Introduction
Spatial averages of stochastic partial differential equations (SPDEs) are known to exhibit macroscopic Gaussian fluctuations, i.e. they satisfy a central limit theorem. For the Kardar-Parisi-Zhang (KPZ) equation with narrow wedge initial data, finite-dimensional convergence is known from [21 ] , but temporal increments require separate control to obtain convergence of paths. The corresponding functional limit theorem has been left open in [21 ] . The goal of this article is to prove the needed tightness on compact time intervals bounded away from zero.
A starting point for central limit theorems for spatial averages is the work of [23 ] on the stochastic heat equation (SHE) with flat initial condition u ( 0 , x ) = 1 u(0, x) = 1 u ( 0 , x ) = 1 σ \sigma σ σ ( u ) = u \sigma(u)=u σ ( u ) = u parabolic Anderson model (PAM):
∂ t u ( t , x ) = 1 2 ∂ x 2 u ( t , x ) + u ( t , x ) ξ ( t , x ) , t > 0 , x ∈ R , u ( 0 , ⋅ ) = 1. (1) \partial_t u(t,x)=\frac{1}{2}\partial_x^2u(t,x)+u(t,x)\xi(t,x),\qquad t>0,x\in\mathbb{R},\qquad u(0,\cdot)=1. \tag*{(1)} ∂ t u ( t , x ) = 2 1 ∂ x 2 u ( t , x ) + u ( t , x ) ξ ( t , x ) , t > 0 , x ∈ R , u ( 0 , ⋅ ) = 1. (1) The mild solution of (1) is a jointly measurable, ( F t ) (\mathcal{F}_t) ( F t )
u ( t , x ) = 1 + ∫ 0 t ∫ R p t − s ( x − y ) u ( s , y ) ξ ( d s , d y ) , u(t,x)=1+\int_0^t\int_{\mathbb{R}}p_{t-s}(x-y)u(s,y)\xi(\mathrm{d}s,\mathrm{d}y), u ( t , x ) = 1 + ∫ 0 t ∫ R p t − s ( x − y ) u ( s , y ) ξ ( d s , d y ) , where the integral is interpreted in the Itô sense, ( F t ) t ≥ 0 (\mathcal{F}_t)_{t\geq0} ( F t ) t ≥ 0 ξ \xi ξ (9) below), and p t ( x ) p_t(x) p t ( x )
p t ( x ) = 1 2 π t exp ( − ∣ x ∣ 2 2 t ) . p_t(x)=\frac{1}{\sqrt{2\pi t}}\exp\left(-\frac{|x|^2}{2t}\right). p t ( x ) = 2 π t 1 exp ( − 2 t ∣ x ∣ 2 ) . The authors of [23 ] studied the asymptotic behaviour as N → ∞ N\to\infty N → ∞ spatial average :
F N ( t ) = ∫ 0 N ( u ( t , x ) − 1 ) d x . F_N(t)=\int_0^N\bigl(u(t,x)-1\bigr)\mathrm{d}x. F N ( t ) = ∫ 0 N ( u ( t , x ) − 1 ) d x . Their study was motivated by the fact that, after proper normalization, F N ( t ) F_N(t) F N ( t ) F N ∗ ( t ) F_N^*(t) F N ∗ ( t ) u N ∗ ( t , x ) u_N^*(t,x) u N ∗ ( t , x ) { u N ∗ ( t , x i ) } i ≥ 1 \{u_N^*(t,x_i)\}_{i\geq1} { u N ∗ ( t , x i ) } i ≥ 1 x i x_i x i [16 ] . By writing the interval [ 0 , N ] [0,N] [ 0 , N ] F N ( t ) F_N(t) F N ( t ) { u ( t , x ) } x ∈ R \{u(t,x)\}_{x\in\mathbb{R}} { u ( t , x ) } x ∈ R
ρ t u ( x ) : = Cov ( u ( t , x ) , u ( t , 0 ) ) = ∫ 0 t p 2 ( t − s ) ( x ) E ∣ u ( s , 0 ) ∣ 2 d s \rho_t^u(x):=\operatorname{Cov}(u(t,x),u(t,0))=\int_0^t p_{2(t-s)}(x)\mathbb{E}|u(s,0)|^2\mathrm{d}s ρ t u ( x ) := Cov ( u ( t , x ) , u ( t , 0 )) = ∫ 0 t p 2 ( t − s ) ( x ) E ∣ u ( s , 0 ) ∣ 2 d s
Var ( F N ( t ) ) = ∫ 0 N ∫ 0 N ρ t u ( x − y ) d x d y = 2 ∫ 0 N ρ t u ( x ) ( N − x ) d x ∼ N K t , \operatorname{Var}(F_N(t))=\int_0^N\int_0^N\rho_t^u(x-y)\mathrm{d}x\mathrm{d}y=2\int_0^N\rho_t^u(x)(N-x)\mathrm{d}x\sim NK_t, Var ( F N ( t )) = ∫ 0 N ∫ 0 N ρ t u ( x − y ) d x d y = 2 ∫ 0 N ρ t u ( x ) ( N − x ) d x ∼ N K t , as N → ∞ N\to\infty N → ∞ K t : = ∫ R ρ t u ( x ) d x = ∫ 0 t E ∣ u ( s , 0 ) ∣ 2 d s < ∞ K_t:=\int_{\mathbb{R}}\rho_t^u(x)\mathrm{d}x=\int_0^t\mathbb{E}|u(s,0)|^2\mathrm{d}s<\infty K t := ∫ R ρ t u ( x ) d x = ∫ 0 t E ∣ u ( s , 0 ) ∣ 2 d s < ∞ E ∣ u ( t , 0 ) ∣ 2 = 2 e t / 4 Φ ( t / 2 ) \mathbb{E}|u(t,0)|^2=2e^{t/4}\Phi(\sqrt{t}/2) E ∣ u ( t , 0 ) ∣ 2 = 2 e t /4 Φ ( t /2 ) Φ \Phi Φ [23 ] using a combination between Stein method and Malliavin calculus, which provides a rate for the convergence in the total variation distance, called a quantitative CLT. Moreover, the authors of [23 ] proved a corresponding functional CLT:
1 N F N ( ⋅ ) → d G ( ⋅ ) in C [ 0 , T ] , \frac{1}{\sqrt{N}}F_N(\cdot)\xrightarrow{d}G(\cdot)\quad\text{in }C[0,T], N 1 F N ( ⋅ ) d G ( ⋅ ) in C [ 0 , T ] , where G G G E [ G ( t ) G ( s ) ] = ∫ 0 t ∧ s E ∣ u ( r , 0 ) ∣ 2 d r \mathbb{E}[G(t)G(s)]=\int_0^{t\wedge s}\mathbb{E}|u(r,0)|^2\mathrm{d}r E [ G ( t ) G ( s )] = ∫ 0 t ∧ s E ∣ u ( r , 0 ) ∣ 2 d r C [ a , b ] C[a,b] C [ a , b ] f : [ a , b ] → R f:[a,b]\to\mathbb{R} f : [ a , b ] → R
Article [23 ] sparked the development of a large area dedicated to similar QCLTs and FCLTs for a variety of equations and noises. We refer the reader to [3 , 4 , 5 , 6 , 7 , 8 , 17 , 18 , 19 , 20 , 27 , 29 , 30 , 31 , 32 ] for a sample of references. The intuition behind these results is the same: the noise may be correlated in space, but as long as the correlation between the big blocks decays fast enough, the resulting variables remain weakly dependent. Moreover, the same intuition applies to a function g g g [12 ] , the authors proved that
1 N d / 2 ∫ [ 0 , N ] d ( g ( u ( t , x ) ) − E [ g ( u ( t , x ) ) ] ) d x → d N ( 0 , c t ) \frac{1}{N^{d/2}}\int_{[0,N]^d}\left(g(u(t,x))-\mathbb{E}[g(u(t,x))]\right)\,dx \xrightarrow{d} N(0,c_t) N d /2 1 ∫ [ 0 , N ] d ( g ( u ( t , x )) − E [ g ( u ( t , x ))] ) d x d N ( 0 , c t ) where u u u g g g [14 ] to locally Lipschitz functions g g g g ( u ) = log u g(u)=\log u g ( u ) = log u h f l a t = log u h^{\mathrm{flat}}=\log u h flat = log u [26 ] .
The KPZ equation is formally written as:
∂ t h ( t , x ) = 1 2 ∂ x 2 h ( t , x ) + 1 2 ( ∂ x h ( t , x ) ) 2 + ξ ( t , x ) , t > 0 , x ∈ R , (2) \partial_t h(t,x)=\frac{1}{2}\partial_x^2h(t,x)+\frac{1}{2}\left(\partial_xh(t,x)\right)^2+\xi(t,x),\quad t>0,x\in\mathbb{R}, \tag*{(2)} ∂ t h ( t , x ) = 2 1 ∂ x 2 h ( t , x ) + 2 1 ( ∂ x h ( t , x ) ) 2 + ξ ( t , x ) , t > 0 , x ∈ R , (2) and models the growth of a randomly evolving surface, for instance a surface formed by particles landing and accumulating at random. Its solution h ( t , x ) h(t,x) h ( t , x ) t t t x x x ( ∂ x h ) 2 (\partial_xh)^2 ( ∂ x h ) 2 [22 ] .
In the present article, we consider the Cole-Hopf transform
h ( t , x ) = log Z ( t , x ) (3) h(t,x)=\log Z(t,x) \tag*{(3)} h ( t , x ) = log Z ( t , x ) (3) of the solution of (PAM) with Dirac delta initial condition:
∂ t Z ( t , x ) = 1 2 ∂ x 2 Z ( t , x ) + Z ( t , x ) ξ ( t , x ) , t > 0 , x ∈ R Z ( 0 , ⋅ ) = δ 0 . (4) \partial_tZ(t,x)=\frac{1}{2}\partial_x^2Z(t,x)+Z(t,x)\xi(t,x),\quad t>0,\ x\in\mathbb{R}\qquad Z(0,\cdot)=\delta_0. \tag*{(4)} ∂ t Z ( t , x ) = 2 1 ∂ x 2 Z ( t , x ) + Z ( t , x ) ξ ( t , x ) , t > 0 , x ∈ R Z ( 0 , ⋅ ) = δ 0 . (4) We say that h = h n w h=h^{\mathrm{nw}} h = h nw δ 0 \delta_0 δ 0 Z ε ( 0 , x ) = 1 2 ε exp ( − ∣ x ∣ ε ) Z_\varepsilon(0,x)=\frac{1}{2\varepsilon}\exp\left(-\frac{|x|}{\varepsilon}\right) Z ε ( 0 , x ) = 2 ε 1 exp ( − ε ∣ x ∣ ) h ( 0 , ⋅ ) = lim ε → 0 + h ε ( 0 , ⋅ ) h(0,\cdot)=\lim_{\varepsilon\to0+}h_\varepsilon(0,\cdot) h ( 0 , ⋅ ) = lim ε → 0 + h ε ( 0 , ⋅ )
h ε ( 0 , x ) = log Z ε ( 0 , x ) = − log ( 2 ε ) − ∣ x ∣ ε . h_\varepsilon(0,x)=\log Z_\varepsilon(0,x)=-\log(2\varepsilon)-\frac{|x|}{\varepsilon}. h ε ( 0 , x ) = log Z ε ( 0 , x ) = − log ( 2 ε ) − ε ∣ x ∣ . Figure 1 shows some profiles of h ε h_\varepsilon h ε ε \varepsilon ε
Figure 1. Narrow wedge profiles h ε ( 0 , x ) = − log ( 2 ε ) − ∣ x ∣ / ε h_\varepsilon(0,x)=-\log(2\varepsilon)-|x|/\varepsilon h ε ( 0 , x ) = − log ( 2 ε ) − ∣ x ∣/ ε ε = 1 \varepsilon=1 ε = 1 0.5 0.5 0.5 0.2 0.2 0.2
Following Walsh [33 ] , we interpret the solution of (1.4) in the mild sense, i.e. we assume that { Z ( t , x ) ; t ≥ 0 , x ∈ R } \{Z(t,x);t\geq0,x\in\mathbb{R}\} { Z ( t , x ) ; t ≥ 0 , x ∈ R } ( F t ) (\mathcal{F}_t) ( F t )
Z ( t , x ) = p t ( x ) + ∫ 0 t ∫ R p t − s ( x − y ) Z ( s , y ) ξ ( d s , d y ) . (5) Z(t,x)=p_t(x)+\int_0^t\int_{\mathbb{R}}p_{t-s}(x-y)Z(s,y)\xi(\mathrm{d}s,\mathrm{d}y). \tag*{(5)} Z ( t , x ) = p t ( x ) + ∫ 0 t ∫ R p t − s ( x − y ) Z ( s , y ) ξ ( d s , d y ) . (5) where the stochastic integral is interpreted in the Itô sense. The normalization below removes the deterministic spatial profile and allows us to use stationarity. Indeed, the process
Z ‾ ( t , x ) = Z ( t , x ) p t ( x ) , x ∈ R , \overline{Z}(t,x)=\frac{Z(t,x)}{p_t(x)},\qquad x\in\mathbb{R}, Z ( t , x ) = p t ( x ) Z ( t , x ) , x ∈ R , is strictly stationary (see Proposition 1.4 of [2 ] ). Moreover, Z ‾ \overline{Z} Z
Z ‾ ( t , x ) = 1 + ∫ 0 t ∫ R p s ( t − s ) t ( y − s t x ) Z ‾ ( s , y ) ξ ( d s , d y ) , \overline{Z}(t,x)=1+\int_0^t\int_{\mathbb{R}}p_{\frac{s(t-s)}{t}}\left(y-\frac{s}{t}x\right)\overline{Z}(s,y)\xi(\mathrm{d}s,\mathrm{d}y), Z ( t , x ) = 1 + ∫ 0 t ∫ R p t s ( t − s ) ( y − t s x ) Z ( s , y ) ξ ( d s , d y ) , using property (3.3) below of the heat kernel. In [13 ] , the authors proved that its spatial average S ‾ N ( t ) = ∫ 0 N ( Z ‾ ( t , x ) − 1 ) d x \overline{S}_N(t)=\int_0^N(\overline{Z}(t,x)-1)\mathrm{d}x S N ( t ) = ∫ 0 N ( Z ( t , x ) − 1 ) d x
1 N log N S ‾ N ( ⋅ ) → d 2 B ( ⋅ ) in C [ 0 , T ] , \frac{1}{\sqrt{N\log N}}\overline{S}_N(\cdot)\xrightarrow{d}\sqrt{2}B(\cdot)\quad\text{in }C[0,T], N log N 1 S N ( ⋅ ) d 2 B ( ⋅ ) in C [ 0 , T ] , where B B B Var ( S ‾ N ( t ) ) ∼ 2 t N log N \operatorname{Var}(\overline{S}_N(t))\sim2tN\log N Var ( S N ( t )) ∼ 2 tN log N [13 ] . Alternatively, this follows from stationarity and the fact that
ρ ‾ t ( x ) : = Cov ( Z ‾ ( t , x ) , Z ‾ ( t , 0 ) ) = ∫ 0 t E ∣ Z ‾ ( s , 0 ) ∣ 2 p 2 s ( t − s ) t ( s t x ) d s ∼ t x as x → ∞ . (6) \overline{\rho}_t(x):=\operatorname{Cov}(\overline{Z}(t,x),\overline{Z}(t,0))=\int_0^t\mathbb{E}|\overline{Z}(s,0)|^2p_{\frac{2s(t-s)}{t}}\left(\frac{s}{t}x\right)\mathrm{d}s\sim\frac{t}{x}\quad\text{as }x\to\infty. \tag*{(6)} ρ t ( x ) := Cov ( Z ( t , x ) , Z ( t , 0 )) = ∫ 0 t E ∣ Z ( s , 0 ) ∣ 2 p t 2 s ( t − s ) ( t s x ) d s ∼ x t as x → ∞. (6) where the second-moment formula (2.31) of [9 ] (with ν = λ = 1 \nu=\lambda=1 ν = λ = 1
m ( s ) : = E ∣ Z ‾ ( s , 0 ) ∣ 2 = 1 + π s e s / 4 Φ ( s / 2 ) . m(s):=\mathbb{E}|\overline{Z}(s,0)|^2=1+\sqrt{\pi s}\,e^{s/4}\Phi\left(\sqrt{s/2}\right). m ( s ) := E ∣ Z ( s , 0 ) ∣ 2 = 1 + π s e s /4 Φ ( s /2 ) . In particular, m ( s ) ≥ 1 m(s) \ge1 m ( s ) ≥ 1 m ( s ) → 1 m(s) \to1 m ( s ) → 1 s ↓ 0 s \downarrow0 s ↓ 0 s / t s/t s / t z = s x / t z = sx/t z = s x / t
∫ 0 ∞ ρ t Z ‾ ( x ) d x = t 2 ∫ 0 t m ( s ) s d s = ∞ . \int_0^\infty\overline{\rho_t^Z}(x)\,\mathrm{d}x = \frac{t}{2}\int_0^t \frac{m(s)}{s}\,\mathrm{d}s = \infty. ∫ 0 ∞ ρ t Z ( x ) d x = 2 t ∫ 0 t s m ( s ) d s = ∞. The covariance tail and variance asymptotic are given by Proposition A.6 below.
There is a natural connection between Z ‾ \overline{Z} Z h h h log Z ‾ ( t , x ) = h ( t , x ) − log p t ( x ) \log\overline{Z}(t,x) = h(t,x) - \log p_t(x) log Z ( t , x ) = h ( t , x ) − log p t ( x )
h ( t , x ) − E [ h ( t , x ) ] = log Z ‾ ( t , x ) − E [ log Z ‾ ( t , x ) ] , x ∈ R h(t,x) - \mathbb{E}[h(t,x)] = \log\overline{Z}(t,x) - \mathbb{E}[\log\overline{Z}(t,x)], \qquad x \in\mathbb{R} h ( t , x ) − E [ h ( t , x )] = log Z ( t , x ) − E [ log Z ( t , x )] , x ∈ R is strictly stationary. It turns out that the spatial average of h h h
S N ( t ) = ∫ 0 N ( h ( t , x ) − E [ h ( t , x ) ] ) d x , (7) S_N(t) = \int_0^N \bigl(h(t,x) - \mathbb{E}[h(t,x)]\bigr)\,\mathrm{d}x, \tag*{(7)} S N ( t ) = ∫ 0 N ( h ( t , x ) − E [ h ( t , x )] ) d x , (7) has a similar asymptotic behaviour to S ‾ N ( t ) \overline{S}_N(t) S N ( t ) [21 ] proved that
Cov ( h ( t , x ) , h ( t , 0 ) ) ∼ t x as x → ∞ , (8) \operatorname{Cov}(h(t,x),h(t,0)) \sim\frac{t}{x} \qquad\text{as } x \to\infty, \tag*{(8)} Cov ( h ( t , x ) , h ( t , 0 )) ∼ x t as x → ∞ , (8) and consequently, Var ( S N ( t ) ) ∼ 2 t N log N \operatorname{Var}(S_N(t)) \sim2tN\log N Var ( S N ( t )) ∼ 2 tN log N N → ∞ N \to\infty N → ∞ [11 ] showed that
Cov ( h f l a t ( t , x ) , h f l a t ( t , 0 ) ) ∼ 2 κ ( t ) π t 3 / 2 ∣ x ∣ − 2 exp ( − x 2 4 t ) , as x → ∞ . \operatorname{Cov}\bigl(h^{\mathrm{flat}}(t,x),h^{\mathrm{flat}}(t,0)\bigr) \sim\frac{2\kappa(t)}{\sqrt{\pi}}t^{3/2}|x|^{-2}\exp\left(-\frac{x^2}{4t}\right), \qquad\text{as } x \to\infty. Cov ( h flat ( t , x ) , h flat ( t , 0 ) ) ∼ π 2 κ ( t ) t 3/2 ∣ x ∣ − 2 exp ( − 4 t x 2 ) , as x → ∞. for a certain constant κ ( t ) > 0 \kappa(t) > 0 κ ( t ) > 0 Var ( S N f l a t ( t ) ) ∼ C t N \operatorname{Var}(S_N^{\mathrm{flat}}(t)) \sim C_tN Var ( S N flat ( t )) ∼ C t N S N f l a t ( t ) S_N^{\mathrm{flat}}(t) S N flat ( t ) (7) with h h h h f l a t h^{\mathrm{flat}} h flat [14 ] , it was already known that S N f l a t S_N^{\mathrm{flat}} S N flat
1 N S N f l a t ( ⋅ ) → d G ( ⋅ ) in C [ 0 , T ] , \frac{1}{\sqrt{N}}S_N^{\mathrm{flat}}(\cdot) \xrightarrow{d} G(\cdot) \quad\text{in } C[0,T], N 1 S N flat ( ⋅ ) d G ( ⋅ ) in C [ 0 , T ] , where G G G
E [ G ( t ) G ( s ) ] = ∫ R Cov ( h f l a t ( t , x ) , h f l a t ( s , 0 ) ) d x . \mathbb{E}[G(t)G(s)] = \int_{\mathbb{R}} \operatorname{Cov}\bigl(h^{\mathrm{flat}}(t,x),h^{\mathrm{flat}}(s,0)\bigr)\,\mathrm{d}x. E [ G ( t ) G ( s )] = ∫ R Cov ( h flat ( t , x ) , h flat ( s , 0 ) ) d x . In addition to (8) , the authors of [21 ] have also proved that as N → ∞ N \to\infty N → ∞
1 N log N S N ( ⋅ ) → f . d . d . 2 B ( ⋅ ) , \frac{1}{\sqrt{N\log N}}S_N(\cdot) \xrightarrow{\mathrm{f.d.d.}} \sqrt{2}B(\cdot), N log N 1 S N ( ⋅ ) f.d.d. 2 B ( ⋅ ) , where → f . d . d . \xrightarrow{\mathrm{f.d.d.}} f.d.d.
The following theorem is the main result of the present article.
Theorem 1.1. For any N ≥ 2 N \ge2 N ≥ 2 S N ( ⋅ ) = { S N ( t ) } t > 0 S_N(\cdot)=\{S_N(t)\}_{t>0} S N ( ⋅ ) = { S N ( t ) } t > 0 γ \gamma γ γ ∈ ( 0 , 1 4 ] \gamma\in(0,\frac{1}{4}] γ ∈ ( 0 , 4 1 ] S N ( ⋅ ) S_N(\cdot) S N ( ⋅ ) N → ∞ N\to\infty N → ∞
1 N log N S N ( ⋅ ) → d 2 B ( ⋅ ) , in C [ ε , T ] , \frac{1}{\sqrt{N\log N}}S_N(\cdot)\xrightarrow{d}\sqrt{2}B(\cdot),\quad\text{in }C[\varepsilon,T], N log N 1 S N ( ⋅ ) d 2 B ( ⋅ ) , in C [ ε , T ] , for any 0 < ε < T 0<\varepsilon<T 0 < ε < T ( B t ) t ≥ 0 (B_t)_{t\ge0} ( B t ) t ≥ 0 → d \xrightarrow{d} d
We state Theorem 1.1 on [ ε , T ] [\varepsilon,T] [ ε , T ] ε \varepsilon ε ε ↓ 0. \varepsilon\downarrow0. ε ↓ 0. Z ‾ − 1 \overline{Z}-1 Z − 1 [13 ] and Proposition 6.1 ibid. combine a quantitative small-time bound on its normalized spatial average with increment estimates whose deterioration near zero is explicitly controlled. Our observable is log Z ‾ − E [ log Z ‾ ] \log\overline{Z}-\mathbb{E}[\log\overline{Z}] log Z − E [ log Z ] Z ‾ \overline{Z} Z [ 0 , T ] [0,T] [ 0 , T ]
The limiting finite-dimensional distributions are already supplied by [21 ] . To prove Theorem 1.1, we establish a moment bound for time increments of the normalized spatial integral, uniformly in the averaging length. The obstruction is that temporal increments of the normalized narrow-wedge SHE are not stationary in space, so the estimates used for flat initial data do not transfer directly; see Remark 4.3 of [21 ] .
As in [21 ] , we will use the Clark-Ocone formula to express X N ( t ) X_N(t) X N ( t ) ξ ( d s , d y ) \xi(\mathrm{d}s,\mathrm{d}y) ξ ( d s , d y ) d x \mathrm{d}x d x [ 0 , N ] [0,N] [ 0 , N ] A ( t , x ; s , y ) q t , x ( y ) \mathcal{A}(t,x;s,y)q_{t,x}(y) A ( t , x ; s , y ) q t , x ( y ) [21 ] , we will not consider the projection of X N ( t ) X_N(t) X N ( t )
The main difficulty is to reduce the increment A ( t 2 , x ; s , y ) q t 2 , x ( y ) − A ( t 1 , x ; s , y ) q t 1 , x ( y ) \mathcal{A}(t_2,x;s,y)q_{t_2,x}(y)-\mathcal{A}(t_1,x;s,y)q_{t_1,x}(y) A ( t 2 , x ; s , y ) q t 2 , x ( y ) − A ( t 1 , x ; s , y ) q t 1 , x ( y ) same variable t 1 t_1 t 1 d x \mathrm{d}x d x x = t 1 u x=t_1u x = t 1 u x = t 2 u x=t_2u x = t 2 u u u u
The key ingredient is the fact that the Malliavin derivative R ( t , x ; s , y ) = D s , y h ( t , x ) \mathcal{R}(t,x;s,y)=D_{s,y}h(t,x) R ( t , x ; s , y ) = D s , y h ( t , x )
The remainder of this article is organized in two sections: in Section 2, we introduce some preliminary material, while in Section 3 we present the proof of Theorem 1.1. Appendix A contains some auxiliary material which is used in the paper.
We end the introduction with a few words about the notation. For any p ≥ 1 p\ge1 p ≥ 1 ∥ ⋅ ∥ p \|\cdot\|_p ∥ ⋅ ∥ p L p ( Ω ) L^p(\Omega) L p ( Ω ) C a , b , … C_{a,b,\ldots} C a , b , … a , b , … a,b,\ldots a , b , …
Preliminaries
In this section, we include some preliminary material which will be used below. We assume that ξ \xi ξ ( Ω , F , P ) (\Omega,\mathcal{F},\mathbb{P}) ( Ω , F , P ) ( F t ) t ≥ 0 (\mathcal{F}_t)_{t\geq0} ( F t ) t ≥ 0 ξ \xi ξ
F t = σ { ξ ( [ 0 , s ] × A ) ; s ∈ [ 0 , t ] , A ∈ B b ( R ) } ∨ N , (9) \mathcal{F}_t=\sigma\{\xi([0,s]\times A);\,s\in[0,t],\,A\in\mathcal{B}_b(\mathbb{R})\}\vee\mathcal{N}, \tag*{(9)} F t = σ { ξ ([ 0 , s ] × A ) ; s ∈ [ 0 , t ] , A ∈ B b ( R )} ∨ N , (9) where B b ( R ) \mathcal{B}_b(\mathbb{R}) B b ( R ) R \mathbb{R} R N \mathcal{N} N P \mathbb{P} P
A random process u = { u ( t , x ) ; t ≥ 0 , x ∈ R } u=\{u(t,x);t\geq0,x\in\mathbb{R}\} u = { u ( t , x ) ; t ≥ 0 , x ∈ R } ( F t ) (\mathcal{F}_t) ( F t ) u ( t , x ) u(t,x) u ( t , x ) F t \mathcal{F}_t F t t > 0 t>0 t > 0 x ∈ R x\in\mathbb{R} x ∈ R u u u ( ω , t , x ) ↦ u ( ω , t , x ) (\omega,t,x)\mapsto u(\omega,t,x) ( ω , t , x ) ↦ u ( ω , t , x ) F ⊗ B ( R + ) ⊗ B ( R ) \mathcal{F}\otimes\mathcal{B}(\mathbb{R}_+)\otimes\mathcal{B}(\mathbb{R}) F ⊗ B ( R + ) ⊗ B ( R )
Malliavin calculus
In this section, we include some basic facts on Malliavin calculus with respect to the isonormal Gaussian process { ξ ( φ ) } φ ∈ H \{\xi(\varphi)\}_{\varphi\in\mathcal{H}} { ξ ( φ ) } φ ∈ H
E [ ξ ( g ) ξ ( h ) ] = ⟨ g , h ⟩ H with H = L 2 ( R + × R ) . \mathbb{E}[\xi(g)\xi(h)]=\langle g,h\rangle_{\mathcal{H}}\quad\text{with}\quad\mathcal{H}=L^2(\mathbb{R}_+\times\mathbb{R}). E [ ξ ( g ) ξ ( h )] = ⟨ g , h ⟩ H with H = L 2 ( R + × R ) . F = f ( ξ ( φ 1 ) , … , ξ ( φ n ) ) , (10) F=f(\xi(\varphi_1),\ldots,\xi(\varphi_n)), \tag*{(10)} F = f ( ξ ( φ 1 ) , … , ξ ( φ n )) , (10) where f ∈ C b ∞ ( R n ) f\in C_b^\infty(\mathbb{R}^n) f ∈ C b ∞ ( R n ) φ i ∈ H \varphi_i\in\mathcal{H} φ i ∈ H n ≥ 1 n\geq1 n ≥ 1 C b ∞ ( R n ) C_b^\infty(\mathbb{R}^n) C b ∞ ( R n ) C ∞ C^\infty C ∞ R n \mathbb{R}^n R n F F F (10) is the H \mathcal{H} H
D t , x F : = ∑ i = 1 n ∂ f ∂ x i ( ξ ( φ 1 ) , … , ξ ( φ n ) ) φ i ( t , x ) , for t > 0 and x ∈ R . D_{t,x}F:=\sum_{i=1}^{n}\frac{\partial f}{\partial x_i}(\xi(\varphi_1),\ldots,\xi(\varphi_n))\varphi_i(t,x),\quad\text{for }t>0\text{ and }x\in\mathbb{R}. D t , x F := i = 1 ∑ n ∂ x i ∂ f ( ξ ( φ 1 ) , … , ξ ( φ n )) φ i ( t , x ) , for t > 0 and x ∈ R . We endow S \mathcal{S} S ∥ F ∥ D 1 , 2 = ( E ∣ F ∣ 2 + E ∥ D F ∥ H 2 ) 1 / 2 \|F\|_{\mathbb{D}^{1,2}}=(\mathbb{E}|F|^2+\mathbb{E}\|DF\|_{\mathcal{H}}^2)^{1/2} ∥ F ∥ D 1 , 2 = ( E ∣ F ∣ 2 + E ∥ D F ∥ H 2 ) 1/2 D D D D 1 , 2 \mathbb{D}^{1,2} D 1 , 2 S \mathcal{S} S ∥ ⋅ ∥ D 1 , 2 \|\cdot\|_{\mathbb{D}^{1,2}} ∥ ⋅ ∥ D 1 , 2
If F ∈ D 1 , 2 F\in\mathbb{D}^{1,2} F ∈ D 1 , 2 F t \mathcal{F}_t F t
D r , z F = 0 for almost every ( r , z ) ∈ ( t , ∞ ) × R , a.s. (11) D_{r,z}F=0\quad\text{for almost every }(r,z)\in(t,\infty)\times\mathbb{R},\ \text{a.s.} \tag*{(11)} D r , z F = 0 for almost every ( r , z ) ∈ ( t , ∞ ) × R , a.s. (11) Itô integral. The Itô integral ∫ 0 t ∫ R g ( s , x ) ξ ( d s , d x ) \int_0^t\int_{\mathbb{R}}g(s,x)\xi(ds,dx) ∫ 0 t ∫ R g ( s , x ) ξ ( d s , d x ) t ≥ 0 t\geq0 t ≥ 0 g g g E ∫ 0 t ∫ R g 2 ( s , x ) d x d s < ∞ \mathbb{E}\int_0^t\int_{\mathbb{R}}g^2(s,x)\,dx\,ds<\infty E ∫ 0 t ∫ R g 2 ( s , x ) d x d s < ∞ t > 0 t>0 t > 0 p ≥ 2 p\geq2 p ≥ 2
∥ ∫ 0 T ∫ R g ( t , x ) ξ ( d t , d x ) ∥ p 2 ≤ z p 2 ∫ 0 T ∫ R ∥ g ( t , x ) ∥ p 2 d t d x , (12) \left\|\int_0^T\int_{\mathbb{R}}g(t,x)\xi(dt,dx)\right\|_p^2\leq z_p^2\int_0^T\int_{\mathbb{R}}\|g(t,x)\|_p^2\,dt\,dx, \tag*{(12)} ∫ 0 T ∫ R g ( t , x ) ξ ( d t , d x ) p 2 ≤ z p 2 ∫ 0 T ∫ R ∥ g ( t , x ) ∥ p 2 d t d x , (12) where z p > 0 z_p>0 z p > 0 p p p
E ( F δ ( u ) ) = E ( ⟨ D F , u ⟩ H ) , ∀ F ∈ D 1 , 2 . (13) \mathbb{E}(F\delta(u)) = \mathbb{E}(\langle DF,u\rangle_{\mathcal{H}}), \qquad\forall F \in\mathbb{D}^{1,2}. \tag*{(13)} E ( F δ ( u )) = E (⟨ D F , u ⟩ H ) , ∀ F ∈ D 1 , 2 . (13) In particular, E [ δ ( u ) ] = 0 \mathbb{E}[\delta(u)] = 0 E [ δ ( u )] = 0 u ∈ Dom ( δ ) u \in\operatorname{Dom}(\delta) u ∈ Dom ( δ ) δ ( u ) \delta(u) δ ( u ) Skorohod integral of u u u g g g E ∫ 0 ∞ ∫ R ∣ g ( s , x ) ∣ 2 d x d s < ∞ \mathbb{E}\int_0^\infty\int_{\mathbb{R}} |g(s,x)|^2\,dx\,ds < \infty E ∫ 0 ∞ ∫ R ∣ g ( s , x ) ∣ 2 d x d s < ∞ g ∈ Dom ( δ ) g \in\operatorname{Dom}(\delta) g ∈ Dom ( δ ) δ ( g ) = ∫ 0 ∞ ∫ R g ( s , x ) ξ ( d s , d x ) \delta(g) = \int_0^\infty\int_{\mathbb{R}} g(s,x)\xi(ds,dx) δ ( g ) = ∫ 0 ∞ ∫ R g ( s , x ) ξ ( d s , d x )
We will use the following results. We refer the reader to [28 ] for more details.
Heisenberg commutation principle: we use the following result (Proposition 1.3.8 of [28 ] ). Suppose that u ∈ D 1 , 2 ( H ) u \in\mathbb{D}^{1,2}(\mathcal{H}) u ∈ D 1 , 2 ( H ) [ 0 , T ] × R [0,T]\times\mathbb{R} [ 0 , T ] × R D t , x u ∈ Dom ( δ ) D_{t,x}u \in\operatorname{Dom}(\delta) D t , x u ∈ Dom ( δ ) ( t , x ) (t,x) ( t , x ) { δ ( D t , x u ) } t , x \{\delta(D_{t,x}u)\}_{t,x} { δ ( D t , x u ) } t , x L 2 ( Ω ; H ) L^2(\Omega;\mathcal{H}) L 2 ( Ω ; H ) δ ( u ) ∈ D 1 , 2 \delta(u) \in\mathbb{D}^{1,2} δ ( u ) ∈ D 1 , 2 ( t , x ) (t,x) ( t , x )
D t , x ( δ ( u ) ) = u ( t , x ) + ∫ 0 T ∫ R D t , x u ( s , y ) ξ ( d s , d y ) . (14) D_{t,x}(\delta(u)) = u(t,x) + \int_0^T\int_{\mathbb{R}} D_{t,x}u(s,y)\xi(ds,dy). \tag*{(14)} D t , x ( δ ( u )) = u ( t , x ) + ∫ 0 T ∫ R D t , x u ( s , y ) ξ ( d s , d y ) . (14) Chain rule: if F ∈ D 1 , 2 F \in\mathbb{D}^{1,2} F ∈ D 1 , 2 φ ∈ C 1 ( R ) \varphi\in C^1(\mathbb{R}) φ ∈ C 1 ( R ) φ ( F ) ∈ D 1 , 2 \varphi(F) \in\mathbb{D}^{1,2} φ ( F ) ∈ D 1 , 2
D φ ( F ) = φ ′ ( F ) D F . (15) D\varphi(F) = \varphi'(F)DF. \tag*{(15)} D φ ( F ) = φ ′ ( F ) D F . (15) Clark-Ocone formula: if F ∈ D 1 , 2 F \in\mathbb{D}^{1,2} F ∈ D 1 , 2 F F F F T \mathcal{F}_T F T
F = E ( F ) + ∫ 0 T ∫ R E [ D t , x F ∣ F t ] ξ ( d t , d x ) . (16) F = \mathbb{E}(F) + \int_0^T\int_{\mathbb{R}}\mathbb{E}[D_{t,x}F\mid\mathcal{F}_t]\xi(dt,dx). \tag*{(16)} F = E ( F ) + ∫ 0 T ∫ R E [ D t , x F ∣ F t ] ξ ( d t , d x ) . (16) Here and below, conditional expectations used as stochastic integrands denote their predictable versions, defined up to equality in L 2 ( Ω × [ 0 , T ] × R ) L^2(\Omega\times[0,T]\times\mathbb{R}) L 2 ( Ω × [ 0 , T ] × R )
Green function of SHE
In this section, we include some background material about a random process G \mathcal{G} G [1 ] and [21 ] .
Let s ≥ 0 s \ge0 s ≥ 0 y ∈ R y \in\mathbb{R} y ∈ R { G ( t , x ; s , y ) ; t > s , x ∈ R } \{\mathcal{G}(t,x;s,y);t>s,x\in\mathbb{R}\} { G ( t , x ; s , y ) ; t > s , x ∈ R }
{ ∂ t G ( t , x ; s , y ) = 1 2 ∂ x 2 G ( t , x ; s , y ) + G ( t , x ; s , y ) ξ ( t , x ) , t > s , x ∈ R , G ( s , ⋅ ; s , y ) = δ y . \begin{cases}
\partial_t\mathcal{G}(t,x;s,y) = \frac{1}{2}\partial_x^2\mathcal{G}(t,x;s,y) + \mathcal{G}(t,x;s,y)\xi(t,x), & t>s,\ x\in\mathbb{R},\\
\mathcal{G}(s,\cdot;s,y) = \delta_y.
\end{cases} { ∂ t G ( t , x ; s , y ) = 2 1 ∂ x 2 G ( t , x ; s , y ) + G ( t , x ; s , y ) ξ ( t , x ) , G ( s , ⋅ ; s , y ) = δ y . t > s , x ∈ R , This means that { G ( t , x ; s , y ) ; t > s , x ∈ R } \{\mathcal{G}(t,x;s,y);t>s,x\in\mathbb{R}\} { G ( t , x ; s , y ) ; t > s , x ∈ R } ( F t ) t ≥ s (\mathcal{F}_t)_{t\ge s} ( F t ) t ≥ s t > s t>s t > s x ∈ R x\in\mathbb{R} x ∈ R 1 1 1
G ( t , x ; s , y ) = p t − s ( x − y ) + ∫ s t ∫ R p t − r ( x − z ) G ( r , z ; s , y ) ξ ( d r , d z ) . (17) \mathcal{G}(t,x;s,y) = p_{t-s}(x-y) + \int_s^t\int_{\mathbb{R}} p_{t-r}(x-z)\mathcal{G}(r,z;s,y)\xi(dr,dz). \tag*{(17)} G ( t , x ; s , y ) = p t − s ( x − y ) + ∫ s t ∫ R p t − r ( x − z ) G ( r , z ; s , y ) ξ ( d r , d z ) . (17) Taking s = 0 s=0 s = 0 y = 0 y=0 y = 0 G ( ⋅ , ⋅ ; 0 , 0 ) \mathcal{G}(\cdot,\cdot;0,0) G ( ⋅ , ⋅ ; 0 , 0 ) Z Z Z [9 ] . Therefore, for any t > 0 t>0 t > 0 x ∈ R x\in\mathbb{R} x ∈ R
G ( t , x ; 0 , 0 ) = Z ( t , x ) . (18) \mathcal{G}(t,x;0,0)=Z(t,x). \tag*{(18)} G ( t , x ; 0 , 0 ) = Z ( t , x ) . (18) Multiplying (2.9) by Z ( s , y ) Z(s,y) Z ( s , y ) F s \mathcal{F}_s F s s s s
G ( t , x ; s , y ) Z ( s , y ) = p t − s ( x − y ) Z ( s , y ) + ∫ s t ∫ R p t − r ( x − z ) G ( r , z ; s , y ) Z ( s , y ) ξ ( d r , d z ) . \mathcal{G}(t,x;s,y)Z(s,y)=p_{t-s}(x-y)Z(s,y)+\int_s^t\int_{\mathbb{R}}p_{t-r}(x-z)\mathcal{G}(r,z;s,y)Z(s,y)\xi(\mathrm{d}r,\mathrm{d}z). G ( t , x ; s , y ) Z ( s , y ) = p t − s ( x − y ) Z ( s , y ) + ∫ s t ∫ R p t − r ( x − z ) G ( r , z ; s , y ) Z ( s , y ) ξ ( d r , d z ) . On the other hand, we import Malliavin differentiability of Z ( t , x ) Z(t,x) Z ( t , x ) [21 ] . The derivative vanishes for s > t s>t s > t 0 < s < t 0<s<t 0 < s < t ( s , y ) (s,y) ( s , y )
D s , y Z ( t , x ) = p t − s ( x − y ) Z ( s , y ) + ∫ s t ∫ R p t − r ( x − z ) D s , y Z ( r , z ) ξ ( d r , d z ) . D_{s,y}Z(t,x)=p_{t-s}(x-y)Z(s,y)+\int_s^t\int_{\mathbb{R}}p_{t-r}(x-z)D_{s,y}Z(r,z)\xi(\mathrm{d}r,\mathrm{d}z). D s , y Z ( t , x ) = p t − s ( x − y ) Z ( s , y ) + ∫ s t ∫ R p t − r ( x − z ) D s , y Z ( r , z ) ξ ( d r , d z ) . The two right-hand sides above are the mild formulation of the same equation, namely
{ ∂ t V ( t , x ) = 1 2 ∂ x 2 V ( t , x ) + V ( t , x ) ξ ( t , x ) , t > s , x ∈ R , V ( s , ⋅ ) = Z ( s , y ) δ y , \begin{cases}
\partial_tV(t,x)=\frac{1}{2}\partial_x^2V(t,x)+V(t,x)\xi(t,x),\quad t>s,\ x\in\mathbb{R},\\
V(s,\cdot)=Z(s,y)\delta_y,
\end{cases} { ∂ t V ( t , x ) = 2 1 ∂ x 2 V ( t , x ) + V ( t , x ) ξ ( t , x ) , t > s , x ∈ R , V ( s , ⋅ ) = Z ( s , y ) δ y , whose random field solution is unique by Theorem 2.4 of [9 ] . The initial datum is F s \mathcal{F}_s F s ( s , ∞ ) (s,\infty) ( s , ∞ ) F s \mathcal{F}_s F s F s \mathcal{F}_s F s 0 < s < t 0<s<t 0 < s < t x ∈ R x\in\mathbb{R} x ∈ R ( s , y ) (s,y) ( s , y )
D s , y Z ( t , x ) = G ( t , x ; s , y ) Z ( s , y ) . (19) D_{s,y}Z(t,x)=\mathcal{G}(t,x;s,y)Z(s,y). \tag*{(19)} D s , y Z ( t , x ) = G ( t , x ; s , y ) Z ( s , y ) . (19) Remark 2.1. The same factorization argument applies to other initial data and semigroups whenever the corresponding mild Green kernel, Malliavin differentiability and uniqueness for the random initial-value equation above are available. These conditions must be checked for each such extension.
The family { G ( t , x ; s , y ) ; 0 ≤ s < t , x , y ∈ R } \{\mathcal{G}(t,x;s,y);0\leq s<t,\ x,y\in\mathbb{R}\} { G ( t , x ; s , y ) ; 0 ≤ s < t , x , y ∈ R } Green function of SHE. We define the scaled Green function of SHE by:
G ‾ ( t , x ; s , y ) = G ( t , x ; s , y ) p t − s ( x − y ) 0 ≤ s < t , x , y ∈ R . \overline{\mathcal{G}}(t,x;s,y)=\frac{\mathcal{G}(t,x;s,y)}{p_{t-s}(x-y)}\qquad0\leq s<t,\ x,y\in\mathbb{R}. G ( t , x ; s , y ) = p t − s ( x − y ) G ( t , x ; s , y ) 0 ≤ s < t , x , y ∈ R . At the diagonal we set G ‾ ( s , x ; s , y ) = 1 \overline{\mathcal{G}}(s,x;s,y)=1 G ( s , x ; s , y ) = 1 s < t s<t s < t
We import the following uniform moment bound from Lemma 3.3 of [11 ] , which follows from Lemma 3.2 of [1 ] together with time-translation and shear invariance. For any k ∈ R k\in\mathbb{R} k ∈ R T > 0 T>0 T > 0
sup 0 ≤ s ≤ t ≤ T x , y ∈ R E [ G ‾ ( t , x ; s , y ) k ] < ∞ . (20) \sup_{\substack{0\leq s\leq t\leq T\\x,y\in\mathbb{R}}}\mathbb{E}\left[\overline{\mathcal{G}}(t,x;s,y)^k\right]<\infty. \tag*{(20)} 0 ≤ s ≤ t ≤ T x , y ∈ R sup E [ G ( t , x ; s , y ) k ] < ∞. (20) By Theorem 2.19.(viii) of [15 ] , G \mathcal{G} G x , y ∈ R x,y \in\mathbb{R} x , y ∈ R
G ( t , x ; s , y ) = ∫ R G ( t , x ; r , z ) G ( r , z ; s , y ) d z . (21) \mathcal{G}(t,x;s,y)=\int_{\mathbb{R}}\mathcal{G}(t,x;r,z)\mathcal{G}(r,z;s,y)\,\mathrm{d}z. \tag*{(21)} G ( t , x ; s , y ) = ∫ R G ( t , x ; r , z ) G ( r , z ; s , y ) d z . (21) Proof of Theorem 1.1
In this section, we present the proof of Theorem 1.1 . We define
X N ( t ) = 1 N log N S N ( t ) . X_N(t)=\frac{1}{\sqrt{N}\log N}S_N(t). X N ( t ) = N log N 1 S N ( t ) . We fix 0 < ε < T 0<\varepsilon<T 0 < ε < T N ≥ 2 N\ge2 N ≥ 2 p ≥ 2 p\ge2 p ≥ 2 α ∈ ( 0 , 1 2 ) \alpha\in\left(0,\frac{1}{2}\right) α ∈ ( 0 , 2 1 ) t 1 , t 2 ∈ [ ε , T ] t_1,t_2\in[\varepsilon,T] t 1 , t 2 ∈ [ ε , T ]
E ∣ X N ( t 2 ) − X N ( t 1 ) ∣ p ≤ C p , α , ε , T ∣ t 2 − t 1 ∣ p α / 2 . (22) \mathbb{E}\lvert X_N(t_2)-X_N(t_1)\rvert^p\le C_{p,\alpha,\varepsilon,T}\lvert t_2-t_1\rvert^{p\alpha/2}. \tag*{(22)} E ∣ X N ( t 2 ) − X N ( t 1 ) ∣ p ≤ C p , α , ε , T ∣ t 2 − t 1 ∣ p α /2 . (22) Choose p > 4 p>4 p > 4 2 / p < α < 1 / 2 2/p<\alpha<1/2 2/ p < α < 1/2 [25 ] ), (22) gives a continuous modification with every Hölder exponent γ < α / 2 − 1 / p \gamma<\alpha/2-1/p γ < α /2 − 1/ p [ ε , T ] [\varepsilon,T] [ ε , T ] [ 1 / m , m ] [1/m,m] [ 1/ m , m ] m ≥ 2 m\ge2 m ≥ 2 ( 0 , ∞ ) (0,\infty) ( 0 , ∞ ) p p p α \alpha α 1 / 2 1/2 1/2
The laws of X N ( ε ) X_N(\varepsilon) X N ( ε ) [21 ] . Together with the uniform increment estimate (22) , whose exponent p α / 2 p\alpha/2 p α /2 [25 ] ) gives tightness in C [ ε , T ] C[\varepsilon,T] C [ ε , T ] 2 B \sqrt{2}B 2 B
The remaining part of the article is dedicated to the proof of (22) .
Clark-Ocone representation
In this section, we give a representation of X N ( t ) X_N(t) X N ( t ) (16) .
Recall definition (3) of h h h t > 0 t>0 t > 0 x ∈ R x\in\mathbb{R} x ∈ R [1 ] and the imported positive and negative moments in (20) imply log Z ( t , x ) ∈ L 2 ( Ω ) \log Z(t,x)\in L^2(\Omega) log Z ( t , x ) ∈ L 2 ( Ω ) (19) , Gaussian factorization and generalized Hölder,
∥ D s , y Z ( t , x ) Z ( t , x ) ∥ 2 ≤ C T p s ( t − s ) / t ( y − s x / t ) , 0 < s < t . \left\lVert\frac{D_{s,y}Z(t,x)}{Z(t,x)}\right\rVert_2\le C_Tp_{s(t-s)/t}(y-sx/t),\qquad0<s<t. Z ( t , x ) D s , y Z ( t , x ) 2 ≤ C T p s ( t − s ) / t ( y − s x / t ) , 0 < s < t . The square of this bound is integrable in ( s , y ) (s,y) ( s , y ) [ s ( t − s ) / t ] − 1 / 2 [s(t-s)/t]^{-1/2} [ s ( t − s ) / t ] − 1/2 ϕ n \phi_n ϕ n log \log log [ 1 / n , n ] [1/n,n] [ 1/ n , n ] ∣ ϕ n ′ ( z ) ∣ ≤ C / z \lvert\phi_n'(z)\rvert\le C/z ∣ ϕ n ′ ( z )∣ ≤ C / z z > 0 z>0 z > 0 ∣ ϕ n ( z ) ∣ ≤ C + ∣ log z ∣ \lvert\phi_n(z)\rvert\le C+\lvert\log z\rvert ∣ ϕ n ( z )∣ ≤ C + ∣ log z ∣ n ≥ 2 n\ge2 n ≥ 2 C c ∞ ( ( 0 , ∞ ) ) C_c^\infty((0,\infty)) C c ∞ (( 0 , ∞ )) 0 ≤ χ n ≤ 1 0\leq\chi_n\leq1 0 ≤ χ n ≤ 1 χ n = 1 \chi_n=1 χ n = 1 [ 1 / n , n ] [1/n,n] [ 1/ n , n ] ϕ n ( z ) = ∫ 1 z χ n ( r ) r − 1 d r \phi_n(z)=\int_1^z\chi_n(r)r^{-1}\,dr ϕ n ( z ) = ∫ 1 z χ n ( r ) r − 1 d r z > 0 z>0 z > 0 ϕ n \phi_n ϕ n z ≤ 0 z\leq0 z ≤ 0 χ n \chi_n χ n ∣ ϕ n ( z ) ∣ ≤ ∣ log z ∣ |\phi_n(z)|\leq|\log z| ∣ ϕ n ( z ) ∣ ≤ ∣ log z ∣ ∣ ϕ n ′ ( z ) ∣ ≤ 1 / z |\phi_n'(z)|\leq1/z ∣ ϕ n ′ ( z ) ∣ ≤ 1/ z z > 0 z>0 z > 0 ϕ n ( Z ( t , x ) ) \phi_n(Z(t,x)) ϕ n ( Z ( t , x )) L 2 ( Ω ) L^2(\Omega) L 2 ( Ω ) L 2 ( Ω ; H ) L^2(\Omega;H) L 2 ( Ω ; H ) log Z ( t , x ) \log Z(t,x) log Z ( t , x ) D Z ( t , x ) / Z ( t , x ) DZ(t,x)/Z(t,x) D Z ( t , x ) / Z ( t , x ) D D D h ( t , x ) ∈ D 1 , 2 h(t,x)\in\mathbb{D}^{1,2} h ( t , x ) ∈ D 1 , 2
Applying the chain rule (15) , followed by the product decomposition (19) of D Z DZ D Z (18) , and the definition of G ‾ \overline{\mathcal{G}} G
D s , y h ( t , x ) = D s , y Z ( t , x ) Z ( t , x ) = G ( t , x ; s , y ) Z ( s , y ) Z ( t , x ) = G ( t , x ; s , y ) G ‾ ( s , y ; 0 , 0 ) G ‾ ( t , x ; 0 , 0 ) = p t − s ( x − y ) p s ( y ) p t ( x ) ⋅ G ‾ ( t , x ; s , y ) G ‾ ( s , y ; 0 , 0 ) G ‾ ( t , x ; 0 , 0 ) . (23) \begin{aligned}
D_{s,y}h(t,x)&=\frac{D_{s,y}Z(t,x)}{Z(t,x)}
=\frac{\mathcal{G}(t,x;s,y)Z(s,y)}{Z(t,x)}
=\frac{\mathcal{G}(t,x;s,y)\overline{\mathcal{G}}(s,y;0,0)}{\overline{\mathcal{G}}(t,x;0,0)}\\
&=\frac{p_{t-s}(x-y)p_s(y)}{p_t(x)}\cdot\frac{\overline{\mathcal{G}}(t,x;s,y)\overline{\mathcal{G}}(s,y;0,0)}{\overline{\mathcal{G}}(t,x;0,0)}. \tag*{(23)}
\end{aligned} D s , y h ( t , x ) = Z ( t , x ) D s , y Z ( t , x ) = Z ( t , x ) G ( t , x ; s , y ) Z ( s , y ) = G ( t , x ; 0 , 0 ) G ( t , x ; s , y ) G ( s , y ; 0 , 0 ) = p t ( x ) p t − s ( x − y ) p s ( y ) ⋅ G ( t , x ; 0 , 0 ) G ( t , x ; s , y ) G ( s , y ; 0 , 0 ) . (23) We use the following property of the heat kernel: for any t , s > 0 t,s>0 t , s > 0 x , y ∈ R x,y\in\mathbb{R} x , y ∈ R
p t ( x ) p s ( y ) = p t + s ( x − y ) p t s t + s ( s x + t y t + s ) , p_t(x)p_s(y)=p_{t+s}(x-y)p_{\frac{ts}{t+s}}\left(\frac{sx+ty}{t+s}\right), p t ( x ) p s ( y ) = p t + s ( x − y ) p t + s t s ( t + s s x + t y ) , (see e.g. Lemma A.4 of [9 ] ). It follows that
p t − s ( x − y ) p s ( y ) p t ( x ) = p s ( t − s ) t ( y − s t x ) = : q s t , x ( y ) . (24) \frac{p_{t-s}(x-y)p_s(y)}{p_t(x)}
=p_{\frac{s(t-s)}{t}}\left(y-\frac{s}{t}x\right)=:q_s^{t,x}(y). \tag*{(24)} p t ( x ) p t − s ( x − y ) p s ( y ) = p t s ( t − s ) ( y − t s x ) =: q s t , x ( y ) . (24) With the Clark-Ocone formula (16) in mind, we write
E [ D s , y h ( t , x ) ∣ F s ] = A ( t , x ; s , y ) q s t , x ( y ) , \mathbb{E}[D_{s,y}h(t,x)\mid\mathcal{F}_s]=\mathcal{A}(t,x;s,y)q_s^{t,x}(y), E [ D s , y h ( t , x ) ∣ F s ] = A ( t , x ; s , y ) q s t , x ( y ) , where
A ( t , x ; s , y ) = E [ G ‾ ( t , x ; s , y ) G ‾ ( s , y ; 0 , 0 ) G ‾ ( t , x ; 0 , 0 ) ∣ F s ] . \mathcal{A}(t,x;s,y)=\mathbb{E}\left[\left.\frac{\overline{\mathcal{G}}(t,x;s,y)\overline{\mathcal{G}}(s,y;0,0)}{\overline{\mathcal{G}}(t,x;0,0)}\right|\mathcal{F}_s\right]. A ( t , x ; s , y ) = E [ G ( t , x ; 0 , 0 ) G ( t , x ; s , y ) G ( s , y ; 0 , 0 ) F s ] . We say that A \mathcal{A} A Clark–Ocone kernel ; see also (22) of [21 ] . Therefore,
h ( t , x ) − E [ h ( t , x ) ] = ∫ 0 t ∫ R A ( t , x ; s , y ) q s t , x ( y ) ξ ( d s , d y ) . h(t,x)-\mathbb{E}[h(t,x)]=\int_0^t\int_{\mathbb{R}}\mathcal{A}(t,x;s,y)q_s^{t,x}(y)\xi(ds,dy). h ( t , x ) − E [ h ( t , x )] = ∫ 0 t ∫ R A ( t , x ; s , y ) q s t , x ( y ) ξ ( d s , d y ) . The moment bound of Lemma 3.1, proved below using only the imported Green-kernel moments, gives the stochastic Fubini integrability condition. Indeed, for fixed t > 0 t>0 t > 0
∫ 0 N ( ∫ 0 t ∫ R ∥ A ( t , x ; s , y ) q s t , x ( y ) ∥ 2 2 d y d s ) 1 / 2 d x ≤ C T N ( ∫ 0 t ( s ( t − s ) t ) − 1 / 2 d s ) 1 / 2 < ∞ . \int_0^N\left(\int_0^t\int_{\mathbb{R}}\|\mathcal{A}(t,x;s,y)q_s^{t,x}(y)\|_2^2\,dy\,ds\right)^{1/2}dx
\leq C_TN\left(\int_0^t\left(\frac{s(t-s)}{t}\right)^{-1/2}ds\right)^{1/2}<\infty. ∫ 0 N ( ∫ 0 t ∫ R ∥ A ( t , x ; s , y ) q s t , x ( y ) ∥ 2 2 d y d s ) 1/2 d x ≤ C T N ( ∫ 0 t ( t s ( t − s ) ) − 1/2 d s ) 1/2 < ∞. Here ∫ R p a ( y ) 2 d y = ( 4 π a ) − 1 / 2 \int_{\mathbb{R}}p_a(y)^2dy=(4\pi a)^{-1/2} ∫ R p a ( y ) 2 d y = ( 4 π a ) − 1/2 x x x L 2 ( [ 0 , N ] × [ 0 , t ] × R × Ω ) L^2([0,N]\times[0,t]\times\mathbb{R}\times\Omega) L 2 ([ 0 , N ] × [ 0 , t ] × R × Ω ) x ∈ [ 0 , N ] x\in[0,N] x ∈ [ 0 , N ] x x x
X N ( t ) = 1 N log N ∫ 0 t ∫ R ( ∫ 0 N A ( t , x ; s , y ) q s t , x ( y ) d x ) ξ ( d s , d y ) . (25) X_N(t)=\frac{1}{\sqrt{N\log N}}\int_0^t\int_{\mathbb{R}}\left(\int_0^N A(t,x;s,y)q_s^{t,x}(y)\,dx\right)\xi(ds,dy). \tag*{(25)} X N ( t ) = N log N 1 ∫ 0 t ∫ R ( ∫ 0 N A ( t , x ; s , y ) q s t , x ( y ) d x ) ξ ( d s , d y ) . (25) Lemma 3.1. For any p ≥ 1 p\ge1 p ≥ 1 T > 0 T>0 T > 0
sup 0 ≤ s < t ≤ T x , y ∈ R ∥ A ( t , x ; s , y ) ∥ p < ∞ . \sup_{\substack{0\le s<t\le T\\x,y\in\mathbb{R}}}\|A(t,x;s,y)\|_p<\infty. 0 ≤ s < t ≤ T x , y ∈ R sup ∥ A ( t , x ; s , y ) ∥ p < ∞. Proof. By Jensen’s inequality for conditional expectation and generalized Hölder’s inequality,
∥ A ( t , x ; s , y ) ∥ p ≤ ∥ G ‾ ( t , x ; s , y ) G ‾ ( s , y ; 0 , 0 ) G ‾ ( t , x ; 0 , 0 ) ∥ p \|A(t,x;s,y)\|_p\le\left\|\frac{\overline{G}(t,x;s,y)\overline{G}(s,y;0,0)}{\overline{G}(t,x;0,0)}\right\|_p ∥ A ( t , x ; s , y ) ∥ p ≤ G ( t , x ; 0 , 0 ) G ( t , x ; s , y ) G ( s , y ; 0 , 0 ) p ≤ ∥ G ‾ ( t , x ; s , y ) ∥ 3 p ∥ G ‾ ( s , y ; 0 , 0 ) ∥ 3 p ∥ G ‾ ( t , x ; 0 , 0 ) − 1 ∥ 3 p . \le\|\overline{G}(t,x;s,y)\|_{3p}\|\overline{G}(s,y;0,0)\|_{3p}\|\overline{G}(t,x;0,0)^{-1}\|_{3p}. ≤ ∥ G ( t , x ; s , y ) ∥ 3 p ∥ G ( s , y ; 0 , 0 ) ∥ 3 p ∥ G ( t , x ; 0 , 0 ) − 1 ∥ 3 p . The conclusion follows by relation (20) . In particular, the inverse factor uses the moment estimate at order − 3 p -3p − 3 p 3 p 3p 3 p
Increment analysis
This section is the core of the proof of Theorem 1.1 , and consists of a detailed analysis of the increments of X N X_N X N
Let 0 < ε ≤ t 1 < t 2 ≤ T 0<\varepsilon\le t_1<t_2\le T 0 < ε ≤ t 1 < t 2 ≤ T (25) , we write
X N ( t 2 ) − X N ( t 1 ) = 1 N log N ∫ t 1 t 2 ∫ R ( ∫ 0 N A ( t 2 , x ; s , y ) q s t 2 , x ( y ) d x ) ξ ( d s , d y ) + 1 N log N ∫ 0 t 1 ∫ R ( ∫ 0 N ( A ( t 2 , x ; s , y ) q s t 2 , x ( y ) − A ( t 1 , x ; s , y ) q s t 1 , x ( y ) ) d x ) ξ ( d s , d y ) = I N ( t 1 , t 2 ) + J N ( t 1 , t 2 ) . (26) \begin{aligned}
X_N(t_2)-X_N(t_1)
={}&\frac{1}{\sqrt{N\log N}}\int_{t_1}^{t_2}\int_{\mathbb{R}}\left(\int_0^N A(t_2,x;s,y)q_s^{t_2,x}(y)\,dx\right)\xi(ds,dy)\\
&+\frac{1}{\sqrt{N\log N}}\int_0^{t_1}\int_{\mathbb{R}}\left(\int_0^N\left(A(t_2,x;s,y)q_s^{t_2,x}(y)-A(t_1,x;s,y)q_s^{t_1,x}(y)\right)\,dx\right)\xi(ds,dy)\\
={}&I_N(t_1,t_2)+J_N(t_1,t_2). \tag*{(26)}
\end{aligned} X N ( t 2 ) − X N ( t 1 ) = = N log N 1 ∫ t 1 t 2 ∫ R ( ∫ 0 N A ( t 2 , x ; s , y ) q s t 2 , x ( y ) d x ) ξ ( d s , d y ) + N log N 1 ∫ 0 t 1 ∫ R ( ∫ 0 N ( A ( t 2 , x ; s , y ) q s t 2 , x ( y ) − A ( t 1 , x ; s , y ) q s t 1 , x ( y ) ) d x ) ξ ( d s , d y ) I N ( t 1 , t 2 ) + J N ( t 1 , t 2 ) . (26) We study separately the moments of I N I_N I N J N J_N J N I N I_N I N
Proposition 3.2. For any p ≥ 2 p\ge2 p ≥ 2 N ≥ 2 N\ge2 N ≥ 2 0 < ε ≤ t 1 < t 2 ≤ T 0<\varepsilon\le t_1<t_2\le T 0 < ε ≤ t 1 < t 2 ≤ T
E ∣ I N ( t 1 , t 2 ) ∣ p ≤ C p , ε , T ( t 2 − t 1 ) p / 2 . \mathbb{E}|I_N(t_1,t_2)|^p\le C_{p,\varepsilon,T}(t_2-t_1)^{p/2}. E ∣ I N ( t 1 , t 2 ) ∣ p ≤ C p , ε , T ( t 2 − t 1 ) p /2 . Proof. By inequality (12) , Minkowski’s inequality, and Lemma 3.1,
∥ I N ( t 1 , t 2 ) ∥ 2 2 ≤ z p 2 N log N ∫ t 1 t 2 ∫ R ∥ ∫ 0 N A ( t 2 , x ; s , y ) q s t 2 , x ( y ) d x ∥ p 2 d y d s \|I_N(t_1,t_2)\|_2^2\le\frac{z_p^2}{N\log N}\int_{t_1}^{t_2}\int_{\mathbb{R}}\left\|\int_0^N A(t_2,x;s,y)q_s^{t_2,x}(y)\,dx\right\|_p^2\,dy\,ds ∥ I N ( t 1 , t 2 ) ∥ 2 2 ≤ N log N z p 2 ∫ t 1 t 2 ∫ R ∫ 0 N A ( t 2 , x ; s , y ) q s t 2 , x ( y ) d x p 2 d y d s ≤ z p 2 N log N ∫ t 1 t 2 ∫ R ( ∫ 0 N ∥ A ( t 2 , x ; s , y ) ∥ p q s t 2 , x ( y ) d x ) 2 d y d s \le\frac{z_p^2}{N\log N}\int_{t_1}^{t_2}\int_{\mathbb{R}}\left(\int_0^N\|A(t_2,x;s,y)\|_p q_s^{t_2,x}(y)\,dx\right)^2\,dy\,ds ≤ N log N z p 2 ∫ t 1 t 2 ∫ R ( ∫ 0 N ∥ A ( t 2 , x ; s , y ) ∥ p q s t 2 , x ( y ) d x ) 2 d y d s ≤ C p , T N log N ∫ t 1 t 2 ∫ R ( ∫ 0 N q s t 2 , x ( y ) d x ) 2 d y d s . (27) \le\frac{C_{p,T}}{N\log N}\int_{t_1}^{t_2}\int_{\mathbb{R}}\left(\int_0^N q_s^{t_2,x}(y)\,dx\right)^2\,dy\,ds. \tag*{(27)} ≤ N log N C p , T ∫ t 1 t 2 ∫ R ( ∫ 0 N q s t 2 , x ( y ) d x ) 2 d y d s . (27) Using definition (24) of q s t , x q_s^{t,x} q s t , x p t ∗ p s = p t + s p_t \ast p_s=p_{t+s} p t ∗ p s = p t + s
∫ R ( ∫ 0 N q s t , x ( y ) d x ) 2 d y = ∫ 0 N ∫ 0 N p 2 s ( t − s ) t ( s t ( u − v ) ) d u d v ≤ ∫ 0 N ∫ R p 2 s ( t − s ) t ( s t ( u − v ) ) d u d v = N t s ≤ N T ε . (28) \begin{aligned}
\int_{\mathbb{R}}\left(\int_0^N q_s^{t,x}(y)\,\mathrm{d}x\right)^2\mathrm{d}y
=\int_0^N\int_0^N p_{\frac{2s(t-s)}{t}}\left(\frac{s}{t}(u-v)\right)\mathrm{d}u\mathrm{d}v \\
&\leq\int_0^N\int_{\mathbb{R}}p_{\frac{2s(t-s)}{t}}\left(\frac{s}{t}(u-v)\right)\mathrm{d}u\mathrm{d}v
=\frac{Nt}{s}\leq\frac{NT}{\varepsilon}. \tag*{(28)}
\end{aligned} ∫ R ( ∫ 0 N q s t , x ( y ) d x ) 2 d y = ∫ 0 N ∫ 0 N p t 2 s ( t − s ) ( t s ( u − v ) ) d u d v ≤ ∫ 0 N ∫ R p t 2 s ( t − s ) ( t s ( u − v ) ) d u d v = s N t ≤ ε N T . (28) Therefore,
∥ I N ( t 1 , t 2 ) ∥ p 2 ≤ C p , T T ε log N ( t 2 − t 1 ) ≤ C p , ε , T ( t 2 − t 1 ) , \lVert I_N(t_1,t_2)\rVert_p^2\leq\frac{C_{p,T}T}{\varepsilon\log N}(t_2-t_1)\leq C_{p,\varepsilon,T}(t_2-t_1), ∥ I N ( t 1 , t 2 ) ∥ p 2 ≤ ε log N C p , T T ( t 2 − t 1 ) ≤ C p , ε , T ( t 2 − t 1 ) , since N ≥ 2 N\geq2 N ≥ 2 p / 2 p/2 p /2
It remains to estimate E ∣ J N ( t 1 , t 2 ) ∣ p \mathbb{E}|J_N(t_1,t_2)|^p E ∣ J N ( t 1 , t 2 ) ∣ p A ( t 1 , x ; s , y ) q s t 2 , x ( y ) \mathcal{A}(t_1,x;s,y)q_s^{t_2,x}(y) A ( t 1 , x ; s , y ) q s t 2 , x ( y ) ∥ A ( t 2 , x ; s , y ) − A ( t 1 , x ; s , y ) ∥ p \lVert\mathcal{A}(t_2,x;s,y)-\mathcal{A}(t_1,x;s,y)\rVert_p ∥ A ( t 2 , x ; s , y ) − A ( t 1 , x ; s , y ) ∥ p ∥ G ‾ ( t 2 , x ; s , y ) − G ‾ ( t 1 , x ; s , y ) ∥ 4 p \lVert\overline{\mathcal{G}}(t_2,x;s,y)-\overline{\mathcal{G}}(t_1,x;s,y)\rVert_{4p} ∥ G ( t 2 , x ; s , y ) − G ( t 1 , x ; s , y ) ∥ 4 p [21 ] . Therefore, we pursue a different method here.
Using the change of variable x = t u x=tu x = t u
∫ 0 N A ( t , x ; s , y ) q s t , x ( y ) d x = ∫ 0 N / t t A ( t , t u ; s , y ) q s t , t u ( y ) d u . (29) \int_0^N\mathcal{A}(t,x;s,y)q_s^{t,x}(y)\,\mathrm{d}x
=\int_0^{N/t}t\mathcal{A}(t,tu;s,y)q_s^{t,tu}(y)\,\mathrm{d}u. \tag*{(29)} ∫ 0 N A ( t , x ; s , y ) q s t , x ( y ) d x = ∫ 0 N / t t A ( t , t u ; s , y ) q s t , t u ( y ) d u . (29) We call u u u ray variable . We apply this for t = t 1 t=t_1 t = t 1 t = t 2 t=t_2 t = t 2 J N J_N J N
J N ( t 1 , t 2 ) = 1 N log N ∫ 0 t 1 ∫ R ∫ 0 N / t 2 t 2 A ( t 2 , t 2 u ; s , y ) q s t 2 , t 2 u ( y ) d u ξ ( d s , d y ) − 1 N log N ∫ 0 t 1 ∫ R ∫ 0 N / t 1 t 1 A ( t 1 , t 1 u ; s , y ) q s t 1 , t 1 u ( y ) d u ξ ( d s , d y ) . (30) \begin{aligned}
J_N(t_1,t_2)
={}&\frac{1}{\sqrt{N\log N}}\int_0^{t_1}\int_{\mathbb{R}}\int_0^{N/t_2}t_2\mathcal{A}(t_2,t_2u;s,y)q_s^{t_2,t_2u}(y)\,\mathrm{d}u\,\xi(\mathrm{d}s,\mathrm{d}y)\\
&-\frac{1}{\sqrt{N\log N}}\int_0^{t_1}\int_{\mathbb{R}}\int_0^{N/t_1}t_1\mathcal{A}(t_1,t_1u;s,y)q_s^{t_1,t_1u}(y)\,\mathrm{d}u\,\xi(\mathrm{d}s,\mathrm{d}y). \tag*{(30)}
\end{aligned} J N ( t 1 , t 2 ) = N log N 1 ∫ 0 t 1 ∫ R ∫ 0 N / t 2 t 2 A ( t 2 , t 2 u ; s , y ) q s t 2 , t 2 u ( y ) d u ξ ( d s , d y ) − N log N 1 ∫ 0 t 1 ∫ R ∫ 0 N / t 1 t 1 A ( t 1 , t 1 u ; s , y ) q s t 1 , t 1 u ( y ) d u ξ ( d s , d y ) . (30) The difficulty is to transform this expression into a form in which the two A \mathcal{A} A same time variable t 1 t_1 t 1
We first show that the moments of J N ( t 1 , t 2 ) J_N(t_1,t_2) J N ( t 1 , t 2 ) (22) , when t 1 , t 2 t_1,t_2 t 1 , t 2 far apart , i.e. t 2 − t 1 > ε 2 t_2-t_1>\frac{\varepsilon}{2} t 2 − t 1 > 2 ε
Proposition 3.3. For any p ≥ 2 p\geq2 p ≥ 2 N ≥ 2 N\geq2 N ≥ 2 0 < ε ≤ t 1 < t 2 ≤ T 0<\varepsilon\leq t_1<t_2\leq T 0 < ε ≤ t 1 < t 2 ≤ T
E ∣ J N ( t 1 , t 2 ) ∣ p ≤ C p , ε , T . (31) \mathbb{E}|J_N(t_1,t_2)|^p\leq C_{p,\varepsilon,T}. \tag*{(31)} E ∣ J N ( t 1 , t 2 ) ∣ p ≤ C p , ε , T . (31) Consequently, if t 2 − t 1 > ε 2 t_2-t_1>\frac{\varepsilon}{2} t 2 − t 1 > 2 ε α > 0 \alpha>0 α > 0
E ∣ J N ( t 1 , t 2 ) ∣ p ≤ C p , α , ε , T ( t 2 − t 1 ) p α / 2 . (32) \mathbb{E}|J_N(t_1,t_2)|^p\leq C_{p,\alpha,\varepsilon,T}(t_2-t_1)^{p\alpha/2}. \tag*{(32)} E ∣ J N ( t 1 , t 2 ) ∣ p ≤ C p , α , ε , T ( t 2 − t 1 ) p α /2 . (32) Proof. We use decomposition (30) , written as J N = : J N ( 2 ) − J N ( 1 ) J_N=:J_N^{(2)}-J_N^{(1)} J N =: J N ( 2 ) − J N ( 1 )
∥ J N ∥ p 2 ≤ 2 ( ∥ J N ( 1 ) ∥ p 2 + ∥ J N ( 2 ) ∥ p 2 ) . \lVert J_N\rVert_p^2\leq2\left(\lVert J_N^{(1)}\rVert_p^2+\lVert J_N^{(2)}\rVert_p^2\right). ∥ J N ∥ p 2 ≤ 2 ( ∥ J N ( 1 ) ∥ p 2 + ∥ J N ( 2 ) ∥ p 2 ) . We treat the two terms by the same method: for i = 1 , 2 i=1,2 i = 1 , 2 (12) and Minkowski’s inequality,
∥ J N ( i ) ∥ p 2 ≤ z p 2 t i 2 N log N ∫ 0 t 1 ∫ R ( ∫ 0 N / t i ∥ A ( t i , t i u ; s , y ) ∥ p s t i , t i u ( y ) d u ) 2 d y d s ≤ C p , T N log N ∫ 0 t 1 ∫ R ( ∫ 0 N / t i q s t i , t i u ( y ) d u ) 2 d y d s . \begin{aligned}
\left\|J_N^{(i)}\right\|_p^2 \le\frac{z_p^2t_i^2}{N\log N}\int_0^{t_1}\int_{\mathbb{R}}\left(\int_0^{N/t_i}\left\|\mathcal{A}(t_i,t_i u;s,y)\right\|p_s^{t_i,t_i u}(y)\,du\right)^2\,dy\,ds \\
&\le\frac{C_{p,T}}{N\log N}\int_0^{t_1}\int_{\mathbb{R}}\left(\int_0^{N/t_i}q_s^{t_i,t_i u}(y)\,du\right)^2\,dy\,ds.
\end{aligned} J N ( i ) p 2 ≤ N log N z p 2 t i 2 ∫ 0 t 1 ∫ R ( ∫ 0 N / t i ∥ A ( t i , t i u ; s , y ) ∥ p s t i , t i u ( y ) d u ) 2 d y d s ≤ N log N C p , T ∫ 0 t 1 ∫ R ( ∫ 0 N / t i q s t i , t i u ( y ) d u ) 2 d y d s . Recalling definition (3.3) of q q q q s t i , t i u ( y ) = p s ( t i − s ) t i ( y − s u ) q_s^{t_i,t_i u}(y)=p_{\frac{s(t_i-s)}{t_i}}(y-su) q s t i , t i u ( y ) = p t i s ( t i − s ) ( y − s u )
∥ J N ( i ) ∥ p 2 ≤ C p , ε , T N log N ∫ 0 t 1 ( N s ∧ N 2 s ) d s . \left\|J_N^{(i)}\right\|_p^2\le\frac{C_{p,\varepsilon,T}}{N\log N}\int_0^{t_1}\left(\frac{N}{s}\wedge\frac{N^2}{\sqrt{s}}\right)\,ds. J N ( i ) p 2 ≤ N log N C p , ε , T ∫ 0 t 1 ( s N ∧ s N 2 ) d s . We analyze the integral above. If t 1 > N − 2 t_1>N^{-2} t 1 > N − 2 [ 0 , N − 2 ] [0,N^{-2}] [ 0 , N − 2 ] [ N − 2 , t 1 ] [N^{-2},t_1] [ N − 2 , t 1 ]
∫ 0 t 1 ( N s ∧ N 2 s ) d s ≤ ∫ 0 N − 2 N 2 s d s + ∫ N − 2 t 1 N s d s = 2 N + N log ( t 1 N 2 ) ≤ C T N log N , \int_0^{t_1}\left(\frac{N}{s}\wedge\frac{N^2}{\sqrt{s}}\right)\,ds\le\int_0^{N^{-2}}\frac{N^2}{\sqrt{s}}\,ds+\int_{N^{-2}}^{t_1}\frac{N}{s}\,ds=2N+N\log(t_1N^2)\le C_TN\log N, ∫ 0 t 1 ( s N ∧ s N 2 ) d s ≤ ∫ 0 N − 2 s N 2 d s + ∫ N − 2 t 1 s N d s = 2 N + N log ( t 1 N 2 ) ≤ C T N log N , whereas if t 1 ≤ N − 2 t_1\le N^{-2} t 1 ≤ N − 2
∫ 0 t 1 ( N s ∧ N 2 s ) d s ≤ ∫ 0 N − 2 N 2 s d s = 2 N ≤ 2 log 2 N log N . \int_0^{t_1}\left(\frac{N}{s}\wedge\frac{N^2}{\sqrt{s}}\right)\,ds\le\int_0^{N^{-2}}\frac{N^2}{\sqrt{s}}\,ds=2N\le\frac{2}{\log2}N\log N. ∫ 0 t 1 ( s N ∧ s N 2 ) d s ≤ ∫ 0 N − 2 s N 2 d s = 2 N ≤ log 2 2 N log N . This proves (31) . Relation (32) follows immediately from (31) .
It remains to estimate E ∣ J N ( t 1 , t 2 ) ∣ p \mathbb{E}|J_N(t_1,t_2)|^p E ∣ J N ( t 1 , t 2 ) ∣ p t 1 , t 2 t_1,t_2 t 1 , t 2 close to each other , i.e. t 1 < t 2 ≤ t 1 + ε 2 t_1<t_2\le t_1+\frac{\varepsilon}{2} t 1 < t 2 ≤ t 1 + 2 ε
We return to decomposition (30) . We write the integral on [ 0 , N / t 1 ] [0,N/t_1] [ 0 , N / t 1 ] [ 0 , N / t 2 ] [0,N/t_2] [ 0 , N / t 2 ] [ N / t 2 , N / t 1 ] [N/t_2,N/t_1] [ N / t 2 , N / t 1 ]
J N ( t 1 , t 2 ) = J N ′ ( t 1 , t 2 ) − J N ′ ′ ( t 1 , t 2 ) , (33) J_N(t_1,t_2)=J_N'(t_1,t_2)-J_N''(t_1,t_2), \tag*{(33)} J N ( t 1 , t 2 ) = J N ′ ( t 1 , t 2 ) − J N ′′ ( t 1 , t 2 ) , (33) where
J N ′ ( t 1 , t 2 ) = 1 N log N ∫ 0 t 1 ∫ R ( ∫ 0 N / t 2 ( t 2 A ( t 2 , t 2 u ; s , y ) q s t 2 , t 2 u ( y ) − t 1 A ( t 1 , t 1 u ; s , y ) q s t 1 , t 1 u ( y ) ) d u ) ξ ( d s , d y ) J_N'(t_1,t_2)=\frac{1}{\sqrt{N\log N}}\int_0^{t_1}\int_{\mathbb{R}}\left(\int_0^{N/t_2}\left(t_2\mathcal{A}(t_2,t_2u;s,y)q_s^{t_2,t_2u}(y)-t_1\mathcal{A}(t_1,t_1u;s,y)q_s^{t_1,t_1u}(y)\right)\,du\right)\xi(ds,dy) J N ′ ( t 1 , t 2 ) = N log N 1 ∫ 0 t 1 ∫ R ( ∫ 0 N / t 2 ( t 2 A ( t 2 , t 2 u ; s , y ) q s t 2 , t 2 u ( y ) − t 1 A ( t 1 , t 1 u ; s , y ) q s t 1 , t 1 u ( y ) ) d u ) ξ ( d s , d y ) J N ′ ′ ( t 1 , t 2 ) = 1 N log N ∫ 0 t 1 ∫ R ( ∫ N / t 2 N / t 1 t 1 A ( t 1 , t 1 u ; s , y ) q s t 1 , t 1 u ( y ) d u ) ξ ( d s , d y ) . J_N''(t_1,t_2)=\frac{1}{\sqrt{N\log N}}\int_0^{t_1}\int_{\mathbb{R}}\left(\int_{N/t_2}^{N/t_1}t_1\mathcal{A}(t_1,t_1u;s,y)q_s^{t_1,t_1u}(y)\,du\right)\xi(ds,dy). J N ′′ ( t 1 , t 2 ) = N log N 1 ∫ 0 t 1 ∫ R ( ∫ N / t 2 N / t 1 t 1 A ( t 1 , t 1 u ; s , y ) q s t 1 , t 1 u ( y ) d u ) ξ ( d s , d y ) . We study separately J N ′ J_N' J N ′ J N ′ ′ J_N'' J N ′′ J N ′ ′ J_N'' J N ′′ A \mathcal{A} A t 2 − t 1 ≤ ε 2 t_2-t_1\le\frac{\varepsilon}{2} t 2 − t 1 ≤ 2 ε J N ′ J_N' J N ′
Proposition 3.4. For any p ≥ 2 p \ge2 p ≥ 2 N ≥ 2 N \ge2 N ≥ 2 0 < ε ≤ t 1 < t 2 ≤ T 0 < \varepsilon\le t_1 < t_2 \le T 0 < ε ≤ t 1 < t 2 ≤ T
E ∣ J N ′ ′ ( t 1 , t 2 ) ∣ p ≤ C p , ε , T ( t 2 − t 1 ) p / 2 . \mathbb{E}\left|J_N^{\prime\prime}(t_1,t_2)\right|^p \le C_{p,\varepsilon,T}(t_2-t_1)^{p/2}. E ∣ J N ′′ ( t 1 , t 2 ) ∣ p ≤ C p , ε , T ( t 2 − t 1 ) p /2 . Proof. As in the proof of Proposition 3.2, by (12) and Minkowski’s inequality,
∥ J N ′ ′ ( t 1 , t 2 ) ∥ p 2 ≤ C p , T N log N ∫ 0 t 1 ∫ R ( ∫ N / t 2 N / t 1 q s t 1 , t 1 u ( y ) d u ) 2 d s d y . \left\|J_N^{\prime\prime}(t_1,t_2)\right\|_p^2 \le\frac{C_{p,T}}{N\log N}\int_0^{t_1}\int_{\mathbb{R}}\left(\int_{N/t_2}^{N/t_1}q_s^{t_1,t_1u}(y)\,\mathrm{d}u\right)^2\mathrm{d}s\mathrm{d}y. ∥ J N ′′ ( t 1 , t 2 ) ∥ p 2 ≤ N log N C p , T ∫ 0 t 1 ∫ R ( ∫ N / t 2 N / t 1 q s t 1 , t 1 u ( y ) d u ) 2 d s d y . We denote ℓ = N t 1 − N t 2 > 0 \ell=\frac{N}{t_1}-\frac{N}{t_2}>0 ℓ = t 1 N − t 2 N > 0
∫ R ( ∫ N / t 2 N / t 1 q s t 1 , t 1 u ( y ) d u ) 2 d y ≤ C T { ℓ s ∧ ℓ 2 s ( t 1 − s ) } . (34) \int_{\mathbb{R}}\left(\int_{N/t_2}^{N/t_1}q_s^{t_1,t_1u}(y)\,\mathrm{d}u\right)^2\mathrm{d}y\le C_T\left\{\frac{\ell}{s}\wedge\frac{\ell^2}{\sqrt{s(t_1-s)}}\right\}. \tag*{(34)} ∫ R ( ∫ N / t 2 N / t 1 q s t 1 , t 1 u ( y ) d u ) 2 d y ≤ C T { s ℓ ∧ s ( t 1 − s ) ℓ 2 } . (34) Indeed, as for (28) , by the semigroup property,
∫ R ( ∫ N / t 2 N / t 1 q s t 1 , t 1 u ( y ) d u ) 2 d y = ∫ N / t 2 N / t 1 ∫ N / t 2 N / t 1 p 2 s ( t 1 − s ) t 1 ( s ( u − v ) ) d u d v ≤ ∫ N / t 2 N / t 1 ∫ R p 2 s ( t 1 − s ) t 1 ( s ( u − v ) ) d u d v = ℓ s . \begin{aligned}
\int_{\mathbb{R}}\left(\int_{N/t_2}^{N/t_1}q_s^{t_1,t_1u}(y)\,\mathrm{d}u\right)^2\mathrm{d}y
=\int_{N/t_2}^{N/t_1}\int_{N/t_2}^{N/t_1}p_{\frac{2s(t_1-s)}{t_1}}\bigl(s(u-v)\bigr)\,\mathrm{d}u\mathrm{d}v \\
&\le\int_{N/t_2}^{N/t_1}\int_{\mathbb{R}}p_{\frac{2s(t_1-s)}{t_1}}\bigl(s(u-v)\bigr)\,\mathrm{d}u\mathrm{d}v=\frac{\ell}{s}.
\end{aligned} ∫ R ( ∫ N / t 2 N / t 1 q s t 1 , t 1 u ( y ) d u ) 2 d y = ∫ N / t 2 N / t 1 ∫ N / t 2 N / t 1 p t 1 2 s ( t 1 − s ) ( s ( u − v ) ) d u d v ≤ ∫ N / t 2 N / t 1 ∫ R p t 1 2 s ( t 1 − s ) ( s ( u − v ) ) d u d v = s ℓ . The second bound follows since p 2 s ( t 1 − s ) t 1 ( s ( u − v ) ) ≤ ( 4 π s ( t 1 − s ) t 1 ) − 1 / 2 ≤ C T s ( t 1 − s ) p_{\frac{2s(t_1-s)}{t_1}}\bigl(s(u-v)\bigr)\le\left(4\pi\frac{s(t_1-s)}{t_1}\right)^{-1/2}\le\frac{C_T}{\sqrt{s(t_1-s)}} p t 1 2 s ( t 1 − s ) ( s ( u − v ) ) ≤ ( 4 π t 1 s ( t 1 − s ) ) − 1/2 ≤ s ( t 1 − s ) C T
It follows that
∥ J N ′ ′ ( t 1 , t 2 ) ∥ p 2 ≤ C p , T N log N ∫ 0 t 1 { ℓ s ∧ ℓ 2 s ( t 1 − s ) } d s . \left\|J_N^{\prime\prime}(t_1,t_2)\right\|_p^2\le\frac{C_{p,T}}{N\log N}\int_0^{t_1}\left\{\frac{\ell}{s}\wedge\frac{\ell^2}{\sqrt{s(t_1-s)}}\right\}\mathrm{d}s. ∥ J N ′′ ( t 1 , t 2 ) ∥ p 2 ≤ N log N C p , T ∫ 0 t 1 { s ℓ ∧ s ( t 1 − s ) ℓ 2 } d s . We write the integral above as the sum of integrals on [ 0 , t 1 2 ] [0,\frac{t_1}{2}] [ 0 , 2 t 1 ] [ t 1 2 , t 1 ] [\frac{t_1}{2},t_1] [ 2 t 1 , t 1 ] I 1 I_1 I 1 I 2 I_2 I 2
We examine I 1 I_1 I 1 t 1 2 > ℓ − 2 \frac{t_1}{2}>\ell^{-2} 2 t 1 > ℓ − 2 [ 0 , t 1 2 ] [0,\frac{t_1}{2}] [ 0 , 2 t 1 ] [ 0 , ℓ − 2 ] [0,\ell^{-2}] [ 0 , ℓ − 2 ] [ ℓ − 2 , t 1 2 ] [\ell^{-2},\frac{t_1}{2}] [ ℓ − 2 , 2 t 1 ]
I 1 ≤ ∫ 0 ℓ − 2 ℓ 2 s ( t 1 − s ) d s + ∫ ℓ − 2 t 1 / 2 ℓ s d s ≤ 2 ε ℓ 2 ∫ 0 ℓ − 2 1 s d s + ℓ [ log ( t 1 2 ) − log ℓ − 2 ] = 2 2 ε ℓ + ℓ log ( t 1 ℓ 2 2 ) ≤ C ε , T ℓ [ 1 + log ( 1 + ℓ ) ] ≤ C ε , T ℓ log N , \begin{aligned}
I_1\le\int_0^{\ell^{-2}}\frac{\ell^2}{\sqrt{s(t_1-s)}}\mathrm{d}s+\int_{\ell^{-2}}^{t_1/2}\frac{\ell}{s}\mathrm{d}s\le\sqrt{\frac{2}{\varepsilon}}\ell^2\int_0^{\ell^{-2}}\frac{1}{\sqrt{s}}\mathrm{d}s+\ell\left[\log\left(\frac{t_1}{2}\right)-\log\ell^{-2}\right] \\
&=2\sqrt{\frac{2}{\varepsilon}}\ell+\ell\log\left(\frac{t_1\ell^2}{2}\right)\le C_{\varepsilon,T}\ell[1+\log(1+\ell)]\le C_{\varepsilon,T}\ell\log N,
\end{aligned} I 1 ≤ ∫ 0 ℓ − 2 s ( t 1 − s ) ℓ 2 d s + ∫ ℓ − 2 t 1 /2 s ℓ d s ≤ ε 2 ℓ 2 ∫ 0 ℓ − 2 s 1 d s + ℓ [ log ( 2 t 1 ) − log ℓ − 2 ] = 2 ε 2 ℓ + ℓ log ( 2 t 1 ℓ 2 ) ≤ C ε , T ℓ [ 1 + log ( 1 + ℓ )] ≤ C ε , T ℓ log N , where for the last inequality, we used the fact that ℓ = N t 2 − t 1 t 1 t 2 ≤ N T ε 2 \ell=N\frac{t_2-t_1}{t_1t_2}\le\frac{NT}{\varepsilon^2} ℓ = N t 1 t 2 t 2 − t 1 ≤ ε 2 N T
1 + log ( 1 + ℓ ) ≤ 1 + log ( 1 + N T ε 2 ) ≤ C ε , T log N . 1+\log(1+\ell)\le1+\log\left(1+\frac{NT}{\varepsilon^2}\right)\le C_{\varepsilon,T}\log N. 1 + log ( 1 + ℓ ) ≤ 1 + log ( 1 + ε 2 N T ) ≤ C ε , T log N . If t 1 2 ≤ ℓ − 2 \frac{t_1}{2}\le\ell^{-2} 2 t 1 ≤ ℓ − 2
I 1 ≤ ℓ 2 ∫ 0 t 1 / 2 d s s ( t 1 − s ) ≤ 2 ε ℓ 2 ∫ 0 ℓ − 2 s − 1 / 2 d s = 2 2 ε ℓ . I_1\le\ell^2\int_0^{t_1/2}\frac{\mathrm{d}s}{\sqrt{s(t_1-s)}}\le\sqrt{\frac{2}{\varepsilon}}\ell^2\int_0^{\ell^{-2}}s^{-1/2}\mathrm{d}s=2\sqrt{\frac{2}{\varepsilon}}\ell. I 1 ≤ ℓ 2 ∫ 0 t 1 /2 s ( t 1 − s ) d s ≤ ε 2 ℓ 2 ∫ 0 ℓ − 2 s − 1/2 d s = 2 ε 2 ℓ . For I 2 \mathcal{I}_2 I 2
I 2 ≤ ∫ t 1 / 2 t 1 ℓ s d s = ℓ ( log t 1 − log t 1 2 ) = ℓ log 2. \mathcal{I}_2 \le\int_{t_1/2}^{t_1} \frac{\ell}{s}\,ds = \ell\left(\log t_1-\log\frac{t_1}{2}\right)=\ell\log2. I 2 ≤ ∫ t 1 /2 t 1 s ℓ d s = ℓ ( log t 1 − log 2 t 1 ) = ℓ log 2. Combining these estimates, we obtain that I 1 + I 2 ≤ C p , ε , T ℓ log N \mathcal{I}_1+\mathcal{I}_2\le C_{p,\varepsilon,T}\ell\log N I 1 + I 2 ≤ C p , ε , T ℓ log N
∥ J N ′ ′ ( t 1 , t 2 ) ∥ p 2 ≤ C p , ε , T ℓ N = C p , ε , T t 2 − t 1 t 1 t 2 ≤ C p , ε , T t 2 − t 1 ε 2 . \left\|J_N''(t_1,t_2)\right\|_p^2\le C_{p,\varepsilon,T}\frac{\ell}{N}=C_{p,\varepsilon,T}\frac{t_2-t_1}{t_1t_2}\le C_{p,\varepsilon,T}\frac{t_2-t_1}{\varepsilon^2}. ∥ J N ′′ ( t 1 , t 2 ) ∥ p 2 ≤ C p , ε , T N ℓ = C p , ε , T t 1 t 2 t 2 − t 1 ≤ C p , ε , T ε 2 t 2 − t 1 . The study of J N ′ ( t 1 , t 2 ) J_N'(t_1,t_2) J N ′ ( t 1 , t 2 )
R ( t , x ; s , y ) : = D s , y h ( t , x ) = G ( t , x ; s , y ) G ( s , y ; 0 , 0 ) G ( t , x ; 0 , 0 ) = q s t , x ( y ) G ‾ ( t , x ; s , y ) G ‾ ( s , y ; 0 , 0 ) G ‾ ( t , x ; 0 , 0 ) , (35) \mathcal{R}(t,x;s,y):=D_{s,y}h(t,x)=\frac{\mathcal{G}(t,x;s,y)\mathcal{G}(s,y;0,0)}{\mathcal{G}(t,x;0,0)}
=\frac{q_s^{t,x}(y)\overline{\mathcal{G}}(t,x;s,y)\overline{\mathcal{G}}(s,y;0,0)}{\overline{\mathcal{G}}(t,x;0,0)}, \tag*{(35)} R ( t , x ; s , y ) := D s , y h ( t , x ) = G ( t , x ; 0 , 0 ) G ( t , x ; s , y ) G ( s , y ; 0 , 0 ) = G ( t , x ; 0 , 0 ) q s t , x ( y ) G ( t , x ; s , y ) G ( s , y ; 0 , 0 ) , (35) for any 0 < s < t 0<s<t 0 < s < t x , y ∈ R x,y\in\mathbb{R} x , y ∈ R
E [ R ( t , x ; s , y ) ∣ F s ] = A ( t , x ; s , y ) q s t , x ( y ) . (36) \mathbb{E}[\mathcal{R}(t,x;s,y)\mid\mathcal{F}_s]=\mathcal{A}(t,x;s,y)q_s^{t,x}(y). \tag*{(36)} E [ R ( t , x ; s , y ) ∣ F s ] = A ( t , x ; s , y ) q s t , x ( y ) . (36) It follows that the integrand which appears in J N ′ ( t 1 , t 2 ) J_N'(t_1,t_2) J N ′ ( t 1 , t 2 )
t 2 A ( t 2 , t 2 u ; s , y ) q s t 2 , t 2 u ( y ) − t 1 A ( t 1 , t 1 u ; s , y ) q s t 1 , t 1 u ( y ) = E [ t 2 R ( t 2 , t 2 u ; s , y ) − t 1 R ( t 1 , t 1 u ; s , y ) ∣ F s ] . t_2\mathcal{A}(t_2,t_2u;s,y)q_s^{t_2,t_2u}(y)-t_1\mathcal{A}(t_1,t_1u;s,y)q_s^{t_1,t_1u}(y)
=
\mathbb{E}[t_2\mathcal{R}(t_2,t_2u;s,y)-t_1\mathcal{R}(t_1,t_1u;s,y)\mid\mathcal{F}_s]. t 2 A ( t 2 , t 2 u ; s , y ) q s t 2 , t 2 u ( y ) − t 1 A ( t 1 , t 1 u ; s , y ) q s t 1 , t 1 u ( y ) = E [ t 2 R ( t 2 , t 2 u ; s , y ) − t 1 R ( t 1 , t 1 u ; s , y ) ∣ F s ] . By the generalized Hölder’s inequality, an immediate consequence of (3.14) and (2.12) is:
∥ R ( t , x ; s , y ) ∥ p ≤ C p , T q s t , x ( y ) for any p ≥ 1. (37) \left\|\mathcal{R}(t,x;s,y)\right\|_p\le C_{p,T}q_s^{t,x}(y)\qquad\text{for any }p\ge1. \tag*{(37)} ∥ R ( t , x ; s , y ) ∥ p ≤ C p , T q s t , x ( y ) for any p ≥ 1. (37) We analyze R \mathcal{R} R By (2.13), for any 0 < s < t 0<s<t 0 < s < t x ∈ R x\in\mathbb{R} x ∈ R
∫ R R ( t , x ; s , y ) d y = 1 G ( t , x ; 0 , 0 ) ∫ R G ( t , x ; s , y ) G ( s , y ; 0 , 0 ) d y = 1. \int_{\mathbb{R}}\mathcal{R}(t,x;s,y)\,dy
=\frac{1}{\mathcal{G}(t,x;0,0)}\int_{\mathbb{R}}\mathcal{G}(t,x;s,y)\mathcal{G}(s,y;0,0)\,dy=1. ∫ R R ( t , x ; s , y ) d y = G ( t , x ; 0 , 0 ) 1 ∫ R G ( t , x ; s , y ) G ( s , y ; 0 , 0 ) d y = 1. By (2.13), R \mathcal{R} R 0 < s < r < t 0<s<r<t 0 < s < r < t x , y ∈ R x,y\in\mathbb{R} x , y ∈ R
R ( t , x ; s , y ) = ∫ R R ( t , x ; r , z ) R ( r , z ; s , y ) d z . (38) \mathcal{R}(t,x;s,y)=\int_{\mathbb{R}}\mathcal{R}(t,x;r,z)\mathcal{R}(r,z;s,y)\,dz. \tag*{(38)} R ( t , x ; s , y ) = ∫ R R ( t , x ; r , z ) R ( r , z ; s , y ) d z . (38) Consequently, for any 0 < s < t 1 < t 2 0<s<t_1<t_2 0 < s < t 1 < t 2 x 1 , x 2 , y ∈ R x_1,x_2,y\in\mathbb{R} x 1 , x 2 , y ∈ R
R ( t 2 , x 2 ; s , y ) − R ( t 1 , x 1 ; s , y ) = ∫ R ( R ( t 1 , w ; s , y ) − R ( t 1 , x 1 ; s , y ) ) R ( t 2 , x 2 ; t 1 , w ) d w . (39) \mathcal{R}(t_2,x_2;s,y)-\mathcal{R}(t_1,x_1;s,y)
=\int_{\mathbb{R}}\bigl(\mathcal{R}(t_1,w;s,y)-\mathcal{R}(t_1,x_1;s,y)\bigr)\mathcal{R}(t_2,x_2;t_1,w)\,dw. \tag*{(39)} R ( t 2 , x 2 ; s , y ) − R ( t 1 , x 1 ; s , y ) = ∫ R ( R ( t 1 , w ; s , y ) − R ( t 1 , x 1 ; s , y ) ) R ( t 2 , x 2 ; t 1 , w ) d w . (39) Note that in the integral above, the two R \mathcal{R} R same first variable t 1 t_1 t 1
p t ( x ) ≤ c p c t ( x ) for any c ≥ 1. (40) p_t(x)\le\sqrt{c}p_{ct}(x)\qquad\text{for any }c\ge1. \tag*{(40)} p t ( x ) ≤ c p c t ( x ) for any c ≥ 1. (40) Note that (3.19) is a consequence of the following identity:
p t ( x ) = c e − c − 1 2 c t x 2 p c t ( x ) for any c > 0. (41) p_t(x)=\sqrt{c}e^{-\frac{c-1}{2ct}x^2}p_{ct}(x)\qquad\text{for any }c>0. \tag*{(41)} p t ( x ) = c e − 2 c t c − 1 x 2 p c t ( x ) for any c > 0. (41) Lemma 3.5. Let p ≥ 1 p \ge1 p ≥ 1 α ∈ ( 0 , 1 2 ) \alpha\in(0,\frac{1}{2}) α ∈ ( 0 , 2 1 ) x 1 , x 2 , y ∈ R x_1,x_2,y \in\mathbb{R} x 1 , x 2 , y ∈ R
∥ R ( t , x 2 ; s , y ) − R ( t , x 1 ; s , y ) ∥ p ≤ C p , α , T ( t − s ) − α 2 ∣ x 1 − x 2 ∣ α ( p 2 s ( t − s ) t ( y − s t x 1 ) + p 2 s ( t − s ) t ( y − s t x 2 ) ) . \left\|\mathcal{R}(t,x_2;s,y)-\mathcal{R}(t,x_1;s,y)\right\|_p \le C_{p,\alpha,T}(t-s)^{-\frac{\alpha}{2}}|x_1-x_2|^\alpha\left(p_{\frac{2s(t-s)}{t}}\left(y-\frac{s}{t}x_1\right)+p_{\frac{2s(t-s)}{t}}\left(y-\frac{s}{t}x_2\right)\right). ∥ R ( t , x 2 ; s , y ) − R ( t , x 1 ; s , y ) ∥ p ≤ C p , α , T ( t − s ) − 2 α ∣ x 1 − x 2 ∣ α ( p t 2 s ( t − s ) ( y − t s x 1 ) + p t 2 s ( t − s ) ( y − t s x 2 ) ) . Proof. By (3.14), R ( t , x 2 ; s , y ) − R ( t , x 1 ; s , y ) = T 1 + T 2 \mathcal{R}(t,x_2;s,y)-\mathcal{R}(t,x_1;s,y)=T_1+T_2 R ( t , x 2 ; s , y ) − R ( t , x 1 ; s , y ) = T 1 + T 2
T 1 = G ‾ ( s , y ; 0 , 0 ) ( G ‾ ( t , x 2 ; s , y ) G ‾ ( t , x 2 ; 0 , 0 ) − G ‾ ( t , x 1 ; s , y ) G ‾ ( t , x 1 ; 0 , 0 ) ) q s t , x 2 ( y ) T_1=\overline{\mathcal{G}}(s,y;0,0)\left(\frac{\overline{\mathcal{G}}(t,x_2;s,y)}{\overline{\mathcal{G}}(t,x_2;0,0)}-\frac{\overline{\mathcal{G}}(t,x_1;s,y)}{\overline{\mathcal{G}}(t,x_1;0,0)}\right)q_s^{t,x_2}(y) T 1 = G ( s , y ; 0 , 0 ) ( G ( t , x 2 ; 0 , 0 ) G ( t , x 2 ; s , y ) − G ( t , x 1 ; 0 , 0 ) G ( t , x 1 ; s , y ) ) q s t , x 2 ( y ) T 2 = G ‾ ( s , y ; 0 , 0 ) G ‾ ( t , x 1 ; s , y ) G ‾ ( t , x 1 ; 0 , 0 ) ( q s t , x 2 ( y ) − q s t , x 1 ( y ) ) . T_2=\overline{\mathcal{G}}(s,y;0,0)\frac{\overline{\mathcal{G}}(t,x_1;s,y)}{\overline{\mathcal{G}}(t,x_1;0,0)}\left(q_s^{t,x_2}(y)-q_s^{t,x_1}(y)\right). T 2 = G ( s , y ; 0 , 0 ) G ( t , x 1 ; 0 , 0 ) G ( t , x 1 ; s , y ) ( q s t , x 2 ( y ) − q s t , x 1 ( y ) ) . By Cauchy–Schwarz, (2.12) and (A.3),
∥ T 1 ∥ p ≤ C p , α , T ∣ x 1 − x 2 ∣ α q s t , x 2 ( y ) . \|T_1\|_p \le C_{p,\alpha,T}|x_1-x_2|^\alpha q_s^{t,x_2}(y). ∥ T 1 ∥ p ≤ C p , α , T ∣ x 1 − x 2 ∣ α q s t , x 2 ( y ) . Using 1 ≤ T α / 2 ( t − s ) − α / 2 1 \le T^{\alpha/2}(t-s)^{-\alpha/2} 1 ≤ T α /2 ( t − s ) − α /2 c = 2 c=2 c = 2
∥ T 1 ∥ p ≤ C p , α , T ( t − s ) − α / 2 ∣ x 1 − x 2 ∣ α p 2 s ( t − s ) t ( y − s x 2 t ) . \|T_1\|_p \le C_{p,\alpha,T}(t-s)^{-\alpha/2}|x_1-x_2|^\alpha p_{\frac{2s(t-s)}{t}}\left(y-\frac{sx_2}{t}\right). ∥ T 1 ∥ p ≤ C p , α , T ( t − s ) − α /2 ∣ x 1 − x 2 ∣ α p t 2 s ( t − s ) ( y − t s x 2 ) . Similarly, generalized Hölder’s inequality, (2.12) and Lemma A.3 imply
∥ T 2 ∥ p ≤ C p , α , T ( t − s ) − α / 2 ∣ x 1 − x 2 ∣ α ( p 2 s ( t − s ) t ( y − s x 1 t ) + p 2 s ( t − s ) t ( y − s x 2 t ) ) . \|T_2\|_p \le C_{p,\alpha,T}(t-s)^{-\alpha/2}|x_1-x_2|^\alpha\left(p_{\frac{2s(t-s)}{t}}\left(y-\frac{sx_1}{t}\right)+p_{\frac{2s(t-s)}{t}}\left(y-\frac{sx_2}{t}\right)\right). ∥ T 2 ∥ p ≤ C p , α , T ( t − s ) − α /2 ∣ x 1 − x 2 ∣ α ( p t 2 s ( t − s ) ( y − t s x 1 ) + p t 2 s ( t − s ) ( y − t s x 2 ) ) . For 1 ≤ p < 2 1 \le p < 2 1 ≤ p < 2
The next lemma gives an estimate for the p p p J N ′ ( t 1 , t 2 ) \mathcal{J}'_N(t_1,t_2) J N ′ ( t 1 , t 2 )
Lemma 3.6. Let p ≥ 1 p \ge1 p ≥ 1 α ∈ ( 0 , 1 2 ) \alpha\in(0,\frac{1}{2}) α ∈ ( 0 , 2 1 ) 0 < t 1 < t 2 ≤ T 0 < t_1 < t_2 \le T 0 < t 1 < t 2 ≤ T t 2 < 2 t 1 t_2 < 2t_1 t 2 < 2 t 1 0 < s ≤ t 1 − ( t 2 − t 1 ) 0 < s \le t_1-(t_2-t_1) 0 < s ≤ t 1 − ( t 2 − t 1 ) u , y ∈ R u,y \in\mathbb{R} u , y ∈ R
∥ t 2 A ( t 2 , t 2 u ; s , y ) q s t 2 , t 2 u ( y ) − t 1 A ( t 1 , t 1 u ; s , y ) q s t 1 , t 1 u ( y ) ∥ p ≤ C p , α , T ( t 2 − t 1 ) α / 2 ( t 1 − s ) − α / 2 ( p 2 s ( t 1 − s ) t 1 ( y − s u ) + p 2 s ( t 2 − s ) t 2 ( y − s u ) ) . \left\|t_2\mathcal{A}(t_2,t_2u;s,y)q_s^{t_2,t_2u}(y)-t_1\mathcal{A}(t_1,t_1u;s,y)q_s^{t_1,t_1u}(y)\right\|_p
\le C_{p,\alpha,T}(t_2-t_1)^{\alpha/2}(t_1-s)^{-\alpha/2}\left(p_{\frac{2s(t_1-s)}{t_1}}(y-su)+p_{\frac{2s(t_2-s)}{t_2}}(y-su)\right). t 2 A ( t 2 , t 2 u ; s , y ) q s t 2 , t 2 u ( y ) − t 1 A ( t 1 , t 1 u ; s , y ) q s t 1 , t 1 u ( y ) p ≤ C p , α , T ( t 2 − t 1 ) α /2 ( t 1 − s ) − α /2 ( p t 1 2 s ( t 1 − s ) ( y − s u ) + p t 2 2 s ( t 2 − s ) ( y − s u ) ) . Proof. Let 0 < s ≤ t 1 − ( t 2 − t 1 ) 0 < s \le t_1-(t_2-t_1) 0 < s ≤ t 1 − ( t 2 − t 1 ) u , y ∈ R u,y \in\mathbb{R} u , y ∈ R
∥ t 2 A ( t 2 , t 2 u ; s , y ) q s t 2 , t 2 u ( y ) − t 1 A ( t 1 , t 1 u ; s , y ) q s t 1 , t 1 u ( y ) ∥ p = ∥ E [ t 2 R ( t 2 , t 2 u ; s , y ) − t 1 R ( t 1 , t 1 u ; s , y ) ∣ F s ] ∥ p ≤ ∥ t 2 R ( t 2 , t 2 u ; s , y ) − t 1 R ( t 1 , t 1 u ; s , y ) ∥ p ≤ t 2 ∥ R ( t 2 , t 2 u ; s , y ) − R ( t 1 , t 1 u ; s , y ) ∥ p + ( t 2 − t 1 ) ∥ R ( t 1 , t 1 u ; s , y ) ∥ p = : A 1 + A 2 . \begin{aligned}
&\left\|t_2\mathcal{A}(t_2,t_2u;s,y)q_s^{t_2,t_2u}(y)-t_1\mathcal{A}(t_1,t_1u;s,y)q_s^{t_1,t_1u}(y)\right\|_p \\
&=\left\|\mathbb{E}\left[t_2\mathcal{R}(t_2,t_2u;s,y)-t_1\mathcal{R}(t_1,t_1u;s,y)\mid\mathcal{F}_s\right]\right\|_p \\
&\le\left\|t_2\mathcal{R}(t_2,t_2u;s,y)-t_1\mathcal{R}(t_1,t_1u;s,y)\right\|_p \\
&\le t_2\left\|\mathcal{R}(t_2,t_2u;s,y)-\mathcal{R}(t_1,t_1u;s,y)\right\|_p+(t_2-t_1)\left\|\mathcal{R}(t_1,t_1u;s,y)\right\|_p=:A_1+A_2.
\end{aligned} t 2 A ( t 2 , t 2 u ; s , y ) q s t 2 , t 2 u ( y ) − t 1 A ( t 1 , t 1 u ; s , y ) q s t 1 , t 1 u ( y ) p = ∥ E [ t 2 R ( t 2 , t 2 u ; s , y ) − t 1 R ( t 1 , t 1 u ; s , y ) ∣ F s ] ∥ p ≤ ∥ t 2 R ( t 2 , t 2 u ; s , y ) − t 1 R ( t 1 , t 1 u ; s , y ) ∥ p ≤ t 2 ∥ R ( t 2 , t 2 u ; s , y ) − R ( t 1 , t 1 u ; s , y ) ∥ p + ( t 2 − t 1 ) ∥ R ( t 1 , t 1 u ; s , y ) ∥ p =: A 1 + A 2 . We examine A 2 A_2 A 2 s s s ( t 2 − t 1 ) 1 − α 2 ( t 1 − s ) α 2 ≤ t 1 − s ≤ T (t_2-t_1)^{1-\frac{\alpha}{2}}(t_1-s)^{\frac{\alpha}{2}}\leq t_1-s\leq T ( t 2 − t 1 ) 1 − 2 α ( t 1 − s ) 2 α ≤ t 1 − s ≤ T
t 2 − t 1 = ( t 2 − t 1 ) α 2 ( t 1 − s ) − α 2 ( t 2 − t 1 ) 1 − α 2 ( t 1 − s ) α 2 ≤ T ( t 2 − t 1 ) α 2 ( t 1 − s ) − α 2 . t_2-t_1=(t_2-t_1)^{\frac{\alpha}{2}}(t_1-s)^{-\frac{\alpha}{2}}(t_2-t_1)^{1-\frac{\alpha}{2}}(t_1-s)^{\frac{\alpha}{2}}\leq T(t_2-t_1)^{\frac{\alpha}{2}}(t_1-s)^{-\frac{\alpha}{2}}. t 2 − t 1 = ( t 2 − t 1 ) 2 α ( t 1 − s ) − 2 α ( t 2 − t 1 ) 1 − 2 α ( t 1 − s ) 2 α ≤ T ( t 2 − t 1 ) 2 α ( t 1 − s ) − 2 α . Using (37) and (40) with c = 2 c=2 c = 2
∥ R ( t 1 , t 1 u ; s , y ) ∥ p ≤ C p , T q s t 1 , t 1 u ( y ) = C p , T p s ( t 1 − s ) t 1 ( y − s u ) ≤ C p , T p 2 s ( t 1 − s ) t 1 ( y − s u ) . \|\mathcal{R}(t_1,t_1u;s,y)\|_p\leq C_{p,T}q_s^{t_1,t_1u}(y)=C_{p,T}p_{\frac{s(t_1-s)}{t_1}}(y-su)\leq C_{p,T}p_{\frac{2s(t_1-s)}{t_1}}(y-su). ∥ R ( t 1 , t 1 u ; s , y ) ∥ p ≤ C p , T q s t 1 , t 1 u ( y ) = C p , T p t 1 s ( t 1 − s ) ( y − s u ) ≤ C p , T p t 1 2 s ( t 1 − s ) ( y − s u ) . Therefore,
A 2 ≤ C p , T ( t 2 − t 1 ) α 2 ( t 1 − s ) − α 2 p 2 s ( t 1 − s ) t 1 ( y − s u ) . (42) A_2\leq C_{p,T}(t_2-t_1)^{\frac{\alpha}{2}}(t_1-s)^{-\frac{\alpha}{2}}p_{\frac{2s(t_1-s)}{t_1}}(y-su). \tag*{(42)} A 2 ≤ C p , T ( t 2 − t 1 ) 2 α ( t 1 − s ) − 2 α p t 1 2 s ( t 1 − s ) ( y − s u ) . (42) To study A 1 A_1 A 1 (39) . By Minkowski’s and Cauchy-Schwarz inequality,
A 1 ≤ T ∫ R ∥ R ( t 1 , w ; s , y ) − R ( t 1 , t 1 u ; s , y ) ∥ 2 p ∥ R ( t 2 , t 2 u ; t 1 , w ) ∥ 2 p d w . A_1\leq T\int_{\mathbb{R}}\|\mathcal{R}(t_1,w;s,y)-\mathcal{R}(t_1,t_1u;s,y)\|_{2p}\|\mathcal{R}(t_2,t_2u;t_1,w)\|_{2p}\,dw. A 1 ≤ T ∫ R ∥ R ( t 1 , w ; s , y ) − R ( t 1 , t 1 u ; s , y ) ∥ 2 p ∥ R ( t 2 , t 2 u ; t 1 , w ) ∥ 2 p d w . We bound these factors using Lemma 3.5, respectively (37) . It follows that
A 1 ≤ C p , α , T ( t 1 − s ) − α 2 ∫ R ∣ w − t 1 u ∣ α ( p 2 s ( t 1 − s ) t 1 ( y − s t 1 w ) + p 2 s ( t 1 − s ) t 1 ( y − s u ) ) × p t 1 ( t 2 − t 1 ) t 2 ( w − t 1 u ) d w = : C p , α , T ( t 1 − s ) − α 2 ( A 1 ′ + A 1 ′ ′ ) . (43) \begin{aligned}
A_1&\leq C_{p,\alpha,T}(t_1-s)^{-\frac{\alpha}{2}}\int_{\mathbb{R}}|w-t_1u|^\alpha\left(p_{\frac{2s(t_1-s)}{t_1}}\left(y-\frac{s}{t_1}w\right)+p_{\frac{2s(t_1-s)}{t_1}}(y-su)\right)\\
&\qquad\times p_{\frac{t_1(t_2-t_1)}{t_2}}(w-t_1u)\,dw\\
&=:C_{p,\alpha,T}(t_1-s)^{-\frac{\alpha}{2}}(A'_1+A''_1). \tag*{(43)}
\end{aligned} A 1 ≤ C p , α , T ( t 1 − s ) − 2 α ∫ R ∣ w − t 1 u ∣ α ( p t 1 2 s ( t 1 − s ) ( y − t 1 s w ) + p t 1 2 s ( t 1 − s ) ( y − s u ) ) × p t 2 t 1 ( t 2 − t 1 ) ( w − t 1 u ) d w =: C p , α , T ( t 1 − s ) − 2 α ( A 1 ′ + A 1 ′′ ) . (43) We study separately A 1 ′ A'_1 A 1 ′ A 1 ′ ′ A''_1 A 1 ′′ A 1 ′ ′ A''_1 A 1 ′′
A 1 ′ ′ = p 2 s ( t 1 − s ) t 1 ( y − s u ) ∫ R ∣ w − t 1 u ∣ α p t 1 ( t 2 − t 1 ) t 2 ( w − t 1 u ) d w = C α p 2 s ( t 1 − s ) t 1 ( y − s u ) ( t 1 ( t 2 − t 1 ) t 2 ) α 2 , \begin{aligned}
A''_1&=p_{\frac{2s(t_1-s)}{t_1}}(y-su)\int_{\mathbb{R}}|w-t_1u|^\alpha p_{\frac{t_1(t_2-t_1)}{t_2}}(w-t_1u)\,dw\\
&=C_\alpha p_{\frac{2s(t_1-s)}{t_1}}(y-su)\left(\frac{t_1(t_2-t_1)}{t_2}\right)^{\frac{\alpha}{2}},
\end{aligned} A 1 ′′ = p t 1 2 s ( t 1 − s ) ( y − s u ) ∫ R ∣ w − t 1 u ∣ α p t 2 t 1 ( t 2 − t 1 ) ( w − t 1 u ) d w = C α p t 1 2 s ( t 1 − s ) ( y − s u ) ( t 2 t 1 ( t 2 − t 1 ) ) 2 α , where C α = E ∣ Z ∣ α C_\alpha=\mathbb{E}|Z|^\alpha C α = E ∣ Z ∣ α Z ∼ N ( 0 , 1 ) Z\sim N(0,1) Z ∼ N ( 0 , 1 )
∫ R ∣ x ∣ α p t ( x ) d x = E ∣ X ∣ α = t α 2 E ∣ Z ∣ α , where X ∼ N ( 0 , t ) . \int_{\mathbb{R}}|x|^\alpha p_t(x)\,dx=\mathbb{E}|X|^\alpha=t^{\frac{\alpha}{2}}\mathbb{E}|Z|^\alpha,\qquad\text{where }X\sim N(0,t). ∫ R ∣ x ∣ α p t ( x ) d x = E ∣ X ∣ α = t 2 α E ∣ Z ∣ α , where X ∼ N ( 0 , t ) . Using the fact that t 1 t 2 ≤ 1 \frac{t_1}{t_2}\leq1 t 2 t 1 ≤ 1
A 1 ′ ′ ≤ C α ( t 2 − t 1 ) α 2 p 2 s ( t 1 − s ) t 1 ( y − s u ) . (44) A''_1\leq C_\alpha(t_2-t_1)^{\frac{\alpha}{2}}p_{\frac{2s(t_1-s)}{t_1}}(y-su). \tag*{(44)} A 1 ′′ ≤ C α ( t 2 − t 1 ) 2 α p t 1 2 s ( t 1 − s ) ( y − s u ) . (44) To evaluate A 1 ′ A'_1 A 1 ′ α > 0 \alpha>0 α > 0 t > 0 t>0 t > 0 x ∈ R x\in\mathbb{R} x ∈ R
∣ x ∣ α p t ( x ) ≤ C α t α / 2 p 2 t ( x ) , |x|^\alpha p_t(x)\leq C_\alpha t^{\alpha/2}p_{2t}(x), ∣ x ∣ α p t ( x ) ≤ C α t α /2 p 2 t ( x ) , where C α = 2 ( 2 α e ) α / 2 C_\alpha=\sqrt{2}\left(\frac{2\alpha}{e}\right)^{\alpha/2} C α = 2 ( e 2 α ) α /2
∣ x ∣ α p t ( x ) = 2 t α / 2 p 2 t ( x ) ( ∣ x ∣ t ) α e − x 2 4 t ≤ 2 t α / 2 p 2 t ( x ) ⋅ K α , |x|^\alpha p_t(x)=\sqrt{2}t^{\alpha/2}p_{2t}(x)\left(\frac{|x|}{\sqrt{t}}\right)^\alpha e^{-\frac{x^2}{4t}}\leq\sqrt{2}t^{\alpha/2}p_{2t}(x)\cdot K_\alpha, ∣ x ∣ α p t ( x ) = 2 t α /2 p 2 t ( x ) ( t ∣ x ∣ ) α e − 4 t x 2 ≤ 2 t α /2 p 2 t ( x ) ⋅ K α , where K α : = sup y > 0 y α e − y 2 / 4 = ( 2 α e ) α / 2 K_\alpha:= \sup_{y>0} y^\alpha e^{-y^2/4} = \left(\frac{2\alpha}{e}\right)^{\alpha/2} K α := sup y > 0 y α e − y 2 /4 = ( e 2 α ) α /2
A 1 ′ ≤ C α ( t 1 ( t 2 − t 1 ) t 2 ) α / 2 ∫ R p 2 t 1 ( t 2 − t 1 ) t 2 ( w − t 1 u ) p 2 s ( t 1 − s ) t 1 ( y − s t 1 w ) d w . A'_1 \le C_\alpha\left(\frac{t_1(t_2-t_1)}{t_2}\right)^{\alpha/2} \int_{\mathbb{R}} p_{\frac{2t_1(t_2-t_1)}{t_2}}(w-t_1u)p_{\frac{2s(t_1-s)}{t_1}}\left(y-\frac{s}{t_1}w\right)\,dw. A 1 ′ ≤ C α ( t 2 t 1 ( t 2 − t 1 ) ) α /2 ∫ R p t 2 2 t 1 ( t 2 − t 1 ) ( w − t 1 u ) p t 1 2 s ( t 1 − s ) ( y − t 1 s w ) d w . To evaluate the last integral, we denote
A = 2 t 1 ( t 2 − t 1 ) t 2 , B = 2 s ( t 1 − s ) t 1 , a = s t 1 . A=\frac{2t_1(t_2-t_1)}{t_2}, \qquad B=\frac{2s(t_1-s)}{t_1}, \qquad a=\frac{s}{t_1}. A = t 2 2 t 1 ( t 2 − t 1 ) , B = t 1 2 s ( t 1 − s ) , a = t 1 s . Note also that p t ( x − a y ) = 1 a p t / a 2 ( x a − y ) p_t(x-ay)=\frac{1}{a}p_{t/a^2}\left(\frac{x}{a}-y\right) p t ( x − a y ) = a 1 p t / a 2 ( a x − y ) t , a > 0 t,a>0 t , a > 0 x , y ∈ R x,y\in\mathbb{R} x , y ∈ R
∫ R p A ( w − t 1 u ) p B ( y − a w ) d w = 1 a ∫ R p A ( x a − t 1 u ) p B ( y − x ) d x = ∫ R p a 2 A ( x − a t 1 u ) p B ( y − x ) d x = p a 2 A + B ( y − a t 1 u ) . \begin{aligned}
\int_{\mathbb{R}}p_A(w-t_1u)p_B(y-aw)\,dw=\frac{1}{a}\int_{\mathbb{R}}p_A\left(\frac{x}{a}-t_1u\right)p_B(y-x)\,dx \\
&=\int_{\mathbb{R}}p_{a^2A}(x-at_1u)p_B(y-x)\,dx=p_{a^2A+B}(y-at_1u).
\end{aligned} ∫ R p A ( w − t 1 u ) p B ( y − a w ) d w = a 1 ∫ R p A ( a x − t 1 u ) p B ( y − x ) d x = ∫ R p a 2 A ( x − a t 1 u ) p B ( y − x ) d x = p a 2 A + B ( y − a t 1 u ) . It follows that
A 1 ′ ≤ C α ( t 2 − t 1 ) α / 2 p a 2 A + B ( y − s u ) = C α ( t 2 − t 1 ) α / 2 p 2 s ( t 2 − s ) t 2 ( y − s u ) . (45) A'_1\le C_\alpha(t_2-t_1)^{\alpha/2}p_{a^2A+B}(y-su)=C_\alpha(t_2-t_1)^{\alpha/2}p_{\frac{2s(t_2-s)}{t_2}}(y-su). \tag*{(45)} A 1 ′ ≤ C α ( t 2 − t 1 ) α /2 p a 2 A + B ( y − s u ) = C α ( t 2 − t 1 ) α /2 p t 2 2 s ( t 2 − s ) ( y − s u ) . (45) since
a 2 A + B = 2 s 2 ( t 2 − t 1 ) t 1 t 2 + 2 s ( t 1 − s ) t 1 = 2 s ( t 2 − s ) t 2 . a^2A+B=\frac{2s^2(t_2-t_1)}{t_1t_2}+\frac{2s(t_1-s)}{t_1}=\frac{2s(t_2-s)}{t_2}. a 2 A + B = t 1 t 2 2 s 2 ( t 2 − t 1 ) + t 1 2 s ( t 1 − s ) = t 2 2 s ( t 2 − s ) . The conclusion follows from (42) , (43) , (44) and (45) .
Proposition 3.7. Let p ≥ 2 p\ge2 p ≥ 2 α ∈ ( 0 , 1 2 ) \alpha\in(0,\frac{1}{2}) α ∈ ( 0 , 2 1 ) N ≥ 2 N\ge2 N ≥ 2 0 < ε ≤ t 1 < t 2 ≤ T 0<\varepsilon\le t_1<t_2\le T 0 < ε ≤ t 1 < t 2 ≤ T t 2 − t 1 ≤ ε 2 t_2-t_1\le\frac{\varepsilon}{2} t 2 − t 1 ≤ 2 ε
E ∣ J N ′ ( t 1 , t 2 ) ∣ p ≤ C p , α , ε , T ( t 2 − t 1 ) α p / 2 . \mathbb{E}|J'_N(t_1,t_2)|^p\le C_{p,\alpha,\varepsilon,T}(t_2-t_1)^{\alpha p/2}. E ∣ J N ′ ( t 1 , t 2 ) ∣ p ≤ C p , α , ε , T ( t 2 − t 1 ) α p /2 . Proof. As in the proof of Proposition 3.2 , by (12) and Minkowski’s inequality,
∥ J N ′ ( t 1 , t 2 ) ∥ p 2 ≤ z p 2 N log N ∫ 0 t 1 ∫ R ( ∫ 0 N / t 2 ∥ t 2 A ( t 2 , t 2 u ; s , y ) q s t 2 , t 2 u ( y ) − t 1 A ( t 1 , t 1 u ; s , y ) q s t 1 , t 1 u ( y ) ∥ p d u ) 2 d y d s . \|J'_N(t_1,t_2)\|_p^2\le\frac{z_p^2}{N\log N}\int_0^{t_1}\int_{\mathbb{R}}\left(\int_0^{N/t_2}\left\|t_2\mathcal{A}(t_2,t_2u;s,y)q_s^{t_2,t_2u}(y)-t_1\mathcal{A}(t_1,t_1u;s,y)q_s^{t_1,t_1u}(y)\right\|_p\,du\right)^2dy\,ds. ∥ J N ′ ( t 1 , t 2 ) ∥ p 2 ≤ N log N z p 2 ∫ 0 t 1 ∫ R ( ∫ 0 N / t 2 t 2 A ( t 2 , t 2 u ; s , y ) q s t 2 , t 2 u ( y ) − t 1 A ( t 1 , t 1 u ; s , y ) q s t 1 , t 1 u ( y ) p d u ) 2 d y d s . We write the integral on [ 0 , t 1 ] [0,t_1] [ 0 , t 1 ] [ 0 , t 1 − ( t 2 − t 1 ) ] [0,t_1-(t_2-t_1)] [ 0 , t 1 − ( t 2 − t 1 )] [ t 1 − ( t 2 − t 1 ) , t 1 ] [t_1-(t_2-t_1),t_1] [ t 1 − ( t 2 − t 1 ) , t 1 ] T 1 T_1 T 1 T 2 T_2 T 2
∥ J N ′ ( t 1 , t 2 ) ∥ p 2 ≤ z p 2 ( T 1 + T 2 ) . \|J'_N(t_1,t_2)\|_p^2\le z_p^2(T_1+T_2). ∥ J N ′ ( t 1 , t 2 ) ∥ p 2 ≤ z p 2 ( T 1 + T 2 ) . We examine T 1 T_1 T 1 t 2 ≤ t 1 + ε 2 < 2 t 1 t_2\le t_1+\frac{\varepsilon}{2}<2t_1 t 2 ≤ t 1 + 2 ε < 2 t 1 α ∈ ( 0 , 1 2 ) \alpha\in(0,\frac{1}{2}) α ∈ ( 0 , 2 1 ) ( a + b ) 2 ≤ 2 ( a 2 + b 2 ) (a+b)^2\le2(a^2+b^2) ( a + b ) 2 ≤ 2 ( a 2 + b 2 )
T 1 ≤ C p , α , ε , T ( t 2 − t 1 ) α N log N { ∫ 0 t 1 − ( t 2 − t 1 ) ( t 1 − s ) − α ∫ R [ ∫ 0 N / t 2 p 2 s ( t 1 − s ) t 1 ( y − s u ) d u ] 2 d y d s T_1\le C_{p,\alpha,\varepsilon,T}\frac{(t_2-t_1)^\alpha}{N\log N}\left\{\int_0^{t_1-(t_2-t_1)}(t_1-s)^{-\alpha}\int_{\mathbb{R}}\left[\int_0^{N/t_2}p_{\frac{2s(t_1-s)}{t_1}}(y-su)\,du\right]^2dy\,ds
\right. T 1 ≤ C p , α , ε , T N log N ( t 2 − t 1 ) α ⎩ ⎨ ⎧ ∫ 0 t 1 − ( t 2 − t 1 ) ( t 1 − s ) − α ∫ R [ ∫ 0 N / t 2 p t 1 2 s ( t 1 − s ) ( y − s u ) d u ] 2 d y d s + ∫ 0 t 1 − ( t 2 − t 1 ) ( t 1 − s ) − α ∫ R [ ∫ 0 N / t 2 p 2 s ( t 2 − s ) t 2 ( y − s u ) d u ] 2 d y d s } ≤ C p , α , ε , T ( t 2 − t 1 ) α N log N ∫ 0 t 1 − ( t 2 − t 1 ) ( t 1 − s ) − α ( N s ∧ N 2 s ) d s . \begin{aligned}
\left.
+\int_0^{t_1-(t_2-t_1)}(t_1-s)^{-\alpha}\int_{\mathbb{R}}\left[\int_0^{N/t_2}p_{\frac{2s(t_2-s)}{t_2}}(y-su)\,du\right]^2dy\,ds\right\} \\
&\le C_{p,\alpha,\varepsilon,T}\frac{(t_2-t_1)^\alpha}{N\log N}\int_0^{t_1-(t_2-t_1)}(t_1-s)^{-\alpha}\left(\frac{N}{s}\wedge\frac{N^2}{\sqrt{s}}\right)\,ds.
\end{aligned} + ∫ 0 t 1 − ( t 2 − t 1 ) ( t 1 − s ) − α ∫ R [ ∫ 0 N / t 2 p t 2 2 s ( t 2 − s ) ( y − s u ) d u ] 2 d y d s ⎭ ⎬ ⎫ ≤ C p , α , ε , T N log N ( t 2 − t 1 ) α ∫ 0 t 1 − ( t 2 − t 1 ) ( t 1 − s ) − α ( s N ∧ s N 2 ) d s . For the last line, positivity permits enlargement of the first spatial integration interval from [ 0 , N / t 2 ] [0,N/t_2] [ 0 , N / t 2 ] [ 0 , N / t 1 ] [0,N/t_1] [ 0 , N / t 1 ]
Note that t 1 − ( t 2 − t 1 ) ≥ ε 2 t_1-(t_2-t_1)\ge\frac{\varepsilon}{2} t 1 − ( t 2 − t 1 ) ≥ 2 ε [ 0 , ε 2 ] [0,\frac{\varepsilon}{2}] [ 0 , 2 ε ] [ ε 2 , t 1 − ( t 2 − t 1 ) ] [\frac{\varepsilon}{2},t_1-(t_2-t_1)] [ 2 ε , t 1 − ( t 2 − t 1 )] A A A B B B t 1 − s ≥ ε / 2 t_1-s\ge\varepsilon/2 t 1 − s ≥ ε /2
A ≤ ( ε 2 ) − α ( ∫ 0 min { ε / 2 , N − 2 } N 2 s d s + 1 { ε / 2 > N − 2 } ∫ N − 2 ε / 2 N s d s ) ≤ ( ε 2 ) − α ( 2 N + 1 { ε / 2 > N − 2 } N log ε N 2 2 ) ≤ C α , ε N log N . \begin{aligned}
A\le\left(\frac{\varepsilon}{2}\right)^{-\alpha}\left(\int_0^{\min\{\varepsilon/2,N^{-2}\}}\frac{N^2}{\sqrt{s}}\,\mathrm{d}s+\mathbf{1}_{\{\varepsilon/2>N^{-2}\}}\int_{N^{-2}}^{\varepsilon/2}\frac{N}{s}\,\mathrm{d}s\right) \\
&\le\left(\frac{\varepsilon}{2}\right)^{-\alpha}\left(2N+\mathbf{1}_{\{\varepsilon/2>N^{-2}\}}N\log\frac{\varepsilon N^2}{2}\right)\le C_{\alpha,\varepsilon}N\log N.
\end{aligned} A ≤ ( 2 ε ) − α ( ∫ 0 m i n { ε /2 , N − 2 } s N 2 d s + 1 { ε /2 > N − 2 } ∫ N − 2 ε /2 s N d s ) ≤ ( 2 ε ) − α ( 2 N + 1 { ε /2 > N − 2 } N log 2 ε N 2 ) ≤ C α , ε N log N . B ≤ ∫ ε / 2 t 1 − ( t 2 − t 1 ) ( t 1 − s ) − α N s d s ≤ 2 N ε ∫ ε / 2 t 1 − ( t 2 − t 1 ) ( t 1 − s ) − α d s ≤ 2 N ε t 1 1 − α 1 − α ≤ C α , ε , T N . B\le\int_{\varepsilon/2}^{t_1-(t_2-t_1)}(t_1-s)^{-\alpha}\frac{N}{s}\,\mathrm{d}s\le\frac{2N}{\varepsilon}\int_{\varepsilon/2}^{t_1-(t_2-t_1)}(t_1-s)^{-\alpha}\,\mathrm{d}s\le\frac{2N}{\varepsilon}\frac{t_1^{1-\alpha}}{1-\alpha}\le C_{\alpha,\varepsilon,T}N. B ≤ ∫ ε /2 t 1 − ( t 2 − t 1 ) ( t 1 − s ) − α s N d s ≤ ε 2 N ∫ ε /2 t 1 − ( t 2 − t 1 ) ( t 1 − s ) − α d s ≤ ε 2 N 1 − α t 1 1 − α ≤ C α , ε , T N . Therefore,
T 1 ≤ C p , α , ε , T ( t 2 − t 1 ) α . T_1\le C_{p,\alpha,\varepsilon,T}(t_2-t_1)^\alpha. T 1 ≤ C p , α , ε , T ( t 2 − t 1 ) α . For T 2 T_2 T 2 ∥ X + Y ∥ p ≤ ∥ X ∥ p + ∥ Y ∥ p \lVert X+Y\rVert_p\le\lVert X\rVert_p+\lVert Y\rVert_p ∥ X + Y ∥ p ≤ ∥ X ∥ p + ∥ Y ∥ p
T 2 ≤ C p , T N log N ∑ i = 1 2 ∫ t 1 − ( t 2 − t 1 ) t 1 ∫ R ( ∫ 0 N / t 2 q s t i , t i ( y ) d u ) 2 d y d s . T_2\le\frac{C_{p,T}}{N\log N}\sum_{i=1}^{2}\int_{t_1-(t_2-t_1)}^{t_1}\int_{\mathbb{R}}\left(\int_0^{N/t_2}q_s^{t_i,t_i} (y)\,\mathrm{d}u\right)^2\mathrm{d}y\,\mathrm{d}s. T 2 ≤ N log N C p , T i = 1 ∑ 2 ∫ t 1 − ( t 2 − t 1 ) t 1 ∫ R ( ∫ 0 N / t 2 q s t i , t i ( y ) d u ) 2 d y d s . We now use again Lemma A.4 with c = 1 c=1 c = 1
T 2 ≤ C p , ε , T N log N ∫ t 1 − ( t 2 − t 1 ) t 1 ( N s ∧ N 2 s ) d s ≤ C p , ε , T log N ∫ t 1 − ( t 2 − t 1 ) t 1 1 s d s ≤ C p , ε , T log N 2 ε ( t 2 − t 1 ) ≤ C p , ε , T ( t 2 − t 1 ) . \begin{aligned}
T_2\le\frac{C_{p,\varepsilon,T}}{N\log N}\int_{t_1-(t_2-t_1)}^{t_1}\left(\frac{N}{s}\wedge\frac{N^2}{\sqrt{s}}\right)\,\mathrm{d}s \\
&\le\frac{C_{p,\varepsilon,T}}{\log N}\int_{t_1-(t_2-t_1)}^{t_1}\frac{1}{s}\,\mathrm{d}s\le\frac{C_{p,\varepsilon,T}}{\log N}\frac{2}{\varepsilon}(t_2-t_1)\le C_{p,\varepsilon,T}(t_2-t_1).
\end{aligned} T 2 ≤ N log N C p , ε , T ∫ t 1 − ( t 2 − t 1 ) t 1 ( s N ∧ s N 2 ) d s ≤ log N C p , ε , T ∫ t 1 − ( t 2 − t 1 ) t 1 s 1 d s ≤ log N C p , ε , T ε 2 ( t 2 − t 1 ) ≤ C p , ε , T ( t 2 − t 1 ) . For the last line, we used the fact that s ≥ t 1 − ( t 2 − t 1 ) ≥ ε − ε 2 = ε 2 s\ge t_1-(t_2-t_1)\ge\varepsilon-\frac{\varepsilon}{2}=\frac{\varepsilon}{2} s ≥ t 1 − ( t 2 − t 1 ) ≥ ε − 2 ε = 2 ε T 1 T_1 T 1 T 2 T_2 T 2
∥ J N ′ ( t 1 , t 2 ) ∥ p 2 ≤ C p , α , ε , T ( ( t 2 − t 1 ) α + ( t 2 − t 1 ) ) ≤ C p , α , ε , T ( t 2 − t 1 ) α . \lVert J_N'(t_1,t_2)\rVert_p^2\le C_{p,\alpha,\varepsilon,T}\left((t_2-t_1)^\alpha+(t_2-t_1)\right)\le C_{p,\alpha,\varepsilon,T}(t_2-t_1)^\alpha. ∥ J N ′ ( t 1 , t 2 ) ∥ p 2 ≤ C p , α , ε , T ( ( t 2 − t 1 ) α + ( t 2 − t 1 ) ) ≤ C p , α , ε , T ( t 2 − t 1 ) α . Indeed, t 2 − t 1 ≤ T t_2-t_1\le T t 2 − t 1 ≤ T 0 < α < 1 0<\alpha<1 0 < α < 1 t 2 − t 1 ≤ T 1 − α ( t 2 − t 1 ) α t_2-t_1\le T^{1-\alpha}(t_2-t_1)^\alpha t 2 − t 1 ≤ T 1 − α ( t 2 − t 1 ) α p / 2 p/2 p /2
Proof of Theorem 1.1. As mentioned at the beginning of the proof of Section 3, it is enough to prove that (3.1) holds. This follows from decompositions (3.5) and (3.12), using: a) Proposition 3.2 for the term I N ( t 1 , t 2 ) I_N(t_1,t_2) I N ( t 1 , t 2 ) J N ( t 1 , t 2 ) J_N(t_1,t_2) J N ( t 1 , t 2 ) t 2 − t 1 > ε / 2 t_2-t_1>\varepsilon/2 t 2 − t 1 > ε /2 J N ′ ′ ( t 1 , t 2 ) J_N''(t_1,t_2) J N ′′ ( t 1 , t 2 ) t 2 − t 1 ≤ ε / 2 t_2-t_1\le\varepsilon/2 t 2 − t 1 ≤ ε /2 J N ′ ( t 1 , t 2 ) J_N'(t_1,t_2) J N ′ ( t 1 , t 2 ) t 2 − t 1 ≤ ε / 2 t_2-t_1\le\varepsilon/2 t 2 − t 1 ≤ ε /2
Acknowledgements
R. M. B. was supported by a grant from the Natural Sciences and Engineering Research Council of Canada. L. C. was partially supported by NSF grant DMS-2443823 and a Collaboration Grant for Mathematicians (#959981) from the Simons Foundation. J. W. was partially supported by a Mitacs Globalink Research Award Abroad (IT52612), which funded his visit to Auburn University.
Auxiliary results
Properties of p t ( x ) p_t(x) p t ( x ) q s t , x ( y ) q_s^{t,x}(y) q s t , x ( y )
In this section, we include some auxiliary results about the increments of p p p q q q
Lemma A.1. Let α ∈ ( 0 , 1 ] \alpha\in(0,1] α ∈ ( 0 , 1 ] t > 0 t>0 t > 0 x , y ∈ R x,y \in\mathbb{R} x , y ∈ R
∫ R ∣ p t ( x − z ) − p t ( y − z ) ∣ 2 d z ≤ 4 − α π t − 1 2 − α ∣ x − y ∣ 2 α . \int_{\mathbb{R}} \lvert p_t(x-z)-p_t(y-z)\rvert^2 \,\mathrm{d}z \le\frac{4^{-\alpha}}{\sqrt{\pi}}t^{-\frac{1}{2}-\alpha}\lvert x-y\rvert^{2\alpha}. ∫ R ∣ p t ( x − z ) − p t ( y − z ) ∣ 2 d z ≤ π 4 − α t − 2 1 − α ∣ x − y ∣ 2 α . Proof. By the semigroup property of the heat kernel, we obtain
∫ R ∣ p t ( x − z ) − p t ( y − z ) ∣ 2 d z = 2 p 2 t ( 0 ) − 2 p 2 t ( x − y ) = 1 π t ( 1 − exp ( − ∣ x − y ∣ 2 4 t ) ) . \int_{\mathbb{R}} \lvert p_t(x-z)-p_t(y-z)\rvert^2 \,\mathrm{d}z = 2p_{2t}(0)-2p_{2t}(x-y) = \frac{1}{\sqrt{\pi t}}\left(1-\exp\left(-\frac{\lvert x-y\rvert^2}{4t}\right)\right). ∫ R ∣ p t ( x − z ) − p t ( y − z ) ∣ 2 d z = 2 p 2 t ( 0 ) − 2 p 2 t ( x − y ) = π t 1 ( 1 − exp ( − 4 t ∣ x − y ∣ 2 ) ) . For any r ≥ 0 r\ge0 r ≥ 0 α ∈ ( 0 , 1 ] \alpha\in(0,1] α ∈ ( 0 , 1 ] 1 − e − r ≤ 1 ∧ r ≤ r α 1-e^{-r}\le1\wedge r\le r^\alpha 1 − e − r ≤ 1 ∧ r ≤ r α
∫ R ∣ p t ( x − z ) − p t ( y − z ) ∣ 2 d z ≤ 1 π t ( ∣ x − y ∣ 2 4 t ) α = 4 − α π t − 1 2 − α ∣ x − y ∣ 2 α . \int_{\mathbb{R}} \lvert p_t(x-z)-p_t(y-z)\rvert^2 \,\mathrm{d}z \le\frac{1}{\sqrt{\pi t}}\left(\frac{\lvert x-y\rvert^2}{4t}\right)^\alpha= \frac{4^{-\alpha}}{\sqrt{\pi}}t^{-\frac{1}{2}-\alpha}\lvert x-y\rvert^{2\alpha}. ∫ R ∣ p t ( x − z ) − p t ( y − z ) ∣ 2 d z ≤ π t 1 ( 4 t ∣ x − y ∣ 2 ) α = π 4 − α t − 2 1 − α ∣ x − y ∣ 2 α . Lemma A.2. Let α ∈ ( 0 , 1 ] \alpha\in(0,1] α ∈ ( 0 , 1 ] t > 0 t>0 t > 0 x , y , z ∈ R x,y,z \in\mathbb{R} x , y , z ∈ R
∣ p t ( x − z ) − p t ( y − z ) ∣ ≤ C α t − α 2 ∣ x − y ∣ α ( p 2 t ( x − z ) + p 2 t ( y − z ) ) . \lvert p_t(x-z)-p_t(y-z)\rvert\le C_\alpha t^{-\frac{\alpha}{2}}\lvert x-y\rvert^\alpha\left(p_{2t}(x-z)+p_{2t}(y-z)\right). ∣ p t ( x − z ) − p t ( y − z )∣ ≤ C α t − 2 α ∣ x − y ∣ α ( p 2 t ( x − z ) + p 2 t ( y − z ) ) . Proof. We refer to the proof of Lemma 3.1 of [10 ] .
Recall definition (3.3) of q q q q q q
Lemma A.3. Let α ∈ ( 0 , 1 ] \alpha\in(0,1] α ∈ ( 0 , 1 ] 0 < s < t 0<s<t 0 < s < t x 1 , x 2 , y ∈ R x_1,x_2,y \in\mathbb{R} x 1 , x 2 , y ∈ R
∣ q s t , x 2 ( y ) − q s t , x 1 ( y ) ∣ ≤ C α ( t − s ) − α 2 ∣ x 1 − x 2 ∣ α ( p 2 s ( t − s ) t ( y − s t x 1 ) + p 2 s ( t − s ) t ( y − s t x 2 ) ) . \lvert q_s^{t,x_2}(y)-q_s^{t,x_1}(y)\rvert\le C_\alpha(t-s)^{-\frac{\alpha}{2}}\lvert x_1-x_2\rvert^\alpha\left(p_{\frac{2s(t-s)}{t}}\left(y-\frac{s}{t}x_1\right)+p_{\frac{2s(t-s)}{t}}\left(y-\frac{s}{t}x_2\right)\right). ∣ q s t , x 2 ( y ) − q s t , x 1 ( y )∣ ≤ C α ( t − s ) − 2 α ∣ x 1 − x 2 ∣ α ( p t 2 s ( t − s ) ( y − t s x 1 ) + p t 2 s ( t − s ) ( y − t s x 2 ) ) . Proof. We apply Lemma A.2. It follows that
∣ q s t , x 2 ( y ) − q s t , x 1 ( y ) ∣ = ∣ p s ( t − s ) t ( y − s t x 1 ) − p s ( t − s ) t ( y − s t x 2 ) ∣ ≤ C α ( s ( t − s ) t ) − α 2 ( s t ∣ x 1 − x 2 ∣ ) α ( p 2 s ( t − s ) t ( y − s t x 1 ) + p 2 s ( t − s ) t ( y − s t x 2 ) ) . \begin{aligned}
\lvert q_s^{t,x_2}(y)-q_s^{t,x_1}(y)\rvert
&=\left\lvert p_{\frac{s(t-s)}{t}}\left(y-\frac{s}{t}x_1\right)-p_{\frac{s(t-s)}{t}}\left(y-\frac{s}{t}x_2\right)\right\rvert\\
&\le C_\alpha\left(\frac{s(t-s)}{t}\right)^{-\frac{\alpha}{2}}\left(\frac{s}{t}\lvert x_1-x_2\rvert\right)^\alpha\left(p_{\frac{2s(t-s)}{t}}\left(y-\frac{s}{t}x_1\right)+p_{\frac{2s(t-s)}{t}}\left(y-\frac{s}{t}x_2\right)\right).
\end{aligned} ∣ q s t , x 2 ( y ) − q s t , x 1 ( y )∣ = p t s ( t − s ) ( y − t s x 1 ) − p t s ( t − s ) ( y − t s x 2 ) ≤ C α ( t s ( t − s ) ) − 2 α ( t s ∣ x 1 − x 2 ∣ ) α ( p t 2 s ( t − s ) ( y − t s x 1 ) + p t 2 s ( t − s ) ( y − t s x 2 ) ) . The conclusion follows since
( s ( t − s ) t ) − α 2 ( s t ) α = ( s t ) α 2 ( t − s ) − α 2 ≤ ( t − s ) − α 2 . \left(\frac{s(t-s)}{t}\right)^{-\frac{\alpha}{2}}\left(\frac{s}{t}\right)^\alpha=\left(\frac{s}{t}\right)^{\frac{\alpha}{2}}(t-s)^{-\frac{\alpha}{2}}\le(t-s)^{-\frac{\alpha}{2}}. ( t s ( t − s ) ) − 2 α ( t s ) α = ( t s ) 2 α ( t − s ) − 2 α ≤ ( t − s ) − 2 α . Lemma A.4. Let c > 0 c>0 c > 0 N ≥ 1 N\ge1 N ≥ 1 0 < ε ≤ t 1 < t 2 ≤ T 0<\varepsilon\le t_1<t_2\le T 0 < ε ≤ t 1 < t 2 ≤ T s ∈ ( 0 , t 1 ) s\in(0,t_1) s ∈ ( 0 , t 1 )
∫ R ( ∫ 0 N / t i p c s ( t i − s ) t i ( y − s u ) d u ) 2 d y ≤ C c , ε , T ( N s ∧ N 2 s ) , for i = 1 , 2. \int_{\mathbb{R}}\left(\int_0^{N/t_i}p_{\frac{cs(t_i-s)}{t_i}}(y-su)\,du\right)^2dy\le C_{c,\varepsilon,T}\left(\frac{N}{s}\wedge\frac{N^2}{\sqrt{s}}\right),\qquad\text{for }i=1,2. ∫ R ( ∫ 0 N / t i p t i cs ( t i − s ) ( y − s u ) d u ) 2 d y ≤ C c , ε , T ( s N ∧ s N 2 ) , for i = 1 , 2. Proof. We denote the integral on the left side by I I I i = 1 , 2 i=1,2 i = 1 , 2
I = ∫ 0 N / t i ∫ 0 N / t i p 2 c s ( t i − s ) t i ( s ( u − v ) ) d u d v ≤ ∫ 0 N / t i ∫ R p 2 c s ( t i − s ) t i ( s ( u − v ) ) d v d u = N t i s ≤ N ε s . (46) I=\int_0^{N/t_i}\int_0^{N/t_i}p_{\frac{2cs(t_i-s)}{t_i}}\bigl(s(u-v)\bigr)\,du\,dv\le\int_0^{N/t_i}\int_{\mathbb{R}}p_{\frac{2cs(t_i-s)}{t_i}}\bigl(s(u-v)\bigr)\,dv\,du=\frac{N}{t_i s}\le\frac{N}{\varepsilon s}. \tag*{(46)} I = ∫ 0 N / t i ∫ 0 N / t i p t i 2 cs ( t i − s ) ( s ( u − v ) ) d u d v ≤ ∫ 0 N / t i ∫ R p t i 2 cs ( t i − s ) ( s ( u − v ) ) d v d u = t i s N ≤ ε s N . (46) This gives the first bound. To obtain the second bound, suppose first that s ∈ ( 0 , ε 2 ) s\in(0,\frac{\varepsilon}{2}) s ∈ ( 0 , 2 ε ) t 2 − s > t 1 − s > t 1 − ε 2 ≥ ε 2 t_2-s>t_1-s>t_1-\frac{\varepsilon}{2}\ge\frac{\varepsilon}{2} t 2 − s > t 1 − s > t 1 − 2 ε ≥ 2 ε
p 2 c s ( t i − s ) t i ( s ( u − v ) ) ≤ p 2 c s ( t i − s ) t i ( 0 ) = 1 4 π c t i s ( t i − s ) ≤ 1 4 π c T ε / 2 ⋅ 1 s , p_{\frac{2cs(t_i-s)}{t_i}}\bigl(s(u-v)\bigr)\le p_{\frac{2cs(t_i-s)}{t_i}}(0)=\frac{1}{\sqrt{4\pi c}}\sqrt{\frac{t_i}{s(t_i-s)}}\le\frac{1}{\sqrt{4\pi c}}\sqrt{\frac{T}{\varepsilon/2}}\cdot\frac{1}{\sqrt{s}}, p t i 2 cs ( t i − s ) ( s ( u − v ) ) ≤ p t i 2 cs ( t i − s ) ( 0 ) = 4 π c 1 s ( t i − s ) t i ≤ 4 π c 1 ε /2 T ⋅ s 1 , and
I ≤ ( N t i ) 2 1 4 π c T ε / 2 ⋅ 1 s ≤ ( N ε ) 2 1 4 π c T ε / 2 ⋅ 1 s . I\le\left(\frac{N}{t_i}\right)^2\frac{1}{\sqrt{4\pi c}}\sqrt{\frac{T}{\varepsilon/2}}\cdot\frac{1}{\sqrt{s}}\le\left(\frac{N}{\varepsilon}\right)^2\frac{1}{\sqrt{4\pi c}}\sqrt{\frac{T}{\varepsilon/2}}\cdot\frac{1}{\sqrt{s}}. I ≤ ( t i N ) 2 4 π c 1 ε /2 T ⋅ s 1 ≤ ( ε N ) 2 4 π c 1 ε /2 T ⋅ s 1 . If s ∈ [ ε 2 , t 1 ) s\in[\frac{\varepsilon}{2},t_1) s ∈ [ 2 ε , t 1 ) 1 s ≤ 2 ε 1 s \frac{1}{s}\le\sqrt{\frac{2}{\varepsilon}}\frac{1}{\sqrt{s}} s 1 ≤ ε 2 s 1
I ≤ N ε s ≤ N 2 ε s ≤ N 2 ε 2 ε 1 s . I\le\frac{N}{\varepsilon s}\le\frac{N^2}{\varepsilon s}\le\frac{N^2}{\varepsilon}\sqrt{\frac{2}{\varepsilon}}\frac{1}{\sqrt{s}}. I ≤ ε s N ≤ ε s N 2 ≤ ε N 2 ε 2 s 1 . Increments of normalized G
In this section, we present a lemma about the increments of G ˉ \bar{G} G ˉ
Lemma A.5. Let p ≥ 2 p\ge2 p ≥ 2 α ∈ ( 0 , 1 2 ) \alpha\in(0,\frac{1}{2}) α ∈ ( 0 , 2 1 ) 0 ≤ s < t ≤ T 0\le s<t\le T 0 ≤ s < t ≤ T x 1 , x 2 , y ∈ R x_1,x_2,y\in\mathbb{R} x 1 , x 2 , y ∈ R
∥ G ˉ ( t , x 2 ; s , y ) − G ˉ ( t , x 1 ; s , y ) ∥ p ≤ C p , T 2 − α π 1 / 4 cos ( π α ) ( t − s ) 1 4 − α 2 ∣ x 1 − x 2 ∣ α , (47) \left\|\bar{G}(t,x_2;s,y)-\bar{G}(t,x_1;s,y)\right\|_p\le C_{p,T}\frac{2^{-\alpha}\pi^{1/4}}{\sqrt{\cos(\pi\alpha)}}(t-s)^{\frac{1}{4}-\frac{\alpha}{2}}|x_1-x_2|^\alpha, \tag*{(47)} G ˉ ( t , x 2 ; s , y ) − G ˉ ( t , x 1 ; s , y ) p ≤ C p , T cos ( π α ) 2 − α π 1/4 ( t − s ) 4 1 − 2 α ∣ x 1 − x 2 ∣ α , (47) and consequently,
∥ G ˉ ( t , x 2 ; s , y ) G ˉ ( t , x 2 ; 0 , 0 ) − G ˉ ( t , x 1 ; s , y ) G ˉ ( t , x 1 ; 0 , 0 ) ∥ p ≤ C α , p , T ∣ x 1 − x 2 ∣ α . (48) \left\|\frac{\bar{G}(t,x_2;s,y)}{\bar{G}(t,x_2;0,0)}-\frac{\bar{G}(t,x_1;s,y)}{\bar{G}(t,x_1;0,0)}\right\|_p\le C_{\alpha,p,T}|x_1-x_2|^\alpha. \tag*{(48)} G ˉ ( t , x 2 ; 0 , 0 ) G ˉ ( t , x 2 ; s , y ) − G ˉ ( t , x 1 ; 0 , 0 ) G ˉ ( t , x 1 ; s , y ) p ≤ C α , p , T ∣ x 1 − x 2 ∣ α . (48) Proof. We first prove (A.2). Dividing (17) by p t − s ( x − y ) p_{t-s}(x-y) p t − s ( x − y ) (24) , we obtain:
G ‾ ( t , x ; s , y ) = 1 + ∫ s t ∫ R p t − r ( x − z ) p r − s ( z − y ) p t − s ( x − y ) G ‾ ( r , z ; s , y ) ξ ( d r , d z ) = 1 + ∫ s t ∫ R q t − s , x − y r − s ( z − y ) G ‾ ( r , z ; s , y ) ξ ( d r , d z ) . \begin{aligned}
\overline{G}(t,x;s,y)=1+\int_s^t\int_{\mathbb{R}}\frac{p_{t-r}(x-z)p_{r-s}(z-y)}{p_{t-s}(x-y)}\overline{G}(r,z;s,y)\xi(\mathrm{d}r,\mathrm{d}z) \\
&=1+\int_s^t\int_{\mathbb{R}}q_{\frac{t-s,x-y}{r-s}}(z-y)\overline{G}(r,z;s,y)\xi(\mathrm{d}r,\mathrm{d}z).
\end{aligned} G ( t , x ; s , y ) = 1 + ∫ s t ∫ R p t − s ( x − y ) p t − r ( x − z ) p r − s ( z − y ) G ( r , z ; s , y ) ξ ( d r , d z ) = 1 + ∫ s t ∫ R q r − s t − s , x − y ( z − y ) G ( r , z ; s , y ) ξ ( d r , d z ) . By inequality (12) , relation (20) , and Lemma A.1 ,
∥ G ‾ ( t , x 2 ; s , y ) − G ‾ ( t , x 1 ; s , y ) ∥ p 2 ≤ C p , T ∫ s t ∫ R ∣ q t − s , x 2 − y r − s ( z − y ) − q t − s , x 1 − y r − s ( z − y ) ∣ 2 d z d r ≤ C p , T 4 1 − α π ∣ x 1 − x 2 ∣ 2 α ∫ s t ( r − s t − s ) 2 α ( ( r − s ) ( t − r ) t − s ) − 1 2 − α d r = C p , T 4 1 − α π ∣ x 1 − x 2 ∣ 2 α ( t − s ) 1 2 − α ∫ s t ( r − s ) α − 1 2 ( t − r ) − α − 1 2 d r = C p , T 4 1 − α π Γ ( α + 1 2 ) Γ ( 1 2 − α ) ∣ x 1 − x 2 ∣ 2 α ( t − s ) 1 2 − α = C p , T 4 1 − α π cos ( π α ) ∣ x 1 − x 2 ∣ 2 α ( t − s ) 1 2 − α . \begin{aligned}
\left\|\overline{G}(t,x_2;s,y)-\overline{G}(t,x_1;s,y)\right\|_p^2
\le C_{p,T}\int_s^t\int_{\mathbb{R}}\left|q_{\frac{t-s,x_2-y}{r-s}}(z-y)-q_{\frac{t-s,x_1-y}{r-s}}(z-y)\right|^2\mathrm{d}z\mathrm{d}r \\
&\le C_{p,T}\frac{4^{1-\alpha}}{\sqrt{\pi}}|x_1-x_2|^{2\alpha}\int_s^t\left(\frac{r-s}{t-s}\right)^{2\alpha}\left(\frac{(r-s)(t-r)}{t-s}\right)^{-\frac{1}{2}-\alpha}\mathrm{d}r \\
&=C_{p,T}\frac{4^{1-\alpha}}{\sqrt{\pi}}|x_1-x_2|^{2\alpha}(t-s)^{\frac{1}{2}-\alpha}\int_s^t(r-s)^{\alpha-\frac{1}{2}}(t-r)^{-\alpha-\frac{1}{2}}\mathrm{d}r \\
&=C_{p,T}\frac{4^{1-\alpha}}{\sqrt{\pi}}\Gamma\left(\alpha+\frac{1}{2}\right)\Gamma\left(\frac{1}{2}-\alpha\right)|x_1-x_2|^{2\alpha}(t-s)^{\frac{1}{2}-\alpha} \\
&=C_{p,T}\frac{4^{1-\alpha}\sqrt{\pi}}{\cos(\pi\alpha)}|x_1-x_2|^{2\alpha}(t-s)^{\frac{1}{2}-\alpha}.
\end{aligned} G ( t , x 2 ; s , y ) − G ( t , x 1 ; s , y ) p 2 ≤ C p , T ∫ s t ∫ R q r − s t − s , x 2 − y ( z − y ) − q r − s t − s , x 1 − y ( z − y ) 2 d z d r ≤ C p , T π 4 1 − α ∣ x 1 − x 2 ∣ 2 α ∫ s t ( t − s r − s ) 2 α ( t − s ( r − s ) ( t − r ) ) − 2 1 − α d r = C p , T π 4 1 − α ∣ x 1 − x 2 ∣ 2 α ( t − s ) 2 1 − α ∫ s t ( r − s ) α − 2 1 ( t − r ) − α − 2 1 d r = C p , T π 4 1 − α Γ ( α + 2 1 ) Γ ( 2 1 − α ) ∣ x 1 − x 2 ∣ 2 α ( t − s ) 2 1 − α = C p , T cos ( π α ) 4 1 − α π ∣ x 1 − x 2 ∣ 2 α ( t − s ) 2 1 − α . The Beta integral is independent of t − s t-s t − s Γ ( z ) Γ ( 1 − z ) = π / sin ( π z ) \Gamma(z)\Gamma(1-z)=\pi/\sin(\pi z) Γ ( z ) Γ ( 1 − z ) = π / sin ( π z ) z = α + 1 2 z=\alpha+\frac{1}{2} z = α + 2 1 π / cos ( π α ) \pi/\cos(\pi\alpha) π / cos ( π α )
For (A.3), we write
G ‾ ( t , x 2 ; s , y ) G ‾ ( t , x 2 ; 0 , 0 ) − G ‾ ( t , x 1 ; s , y ) G ‾ ( t , x 1 ; 0 , 0 ) = G ‾ ( t , x 2 ; s , y ) − G ‾ ( t , x 1 ; s , y ) G ‾ ( t , x 2 ; 0 , 0 ) + G ‾ ( t , x 1 ; s , y ) G ‾ ( t , x 1 ; 0 , 0 ) − G ‾ ( t , x 2 ; 0 , 0 ) G ‾ ( t , x 1 ; 0 , 0 ) G ‾ ( t , x 2 ; 0 , 0 ) , \begin{aligned}
\frac{\overline{G}(t,x_2;s,y)}{\overline{G}(t,x_2;0,0)}-\frac{\overline{G}(t,x_1;s,y)}{\overline{G}(t,x_1;0,0)} \\
&=\frac{\overline{G}(t,x_2;s,y)-\overline{G}(t,x_1;s,y)}{\overline{G}(t,x_2;0,0)}+\overline{G}(t,x_1;s,y)\frac{\overline{G}(t,x_1;0,0)-\overline{G}(t,x_2;0,0)}{\overline{G}(t,x_1;0,0)\overline{G}(t,x_2;0,0)},
\end{aligned} G ( t , x 2 ; 0 , 0 ) G ( t , x 2 ; s , y ) − G ( t , x 1 ; 0 , 0 ) G ( t , x 1 ; s , y ) = G ( t , x 2 ; 0 , 0 ) G ( t , x 2 ; s , y ) − G ( t , x 1 ; s , y ) + G ( t , x 1 ; s , y ) G ( t , x 1 ; 0 , 0 ) G ( t , x 2 ; 0 , 0 ) G ( t , x 1 ; 0 , 0 ) − G ( t , x 2 ; 0 , 0 ) , and apply generalized Hölder’s inequality. Since (20) holds for every k ∈ R k\in\mathbb{R} k ∈ R G ‾ ( t , x i ; 0 , 0 ) − 1 \overline{G}(t,x_i;0,0)^{-1} G ( t , x i ; 0 , 0 ) − 1 s = 0 s=0 s = 0 t 1 4 − α 2 t^{\frac{1}{4}-\frac{\alpha}{2}} t 4 1 − 2 α T 1 4 − α 2 T^{\frac{1}{4}-\frac{\alpha}{2}} T 4 1 − 2 α
Covariance of the normalized stochastic heat equation
The covariance asymptotic below follows from [21 Remark 4.4] , applied to the identity function. The variance asymptotic is [13 Proposition 4.1] . We include a short direct proof for completeness.
Proposition A.6 (Normalized-SHE covariance). Recall ρ ‾ t Z ‾ ( x ) \overline{\rho}_t^{\overline{Z}}(x) ρ t Z ( x ) (6) . For every fixed t > 0 t>0 t > 0
ρ ‾ t Z ‾ ( x ) ∼ t ∣ x ∣ as ∣ x ∣ → ∞ and Var ( S ‾ N ( t ) ) ∼ 2 t N log N as N → ∞ . \overline{\rho}_t^{\overline{Z}}(x)\sim\frac{t}{|x|}\quad\text{as }|x|\to\infty\quad\text{and}\quad\operatorname{Var}(\overline{S}_N(t))\sim2tN\log N\quad\text{as }N\to\infty. ρ t Z ( x ) ∼ ∣ x ∣ t as ∣ x ∣ → ∞ and Var ( S N ( t )) ∼ 2 tN log N as N → ∞. Proof. Put m ( s ) = E ∣ Z ( s , 0 ) ∣ 2 m(s)=\mathbb{E}|Z(s,0)|^2 m ( s ) = E ∣ Z ( s , 0 ) ∣ 2 Z ‾ ( s , x ) = G ‾ ( s , x ; 0 , 0 ) \overline{Z}(s,x)=\overline{G}(s,x;0,0) Z ( s , x ) = G ( s , x ; 0 , 0 ) [11 ] , with noise strength one, imply m ( s ) → 1 m(s)\to1 m ( s ) → 1 s ↓ 0 s\downarrow0 s ↓ 0 M t : = sup 0 < s ≤ t m ( s ) < ∞ M_t:=\sup_{0<s\leq t}m(s)<\infty M t := sup 0 < s ≤ t m ( s ) < ∞ x > 0 x>0 x > 0 v = t ( t − s ) / s v=t(t-s)/s v = t ( t − s ) / s v = x 2 w v=x^2w v = x 2 w
ρ ‾ t Z ( x ) = ∫ 0 t m ( s ) p 2 s ( t − s ) / t ( s x / t ) d s . \overline{\rho}_t^Z(x)=\int_0^t m(s)p_{2s(t-s)/t}(sx/t)\,\mathrm{d}s. ρ t Z ( x ) = ∫ 0 t m ( s ) p 2 s ( t − s ) / t ( s x / t ) d s . x ρ ‾ t Z ( x ) = t ∫ 0 ∞ x 2 t + x 2 w m ( t 2 t + x 2 w ) p 2 w ( 1 ) d w . x\overline{\rho}_t^Z(x)=t\int_0^\infty\frac{x^2}{t+x^2w}m\left(\frac{t^2}{t+x^2w}\right)p_{2w}(1)\,\mathrm{d}w. x ρ t Z ( x ) = t ∫ 0 ∞ t + x 2 w x 2 m ( t + x 2 w t 2 ) p 2 w ( 1 ) d w . The second integrand converges to t p 2 w ( 1 ) / w tp_{2w}(1)/w t p 2 w ( 1 ) / w t M t p 2 w ( 1 ) / w tM_tp_{2w}(1)/w t M t p 2 w ( 1 ) / w ∫ 0 ∞ p 2 w ( 1 ) w − 1 d w = 1 \int_0^\infty p_{2w}(1)w^{-1}\,\mathrm{d}w=1 ∫ 0 ∞ p 2 w ( 1 ) w − 1 d w = 1 z = 1 / ( 2 w ) z=1/(2\sqrt{w}) z = 1/ ( 2 w )
By stationarity, Var ( S N ( t ) ) = 2 ∫ 0 N ( N − x ) ρ ‾ t Z ( x ) d x \operatorname{Var}(S_N(t))=2\int_0^N(N-x)\overline{\rho}_t^Z(x)\,\mathrm{d}x Var ( S N ( t )) = 2 ∫ 0 N ( N − x ) ρ t Z ( x ) d x m ( t ) − 1 m(t)-1 m ( t ) − 1 [ 0 , R ] [0,R] [ 0 , R ] O ( N ) O(N) O ( N ) δ ∈ ( 0 , t ) \delta\in(0,t) δ ∈ ( 0 , t ) R > 0 R>0 R > 0 ( t − δ ) / x (t-\delta)/x ( t − δ ) / x ( t + δ ) / x (t+\delta)/x ( t + δ ) / x ∫ R N ( N − x ) x − 1 d x = N log N + O R ( N ) \int_R^N(N-x)x^{-1}\,\mathrm{d}x=N\log N+O_R(N) ∫ R N ( N − x ) x − 1 d x = N log N + O R ( N ) N log N N\log N N log N 2 ( t − δ ) 2(t-\delta) 2 ( t − δ ) 2 ( t + δ ) 2(t+\delta) 2 ( t + δ ) δ ↓ 0 \delta\downarrow0 δ ↓ 0
