = dx, (dx)= dx the Lebesgue measure. 0 Mt (ii) if Z0 = δ0(x),then: (dx)= p(t, x) dx Mt · 14 (iii) if Z0 = dx: t (dx dx )= p(2s, x , x )ds dx dx (7.18) Mt 1 2 1 2 · 1 2 Z0 (iv) if Z0 = δ0(x): t t(dx1dx2)= p(t s, 0,y)p(2s, x1, x2) dy ds dx1 dx2 (7.19) M R − · Z0 Z 2 where dx, dx1dx2 denote the Lebesgue measures on R and R , respectively. (v) if t1 t1 (dx dx )= p(2s + t t , x , x )ds dx dx (7.20) Mt1t2 1 2 2 − 1 1 2 · 1 2 Z0 (vi) if t1 t1 t1t2 (dx1dx2)= p(t1 s, 0,y)p(s,y,x1)p(t2 t1 + s,y,x2) dyds dx1 dx2 (7.21) M R − − · Z0 Z 2 where dx, dx1dx2 denote the Lebesgue measures on R and R , respectively. Proof. Consider SBM Z with Z = µ. Then for λ R and φ C (Rd): t 0 ∈ + ∈ + E [exp( λ φ, Z )] = exp [ u(λ, t),µ ] =: exp( F (λ)) µ − h ti − h i − where, u(λ, t) satisfies the equation: u˙(t) = ∆u(t) u2(t) − (7.22) u(λ, 0) = λφ Then the first moment is computed according to ∂eF (λ) E( φ(x)Z (dx)) = = F ′(0) t ∂λ Z λ=0 since F (0) = 0. F (λ) will be developed as a Taylor series below. First rewrite ( 7.22) as a Volterra integral equation of the second kind, namely: t 2 u(t)+ Tt−s(u (s)) ds = Tt(λφ) Z0 whose solution is given by its Neumann series: n u(t)= T (λφ)+ ( 1)kTk(T (λφ)) (7.23) t − t Xk=1 where the operators Tt and T are defined by: 15 Tt(λφ(x)) = λ p(t,x,y)φ(y) dy R Z t t 2 2 2 T(Tt(λφ)) = Tt−s[(Ts(λφ)) ] ds = λ Tt−s[(Tsφ) ] ds (7.24) − − Z0 Z0 t 2 = λ2 p(t s,x,y) p(s,y,z)φ(z) dz dy ds. − R − R Z0 Z Z We obtain F (λ)= ∞ ( 1)n+1c λn, in particular: n=1 − n P c1 = Ttφµ(dx)= p(t,x,y)φ(y) dy µ(dx) (7.25) R R R Z Z Z t 2 c2 = p(t s,x,y) p(s,y,z)φ(z) dz dy ds µ(dx) R R − R Z Z0 Z Z (7.26) t = p(t s,x,y)p(s,y,z1)p(s,y,z1)φ(z1)φ(z2) dz1 dz2 dy µ(dx) ds R4 − Z0 Z t c4 = p(t s, x, w) R 0 − Z Z 2 s 2 (7.27) p(s s1,w,y) p(s,y,z)φ(z) dz dy ds1 ds dwµ(dx) R − R "Z0 Z Z # and so on. Taking now d = 1, Z = dx, a given Borel set A (R) and φ = I , we obtain from ( 7.25): 0 ∈B A E A, Zt = p(t,x,y)φ(y)dy dx = p(t,x,y) dx IA(y) dy h i R R R R Z Z Z Z = dy ZA i.e. E Z (dx) is the Lebesgue measure. { t } The covariance measure E(Zt(dx1)Zt(dx2)) can now be computed applying the same procedure to φ = λ I + λ I in which case we conclude that u(t),µ = F (λ , λ ) is again a Taylor series in λ , and λ , 1 A1 2 A2 − h i 1 2 1 2 with constant coefficient equal to zero, and only the coefficient of λ1λ2 has to be computed: t 2 T(Ttφ)= Tt−s[(Tsφ) ] ds − 0 Z t 2 I 2 I I 2 I 2 = Tt−s[λ1(Ts A1 ) +2λ1λ2Ts A1 Ts A2 + λ2(Ts A2 ) ] ds − 0 Z t t (7.28) 2 I 2 I I = λ1 Tt−s[(Ts A1 ) ] ds 2λ1λ2 Tt−s[Ts A1 Ts A2 ]ds − 0 − 0 · Z t Z λ2 T [(T I )2] ds − 2 t−s s A2 Z0 So: 2 1 ∂ F (λ1, λ2) E(Zt(A1)Zt(A2)) = 2 ∂λ1∂λ2 λ1=λ2=0 (7.29) t I I = Tt−s[Ts A1 Ts A2 ] µ(dx) ds. R · Z0 Z 16 In order to obtain the density for the measure E(Zt(dx1)Zt(dx2)), we write in detail the last integral as follows: t p(t s,x,y) p(s,y,z1)dz1 p(s,y,dz2)dz2 µ(dx) ds dy R R − · Z0 Z Z ZA1 ZA2 applying Fubini yields: t p(t s,x,y)p(s,y,z1)p(s,y,z2) dy µ(dx) ds dz1 dz2 R R − ZA1ZA2Z0 Z Z from which, we conclude that E(Zt(dx1)Zt(dx2)) has the following density w.r.t. Lebesgue measure: t p(t s,x,y)p(s,y,z1)p(s,y,z2) dy µ(dx) ds (7.30) R R − Z0 Z Z assuming µ(dx) is the Lebesgue measure in R and dx dx is the Lebesgue measure in R2 one obtains the 1 · 2 second moment measure (dx dx ) defined as: Mt 1 2 . t (dx dx ) = p(2s, x , x )ds dx dx (7.31) Mt 1 2 1 2 · 1 2 Z0 For the case t = t , assume t E φ Z φ Z = 1 t1 · 2 t2 Z Z = E E φ Z φ Z σ(Z ):0 r t 1 t1 · 2 t2 | r ≤ ≤ 1 Z Z = E φ1Zt E φ2Zt Z 1 · 2 |F t1 Z Z = E φ Z T φ Z 1 t1 · t2−t1 2 t1 Z Z = E φ Z T (φ )Z 1 t1 · t2−t1 2 t1 Z Z = E[ φ ,Z T (φ ),Z ] h 1 t1 i h t2−t1 2 t1 i I I so, replacing in ( 7.29) A2 by Tt2−t1 ( A2 ) and performing the same analysis, we can define following measure on R2 t1 (dx dx )= p(2s + t t , x , x )ds dx dx . Mt1t2 1 2 2 − 1 1 2 · 1 2 Z0 Finally, note that if Z0 = δ0(x), then (7.30) yields: t1 t1t2 (dx1dx2)= p(t1 s, 0,y)p(s,y,x1)p(t2 t1 + s,y,x2) dy ds dx1dx2 M R − − · Z0 Z Remark 7.1. The above procedure can also be used for any α (0, 2],β = 1, any dimension d 1 and any ∈ ≥ C0 semigroup St with probability transition function p(t,x,y). 17 7.3 Proof of Theorem 6.2 We can now use the results of the last subsection to determine properties of the solutions of (5.7). (i) It follows from the form of the characteristic function (given by (5.12) with β = 1 in (5.13)) that the log of the characteristic function of λφ, Xt is not a quadratic in λ and in fact the fourth cumulant is positive so that the random field is leptokurtic.h i (ii) Recall that t X(t, x)= p(t s,x,y)WZs (ds, dy) R − Z0 Z Lemma 7.2. t 2 E 4 E Zs(dy) [ Xt L ]= C ds k k 2 R √t s Z0 Z − (7.32) +2 p2(2t s s ,y ,y ) E (Z (dy )Z (dy )) ds ds . − 1 − 2 1 2 s1 1 s2 2 1 2 ZD2 Proof. Using the shorthand p = p(t s,x,y), dW s = W (ds, dy); p (x) = p(t s ,x,y ), p = p(t − Zs i − i i ij − si 2 2 si, xj ,yi), dW = WZ (dsi, dyi), for ,i =1, 4, j =1, 2 as well as 1 = [0,t] R, 2 = [0,t] R , 4 = si ··· D × D × D [0,t]4 R4, and noting that p = p (x ), we first compute: × ij i j 2 t 2 E 4 E [ Xt L Zs, 0 s t]= p(t s,x,y)WZs (ds, dy) dx k k 2 | ≤ ≤ R R − "Z Z0 Z # 2 2 s s = E p dW dx1 p dW dx2 R R "Z ZD1 # "Z ZD1 # s1 s2 = E p1(x1) dW p2(x1) dW dx1 R · Z ZD1 ZD1 s3 s4 p3(x2) dW p4(x2) dW dx2 · R · Z ZD1 ZD1 s1 s2 s3 s4 = E p11p21 dW dW dx1 p32p42 dW dW dx2 R R Z ZD2 Z ZD2 s1 s2 s3 s4 = E p11p21p32p42 dW dW dW dW dx1dx2 R2 Z ZD4 s1 s2 s3 s4 = p11p21p32p42 E(dW dW dW dW )dx1dx2 R2 Z ZD4 = I1 + I2 + I3. Each one of the above integrals can be evaluated using the independence of the increments of the Wiener process, according to the following cases: Case 1: s = s s = s , then: p p = p2 = p2(x ), p p = p2 = p2(x ), and E(dW s1 dW s2 dW s3 dW s4 )= 1 2 ∧ 3 4 11 21 11 1 1 32 42 32 3 2 Zs1 (dy1)ds1Zs3 (dy3)ds3, hence: 18 I 2 2 1 = p1(x1)p3(x2) Zs1 (dy1)ds1Zs3 (dy3)ds3dx1dx2 R2 Z ZD2 2 2 = p1(x1) Zs1 (dy1)ds1dx1 p3(x2) Zs3 (dy3)ds3dx2 R R Z ZD1 Z ZD1 2 2 = p Zs(dy)dsdx R Z ZD1 t 2 2 = p dx Zs(dy)ds R R Z0 Z Z 2 t Z (dy) = C s ds R √t s Z0 Z − Case 2: s = s s = s , then: p p = p p = p (x )p (x ), p p = p p = p (x )p (x ), and 1 3 ∧ 2 4 11 32 11 12 1 1 1 2 21 42 21 22 2 1 2 2 E s1 s2 s3 s4 (dW dW dW dW )= Zs1 (dy1)ds1Zs2 (dy2)ds2, hence: I 2 = p1(x1)p1(x2)p2(x1)p2(x2) Zs1 (dy1)ds1Zs2 (dy2)ds2dx1dx2 R2 Z ZD2 = p1(x1)p2(x1) dx1 p1(x2)p2(x2) dx2 Zs1 (dy1)ds1Zs2 (dy2)ds2dx1 R · R ZD2 Z Z = p2(2t s s ,y ,y ) Z (dy )ds Z (dy )ds − 1 − 2 1 2 s1 1 1 s2 2 2 ZD2 Case 3: s = s s = s , then: p p = p p = p (x )p (x ), p p = p p = p (x )p (x ), and 1 4 ∧ 2 3 11 42 11 12 1 1 1 2 21 32 21 22 2 1 2 2 E s1 s2 s3 s4 (dW dW dW dW )= Zs1 (dy1)ds1Zs2 (dy2)ds2, and we get the same result as before, namely: I = p2(2t s s ,y ,y ) Z (dy )ds Z (dy )ds 3 − 1 − 2 1 2 s1 1 1 s2 2 2 ZD2 Putting everything together, yields: 4 E[ Xt Zs, 0 s t] k kL2 | ≤ ≤ t 2 (7.33) Zs(dy) 2 = C ds +2 p (2t s1 s2,y1,y2) Zs1 (dy1)ds1Zs2 (dy2)ds2 R √t s − − Z0 Z − ZD2 So: 4 4 E[ Xt ]= E E[ Xt Zs, 0 s t] k kL2 k kL2 | ≤ ≤ t 2 (7.34) E Zs(dy) 2 E = C ds +2 p (2t s1 s2,y1,y2) (Zs1 (dy1)Zs2 (dy2)) ds1ds2. R √t s − − Z0 Z − ZD2 We now return to the proof of Theorem 6.2 19 Proof. The first term on the right of 7.32 is evaluated below 2 t Z (dx) E s ds = R √t s Z0 Z − t t E 1 = Zs1 (dx1)Zs2 (dx2) ds1 ds2 " 0 0 (t s1)(t s2) R R ! # Z Z − − Z Z (7.35) t s2 p 1 E = [Zs1 (dx1)Zs2 (dx2)] ds1 ds2 [(t s )(t s )]1/2 R R Z0 Z0 1 2 Z Z t t − − 1 E + [Zs1 (dx1)Zs2 (dx2)] ds1 ds2 [(t s )(t s )]1/2 R R Z0 Zs2 − 1 − 2 Z Z Assume now Z = δ (x), and s s , using ( 7.21), one obtains: 0 0 1 ≤ 2 s1 s1 E [Zs1 (dx1)Zs2 (dx2)] = p(s1 s, 0,y) dy ds = ds = s1 R R R − Z Z Z0 Z Z0 and similarly, for s s : 2 ≤ 1 E [Zs1 (dx1)Zs2 (dx2)] = s2 R R Z Z so that ( 7.35) can be written as: t 2 t s2 t t E Zs(dx) s1ds1ds2 s2ds1ds2 ds = 1/2 + 1/2 R √t s [(t s1)(t s2)] [(t s1)(t s2)] Z0 Z − Z0 Z0 − − Z0 Zs2 − − the first of the above integrals one can be directly done: 1 s2 s ds 1 4 2 1 1 = t3/2 + (t s )3/2 2t(t s )1/2 (t s )1/2 (t s )1/2 (t s )1/2 3 3 − 2 − − 2 − 2 Z0 − 1 − 2 1 4 2 t3/2 + (t s )3/2 ≤ (t s )1/2 3 3 − 2 − 2 4 t3/2 2 = + . 3 (t s )1/2 3 − 2 Hence: t s2 s ds ds 4 2 1 1 2 t2 + t2 =2t2 [(t s )(t s )]1/2 ≤ 3 3 Z0 Z0 − 1 − 2 similarly, for the second integral: s t ds 2 1 = s (t s )1/2 (t s )1/2 2 − 2 Zs2 − 1 and: t t s ds ds t2 2 1 2 . [(t s )(t s )]1/2 ≤ 2 Z0 Zs2 − 1 − 2 Both inequalities together imply: t 2 Zs(dx) 2 E ds C1t (7.36) R √t s ≤ Z0 Z − Similarly, we estimate below the second integral of ( ??): 20 p2(2t s s , x , x ) E (Z (dx )Z (dx )) ds ds = − 1 − 2 1 2 s1 1 s2 2 1 2 ZD2 t s2 2 E = p (2t s1 s2, x1, x2) (Zs1 (dx1)Zs2 (dx2)) ds1ds2 R R − − Z0 Z0 Z Z t t 2 E + p (2t s1 s2, x1, x2) (Zs1 (dx1)Zs2 (dx2)) ds1ds2 R R − − Z0 Zs2 Z Z when 0 s s . Using ( 7.21) with 0 s s s t, we obtain the following estimate for: ≤ 1 ≤ 2 ≤ ≤ 1 ≤ 2 ≤ I . 2 E 1(t,s1,s2) = p (2t s1 s2, x1, x2) (Zs1 (dx1)Zs2 (dx2)) R R − − Z Z √2 s2 1 e= ds √2t s1 s2 0 (2t s s )+4s − − Z − 1 − 2 √2 s2 1 p ds ≤ √2t s1 s2 √4s − − Z0 s C 2 ≤ 1 2t s s r − 1 − 2 so, we obtain: t s2 t s2 s I = I (t,s ,s ) ds ds C 2 ds ds 1 1 1 2 1 2 ≤ 1 2t s s 1 2 Z0 Z0 Z0 Z0 r 1 2 t − − e = C √s 2√2t s 2√2t 2s ds (7.37) 1 2 − 2 − − 2 2 Z0 t C √s √tds C t2. ≤ 2 2 2 ≤ 3 Z0 When 0 s s , using ( 7.21) with 0 s s s t, and following the same steps above for the ≤ 2 ≤ 1 ≤ ≤ 2 ≤ 1 ≤ integral: I . 2 E 2 = p (2t s1 s2, x1, x2) (Zs2 (dx1)Zs1 (dx2)) R R − − Z Z t e C ≤ 1 2t s s r − 1 − 2 therefore: t s2 t t t I = I (t,s ,s ) ds ds C ds ds 2 2 1 2 1 2 ≤ 4 2t s s 1 2 Z0 Z0 Z0 Zs2 r − 1 − 2 t e = C √t 2√2t 2s 2√t s ds (7.38) 5 − 2 − − 2 2 Z0 t C √t√tds = C t2. ≤ 6 2 6 Z0 Gathering ( 7.36), ( 7.37) and ( 7.38) yields: 4 2 E[ Xt ] Ct . k kL2 ≤ In the same way we can obtain the estimates 4 2 E[ Xt Xs ] C(t s) . k − kL2 ≤ − for the increments which shows the continuity of the paths using Kolmogorov’s criteria. 21 8 Comments and Open Problems 1. 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