APPROXIMATE SOLUTIONS TO SECOND ORDER PARABOLIC

EQUATIONS II: TIME-DEPENDENT COEFFICIENTS

WEN CHENG, ANNA MAZZUCATO, AND VICTOR NISTOR

Abstract. We consider second order parabolic equations with coefficients

that vary both in space and in time (non-autonomous). We derive closed-form approximations to the associated fundamental solution by extending

the Dyson-Taylor commutator method that we recently established for au-

tonomous equations. We establish error bounds in Sobolev spaces and showthat by including enough terms, our approximation can be proven to be accu-

rate to arbitrary high order in the short-time limit. We show how our method

extends to give an approximation of the solution for any fixed time and withinany given tolerance. Some applications to option pricing are presented. In

particular, we perform several numerical tests, and specifically include results

on Stochastic Volatility models.

Contents

Introduction 11. Preliminaries on evolution systems 52. Perturbative expansions 143. Dilation of the operator 194. Error Analysis 255. Invariance under affine transformations 326. Applications 35References 45

Introduction

The aim of the paper is to provide accurate, efficiently computable approxima-tions for the Green’s function or fundamental solution of parabolic equations inRN , with variable coefficients depending both on space and time.

More precisely, we consider operators of the form ∂t − L, where

(0.1) L =

N∑i,j

aij(t, x)∂i∂j +

N∑i

bi(t, x)∂i + c(t, x),

with x = (x1, ..., xN ) ∈ RN and ∂k := ∂∂xk

. The coefficients aij , bi, and c areassumed smooth and all their derivatives are assumed to be uniformly bounded.

Date: June 10, 2011.W.C. and V.N. were partially supported by the NSF Grants DMS-0713743, OCI-0749202,

and DMS-1016556. A.M. was partially supported by NSF grants DMS 0708902, 1009713, and1009714. Manuscripts available from http://www.math.psu.edu/nistor/ .

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(We write aij , bj , c ∈ C∞b(R+ × RN

).) Without loss of generality we can assume

that aij = aji as well. These type of equations are sometimes referred to as Fokker-Planck equations and they arise naturally in many contexts, in particular in statisti-cal mechanics and probability. An important example is given by heat equations onmanifolds with geometry evolving in time. In this case, L(t) = ∆g(t) is the Laplace-Beltrami operator associated to a time-dependent metric g. In a non-relativisticcontext, such metris appear for instance in covariant formulations of continuummechanics [24].

We impose a uniform strong ellipticity condition on the operators L(t), i.e., thereexists a constant γ > 0, such that

(0.2)∑

aij(t, x)ξiξj ≥ γ‖ξ‖2, ∀t ≥ 0, x, ξ ∈ RN ,

and we consider the initial value problem (IVP)

(0.3)

{∂tu(t, x)− L(t)u(t, x) = g(t, x) in (0,∞)× RN ,u(0, x) = f(x), on {0} × RN ,

where u, f , and g are in suitable spaces. Even when the initial value problem(0.3) is well posed, it is difficult to obtain exact solution formulas. Only in specialcases, for example when L(t) is a constant-coefficient operator, the solution canbe obtained explicitly in terms of the initial data. Such explicit representationof the solution is not available in general for variable-coefficient operators. Thedependence of the coefficients on time poses additional difficulties as discussed laterin this Introduction.

Several methods are available to approximate the exact solution: numerically,asymptotically, and even analytically. However, the majority of these methods areslow. Our goal is to derive approximate closed-form solutions for the IVP (0.3)that are accurate, yet efficiently computable for initial data that usually arises inpractice. A fast solution method is crucial when calibrating unknown parameters,especially in the Baeysian inference framework.

We first develop a general approach to approximate the Green’s function orfundamental solution for the operator ∂t − L, which then leads to an approximateformula for the solution of (0.3) by convolution with the initial data. The conditionsgiven above on the coefficients of L together with the strong ellipticity condition onthe operator ensures the well-posedness of the IVP in Sobolev and other functionspaces (see for example [1, 18, 23] and references therein). In addition, the solutionoperator forms a so-called evolution system, informally a generalization of the semi-group etL, which gives the solution operator in the case L is independent of time.(See Definition 1.10.) We will denote the evolution system by U(t, r), 0 ≤ r ≤ t,throughout the paper.

The approximation scheme for the Green’s function is an extension to the caseof time-dependent coefficients of a method recently introduced by the authors andtheir collaborators in [5]. This method, which will be referred to in the paper as theDyson-Taylor commutator method, combines known techniques in a novel way andis based on a suitable local parabolic rescaling, Taylor expansion of the coefficients,and Duhamel’s formula, together with exact commutator formulas. It is moreelementary than those found in the literature, yet accurate to arbitrary order intime in the short-time limit. When L is independent of t, global error estimatesas t → 0+ on the operator norm of the approximate solution operator in Sobolev

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spaces W r,p were derived in [5]. The Sobolev spaces can be exponentially weighted,a setting of interest for certain applications to probability (see the discussion in[5] and Definition 1.19). Deriving such estimates is more challenging here, since itis more difficult to obtain the needed mapping properties for an evolution systemthan for a semigroup, in particular because the equation generally is not invariantunder time rescaling. We use these more general result to obtain new applicationsto stochastic volatility models.

The main result of this paper is the Theorem 0.1 below. We will collectivelydenote by Lγ the class of operators L of the form (0.1) satisfying the ellipticitycondition (0.2) (this class is formally introduced in Definition 1.1). We also needto introduce the concept of an admissible function, that is a function z = z(x, y)such that z(x, x) = x and all its derivatives ∂αz are bounded for α 6= 0. The pint zis the center for the dilations used in approximating the Green’s function GLt (x, y).One can choose z(x, y) = z, but our results are more general. This is a key noveltyof our method. In current work, we are investigating how different choices of zinfluence the accuracy of the approximation in examples of practical interest [3].Lastly, we denote by W r,p

a,z the Sobolev space with weight exp(a(1 + |x − z|2)1/2),formally introduced in Definition 1.19. The reason for introducing the weightedSobolev spaces W r,p

a,z is that we want the typical initial conditions that arrise inapplications (such as pay-offs of continent claims) to be covered by our results.

Theorem 0.1. Let m be a positive integer, L an operator in Lγ , and z = z(x, y) anadmissible function. Then L generates an evolution system U(t, t′) in the Sobolevspace W r,p

a,z (RN ), r ∈ R+, 1 < p < ∞, a ∈ R. Furthermore, the Green’s functionfor ∂t − L has the asymptotic expansion

GLt (x, y) := G[m,z]t (x, y) + sm+1Et,zm (x, y)

where G[m,z]t (x, y) is the mth order approximate Green’s function given by

GLt (x, y) = s−N(eL0(z + s−1(x− z), z + s−1(y − z))

+

m∑`=1

s`Λ`z(z + s−1(x− z), z + s−1(y − z)))

with Λ`z defined in Equations (3.19) and (3.18) and Lemma 3.10 and the errorterm sm+1Et,zm (x, y) is small in the following sense. For any f ∈W k,p

a,z , a, k, r ∈ R,

1 < p <∞, define the error operator E [m,z]t by

E [m,z]t f(x) =

∫Et,zm (x, y)f(y)dy,

then

‖E [m,z]t f‖W r+k,p

a,z≤ CL,m,a,z,pt−r/2‖f‖Wk,p

a,z,

for any t ∈ [0, T ], s > 0, where the constant CL,m,a,z,p does not depend on t ∈ [0, T ].

There is a well-established literature on deriving asymptotic formulas for thefundamental solution of parabolic equations in the short-time regime, but mostly forthe autonomous case. A fundamental approach consists in viewing L as a Laplace-Beltrami operator on a manifold with metric tensor g given by [aij ]

−1, plus lower-order terms. Asymptotic formulas for heat kernels on manifolds are discussed, forexample, in [2, 14, 15, 19, 21, 29, 29, 30], (see also [9, 20, 27] for a pseudo-differential

3

operator perspective). Several of these results are for compact manifolds, and theytranslate to local estimates on noncompact, complete manifolds. We remark thatthe error bounds in our main theorem are global in space.

Minakshisundaram-Pleijel [21] in particular obtained the following asymptoticexpansion

(0.4) Gt(x, y) = (4πt)−N2 e−

d(x,y)2

4t

(G(0)(x, y) + G(1)(x, y)t+ G(2)(x, y)tn + . . .

),

as t→ 0+, where d(x, y) is the geodesic distance between x and y in the metric g,and G(j)(x, y) are smooth functions in x and y. Greiner [9] constructed the sametype of expansion as a parametrix for ∂t − L via a Volterra series. These types ofasymptotic formulas have proven very successful from a theoretical point of view, forexample in estimating eigenvalues for the operator L and obtaining trace formulas(see the articles cited above). In applications to probability, Henry-Labordere [12]used the asymptotic expansion (0.4) to compute the implied volatility in stochasticvolatility models. This geometric approach has proven less successful in practicalimplementations, given that in general there are no exact formulas for the geodesicdistance d(x, y), which therefore needs to be computed numerically or otherwiseestimated, usually asymptotically for x ∼ y.

Pseudo-differential calculus gives a related parametrix expansion for ∂t−L, usingtransport equations for amplitudes:

(0.5) GL(t, x, y) ∼∑j≥0

tj−n2 pj

(x, t−

12 (x− y)

)e−

(x−y)TA(x)−1(x−y)4t ,

as t → 0+, where pj(x,w) is a polynomial of degree j in w, and A(x) := [aij(x)].(See Taylor [27, Chapter 7, Section 13] for a derivation and for error estimateson compact manifolds.) This expansion using pseudodifferential operators is akinto semiclassical asymptotics for Schrodinger operators, which are sometimes ap-proached using the Wentzel-Kramers-Brillouin (WKB) method. In this context,the WKB method gives

(0.6) Gt(x, y) =1

(2πt)N/2exp

(− a(x, y)2

2t+∑k≥0

ck(x, y)tk),

with a(x, y), ck(x, y) are to be determined. For details about this method, see forexample Kampen [15].

Finally, we mention the Lie group approach based on the groups on invarianttransformations for the differential equations. In some cases, it is possible to findthe exact fundamental solutions by reducing the equation to an equivalent one thatcan be solved exactly (see [16, 17] for applications). For a further discussion, werefer the reader to Olver’s monograph [22]. A main difficulty of this approach isthat usually a large system of ODEs for the generators of the group needs to besolved.

In this paper, we shall use the so called Dyson-Taylor commutator method toderive an asymptotic series of the Green’s function of (0.3) similar to (0.4) and (0.5),but based on elementary methods, algorithmically computable, and of arbitraryhigh accuracy (if enough terms are included in the approximation). Furthermore,the numerical tests we perform show that our methods works even for certain classesof PDEs not satisfying our assumptions of regularity and uniform ellipticity. Theseequations are usually degenerate and their coefficients unbounded. Examples arises

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in probability and financial mathematics, for instance the well-known Black-Scholesequation

∂tu(t, x) =σ2x2

2u(t, x) + rx∂xu(t, x)− ru(t, x).

(See [3] for numerical results for the Black-Scholes equation and other models in1D.) In a forthcoming paper, we plan to extend extend our method to degenerateparabolic equations (typically with polynomial coefficients) and will justify the useof our method for the examples in this paper using a partition of unity and geometricproperties of our equation, as in [3].

The paper is organized as follows. In Section 1, we present some preliminaryresults on non-autonomous, second order strongly elliptic operators L ∈ Lγ time-dependent coefficients. For such operators, we study the existence and mappingproperties of the solution operator (or Green function) of the parabolic equation(∂t − L(t))u(t, x) = 0 and establish estimates needed in our later error analysis.In Section 2, we consider a time-independent strongly elliptic operator L0 ∈ Lγperturbed by another possibly time-dependent operator V , and study the mappingproperties of the evolution system (solution operator) generated by L := L0 +V . As a particular case, we apply interpolation theory to extend and refine themapping properties obtained in Section 1. Section 3.1 continues the discussion ofSection 2, in this section, using a parabolic scaling argument, we transform ourproblem into another equivalent problem. Taylor expansion allows us to split theoriginal operator into the sum of a time-independent operator and another time-dependent operator. After an iterative procedure, we obtain a formal expansionof the evolution system of our equation (∂t − L(t))u(t, x) = 0. In section 4, wejustify the convergence of the series of the evolution system obtained in Subsection3.1. In Section 5, we show that our asymptotic expansion is invariant under affinetransformations. In the last section, Section 6, we address some issues that arisein then numerical implementations of our approach. In particular, we extend ourresults from small time to any time by a bootstrap scheme. We also illustrate theDyson-Taylor commutator method by a concrete example, and present importantapplications of our approach to stochastic volatility models that appear in optionpricing theory.

0.1. Acknowledgments. We would like to thank Radu Constantinescu, NicolaCostanzino, John Liechty, Jim Gatheral, and Christoph Schwab for useful discus-sions. Also, Victor Nistor acknowledges support from Hausdorff Mathematical In-stitute in Bonn, where part of this work has been done. Anna Mazzucato and WenCheng acknowledge the hospitality and support of the Institute for Mathemtics andits Applications (IMA) at the University of Minnesota, where part of this work wasconducted. The IMA is partially funded by the National Science Foundation andthe University of Minnesota.

1. Preliminaries on evolution systems

We begin by describing in more details the class of operators to which our mainresults applies and some of their main properties.

We first introduce some notations and recall several definitions. In what follows,we denote the inner product on RN by

(1.1) (u, v) =

∫RN

u(x)v(x)dx.

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Let us denote

(1.2) 〈ξ〉 := (1 + |ξ|2)1/2

and let

(1.3) u(ξ) =

∫RN

e−ıξ·xu(x)dx

be the Fourier transform of u. We also recall the definition of some basic factsabout Lp-based Sobolev spaces W r,p(RN ): for any 1 < p <∞, r ∈ R, we define

(1.4) W r,p = W r,p(RN ) := {u : RN → C , 〈ξ〉ru ∈ Lp(RN )}

= {u : RN → C , (1−∆)r/2u ∈ Lp(RN )},

If r ∈ Z+, then W r,p(RN ) = {u : RN → C, ∂αu ∈ Lp(RN ), |α| ≤ r}, so we recoverthe usual definition.

Since the dimension N is fixed throughout the paper, we will usually writeW r,p for W r,p(RN ). Similarly, we shall often write Lp instead of Lp(RN ). When

1 < p <∞, the dual of W r,p is the Sobolev space W−r,p′

with 1/p+ 1/p′ = 1.Let

(1.5) C∞b (R+ × RN ) := {f : R+ × RN → C, ∂αf bounded for all α }.

We remark that, if f ∈ C∞b (R+ ×RN ), then f is Holder continuous with respect tothe time variable t uniformly in the space variable x ∈ RN .

Definition 1.1. We shall denote by L the set of second-order differential operatorsL(t) of the form

(1.6) L(t) :=

N∑i,j=1

aij(t, x)∂i∂j +

N∑k=1

bk(t, x)∂k + c(t, x),

where the matrix (aij) is symmetric and aij , bk, c ∈ C∞b (R+ ×RN ) are real valued.We shall denote by Lγ the subset of operators L(t) ∈ L satisfying the uniform strongellipticity estimate (0.2) with the ellipticity constant γ

Let A : D(A)→ X be a closed linear operator defined on a subspace D(A) ⊂ Xof the Banach space X. We shall denote by ρ(A) the resolvent set of A, namely theset

(1.7) ρ(A) := {λ ∈ C, λ−A : D(A)→ X is bijective }.

We next recall the definition of a sectorial operator [18].

Definition 1.2. A closed operator A : D(A)→ X, where D(A) is a linear subspaceof the Banach space X is called sectorial if there are constants ω ∈ R, θ ∈ (π/2, π),and M > 0 such that{

ρ(A) ⊃ Sθ,ω := {λ ∈ C : λ 6= ω, |arg(λ− ω)| < θ}‖R(λ,A)‖ ≤ M

|λ−ω| ,∀λ ∈ Sθ,ω,

where ρ(A) is the resolvent set of A.

Later on, we will give a sufficient condition to guarantee that L(t) is sectorialand use sectoriality to obtain mapping properties for the evolution system U(t, r)that L(t) generates.

6

1.1. Properties of the class Lγ. Elliptic pseudo-differential operators, in partic-ular elements of the class Lγ , generate equivalent norms in Sobolev spaces. (See[26, 27, 31] for definition and basic properties of pseudodifferential operators.)

Proposition 1.3. Let m ≥ 0 and Q ∈ Ψm1,0(RN ) be a uniformly elliptic operator

and 1 < p <∞. Then the following two norms are equivalent

(1.8) ‖u‖Wm,p ∼ ‖u‖Lp + ‖Qu‖Lp .

We sketch the proof for the reader’s convenience. Recall that Wm,p = Wm,p(RN )and Lp = Lp(RN ).

Proof. Since Q ∈ Ψm1,0(RN ), Q is a bounded operator from Wm,p to Lp by [26], and

hence there exists C1 such that

‖u‖Lp + ‖Qu‖Lp ≤ C1‖u‖Wm,p

On the other hand, since Q is uniformly elliptic, there exists a pseudodifferentialoperator R ∈ Ψ−m1,0 (RN ) such that I = RQ − S, where I is the identity operator,

and S ∈ Ψ−∞(RN ). Thus by mapping properties of pseudodifferential operators,we have

‖u‖Wm,p ≤ ‖RQu‖Wm,p + ‖Su‖Wm,p ≤ C(‖Qu‖Lp + ‖u‖Lp)

The proof is complete. �

We then obtain the following.

Corollary 1.4. Suppose L ∈ Lγ , 1 < p < ∞, and m is a nonnegative integer.Then for each t the following two norms are equivalent

(1.9) ‖u‖W 2m,p ∼ ‖u‖Lp + ‖Lm(t)u‖Lp .

Proof. One can easily check that if L ∈ Lγ , then Lm(t) is uniformly elliptic. Anapplication of the above proposition then completes the proof. �

The constants implicit in Equation (1.8) and Equation (1.9) above are uniformin the class Lγ .

Next we show that if L ∈ Lγ , then L(t) is Holder continuous in t and sectorialfor each t ∈ [0, T ] between suitable Sobolev spaces. Their properties in turn givethe needed mapping bounds for the evolution system. (See [1, 18, 23] for instance.)We split the proof into several propositions.

Proposition 1.5. Suppose L ∈ Lγ , then for any k ∈ N.

L(t) : W k+2,p →W k,p

is Holder continuous in t of exponent α = 1.

Proof. For any 0 ≤ t1 ≤ t2 ≤ T , we have

L(t2)− L(t1) =∑

(aij(t2, x)− ai,j(t1, x)) ∂i∂j

+∑

(bi(t2, x)− bi(t1, x)) ∂i + c(t2, x)− c(t1, x)

We also notice that for any multi-index β with |β| ≤ k, by our assumption on thecoefficients of the operator L,

‖aij(t2, x)− aij(t1, x)‖W |β|,∞ ≤ C(t2 − t1)7

similarly,

‖bi(t2, x)− bi(t1, x)‖W |β|,∞ ≤ C(t2 − t1)

‖c(t2, x)− c(t1, x)‖W |β|,∞ ≤ C(t2 − t1)

Therefore, for any u ∈W k+2,p,

‖(L(t2)− L(t1))u‖Wk,p ≤ ‖∑

(aij(t2, x)− aij(t1, x))∂2

∂xi∂xju‖Wk,p

+‖∑

(bi(t2, x)− bi(t1, x))∂

∂xiu‖Wk,p + ‖(c(t2, x)− c(t1, x))u‖Wk,p

≤∑|β|≤k

∑i,j

‖aij(t2, x)− aij(t1, x)‖W |β|,∞‖u‖Wk+2,p

+∑|β|≤k

∑i

‖bi(t2, x)− bi(t1, x)‖W |β|,∞‖u‖Wk+2,p

+∑|β|≤k

‖c(t2, x)− c(t1, x)‖W |β|,∞‖u‖Wk+2,p ≤ C(t2 − t1)‖u‖Wk+2,p

i.e., ‖L(t2)− L(t1)‖Wk+2,p→Wk,p ≤ C(t2 − t1). The proof is complete. �

The following well known proposition gives a sufficient condition to guaranteethat an operator is sectorial.

Proposition 1.6. Let A : D(A) ⊂ X → X be a linear operator such that ρ(A)contains a half plane {λ ∈ C : Reλ ≥ ω}, and

‖λR(λ,A)‖L(X) ≤M,Reλ ≥ ωwith ω ∈ R,M > 0. Then A is sectorial.

Proof. See [18], page 43. �

Our goal is to prove that L(t) : W 2k+2,p = W 2k+2,p(RN ) → W 2k,p is sectorialfor any non-negative integer k. The following special case is well known in theliterature (see Lunardi [18], page 73 for example).

Proposition 1.7. Let L ∈ Lγ . For each t, L(t) defines a continuous map W 2,p →Lp and its resolvent (λ−L(t))−1, λ ∈ ρ(L(t)), satisfy the conditions of Proposition(1.6), thus L(t) is sectorial from W 2,p to Lp.

Proof. Let us fix t0 and write L = L(t0). By uniform, strong ellipticity, if λ is in theresolvent set of L(t), then Reλ > 0. Hence we can write λ = r2eiθ with θ ∈ [−π2 ,

π2 ].

We now define an augmented, auxiliary operator in N + 1 variables:

Lθ = L+ eiθ∂2s .

One can easily check that Lθ is still a secon-order, elliptic, pseudo-differential opera-tor on RN+1. Therefore, by Equation (1.8), we have the following norm equivalence

(1.10) ‖v‖W 2,p ∼ ‖v‖Lp + ‖Lθv‖Lp

for any v ∈W 2,p(RN+1).We next introduce a smooth cut-off function ϕ ∈ C∞c (R) such that ϕ(s) = 1 for

|s| ≤ 12 and ϕ(s) = 0 for |s| ≥ 1. For any u ∈W 2,p(RN ) and r > 0, we set

v(x, s) = ϕ(s)eirsu(x).8

A direct computation yields

Lθv = ϕ(s)eirs(L− r2eiθ)u+ ei(θ+rs)(ϕ′′(s) + 2irϕ

′(s))u.

Therefore, the norm equivalence (1.10) implies that

‖v‖W 2,p ≤ C (‖v‖Lp + ‖Lθv‖Lp)

= C(‖v‖Lp + ‖ϕeirs(L− r2eiθ)u+ ei(θ+rs)(ϕ

′′+ 2irϕ

′)u‖Lp

)≤ C

(‖u‖Lp

(‖ϕ‖Lp + 2r‖ϕ′‖Lp) + ‖ϕ

′′‖Lp)

+ ‖L− r2eiθu‖Lp)

≤ C(

(1 + r)‖u‖Lp + ‖L− r2eiθu‖Lp)

(1.11)

On the other hand, by our construction of ϕ, we have

(1.12) ‖v‖pW 2,p ≥∫RN×(−1/2,1/2)

|∂2s (ueirs)|pdxds = r2p‖u‖pLp .

Combining the inequalities (1.11) and (1.12), we obtain

r2‖u‖Lp ≤ ‖v‖W 2,p ≤ C(

(1 + r)‖u‖Lp) + ‖(L− r2eiθ)u‖Lp).

Then set λ = r2eiθ and choose r such that r2/2 ≥ C(1 + r) to obtain

‖λu‖Lp ≤ C‖(L− λ)u‖Lp .

This gives

‖λR(λ, L)‖Lp→Lp ≤ C.On the other hand, it is well known that the resolvent set ρ(L) contains a half

plane (see [18] page 73 for example). Therefore, L satisfies all the conditions inProposition (1.6) and thus is sectorial. The proof is complete. �

We have next the following generalization.

Lemma 1.8. If L ∈ Lγ , then for each t and k, L(t) defines a continous mapW 2k+2,p → W 2k,p with the property that the the resolvent set of L(t) contains ahalf plane {λ ∈ C : Reλ ≥ ω}.

Proof. When k = 0, the result is obvious by Proposition (1.7). Choose any λ inρLp(L(t)), the resolvent of L(t) : W 2,p → Lp and denote by Rλ = (λ − L(t))−1 :Lp → W 2,p. Then Rλ is bounded by the definition of the resolvent set. We claimthat RλW

2k,p ⊂W 2k+2,p and that the resulting linear operator W 2k,p →W 2k+2,p

is bounded.Indeed, this is proved as follows. First, let us notice that RλL(t)f = f = RλL(t)

for f ∈ D(L(t)) = W 2,p. Let now f ∈ W 2k,p. Then Lk+1(t)Rλf = Lk(t)f ∈ Lp.By the norm equivalence (1.8), we conclude that Rλf ∈ W 2k+2,p and hence ourclaim holds.

On the other hand, it is obvious that (λ − L(t))W 2k+2,p ⊂ W 2k,p. So actuallywe have RλW

2k,p = W 2k+2,p and hence Rλ : W 2k,p → W 2k+2,p and (λ − L(t)) :W 2k+2,p → W 2k,p are inverses to each other. This fact tells us that ρLp(L(t)) iscontained in the resolvent set of the operator L(t) : W 2k+2,p →W 2k,p for k > 0. �

Finally, we prove that for any L ∈ Lγ and any t, the operator L(t) is sectorialbetween suitable spaces.

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Lemma 1.9. If L ∈ Lγ , then for each t and k, the operator L(t) : W 2k+2,p →W 2k,p

is sectorial.

Proof. For each fixed t = t0, since the sectorial property does not rely on t, we dropthe time dependence and simply write L0 = L(t0). We shall apply proposition (1.6)to prove that L0 is sectorial between W 2k+2,p and W 2k,p. For any u ∈ W 2k,p andλ ∈ ρ(L0), then by Lemma (1.8), R(λ, L0)u ∈ W 2k,p. Therefore, using the normequivalence (1.8) twice, we obtain

‖λR(λ, L0)u‖W 2k,p ≤ C(‖λR(λ, L0)u‖Lp + ‖λLk0R(λ, L0)u‖Lp)

= C(‖λR(λ, L0)u‖Lp + ‖λR(λ, L0)Lk0u‖Lp)

≤ C(‖u‖Lp + ‖Lk0u‖Lp) ≤ C‖u‖W 2k,p ,

that is,

‖λR(λ, L0)‖W 2k,p→W 2k,p ≤ C.In the second equality, as in Lemma (1.8), we used the same fact that if u ∈ D(Lk0),then R(λ, L0)Lk0 = Lk0R(λ, L0). (See [6],chapter 7). We also applied Proposition(1.7) that λR(λ, L0) is bounded from Lp to Lp.

Then by Lemma (1.8) and Proposition (1.6), L0 : W 2k+2,p →W 2k,p is sectorial.The proof is complete. �

1.2. Existence and properties of the evolution system. If L(t) = L0, thatis, if all the coefficients are time independent, then under some hypothesis L willgenerate a semigroup etL, it can be considered as the solution operator. However,in the nonautonomous case, the solution operator is not a semigroup, instead it isan evolution system, if it exists.

Definition 1.10. A two parameter family of bounded linear operators U(t, r) onX, 0 ≤ r ≤ t ≤ T , is called an evolution system if the following two conditions aresatisfied

(1) U(r, r) = I, U(t, r)U(r, s) = U(t, s) for 0 ≤ s ≤ r ≤ t ≤ T ,(2) (t, r)→ U(t, r) is strongly continuous for 0 ≤ r ≤ t ≤ T .

We now introduce the standard assumption made in the literature that gives theexistence of the evolution system. See for instance [1, 23], or [18], page 212.

Definition 1.11. A family of strongly elliptic operators L(t) : D(L(t)) ⊂ X → X,t ∈ [0, T ], will be called uniformly sectorial if the following conditions are satisfied:

(1) The domains D(t) are independent of t. Denote the common domain by D.Then D is dense in X.

(2) We can endowe D with a Banach space norm such that D → X is contin-uous and L(t) is Holder continues, i.e., there exists α ∈ (0, 1], such that

‖L(t)− L(r)‖D→X ≤ C|t− r|α,

(3) There are constants ω ∈ R, θ ∈ (π/2, π),M > 0 such that, for any t ∈ [0, T ]{ρ(L(t)) ⊃ Sθ,ω = {λ ∈ C : λ 6= ω, |arg(λ− ω)| < θ}‖R(λ, L(t))‖ ≤ M

|λ−ω| ,∀λ ∈ Sθ,ω

For later use, we recall the following useful result [1, 18, 23].10

Theorem 1.12. Suppose L is uniformly sectorial, then L generates an evolutionsystem U(t, r), and for any 0 ≤ r ≤ t ≤ T

‖U(t, r)‖X ≤ C , ‖U(t, r)‖X→D ≤C

t− r,

‖∂tU(t, r)‖ = ‖L(t)U(t, r)‖X ≤C

t− r,

‖L(t)U(t, r)‖D→X ≤ C , ∂sU(t, s) = U(t, s)L(s).

Proof. See, for example, Lunardi[18], Corollary 6.1.8, page 219. �

Proposition (1.5) and Lemma (1.9) show that Lγ consists of uniformly sectorialoperators.

Corollary 1.13. Suppose L ∈ Lγ . Then L generates an evolution system U(t, r),0 ≤ r ≤ t ≤ T , and for any real k, l ≥ 0, and 1 < p <∞, we have

‖U(t, r)‖Wk,p→Wk,p ≤ C , ‖U(t, r)‖Wk,p→Wk+2,p ≤ C

(t− r),

‖L(t)U(t, r)‖Wk+2,p→Wk,p ≤ C , and ‖L(t)U(t, r)‖Wk,p→Wk,p ≤ C.

Proof. The proof is mainly an application of a duality argument and space in-terpolation. First, by Theorem (1.12), for any nonnegative even integer k and1 < p < ∞, the above inequalities are true. Next we apply the duality method topass the mapping properties to negative Sobolev spaces. Note that U(t, s) satisfiesthe equation

(1.13)

{U(t, t) = 1

∂tU(t, r) = L(t)U(t, r).

We define the adjoint operator V (t, r) = U(T − r, T − t)∗, then

V (t, t) = U(T − t, T − t)∗ = 1

and

V (t, s)V (s, r) = U(T − s, T − t)∗U(T − r, T − s)∗

= [U(T − r, T − t)]∗ = V (t, r).

Therefore, V (t, r) is also an evolution system, and moreover,

∂tV (t, r) = ∂tU(T − r, T − t)∗ = ∂tU(T − r, T − t)]∗

= [U(T − r, T − t)L(T − t)]∗ = L∗(T − t)U(T − r, T − t)∗

= L∗(T − t)V (t, r).

(1.14)

i.e., ∂tV (t, r) = L∗(T − t)V (t, r). According to our assumptions, the family L∗(T −t) is of the same type as L(t). Thus the evolution system should satisfy the samemapping properties with U(t, r). Then for any positive k, we have

‖U(t, r)‖W−k,p→W−k,p = ‖U(t, r)∗‖Wk,q→Wk,q

= ‖V (T − r, T − t)‖Wk,q→Wk,q ≤ C(1.15)

and

‖U(t, r)‖W−k−2,p→W−k,p = ‖U(t, r)∗‖Wk,q→Wk+2,q

= ‖V (T − r, T − t)‖Wk,q→Wk+2,q ≤ C

(T − r)− (T − t)=

C

t− r,

(1.16)

11

where q is the conjugate of p, i.e., 1p + 1

q = 1.

At last, we apply the spaces interpolation technique to obtain mapping propertiesbetween non-integer Sobolev spaces. For any ` ∈ R, assume k ≤ ` < k + 2 where kis an even integer. Then by the complex interpolation,

W `,p =(W 2k,p′ ,W 2k+2,p′′

)[θ],W `+2,p =

(W 2k+2,p′ ,W 2k+4,p′′

)[θ],

where

` = (1− θ) · 2k + θ · (2k + 2)

1

p=

1− θp′

+1

p′′.

Therefore,

‖U(t, r)‖W `,p→W `,p ≤ C1−θ1 Cθ2 ≤ C,

‖U(t, r)‖W `,p→W `+2,p ≤(

C1

t− r

)1−θ (C2

t− r

)θ≤ C

t− r.

�

Since our approximation is asymptotic near zero, without loss of generality,henceforth we will assume T = 1. We also mention that throughout this paperC is a generic constant, it may be different at different appearance.

Let us now return to the study of the initial value problem (0.3), in the literaturethere are several types of solutions (mild, classical, strong) for (0.3). Thus we needto clarify what we mean by a solution of (0.3).

We shall use the following notion of solution, see [18], page 123-124.

Definition 1.14. Let g ∈ C([0,∞), X). By a strong solution in X of (0.3) wemean a function

(1.17) u ∈ C([0,∞), X) ∩ C1((0,∞), X) ∩ C((0,∞),D(L(t))),

such that ∂tu(t) = L(t)u(t) + g(t) in X, for t > 0, and u(0) = f ∈ X.

We are also interested in the case that f is in a larger space, because, in concreteapplications, the initial data f may be unbounded, even not Lp−integrable. Anexample is provided by the payoff function of a European call option. To includesuch cases, we therefore introduce exponentially weighted Sobolev spaces. Given afixed point z ∈ RN , we set

(1.18) 〈x〉z := 〈x− z〉 = (1 + |x− z|2)1/2

and define W k,pa,z (RN ) for k ∈ Z+, a ∈ R, 1 < p <∞, by

(1.19) W k,pa,z = W k,p

a,z (RN ) := e−a〈x〉zW k,p(RN )

= {u : RN → C, ∂αx(ea〈x〉zu(·)

),∈ Lp(RN ), |α| ≤ k}, if k ∈ Z+,

with norm

‖u‖pWk,pa,z

:= ‖ea〈x〉zu‖pWk,p =

∑|α|≤k

‖∂αξ(ea〈x〉zu(x)

)‖pLp .

The parameter z will be called the weight center. The space W k,pa,z is independent

of z as a vector space, but the norm does depend on z. The reason for introducingz is to obtain uniform constants indepenent of the norms used.

12

Let us consider the operator La(t) = ea<x>zL(t)e−a<x>z . We notice that prov-ing a result for L(t) acting between the weighted Sobolev spaces W k,p

a,z is the samething as proving the corresponding result for La(t) acting between the weighted

Sobolev spaces W k,p = W k,p0,z . But in order to pass from the conjugated operator

La(t) to the ordinary operator L(t), we require that L(t) and La(t) have the sameproperties.

Lemma 1.15. If L(t) ∈ Lω and a ∈ R, then La(t) ∈ Lω.

Proof. Suppose u ∈W k,p and L(t) ∈ Lω. Denote γ(x) = e−a<x>z . Then

La(t)u = ea<x>zL(t)e−a<x>zu

= γ−1(x)

[∑aij(t, x)

∂2

∂xi∂xj+∑

bi(t, x)∂

∂xi+ c(t, x)

]γ(x)u(x)

= γ−1(x)[∑

aij(t, x) (γ(x)∂iju(x) + ∂iγ(x)∂ju(x) + ∂jγ(x)∂iu(x)

+∂ijγ(x)u(x)) +∑

bi(t, x) (∂iγ(x)u(x) + γ(x)∂iu(x)) + c(t, x)γ(x)u(x)]

= L(t)u+ γ−1(x)

∑ 2aij(t, x)∂iγ(x)∂j +

∑i,j

∂i∂jγ(x)

+∑

bi(t, x)∂iγ(x))]u(x).

Notice that La(t)− L(t) is a first order differential operator whose coefficients aresmooth with all their derivatives uniformly bounded. Therefore, La(t) satisfies allthe assumptions that we make for L(t). So La(t) ∈ Lω. �

Therefore, by the above lemma (1.15) and our foregoing discussion, we mayreduce our arguments to the case a = 0 and z is arbitrary. In particular, L(t) :W k+2,pa,z →W k,p

a,z is well defined and continuous for any a, since this is true for a = 0.

Lemma 1.16. If L ∈ Lγ , γ > 0, and U(t, r) is the resulting evolution system.Then

‖U(t, r)− I‖Wk+2,pa,z →Wk,p

a,z≤ C|t− r|,

for a constant C independent of t, r, and z. In particular, if we fix r, then

[r,∞) 3 t→ U(t, r) ∈ B(W k+2,pa,z ,W k,p

a,z )

defines a continuous operator.

Proof. Notice that if f ∈W k+2,pa,z , then ∂

∂tU(t, r)f = L(t)U(t, r)f , and hence

‖(U(t, r)− I)f‖Wk,pa,z≤∫ t

r

‖L(τ)U(τ, r)f‖Wk,pa,zdτ

≤∫ t

r

‖U(τ, r)f‖Wk+2,pa,z

dτ ≤ C|t− r|‖f‖Wk+2,pa,z

,

by Theorem (1.12). Then, assuming t1 ≥ t2 ≥ r, we obtain

‖U(t1, r)f − U(t2, r)f‖Wk,pa,z≤ ‖(U(t1, t2)− I)U(t2, r)f‖Wk,p

a,z

≤ C|t1 − t2|‖U(t2, r)f‖Wk+2,pa,z

≤ C|t1 − t2|‖f‖Wk+2,pa,z

.

This completes the proof of the second part. �

13

2. Perturbative expansions

Let us assume now that L,L0 ∈ Lγ , with γ > 0 fixed, and with L0 time inde-pendent. We shall write

(2.1) L(t) = L0 + V (t).

In this section we study the effect of above splitting of the operator L(t). Moreprecisely, we shall investigate the classical question of relating the evolution systemU(t, s) generated by L to the semigroup etL0 generated by L0 [18, 1].

2.1. Analytic semigroups. Let A be sectorial, more precisely, to fix notation,let us assume that the resolvent set ρ(A) contains a sector Sθ,ω = {λ ∈ C : λ 6=ω, |arg(λ−ω)| < θ} and that ‖R(λ, L(t))‖ ≤ M

|λ−ω| ,∀λ ∈ Sθ,ω. For such an operator,

we can define the integral

etA =1

2πi

∫ω+γr,η

etλR(λ,A)dλ, t > 0,

where r > 0, η ∈ (π/2, θ) and the integral curve

γr,η = {λ ∈ C : |arg(λ)| = η, |λ| ≥ r} ∪ {λ ∈ C : |arg(λ)| ≤ η, |λ| = r}

is oriented counterclockwise. We also set e0Ax = x,∀x ∈ X. Then the family ofoperators {etA} is said to be the analytic semigroup generated by A.

The following proposition concerning mapping properties of the semigroup gen-erated by L0 is taken from [5].

Proposition 2.1. Assume L0 ∈ Lγ is time independent, as above, then etL0 isbounded on [0, 1] and

‖etL0f‖W r,pa,z (RN ) ≤ C(r, s)t(s−r)/2‖f‖W s,p

a,z (RN ), r ≥ s,

with C(r, s) independent of t.

An immediate consequence of the above result is

Corollary 2.2. Let s, r ∈ R be arbitrary and L0 ∈ Lγ be time independent, asabove. We then have that the map

(0,∞) 3 t→ etL0 ∈ B(W s,pa,z ,W

r,pa,z )

is infinitely many times differentiable.

Proof. Notice that ∂kt etL0 = etL0Lk0 , so it suffices to show that the map (0,∞) 3

t → etL0 ∈ B(W s−2k,pa,z ,W r,p

a,z ) is continuous. Let t ≥ δ > 0. Then eδL0 maps

W s−2k,pa,z to W r+2,p

a,z continuously, by Proposition 2.1. Writing etL0 = e(t−δ)L0eδL0

and using the continuity of [δ,∞) 3 t → e(t−δ)L0 ∈ B(W r+2,pa,z ,W r,p

a,z ), again byproposition (2.1), we obtain the result. �

2.2. Mapping properties of U(t, s). We now proceed to develop for the evolutionsystem U(t, s), generated by L ∈ Lγ , similar mapping properties to the mappingproperties of the semigroup etL0 obtained in Proposition (2.1). We start by studyingthe perturbation of evolution systems. If we fix r = 0, our evolution system U(t, s)generated by L becomes a one parameter evolution system. We donote U(t) =

14

U(t, 0). Recall that L0 generates an analytic semigroup. Now consider the followingequation

(2.2)

{∂tu(t, x)− L0u(t, x) = g(t, x) in (0,∞)× RN

u(0, x) = f(x), on {0} × RN ,

Then we have

Lemma 2.3. If g ∈ L1([0, 1], Lp)⋂C((0, 1], Lp), and u(t, x) ∈ Lp is a classical

solution to (2.2), then

u(t, x) = etL0f +

∫ t

0

e(t−τ)L0g(τ)dτ, 0 ≤ t ≤ 1

for any initial data f ∈ Lp. Moreover, assume L(t) generates an evolution sys-tem U(t, r) and U(t) = U(t, 0) is the one parameter system, then the classicalLp−solution to the equation (0.3) is given by

(2.3) u(t) = U(t)f = etL0f +

∫ t

0

e(t−τ)L0V (τ)u(τ)dτ

for any f ∈ Lp.

Proof. Define h(τ) = e(t−τ)L0u(τ), 0 ≤ τ ≤ t. Since u(t, x) is a classical solution,h(τ) is continuously differentiable when τ > 0,

h(0) = etL0f, h(t) = u(t),

and

h′(τ) = −L0e(t−τ)L0u(τ) + e(t−τ)L0u′(τ) = e(t−τ)L0g(τ), 0 < τ < t.

Integrating it from ε to t− ε, we have

eεL0u(t− ε) = e(t−ε)L0u(ε) +

∫ t−ε

ε

e(t−τ)L0g(τ)dτ.

Sending ε to zero completes the proof of the first part. For the second part, we firstassume that f ∈W 2,p, since L(t) generates an evolution system, suppose u(t, x) isthe classical solution of (0.3). Define

g(t) = V u(t, x) = (L(t)− L0)u(t, x) = ut(t, x)− L0U(t)f.

On the one hand, U(t) : W 2,p →W 2,p is continuous and bounded, and L0 : W 2,p →Lp is bounded, so L0U(t)f is continuous and Lp−integrable. On the other hand,obviously ut(x) is continuous by definition of the solution and is also integrable.Therefore, By the result of the first part of the lemma, u(t, x) has the form (2.3).For the general case when f ∈ Lp, since W 2,p is dense in Lp, We only need to showthat the right hand side of (2.3) is bounded in the Lp norm. (We make a note herethat in this case the right hand side is not necessarily in L1((0, T ), Lp), we shallonly apply the density argument, not the first part of the Lemma.) This is true byapplying the mapping properties of the semigroup etL0 (Proposition 2.1) and the

15

evolution system U(t) (Corollary 1.13), that is,

‖∫ t

0

e(t−τ)L0V (τ)U(τ)dτ‖Lp→Lp

≤∫ t

0

‖e(t−τ)L0‖W−1,p,Lp‖V (τ)‖W 1,p→W−1,p‖U(τ)‖Lp→W 1,pdτ

≤∫ t

0

1√t− τ

1√τdτ <∞

The proof is complete. �

Lemma 2.4. (Mapping properties of U(t, r)) Suppose U(t, r) is the two parameterevolution system introduced before, and k ≤ r, t > 0, a ∈ R. Then

‖U(t1, t2)‖Wk,pa,z→W r,p

a,z≤ C(t1 − t2)(k−r)/2.

Proof. As discussed before, we only need to prove the case a = 0. We also omitp from the notation, because it is the same, hence we write W k,p = W k. We firstassume that k ≤ r < k + 2, then by Corollary 1.13 and Proposition 2.1, startingfrom equation (2.3) we have

‖U(t1, t2)‖Wk→W r ≤ ‖e(t1−t2)L0‖Wk→W r

+

∫ t1−t22

0

‖e(t1−t2−τ)L0‖Wk−2→W r‖V ‖Wk→Wk−2‖U(τ + t2, t2)‖Wk→Wkdτ

+

∫ t1−t2

t1−t22

‖e(t1−t2−τ)L0‖Wk→W r‖V ‖Wk+2→Wk‖U(τ + t2, t2)‖Wk→Wk+2dτ

≤ C

((t1 − t2)

k−r2 +

∫ t1−t22

0

(t1 − t2 − τ)k−2−r

2 dτ +

∫ t1−t2

t1−t22

(t1 − t2 − τ)k−r2dτ

τ

)≤ C(t1 − t2)(k−r)/2

that is

‖U(t1, t2)‖Wk→W r ≤ C(t1 − t2)(k−r)/2, t1 ≥ t2 ≥ 0.

For the general case, let δ = r−km , where m is an integer and m > r−k

2 . Then byour above argument, for j = 1, 2, · · · ,m

‖U(t1 − (j − 1)t1 − t2m

, t1 − jt1 − t2m

)‖Wk+(j−1)δ→Wk+jδ ≤ C(t1 − t2m

) k−r2m

Therefore,

‖U(t1, t2)‖Wk→W r ≤ C(t1 − t2m

)m k−r2m

= C(t1 − t2)(k−r)/2,

where C depends on k, r, p, and a. �

In particular, consider the one parameter evolution system U(t), then the mean-ing of Proposition 2.1 and Lemma 2.4 is that both etL0 and U(t) are smoothingoperators as long as t ≥ δ > 0, i.e., they map a function from any Sobolev space toanother Sobolev space continuously. Moreover, they do not decrease the regularityof this function for any t ≥ 0. The worst case is that t = 0, and etL0 and U(t)become the identity operator. This fact will be useful in later applications. Another

16

consequence of Lemma (2.4) is that the Green function for any L ∈ Lγ exists bySchwartz Kernel Theorem, i.e., there exists GLt (x, y) ∈ C∞((0,∞)×RN ×RN ) suchthat

(2.4) U(t)f(x) =

∫RNGLt (x, y)f(y)dy

and explicitly, we have

GLt (x, y) =< δx, U(t)δy >

where < ·, · > is the duality pairing between C∞(RN ) and compactly supporteddistributions E ′(RN ), δz is the Dirac delta distribution. As we mentioned before,one purpose of this paper is to approximate GLt (x, y).

As a consequence of Lemma (2.4), we also have the following corollary similarto Corollary (2.2)

Corollary 2.5. If L(t) ∈ Lγ , and U(t, r), t ≥ r ≥ 0 is the resulting two parameterevolution system, then

(r,+∞) 3 t→ U(t, r) ∈ B(W s,pa,z ,W

m,pa,z )

is infinitely many times differentiable for any s and m.

Proof. For any positive integer k, it is easy to show that formally ∂kt U(t, r) =h(L(t), ∂tL(t))U(t, r) where h(L(t), ∂tL(t)) is a 2k order differential operators withsmooth and bounded coefficients. For any fixed δ with t ≥ δ > r,U(δ, r) is a continu-ous map from W s,p

a,z to Wm+2k+2,pa,z by lemma (2.4). Moreover, U(t, δ) is continuous

from Wm+2k+2,pa,z to Wm+2k,p

a,z by lemma (1.16). Lastly, clearly h(L(t), ∂tL(t)) ∈B(W r+2k,p

a,z ,W r,pa,z ) is also continuous. Therefore, combining the three operators and

using the definition of evolution system we conclude that ∂kt U(t, r) is continuousfrom W s,p

a,z to W r,pa,z �

Next we proceed to expand the operator U(t) at t = 1. For each k ∈ Z+, wedenote

Σk := {τ = (τ0, τ1, . . . , τk) ∈ Rk+1, τj ≥ 0,∑

τj = 1}

' {σ = (σ1, . . . , σk) ∈ Rk, 1 ≥ σ1 ≥ σ2 ≥ . . . σk−1 ≥ σk ≥ 0}

the standard unit simplex of dimension k. The bijection above is given by σj =τj + τj+1 + . . .+ τk. Using this bijection, for any operator-valued function f of RNwe can write∫

Σk

f(τ)dτ =

∫ 1

0

∫ σ1

0

. . .

∫ σk−1

0

f(1− σ1, σ1 − σ2, . . . , σk−1 − σk, σk)dσk . . . dσ1

Throughout, operator-valued integrals are taken in the sense of Bochner.

Lemma 2.6. Let Lj ∈ Lγ and let Vj be such that e−bj〈x〉Vj ∈ L, j = 1, . . . , k, forsome b = (b1, . . . , bk) ∈ Rk+, k ∈ Z+. Then

Φ(τ) = eτ0L0V1eτ1L1 . . . eτk−1Lk−1VkE(τk), τ ∈ Σk

defines a continuous function Φ : Σk → B(W s,pa,z (RN ),W r,p

a−|b|(RN )) for any a ∈ R

r ≥ s, and 1 < p <∞, where either E(τk) = eτkLk or E(τk) = U(τk) = U(τk, 0).

In this lemma, we used the standard multi-index notation |b| =∑kj=1 bj .

17

Proof. Our proof is based on the fact that both eτL and U(τ) are smoothing opera-tors when τ > 0, and they have the same type mapping properties (see Proposition2.1 and Lemma 2.4). It suffices to prove that Φ is continuous on each of the setsVj := {τj > 1/(k+2)}, j = 0, . . . , k, since they cover Σk. Let us assume that j = 0,for the simplicity of notation.(If j = k, it is slightly different. We will indicatelater.)

By assumption and by Proposition 2.1 and Lemma 1.16, each of the functions

[0,∞) 3 τj → VjeτjLj ∈ B(W rj+4,p

cj ,Wrj ,pcj−bj ), 1 ≤ j < k,

[0,∞) 3 τk → VkE(τk) ∈ B(W rk+4,pck

,W rk,pck−bk)

is continuous. For a suitable choice of cj and rj (more precisely, cj = cj+1 − bj+1,ck = a, rj = rj+1 − 4, rk = s), we obtain that the map

[0,∞)k 3 (τj) =: τ ′ → Ψ(τ ′) := V1eτ1L1 ...Vke

τkLk ∈ B(W s,pa ,W s−4k,p

a−|b| )

is continuous.Corollary 2.2 gives that the map τ0 → eτ0L0 ∈ B(W s−4k,p

a−|b| ,W r,pa−|b|) is continuous

for τ0 ≥ 1/(k+2)(If j = k, we shall use Corollary (2.5)). This proves the continuityof Φ on V0 and completes the proof of the lemma. �

In particular,in the above lemma if Lj = L0, j = 1, 2, · · · , k, namely, a secondorder strongly elliptic constant-coefficient operator, and the coefficients of Vj are ofpolynomial growth, an immediate result of lemma (2.6) is

Corollary 2.7. If L(t) is defined by (0.1), L0 is a second order strongly ellipticconstant-coefficient operator, and the coefficients of Lj are polynomials in x. Thenfor some b = (b1, . . . , bk) ∈ Rk+, k ∈ Z+,

Φ(τ) = eτ0L0L1eτ1L0 . . . eτk−1L0LkE(τk), τ ∈ Σk,

defines a continuous function Φ : Σk → B(W s,pa,z (RN ),W r,p

a−|b|(RN )) for any a ∈ R

r ≥ s, and 1 < p <∞, where either E(τk) = eτkLk or E(τk) = U(τk).

Later on, in the expansion of the Green’s function of L(t), the operators will fitin the conditions of the above corollary.

Proposition 2.8. Let d ∈ Z+, and L(t) is split as in equation (2.2), then

U(1) = eL0 +

∫Σ1

eτ0L0V (τ1)eτ1L0dτ

+

∫Σ2

eτ0L0V (τ1)eτ1L0V (τ2)eτ2L0dτ + · · ·+

+

∫Σd

eτ0L0V (τ1)eτ1L0 . . . eτd−1L0V (τd)eτdL0dτ

+

∫Σd+1

eτ0L0V (τ1)eτ1L0 . . . eτdL0V (τd+1)U(τd+1)dτ,

(2.5)

and each integral is a well-defined Bochner integral.

The positive integer d will be called the iteration level of the approximation.Later on, V will be replaced by a Taylor approximation of L, so that V will havepolynomial coefficients in x and t.

18

Proof. Recall that Lemma 2.3 reads

U(t)− etL0 =

∫ t

0

e(t−τ)L0V (τ)U(τ)dτ.

Setting t = 1 gives the desired result for d = 0. The result for any d then followsby induction using the above formula repeatedly. Explicitly,

(2.6) U(1) = eL0 +

∫Σ1

e(1−σ1)L0V (σ1)eσ1L0dσ

+

∫Σ2

e(1−σ1)L0V (σ1)e(σ1−σ2)L0V (σ2)eσ2L0dσ

+ · · ·+∫

Σd−1

e(1−σ1)L0V (σ1) . . . e(σd−2−σd−1)L0V (σd−1)U(σd−1)dσ

= eL0 +

∫Σ1

e(1−σ1)L0V (σ1)eσ1L0dσ +

∫Σ2

e(1−σ1)L0V (σ1)e(σ1−σ2)L0V (σ2)eσ2L0dσ

+ · · ·+∫

Σd−1

e(1−σ1)L0V (σ1) . . . V (σd−1)eσd−1L0dσ

+

∫Σd−1

∫ σd−1

0

e(1−σ1)L0V (σ1) . . . V (σd−1)e(σd−1−σd)L0V (σd)U(σd)dσdσd

= eL0 +

∫Σ1

e(1−σ1)L0V (σ1)eσ1L0dσ +

∫Σ2

e(1−σ1)L0V (σ1)e(σ1−σ2)L0V (σ2)eσ2L0dσ

+ · · ·+∫

Σd

e(1−σ1)L0V (σ1)e(σ1−σ2)L0 . . . e(σd−1−σd)L0V (σd)U(σd)dσ,

where each integral is well defined as a Bochner integral by the Lemma. �

3. Dilation of the operator

For any function v(t, x) and f(x), we choose an arbitrary but fixed basepoint zand dilate them in the following sense

vs(t, x) = v(s2t, z + s(x− z))fs(x) = f(z + s(x− z)).

For the operator L, we set

(3.1) Ls =∑

asij(s2t, z + s(x− z))∂i∂j + s

∑bsi (s

2t, z + s(x− z))∂i+ s2cs(s2t, z + s(x− z)).

It is not difficult to show that if u(t, x) is a solution of the equation (0.3), thenus(t, x) is a solution of the following equation

(3.2)

{∂tu

s(t, x)− Lsus(t, x) = 0 in (0,∞)× RN

us(0, x) = fs(x), f ∈ C∞(Rn) on {0} × RN ,

Clearly, Ls satisfies all the conditions we assumed above. Denote the evolutionsystem generated by Ls by UL

s

(t). Also, let GLt (x, y) and GLst (x, y) be the Greenfunctions or fundamental solutions for the operator ∂t−L and ∂t−Ls respectively.We want to relate GLt (x, y) and GLst (x, y).

19

Lemma 3.1. Let z be a fixed but arbitrary point in RN . Then for any s > 0, wehave

GLt (x, y) = s−NGLs

s−2t(z +x− zs

, z +y − zs

).

In particular, when s =√t,

(3.3) GLt (x, y) = t−N/2GL√t

1 (z +x− z√

t, z +

y − z√t

).

Proof. Without loss of generality, we assume z = 0. On one hand, by definition ofGreen’s function,

us(t, x) =

∫RNGL

s

t (x, y)fs(y)dy =

∫RNGL

s

t (x, y)f(sy)dy

= s−N∫RNGL

s

t (x,y

s)f(y)dy

On the other hand,

us(t, x) = u(s2t, sx) =

∫RNGLs2t(sx, y)f(y)dy.

Therefore,

s−N∫RNGL

s

t (x,y

s)f(y)dy =

∫RNGLs2t(sx, y)f(y)dy,

which implies,

s−NGLs

t (x,y

s) = GLs2t(sx, y).

After a change of variable, we get

GLt (x, y) = s−NGLs

s−2t(x

s

y

s).

�

With this lemma in hand, in order to approximate GLt (x, y), it suffices to ap-

proximate GL√t

1 (x, y). We shall apply the perturbation technique illustrated asfollows.

3.1. Formal expansion of the operator Ls. Suppose Ls is given by (3.1). Thenwe Taylor-expand it with respect to the parameter s to obtain

Ls = L0 +

n∑m=1

smLm + V sn+1,

where

(3.4) Lm =1

m!

(dm

dsmLs)∣∣∣∣

s=0

.

The operators Lm are independent of s. However, V sn+1 does depend on s, and allof the terms depend on z even it does not appear in the notation. We also useanother way to denote V sn+1, that is,

V sn+1 = sn+1Ls,zn+1.20

We shall look for a general formula for Lm. For a function f(t, x) smooth enough,we can Taylor expand it around (0, z) as

f(t, x) =

∞∑l=0

∞∑k=0

tl(x− z)k

l!k!

∂l

∂tl∂k

∂xkf(0, z)

Therefore,

f(s2t, z + s(x− z)) =

∞∑l=0

∞∑k=0

(s2t)lsk(x− z)k

l!k!

∂l

∂tl∂k

∂xkf(0, z)

(3.5)1

m!

dm

dsmf(s2t, z + s(x− z))

∣∣∣∣s=0

=

[m2 ]∑l=0

tl(x− z)m−2l

l!(m− 2l)!

∂l

∂tl∂m−2l

∂xm−2lf(0, z)

Combine this with (3.4), we can explicitly write Lm as

Lm =

d∑i,j=1

[m2 ]∑l=0

tl(x− z)m−2l

l!(m− 2l)!

∂l

∂tl∂m−2l

∂xm−2laij(0, z)

∂2

∂xi∂xj

+

d∑i

[m−12 ]∑l=0

tl(x− z)m−1−2l

l!(m− 1− 2l)!

∂l

∂tl∂m−1−2l

∂xm−1−2lbi(0, z)

∂

∂xi

+

[m−22 ]∑l=0

tl(x− z)m−2−2l

l!(m− 2− 2l)!

∂l

∂tl∂m−2−2l

∂xm−2−2lc(0, z)

(3.6)

where m ≥ 2. So Lm is a second order differential operator with polynomialcoefficients with degree at most m with respect to x− z and [m2 ] with respect to t.We can write the first few terms in the Taylor expansion explicitly,

L0 =

d∑i,j=1

aij(0, z)∂2

∂xi∂xj

L1 =

d∑i,j=1

(x− z)∇ai,j(0, z)∂2

∂xi∂xj+

d∑i=1

bi(0, z)∂

∂xi

L2 =

d∑i,j=1

(1

2(x− z)T∇2aij(0, z)(x− z) + t∂taij(0, z)

)∂2

∂xi∂xj

+

d∑i=1

((x− z)∇bi(0, z))∂

∂xi+ c(0, z)

Remark 3.2. Clearly, L0 is an operator with constant coefficients. By our assump-tion (0.2), L0 generates an analytic semigroup, explicitly, we have

(3.7) etL0 =1√

(4πt)N det(A0)e−

(x−y)t(A0)−1(x−y)4t ,

where A0 := A(0, z) = [aij(0, z)] and N is the dimension.

21

3.2. Asymptotic expansion of the evolution system. Recall proposition 2.8,we want to explicitly express U(t) in a nice way, and now Ls−L0 =

∑nm=1 s

mLm+V sn+1 will serve the role as V does in proposition 2.8, and L0 is what we introducedabove.

To compute the integrals in the above lemma, we need

Lemma 3.3. (Baker-Campbell-Hausdorf formula) A and B are two operators, then

[eA, B] =

([A,B] +

1

2![A, [A,B]] +

1

3![A, [A, [A,B]]] + . . .

)eA.(3.8)

In general this formula is an infinite series. But in our later application, it willbe a finite series. This formula tells us how to commute eA and B:

eAB =

(B + [A,B] +

1

2![A, [A,B]] +

1

3![A, [A, [A,B]]] + . . .

)eA.

If we apply this repeatedly in proposition 2.8, we can reduce the integrals.Now let’s give some definitions as those in [5].

Definition 3.4 (Spaces of Differentials). For any nonnegative integers a, b we de-note by D(a, b) the vector space of all differentiations of degree at most a and orderat most b. We extend this definition to negative indices by defining D(a, b) = {0}if either a or b is negative. By degree of A we mean the highest power of thepolynomials appearing as coefficients in A.

Definition 3.5 (Adjoint Representation). For any two differentiations A1 ∈ D(a1, b1)and A2 ∈ D(a2, b2) we define adA1(A2) by

(3.9) adA1(A2) := [A1, A2] = A1A2 −A2A1

and for any integer j ≥ 1 we define adjA1(A2) recursively by

(3.10) adjA1(A2) := adA1(adj−1

A1(A2))

Proposition 3.6. Suppose A1 ∈ D(a1, b1) and A2 ∈ D(a2, b2). Then for any inte-ger k ≥ 1,

adkA1(A2) ∈ D(k(a1 − 1) + a2, k(b1 − 1) + b2).(3.11)

Proof. We first notice that

adA1(A2) ∈ D(a1 − 1 + a2, b1 − 1 + b2).(3.12)

Next, from (3.10) we have

adkA1(A2) = adA1(adA1(adA1(adA1(. . . ))))(3.13)

so that an application of (3.12) k times yields the result. �

Lemma 3.7. Let m, k be fixed integers ≥ 1. Let Lz0 ∈ D(0, 2) be the constantcoefficient operator and Lzm ∈ D(m, 2) be the operator given above, Then,

adkL0(Lm) ∈ D(m− k, k + 2).(3.14)

In particular,

adkL0(Lm) = 0 ∀k > m.(3.15)

22

Proof. Applying Lemma 3.7 we see that adkLz0 (Lzm) ∈ D(m − k,m + 2). If k > m,

then by definition D(m− k,m+ 2) = {0} and we obtain (3.15). �

Lemma 3.8. Let L0 and Lm be defined above, then for any θ ∈ (0, 1),

e(1−θ)L0Lm(θ) = Pm(θ, x− z, ∂)e(1−θ)L0

where

Pm(θ, x− z, ∂) := Lm(θ) +

m∑i=1

(1− θ)i

i!adiL0

(Lm(θ)) ∈ D(m,m+ 2)

is a finite sum of terms with the form a(z)(1− θ)iθj(x− z)k∂αx , in which a(z) andall its derivatives are bounded, α is a multi-index.

Proof. Setting A = (1−θ)L0 and B = Lm(θ) in Baker-Campbell-Hausdorf formulayields the results. �

Next, we shall rewrite equation (2.6) in a more computable and explicit form. Inabuse of notations, it is convenient to write Ls,zn+1 = Ln+1. Recall that Lm = Lm(t)

is a function of t, thus so is V . Plug V =∑n+1m=1 s

mLm(t) into (2.6) and expand it,we obtain

U(1) = eL0 +

d∑k=1

∑1≤αi≤n+1

∫Σk

e(1−σ1)L0sα1Lα1(σ1)e(σ1−σ2)L0 . . . e(σk−1−σk)L0sαkLαk(σk)eσkL0dσ

+∑

1≤αi≤n+1

∫Σd+1

e(1−σ1)L0sα1Lα1(σ1)e(σ1−σ2)L0 . . . e(σd−σd+1)L0sαd+1Lαd+1

(σd+1)U(τd+1)dσ,

= eL0 +

d∑k=1

∑1≤αi≤n+1

sα1+···+αk∫

Σk

e(1−σ1)L0Lα1(σ1)e(σ1−σ2)L0 . . . e(σk−1−σk)L0Lαk(σk)eσkL0dσ

+∑

1≤αi≤n+1

sα1+···+αd+1

∫Σd+1

e(1−σ1)L0Lα1(σ1)e(σ1−σ2)L0 . . . e(σd−σd+1)L0Lαd+1

(σd+1)U(σd+1)dσ,

(3.16)

To simplify the above formula, we first introduce the notations as follows

Definition 3.9. For any integers 1 ≤ k ≤ d + 1 and `, we shall denote by Ak,`the set of multi-indexes α = (α1, α2, . . . , αk) ∈ Nk, such that |α| :=

∑αj = `.

Furthermore, we denote A` :=⋃`k=1 Ak,`. For symmetry, it will be convenient to

set A0 = {∅}.

Clearly, since αi ≥ 1, the set Ak,` is empty if ` < k. The meaning of ` is that ofthe corresponding power of s and the meaning of k is that of the expansion order.For each α ∈ Ak,`, we denote if k < d+ 1

Λα,z =

∫Σk

e(1−σ1)L0Lα1(σ1)e(σ1−σ2)L0 . . . e(σk−1−σk)L0Lαk(σk)eσkL0dσ

23

if k = d+ 1,

Λα,z =

∫Σd+1

e(1−σ1)L0Lα1(σ1)e(σ1−σ2)L0 . . . e(σd−σd+1)L0Lαd+1(σd+1)U(σd+1)dσ

=

∫Σd+1

Pα1(σ1, x− z, ∂)e(1−σ2)L0 · · · e(σd−σd+1)L0Lαd+1(σd+1)U(σd+1)dσ

= · · · · · ·

=

∫Σd+1

Pα1(σ1, x− z, ∂) · · ·Pαd+1(σd, x− z, ∂)e(1−σd+1)L0U(σd+1)dσ

A simple but useful lemma about Λα,z is the following, which we record for lateruse

Lemma 3.10. For any given multi-index α ∈ Ak,` with k ≤ d, then

Λα,z = Pα(x, z, ∂)eL0

where the product is the composition of operators and Pα(x, z, ∂) is a differentialoperator of order 2k + ` and polynomial degree≤ ` = |α|. More precisely, we have

Pα(x, z, ∂) =

∫Σk

Pα1(σ1, x− z, ∂)Pα2(σ2, x− z, ∂) · · ·Pαk(σk, x− z, ∂)dσ

=∑|β|≤`

∑|γ|≤`+2k

aβ,γ(z)(x− z)β∂γx(3.17)

where aβ,γ(z) ∈ C∞b (R).

Proof. Applying Lemma (3.8) repeatedly, we have

Λα,z =

∫Σk

e(1−σ1)L0Lα1(σ1)e(σ1−σ2)L0 . . . e(σk−1−σk)L0Lαk(σk)eσkL0dσ

=

∫Σk

Pα1(σ1, x− z, ∂)e(1−σ2)L0 · · · e(σk−1−σk)L0Lαk(σk)eσkL0dσ

= · · · · · ·

=

∫Σk

Pα1(σ1, x− z, ∂)Pα2(σ2, x− z, ∂) · · ·Pαk(σk, x− z, ∂)eL0dσ

=

(∫Σk

Pα1(σ1, x− z, ∂)Pα2

(σ2, x− z, ∂) · · ·Pαk(σk, x− z, ∂)dσ

)eL0 .

where σj = τj + τj+1 + · · ·+ τk. By Lemma (3.8) and Lemma (3.7), we know thateach operator Pαi(σi, x − z, ∂) ∈ D(αi, αi + 2), i = 1, 2, · · · , k. Thus Pα(x, z, ∂) ∈D(|α|, |α|+2k) = D(`, `+2k). Also notice that each Pαi(σi, x−z, ∂) has polynomialcoefficients in σi, so the integration with respect to σ will be exact, and Pα(x, z, ∂)is of the desired form. The proof is complete. �

We also set

(3.18) Λk,`z =∑

α∈Ak,`

Λα,z

and

(3.19) Λ`z =

min(`,d+1)∑k=1

Λk,`z

For convenience, let Λ0z = eL0 .

24

Now we record the above calculation as the following main theorem of this section

Theorem 3.11. Let M = (d + 1)(n + 1). Suppose U(t) is the one parameterevolution system, and d ≥ n, then it has the expansion

(3.20) U(1) = eL0 +

m∑`=1

s`Λ`z + sm+1Es,zd,n,

where Es,zd,n =∑M`=m+1 s

`−m−1Λ`z is the error term. Recall that d is the iteration

level and n is the expansion order of L(t).

Proof. The proof is straightforward. Rewrite (3.16) with the above notations, itbecomes

(3.21) U(1) =

M∑`=0

min(d+1,`)∑k=1

∑α∈Ak,`

s`Λα,z.

Picking up the terms with powers of s less than m + 1, and putting all the otherhigher order terms in the last term Es,zd,n completes the proof. �

Remark 3.12. If ` ≤ n, we mention that Λα,z and Λ`z are both independent of d,the iteration level, as long as d ≥ n. Moreover, Λ`z is independent of n also as longas n ≥ max(αi). These facts will be useful in our error analysis. If these conditionsare satisfied, Es,zd,n will depend only on m, and then we will write also Es,zm = Es,zd,n.

4. Error Analysis

In this section we shall mainly apply the pseudodifferential operator techniquesto justify that our approximation yields accurate solution to arbitrary prescribedorder in time. For all relevant properties of pseudodifferential operators, we referto [26]. We start from the operator Lm in the expansion (3.4). As we mentionedbefore the differential operators Lm, 0 ≤ m ≤ n + 1, are second order differentialoperator with polynomial coefficients. Moreover, Lm has coefficients of degree atmost m in x−z. An immediate consequence of this fact is recorded in the followinglemma.

Lemma 4.1. The family

{〈x〉−jz Lzj , 〈x〉−n−1z Ls,zn+1; s ∈ (0, 1], z ∈ RN , j = 0, . . . , n+ 1}

defines a bounded subset of L.

Recall that for convenience, we also denote Ls,zn+1 by Lzn+1, which actually de-pends on s and the dilation center z as well.

Lemma 4.2. For each given ε > 0, the family

{e−ε<z−w>e−ε〈x〉wLzj , s ∈ (0, 1], z ∈ RN , j = 0, . . . , n+ 1}

is a bounded subset of L.

Proof. If w = z, then the desired result follows directly from Lemma 4.1 and thesimple observation that 〈x〉jze−ε〈x〉z ≤ C, with C independent of z and j.

25

If w 6= z, then

< x− z > − < x− w >=√

1 + |x− z|2 −√

1 + |x− w|2

=(|x− z| − |x− w|)(|x− z|+ |x− w|)√

1 + |x− z|2 +√

1 + |x− w|2

≤ |w − z| ≤< w − z > (triangle inequality).

(4.1)

Therefore eε(<x−z>−<x−w>−<w−z>) ≤ 1, and the family

eε(<x−z>−<x−w>−<w−z>)e−ε<x>zLzj = e−ε<z−w>e−ε〈x〉wLzj

is bounded for s ∈ (0, 1] and j = 0, 1, 2, · · · , n+ 1 as claimed. �

Lemma 2.6 and lemma (4.2) then give

Corollary 4.3. For any α1, α2, · · · , αk with∑ki=1 αi = `, the operators

Λα,` =

∫Σk

eτ0L0Lα1(τ1)eτ1L0 · · · eτk−1L0Lαk(τk)eτkL0dτ, k ≤ d

and

Λα,` =

∫Σd+1

eτ0L0Lα1(τ1)eτ1L0 · · · eτdL0Lαd+1(τd+1)U(τd+1)dτ

are bounded linear operators from W s,pa,z to W r,p

a−ε for any z ∈ RN , r, s ∈ R, 1 < p <∞, and ε > 0. Moreover, we have that

‖Λα,`‖W s,pa,z→W r,p

a−ε≤ Cs,r,p,a,εekε<z−w>,

for a constant Cs,r,p,a,ε that does not depend on z. In particular, each Λα,` is anoperator with smooth kernel Λα,`(x, y).

Therefore, the above corollary gives

(4.2) Λα,`f(x) =

∫RN

Λα,`(x, y)f(y)dy.

From now on, we shall denote by T (x, y) the kernel of an operator T with smoothkernel. Then in terms of kernels, theorem (3.11) becomes

U(1)(x, y) = eL0(x, y) +

m∑`=1

s`Λ`z(x, y) + sm+1Es,zd,n(x, y)

By lemma (3.1), if we do the substitution x = z+s−1(x−z) and y = z+s−1(y−z)in the above equation, we have

GLt (x, y) = s−N(eL0(z + s−1(x− z), z + s−1(y − z))

+

m∑`=1

s`Λ`z(z + s−1(x− z), z + s−1(y − z))

+ sm+1Es,zd,n(z + s−1(x− z), z + s−1(y − z)))

=: G[m,z]t (x, y) + sm+1Et,zd,n(x, y),

(4.3)

where s =√t, and recall GLt (x, y) is the Green function of the operator ∂t − L(t).

We can compute G[m,z]t (x, y) explicitly, then

sm+1Et,zd,n(x, y) = GLt (x, y)− G[m,z]t (x, y)

26

is the error term which we need to bound. We define the error operator as

(4.4) E [m,z]t,d,n f(x) =

∫Et,zd,n(x, y)f(y)dy

In abuse of notations, if G[m,z]t denotes the operator with kernel G[m,z]

t (x, y), then

U(t, 0) = G[m,z]t + sm+1E [m,z]

t,d,n .

For d and n large, E [m,z]t,d,n is independent of d and n, so from now on we shall drop

d and n from the notation and write simply E [m,z]t .

By the definition of the error operator (4.4) and equation (3.21), we have

(4.5) E [m,z]t =

M∑`=m+1

min(d+1,`)∑k=1

∑α∈Ak,`

s`−m−1Λα,z

Later on we shall estimate the error by splitting E [m,z]t into two parts, namely

(4.6) E [m,z]t =

h∑`=m+1

∑k=1

∑α∈Ak,`

s`−m−1Λα,z + sh−mE [h,z]t .

In all the above formulas, we do not specify the dilation center z, which in generalmay be a function of x and y: z = z(x, y). For our error analysis, we need to specifythe dilation center

Definition 4.4. A function z : R2N → RN will be called admissible if

(i) z(x, x) = x, for all x ∈ RN .(ii) All derivatives of z are bounded.

A typical example is z(x, y) = λx + (1 − λ)y, for some fixed parameter λ. Asimple application of the mean value theorem gives that 〈z − x〉 ≤ C〈y − x〉 forsome constant C > 0. From the application point of view, z(x, y) = x will give usthe simplest formula to approximate the Green function [3]. However, as discussedin that paper, this may not be the best choice

4.1. Bounds for the principle term. In this subsection, we consider the desiredterm

G[m,z]t (x, y) =

m∑`=0

s`Λ`z(z + s−1(x− z), z + s−1(y − z))

and we shall fix the function z = z(x, y) which is admissible. Recall that

Λ`z =

min(d+1,`)∑k=1

∑α∈Ak,`

s`Λα,z

We treat each operator Λα,z in one time. Define the operator

(4.7) Ls,αf(x) = s−N∫RN

Λα,z(z + s−1(x− z), z + s−1(y − z))f(y)dy,

We will show below that for an admissible function z, and α = (α1, . . . , αk) ∈Ak,`, k ≤ n, αi ≤ n, the operator Ls,α is a pseudodifferential operator whose sym-bol is well behaved. We shall then use symbol calculus to derive the desired errorestimates. Let’s first recall some standard definitions and results from pseudodif-ferential calculus. Let m ∈ R. We define Sm1,0 to be the set of all functions σ(x, ξ)

27

in C∞(RN × RN ) such that for any two multi-indices α and β, there is positiveconstant Cα,β , depending on α and β only, such that∣∣∣(Dα

xDβξ σ(x, ξ)

)∣∣∣ ≤ Cα,β(1 + |ξ|)m−|β|

then we call any function σ in⋃m∈R S

m1,0 a symbol, and we denote S−∞ =

⋂m∈R S

m1,0.

Any operator whose symbol in S−∞ is a smoothing operator. Now if σ(x, ξ) is asymbol. Then the pseudodifferential operator σ(x,D) associated to σ(x, ξ) is de-fined by

(σ(x,D)ψ)(x) =1

(2π)N/2

∫RN

eix·ξσ(x, ξ)ψ(ξ)dξ

where D = 1i ∂ and

(4.8) Fψ(x) = ψ(ξ) :=

∫RN

e−ıξ·xψ(x)dx

the usual Fourier transform of ψ. Next let’s relate the operator σ(x,D) with itsdistributional kernel, actually we can recover one from the other under some condi-tions. Denote by F2 the Fourier transform in the second variable of a function of twovariables. For σ(x, ξ) ∈ S−∞, the operator σ(x,D) is smoothing with distributionkernel

σ(x,D)(x, y) = (2π)−N∫RN

eı(x−y)·ξσ(x, ξ)dξ = (F−12 σ)(x, x− y).

Let us denote by K a smooth function on RN × RN , if the integral (smoothing)operator defined by K is in fact a pseudodifferential operator σ(x,D), then we canrecover σ from K by the formula F−1

2 σ(x, y) = K(x, x− y), so

(4.9) σ(x, ξ) =

∫RN

e−ıξ·yK(x, x− y)dy.

Concerning the class S−∞, the following result is also standard and we are goingto use it later on.

Lemma 4.5. (i) The Fourier transform in the second variable establishes an iso-morphism F2 : S−∞ := S−∞(RN × RN )→ S−∞.

(ii) Multiplication defines a continuous map Sm(1,0) × S−∞ → S−∞.

For more about pseudodifferential calculus, we refer to the works of Taylor[27, 26] and Wong [31].

With this tool in hand, we move on to do the analysis. Recall that the func-

tion G(z;x) = (4π)−N/2 det(A(z))−1/2e−xTA(z)−1x/4 introduced in Equation (3.7):

Then the distribution kernel of eLz0 is given by

(4.10) eLz0 (x, y) = G(z;x− y),

A direct computation gives the following lemma, which coincides with the fact thateL

z0 is a convolution operator.

Lemma 4.6. Let z ∈ RN be a parameter and let us consider the operator T =a(z)(x − z)β∂γxeL

z0 , where β and γ are multi-indices and a ∈ C∞b (RN ). Then the

distribution kernel of T is given by

T (x, y) = a(z)(x− z)β(∂γxG)(z;x− y).

28

The next theorem characterizes the symbol of Ls,α.

Theorem 4.7. Let α ∈ Ak,`, k ≤ n, αi ≤ n. Assume that z : RN × RN satisfiesz(x, x) = x and ∂αz is bounded for all α 6= 0. Then there exists a uniformly boundedfamily {%s}s∈(0,1] in S−∞ such that

Ls,α = σs(x,D) := %s(x, sD), σs(x, ξ) = %s(x, sξ).

Proof. By Lemma 3.10, we know that Λα,z is a finite sum of terms of the form

a(z)(x−z)β∂γxeLz0 . We recall that a(z) is a function that itself and all its derivatives

are bounded. Suppose kz(x, y) is the distribution kernel of a(z)(x− z)β∂γxeLz0 and

letKs(x, y) := s−Nkz(z + s−1(x− z), z + s−1(y − z)), z = z(x, y).

By abuse of notation, we shall denote also by Ks the integral operator defined byKs. It suffices to prove our theorem for Ks. Namely, it is enough to show thatthere exists a uniformly bounded family {%s}s∈(0,1] in S−∞ such that

Ks = %s(x, sD).

By lemma 4.6, we have that the distribution kernel of ∂γxeLz0 is of the form

ς(z, x− y) for some ς ∈ S−∞. More precisely ς(z, x) is the Fourier tranform of thefunction (ıξ)γeξ·A(z)·ξ. This gives

Ks(x, y) = a(z(x, y))s−|β|−N (x− z(x, y))βς(z(x, y), s−1(x− y)) =:

a(z)s−|β|−N (x− z)βς(z, s−1(x− y)), z = z(x, y).

Then by (4.9), the symbol of Ks, σs(x, ξ) is given by

σs(x, ξ) =

∫RN

e−ıy·ξa(z)s−|β|−N (x− z)βς(z, s−1y)dy, z = z(x, x− y).

Let us substitute y with sy and let us denote

%s(x, ξ) =

∫RN

e−ıy·ξa(z)s−|β|(x− z)βς(z, y)dy, z = z(x, x− sy).

Then σs(x, ξ) = %s(x, sξ), so we need to show that %s is a bounded family in S−∞.Notice that a(z) ∈ S0

(1,0) and s−1(xj − zj(x, x − sy)) ∈ S0(1,0) and they form

bounded families for s ∈ [0, 1], then by Lemma (4.5) the proof is complete. �

The next lemma is obvious

Lemma 4.8. Let %(x, ξ) be a symbol in S−∞, then sk%(x, sξ) is a symbol in S−k1,0

uniformly bounded in (0, 1] with respect to s.

Proof. Denote ∂1 and ∂2 the derivatives of %(x, ξ) with respect to the first and

second variable respectively. Since %(x, ξ) ∈ S−∞, of course %(x, ξ) ∈ S−k1,0 , thus forany α and β we have

|∂αx ∂βξ %(x, ξ)| ≤ C(1 + |ξ|)−k−|β|.

Therefore,

|∂αx ∂βξ (sk%(x, sξ))| = |sk+β∂α1 ∂

β2 %(x, sξ)|

≤ Csk+β(1 + |sξ|)−k−|β| ≤ C(1 + |ξ|)−k−|β|

where C does not depend on s. Thus sk%(x, sξ) is uniformly bounded in S−k1,0 for

s ∈ (0, 1]. �29

We now obtain the main result of this subsection, the desired refined mappingproperty estimate by standard results from pseudodifferential operators theory.

Theorem 4.9. For any 1 < p <∞, any r ∈ R,

(4.11) sk‖Ls,α‖W r,p→W r+k,p ≤ Ck,r,p,for a constant Ck,r,p independent of t ∈ (0, 1].

From (4.3), we immediately obtain the desired estimate on the principal part ofthe asymptotic expansion.

Corollary 4.10. For each 1 < p <∞, r ∈ R, and any f ∈W r,p∫RNG[m,z]t (x, y)f(y) dy,

is uniformly bounded in W r,p for t ∈ (0, 1].

4.2. Bounds for the error term. In this subsection, we shall bound the errorterm Et,zd,n, which is the sum of two kinds of operators. The first is the one wediscussed in the last subsection, which is actually a pseudodifferential operatorand behaves well(Theorem (4.9)). The second is the operator Λα,l with either

α ∈ An+1,` or for some αi = n+ 1. In order to bounded Et,zd,n, it suffices to obtainmapping properties of the latter operator. In this case, the operator will dependon t also. Generally, we do not know whether Λα,l is a pseudodifferential operatoror not. However, we are going to show that Λα,l also behaves well, and has asimilar mapping property with Theorem (4.9) but a little bit rougher. It turns outthat this rough estimate is enough to give us the desired error control. In steadof pseudodifferential calculus applied in the last subsection, the main technique weshall use is the so called Riesz’s Lemma. (See for example [28].)

Lemma 4.11. (Riesz) Assume K is an integral operator with kernel k(x, y), thatis,

Ku(x) =

∫X

k(x, y)u(y)dµ(y),

where (X,µ) is a mearsure space. If k(x, y) is measurable on X ×X and

(4.12)

∫X

|k(x, y)|dµ(x) ≤ C1,

∫X

|k(x, y)|dµ(y) ≤ C2

for all y and for all x respectively. Then K is a bounded operator on Lp(X,µ) foreach p ∈ [1,∞]. Moreover,

‖K‖ ≤ C1/p1 C

1/q2 ,

where q is the conjugate of p.

The main result of this subsection is as follows

Theorem 4.12. Let z be admissible, r ≥ 0. Then we have

(4.13) sk+r‖Ls,α‖W r,p→W r+k,p ≤ Ck,r,p.

Note that the main difference between Theorem (4.9) and Theorem (4.12) is theadditional r. This result is rougher, but as we mentioned before, it is enough togive us the main theorem of this paper.

Before we prove this theorem, we first recall some notations and prove a lemma.We shall denote by W r,p

a = W r,pa,w as before, where w is the center of the weight

30

〈x〉w = 〈x − w〉 used to define our exponentially weighted Sobolev spaces (1.19).We shall write Lpa,z = W 0,p

a,z . The following lemma is a special case of Theorem(4.12).

Lemma 4.13. Assume that z : RN×RN is admissible. For any α, any 1 < p <∞,k ∈ Z+, r ≥ 0, and a ∈ R

(4.14) sk‖Ls,α‖Lpa→Wk,pa≤ Ck,p,

for a constant Ck,p independent of t ∈ (0, 1], independent of a in a bounded set, andindependent of the center of the weight that defines the weighted Sobolev spaces.

Proof. The proof will be mainly an application of the Riesz’s Lemma. Because ofthe reason we mentioned before, we may assume that a = 0. Recall that

Ls,α(x, y) = s−NΛα,z(z + s−1(x− z), z + s−1(y − z))

is the kernel of the operator Ls,α, where z = z(x, y). Then by Riesz’s Lemma itsuffices to show that for any multi-index γ with |γ| ≤ k,

(4.15)

∫RN

s|γ||∂γxLs,α(x, y)|dy ≤ C1,

∫RN

s|γ||∂γxLs,α(x, y)|dx ≤ C2,

where the constants C1 and C2 should be independent of x and y respectively.Generally, we need to estimate the growth rate of s−N∂γxΛα,z(z + s−1(x − z), z +s−1(y − z)). with respect to x and y. We need to use weighted Sobolev spacesintroduced in (1.19). Recall that the mapping properties between the weightedSobolev spaces are uniform in terms of the weight center, thus we can choose z asthe weight center. Notice that ∂γxLs,α(x, y) is the sum of terms of the form

(4.16) s−N−j∂βx∂β′

z ∂β′′

y Λα,z(z + s−1(x− z), z + s−1(y − z)) · ξ(z),

where j ≤ |γ| and ξ(z) is the product of derivatives of z with respect to x, it isbounded as z is admissible. While

|∂βx∂β′

z ∂β′′

y Λα,z(x, y)| = | < ∂βδx, ∂β′

z Λα,z∂β′′

δy > |

≤ C‖∂βδx‖H−q−a−ε‖∂β′

z Λα,z‖H−q−a→Hq−a−ε‖∂β′′

δy‖H−q−a .(4.17)

Next we shall estimate the three norms at the right hand side of the above estimate.For each multi-index β, ∂βx ∈ H−q(RN ) as long as q > N + |β|. Therefore, if wechoose z as the base point and q > N + |β|. Then for all a ∈ R and ε > 0

‖∂βδx‖H−q−a−ε := ‖e−(a+ε)<x−z>∂βδx‖H−q ≤ Ce−(a+ε)<x−z>

and

‖∂β′′

δy‖H−q−a := ‖e−a<y−z>∂β′′

δy‖H−q ≤ Ce−a<y−z>

For the second term ‖∂β′z Λα,z‖H−q−a→Hq−q−ε , since all the coefficients and their deriva-

tives of L(t) are bounded, ∂β′

z Λα,z will satisfy the same mapping properties as Λα,z.Thus by Corollary (4.3),

‖∂β′

z Λα,z‖H−q−a→Hq−a−ε ≤ Ceε<z−x>

Now get back to (4.17), we have

|∂βx∂β′

z ∂β′′

y Λα,z(x, y)| ≤ Ceε<z−x>−a<y−z>−(a+ε)<x−z> = Ce−a<y−z>−a<x−z>

31

Therefore, we obtain

|s−N−j∂βx∂β′

z ∂β′′

y Λα,z(z + s−1(x− z), z + s−1(y − z)) · ξ(z)|

≤ Cs−N−|γ|e−a<s−1(y−z)>−a<s−1(x−z)>

≤ Cs−N−|γ|e−a<s−1(y−x)>

(4.18)

In the last inequality, we have used the triangle inequality < y−z > + < x−z >≥<y−x >. Then after the change of variable λ = y−x

s , we find that (4.15) holds. Theproof is complete. �

Proof of Theorem (4.12): Notice that W r,p ⊂ Lp for any r ≥ 0 (for non-integer r,it is a consequence of the interpolation argument). Then if we consider Ls,α as anoperator from Lp to W r+k,p in stead of from W r,p to W r+k,p, the result followsdirectly from Lemma (4.13). 2

Recall that E [m,z]t is the sum of two kinds of operators we mentioned before, then

a direct corollary of Theorem (4.12) and Theorem (4.9) is the following

Corollary 4.14. Assume z is admissible and r ≥ 0, then

(4.19) ‖E [m,z]t ‖W r,p→W r+k,p ≤ Cs−r−k.

Surprisingly, it turns out that the r at the right hand side of equation (4.19) isredundant, we can get rid of it to obtain a more refined estimate.

Theorem 4.15. Assume z is admissible and r ≥ 0, then E [m,z]t satisfies the fol-

lowing mapping property

(4.20) ‖E [m,z]t ‖W r,p→W r+k,p ≤ Cs−k

Proof. Recall that as long as d ≥ n > m, G[m]t (x, y) does not depend on d and n.

Thus as the difference

E [m,z]t (x, y) = U(1)(z +

x− z√t, z +

y − z√t

)− G[m,z]t (x, y)

also does not depend on n and d. In the expansion (3.21), we expand it to muchmore terms, specifically, such that M ≥ m + r − 1. Then by Theorem (4.9) andCorollary (4.14)

‖E [m,z]t ‖W r,p→W r+k,p ≤

M∑`=m+1

s`−m−1∑

k=m+1

∑α∈Ak,`

‖Lα,z‖W r,p→W r+k,p

+ sM+1−m‖E [M,z]t ‖W r,p→W r+k,p ≤ Cs−k(1 + sM+1−ms−r) ≤ Cs−k.

This completes the proof. �

This completes the proof of the main Theorem (0.1).

5. Invariance under affine transformations

Recall that in Section 3 we considered a new equation (3.2) obtained by parabol-ically scaling the original equation (0.3). As in Lemma 3.1, the Green functions tothe original equation and the dilated equation are explicity related to each other,and this precise relation is one of the key steps of our Dyson-Taylor commutator

32

method. In this section, we investigate the role of affine transformations on ourapproximations, which is needed in some applications.

Throughout this section, we assume that

F (ξ) = Dξ +Q, D ∈ RN×N, Q ∈ RN

is a linear transformation with det(D) 6= 0. For any function u(t, x), we also defineits transformation as uF (t, x) = u(t, F (x)). Therefore, if u(t, x) is the solution tothe equation (0.3) with g = 0. Then it is easy to see that uF (t, x) satisfies thefollowing equation

(5.1)

{∂tu

F (t, x)− LF (t)uF (t, x) = 0 in (0,∞)× RN ,uF (0, x) = fF (x), on {0} × RN ,

where LF (t) is defined as (L(t)u(t, x))F = LF (t)uF (t, x). Assume GLt (x, y) and

GLFt (x, y) are the Green functions to equations (0.3) and (5.1). We first explore therelationship between these two kernels. On one hand, by definition,

u(t, x) =

∫GLt (x, y)f(y)dy.

Therefore,

uF (t, x) = u(t, F (x)) =

∫GLt (F (x), y)f(y)dy.

On the other hand,

uF (t, x) =

∫GL

F

t (x, y)fF (y)dy =

∫GL

F

t (x, y)f(F (y))dy

= det (D−1)

∫GL

F

t (x, F−1(y))f(y)dy.

Comparison of the above two equations leads to

GLt (F (x), y) = det (D−1)GLF

t (x, F−1(y)).

After a change of variable we obtain

(5.2) GLt (x, y) = det (D−1)GLF

t (F−1(x), F−1(y)).

Similar to Lemma 3.1, we also have

Proposition 5.1. Suppose GLt ,GLs

t ,GLF,st are the Green’s functions to the operatorL,Ls, LF,s respectively. Then they are related by

GLt (x, y) = s−NGLs

s−2t

(z +

x− zs

, z +y − zs

)= s−NGL

F,s

s−2t

(F−1(z) +

F−1(x)− F−1(z)

s, F−1(z) +

F−1(y)− F−1(z)

s

) ∣∣D−1∣∣ .

Proof. For convenience, we denote v(t, ξ) = uF (t, x) = u(t, F (x)), ξ = F (x). ByLemma 3.1

us(t, x) = s−N∫GL

s

t

(x, z +

y − zs

)f(y)dy

vs(t, ξ) = s−N∫GL

F,s

t (ξ, z +y − zs

)f(F (y))dy.

33

Therefore,

vs(t, F−1(x)) = s−N∫GL

F,s

t (F−1(x), z +y − zs

)f(F (y))dy

= v(s2t, z + s(F−1(x)− z)) = u(s2t, F (z + s[F−1(x)− z]))= u(s2t,D(z + s[D−1(x−Q)− z]) +Q) = u(s2t, [Dz +Q] + s(x− [Dz +Q]))

= us(t, x; z = Dz +Q) = us(t, x; z = F (z))

= s−N∫GL

s

t

(x, z +

y − zs

)f(y)dy

∣∣z=F (z) ,

which gives

s−N∫GL

F,s

t (F−1(x), z +y − zs

)f(F (y))dy

= s−N∫GL

s

t

(x, z +

y − zs

)f(y)dy

∣∣z=F (z) .

After a change of variable, this relation reads

s−N∫GL

F,s

t (F−1(x), z +F−1(y)− z

s)f(y)dy

∣∣D−1∣∣

= s−N∫GL

s

t

(x, z +

y − zs

)f(y)dy

∣∣z=F (z)

Thus we must prove

(5.3) GLF,s

t (F−1(x), F−1(z) +F−1(y)− F−1(z)

s)∣∣D−1

∣∣ = GLs

t

(x, z +

y − zs

),

which follows from Lemma 3.1, where we have shown that

(5.4) GLs2t(z + s(x− z), y) = s−NGLs

t (x, z +y − zs

)

A simple algebra on equations (5.3) and (5.4) yields our result. �

Note that LF still satisfies all of our assumptions, therefore by our Main Theorem0.1 we have the following asymptotic expansion

(5.5) GLF

t (ξ, y) = G[m,z],LF

t (ξ, y) + sm+1Et,zd,n(ξ, y)

and if we define the error operator as

E [t,z]m f(x) =

∫Et,zd,n(ξ, y)f(y)dy,

we have

(5.6) ‖E [t,z]m f‖W r+k,p

a,z≤ Ct−r/2‖f‖Wk,p

a,z,

but we know that GLt (x, y) and GLFt (ξ, y) are related by

(5.7) GLt (x, y) = GLF

t (F−1(x), F−1(y))|D−1|.

Combining (5.5) and (5.7) yields(5.8)

GLt (x, y) = G[m,z],LF

t (F−1(x), F−1(y))|D−1|+ sm+1Et,zd,n(F−1(x), F−1(y))|D−1|.34

Let us now define the operator

Et,zm f(x) =

∫Et,zd,n(F−1(x), F−1(y))|D−1|f(y)dy.

A simple calculation shows that Et,zm satisfies the same inequality as Et,zm does in(5.6) with a difference constant C. It is worth noting that it is crucial to assumedet(D) 6= 0.

The conclusion is that the approximation introduced in Theorem 0.1 is invariantunder affine transformations.

6. Applications

In this section, we shall numerically test our short time approximation of theGreen function of parabolic equations. We then extend our method to approxi-mate solutions of parabolic equations for large time. We also give concrete, simpleexamples to illustrate our approach.

6.1. Numerical implementation. Given the approximation of the Green func-

tion G[m]t (x, y), then

∫RN G[m]

t (x, y)f(y)dy is an approximation of the solution to thenon-autonomous Cauchy problem (0.3) with g = 0. In practice we do not haveexplicit formulas for the integral in general, thus numerical integration is required.In this case two other sources of errors will be introduced. One is the numericalquadrature error, and the other is the truncation error, for the integration domainis infinite. In the following of this subsection, we aim to control the overall error.

It is reasonable to assume that one needs to compute the the numerical solutionof u(t, x) on the interval I = [x, x]. As mentioned before, we need to truncate thedomain first. Also, we split the integral into two parts,

(6.1)

∫G[m]t (x, y)f(y)dy =

∫|y|<d

G[m]t (x, y)f(y)dy +

∫|y|>d

G[m]t (x, y)f(y)dy

for the first part we shall apply numerical quadrature rules. The second part issimply ignored and considered as the truncation error. Since the approximated

Green’s function G[m]t (x, y) decays exponentially, for fixed I, we can always choose

d big enough such that the truncation error is arbitrarily small, i.e.,

(6.2)

∣∣∣∣∣∫|y|>d

G[m]t (x, y)f(y)dy

∣∣∣∣∣ < ε,

where ε is small and dependent on d.

Define Itf as the function obtained by numerical integration of∫G[m]t (x, y)f(y)dy.

For the integral∫|y|<d G

[m]t (x, y)f(y)dy we can choose a kth order quadrature rule

with mesh size h. Then for any x ∈ I,

(6.3) ‖Itf − G[m]t f‖L∞(I) < ε+ Chk.

Combining this estimate with Equation (6.2) and the Sobolev embedding theorem,we obtain, for p big enough,

‖(It − etL)f‖L∞(I) ≤ ‖(It − G[m]t )f‖L∞(I) + ‖(G[m]

t − etL)f‖L∞(RN)

≤ ε+ Chk + C3‖(G[m]t − etL)f‖Wk,p

a,z (RN)

≤(ε+ Chk + Ct(m+1)/2

)‖f‖Wk,p

a,z (RN).

(6.4)

35

We formulate our discussion as the following

Theorem 6.1. Consider the non-autonomous Cauchy problem (0.3) with g = 0,the operator It is defined as above. Then for any interval I = [x, x] and a kth orderquadrature rule, we have

‖(It − etL)f‖L∞(I) ≤(ε+ Chk + Ct(m+1)/2

)‖f‖Wk,p

a,z (RN),

where ε is dependent on I and can be arbitrarily small, C is constant independentof t and h.

6.2. The bootstrap scheme. Note that our approximation is only accurate pro-vided t is small. In applications, it is not necessarily in this case, i.e., , we wouldhave relatively large t. Even if t is small, to improve the accuracy we need tocompute high order approximations of the Green function. However, this is notalways feasible in practice. As the order of accuracy goes up, computing the Greenfunction becomes more expensive. In this subsection, we shall introduce the boot-strap scheme to extend our results to relatively large t. We also show that highprecision can be obtained by using as low as a second order approximation of theGreen function.

Before we start our analysis, we first give some preliminaries. In our error esti-mate, we need the following useful lemmas:

Lemma 6.2. Assume U(t, s) is the evolution system generated by the operator L(t)on X with norm ‖ · ‖. Then we can find an equivalent norm ‖| · ‖|t which is timedependent, such that

‖|U(t, s)x‖|t ≤ eω(t−s)‖|x‖|s,i.e., U(t, s) is a contraction between ‖| · ‖|t and ‖| · ‖|s.

Proof. By Lemma 2.4, ‖U(t, s)‖ ≤Meω(t−s). Set V (t, s) = e−ω(t−s)U(t, s), then itis clear that V (t, s) is uniformly bounded by M . We define a new norm as

(6.5) ‖x‖Xs = ‖|x‖|s = supt≥s‖V (t, s)f‖.

Obviously, we have ‖x‖ ≤ ‖|x‖| ≤M‖x‖. Thus ‖| · ‖| is equivalent to ‖ · ‖ on X.Note that by our definition

‖|V (t, s)x‖|t = supr≥t‖V (r, t)V (t, s)x‖ = sup

r≥t‖V (r, s)x‖

≤ supr≥s‖V (r, s)x‖ = ‖x‖s.

Plugging in V (t, s) = e−ω(t−s)U(t, s), we obtain the desired estimate

(6.6) ‖|U(t, s)x‖|t ≤ eω(t−s)‖|x‖|s.

�

Thereafter, we denote ‖| · ‖|t,s = ‖ · ‖Xs→Xt . Next, we shall investigate the effectof the bootstrap strategy in estimating the solution to the equation (0.3) with g = 0.By the evolution property, U(t, 0) = U(t, n−1

n t)U(n−1n t, n−2

n t) · · · · · ·U( tn , 0). Then

in the bootstrap scheme, we approximate U(k+1n t, kn t) by G[m]

t/n. We need to bound

the error U(t, 0)−(G[m]t/n

)nin some norm.

36

Theorem 6.3. Assume that U(t, s) is the evolution system generated by L(t), and

G[m]t is the mth order approximation of the Green function of L(t). Then

‖|U(t, 0)−(G[m]t/n

)n‖|t,0 ≤M

t(m+1)/2

n(m−1)/2eωt.

Proof. Notice that we have the identity

(6.7) U(t, 0)−(G[m]t/n

)n=

n−1∑k=0

U(t,k + 1

n)

(U(k + 1

nt,k

nt)− G[m]

t/n

)(G[m]t/n

)k.

Therefore,

(6.8) ‖|U(t, 0)−(G[m]t/n

)n‖|t,0

= ‖|n−1∑k=0

U(t,k + 1

n)

(U(k + 1

nt,k

nt)− G[m]

t/n

)(G[m]t/n

)k‖|t,0

≤n−1∑k=0

‖|U(t,k + 1

n)‖|t, k+1

n‖|(U(k + 1

nt,k

nt)− G[m]

t/n

)‖| k+1

n , kn‖|(G[m]t/n

)k‖| kn ,0

By Lemma 6.2 and our main Theorem 0.1, we have

(6.9) ‖|U(k + 1

nt,k

nt)− G[m]

t/n‖| k+1n , kn

≤M‖U(k + 1

nt,k

nt)− G[m]

t/n‖ ≤M(t

n

)m+12

.

The triangle inequality and Lemma 6.2 then lead to

‖|G[m]t/n‖| k+1

n , kn≤ eω t

n +M

(t

n

)(m+1)/2

Therefore,

(6.10) ‖|(G[m]t/n

)k‖| kn ,0≤

[eω

tn +M

(t

n

)(m+1)/2]k

Using Equations (6.9) and (6.10) in (6.8), we obtain

‖|U(t, 0)−(G[m]t/n

)n‖|t,0 ≤

n−1∑k=0

eωn−k−1n tM

(t

n

)(m+1)/2[eω

tn +M

(t

n

)(m+1)/2]k

≤Mn

(t

n

)(m+1)/2

eωt

[eω

tn +M

(t

n

)(m+1)/2]n

≤M t(m+1)/2

n(m−1)/2eωt

(6.11)

In the last inequality we used the fact that the limit

limn→∞

(eω

tn +M

( tn

)(m+1)/2)n

exists and it grows at most exponentially with respect to t. �

37

Remark 6.4. Theorem 6.3 is of great interest in practice. It says that we do notneed to compute high order approximations of the Green function which in generalbecomes more complicated as the order goes higher, but a second order approxima-tion (m = 2) will be enough. As the number of bootstrap steps n goes to infinity,the error will go to zero. Thus one can use many steps to reduce the error.

Remark 6.5. If the operator L(t) is independent of t, then it generates a semigroup.In this case, we are able to make the equivalent norm independent of time. Actually,if ‖etL‖ ≤ Meωt, then we can define V (t) = e−ωtetL and a new norm ‖|x‖| =supt≥0 ‖V (t)x‖. It is easy to check that this new norm is equivalent to the originalnorm and V (t) is a contraction semigroup under the norm ‖| · ‖|.

6.3. An example. In this subsection, we illustrate our Dyson-Taylor commutatormethod and its accuracy by a simple example. We shall consider the operator

(6.12) L = a(t)∂2x + b(t)∂x + c(t), t ∈ [0,∞), x ∈ R.

and assume that L satisfies our general conditions. On one hand, we can applyour method to find an explicit formula of the nth order approximation of the Greenfunction of the operator L, and thus obtain an approximation of the solution of theequation

(6.13)

{∂tu(t, x) = Lu(t, x), t > 0, x ∈ R,u(0, x) = f(x)

On the other hand, notice that the coefficients of L are space-independent, andhence we can use the Fourier transfom to find the solution to the above equationexplicitly. This enables us to compare our approximation with the true solution.

Without loss of generality, we can assume that c(t) = 0. Otherwise, we apply

the change of variable v(t, x) = e−∫c(t)dtu(t, x), then v(t, x) will satisfy the same

equation with c(t) = 0.Following the procedures in Section 3, we first dilate this operator to get

(6.14) Ls,z = a(s2t)∂2x + sb(s2t)∂x.

The Taylor expansion of Ls,z with respect to the parameter s reads

(6.15) Ls,z =∞∑m=0

smLm,

where by (3.6)

L0 = a(0)∂2x

and

(6.16) Lm =

{tk

k!∂kt a(0)∂2

x, if m = 2ktk

k!∂kt b(0)∂x, if m = 2k + 1

we also denote Lm = tkLm.Note that for every operator Lm, the coefficients are independent of the space

variable x. Thus the semigroup eτL0 commutes with Lm for every m. In this case,formula (3.16) can be greatly simplified. As a matter of fact,

38

U(1, 0) = eL0 + sL1eL0 + s2

(1

2L2

1 +1

2L2

)eL0

+ s3

(1

6L3

1 +1

2L1L2 +

1

2L3

)eL0

+ s4

(1

24L4

1 +1

2L1L3 +

1

8L2

2 +1

3L4

)eL0 + · · · ,

(6.17)

where eL0 = 1

2√πa(0)

e−(x−y)24a(0) .

Here are now some details for (6.17):Note: L0, L1 are independent on time, but L2, L3 are dependent on time by

(3.16),

U(1, 0) = eL0 + s

∫ 1

0

e(1−σ)L0L1eσL0dσ

+ s2

[∫ 1

0

∫ σ1

0

e(1−σ1)L0L1e(σ1−σ2)L0L1e

σ2L0dσ2dσ1 +

∫ 1

0

e(1−σ)L0L2(σ)eσL0dσ

]+ s3

[∫ 1

0

∫ σ1

0

∫ σ2

0

e(1−σ1)L0L1e(σ1−σ2)L0L1e

(σ2−σ3)L0L1eσ3L0dσ3dσ2dσ1

+

∫ 1

0

∫ σ1

0

e(1−σ1)L0L1e(σ1−σ2)L0L2(σ2)eσ2L0dσ2dσ1

+

∫ 1

0

∫ σ1

0

e(1−σ1)L0L2(σ1)e(σ1−σ2)L0L1eσ2L0dσ2dσ1 +

∫ 1

0

e(1−σ)L0L3(σ)eσL0dσ

]+O(s4)

= eL0 + sL1eL0 + s2

(L2

1

∫ 1

0

∫ σ1

0

dσ2dσ1 + L2

∫ 1

0

σdσ

)+ s3

(L3

1

∫ 1

0

∫ σ1

0

∫ σ2

0

dσ3dσ2dσ1 + L1L2

∫ 1

0

∫ σ1

0

σ2dσ2dσ1 + L1L2

∫ 1

0

∫ σ1

0

σ1dσ2dσ1

+L3

∫ 1

0

σdσ

)+O(s4)

= (6.17)

(6.18)

Next, as we mentioned above, we use Fourier Transform to find the Green’sfunction to equation (6.13). Applying the Fourier Transform to (6.13), it thenreads

∂tu(t, ξ) =(iξb(t)− ξ2a(t)

)u(t, ξ), u(0) = f(ξ),

where u = F(u) is the Fourier Transform. The above ODE is easy to solve. Denote

α(t) =∫ t

0a(τ)dτ, β(t) =

∫ t0b(τ)dτ , then

u(t, ξ) = e∫ t0 (iξb(τ)−ξ2a(τ))dτ f(ξ) = e(iξβ(t)−ξ2α(t))f(ξ).

39

Therefore,

u(t, x) = F−1(u(t, ξ)) =1√2π

∫e(iξ(β(t)+x)−ξ2α(t))f(ξ)dξ

=1

2π

∫ ∫e(iξ(β(t)+x−y)−ξ2α(t))f(y)dydξ

=1

2√πα(t)

∫e−

(β(t)+x−y)24α(t) f(y)dy,

which gives the exact Green’s function

GLt (x, y) =1

2√πα(t)

e−(β(t)+x−y)2

4α(t) .

By Lemma 3.1

U(1, 0; z = x)(x, y) = sGLt (x, x+ s(y − x)), s =√t.

actually, in this case the Green function does not depend on z.Our Taylor-Commutator method suggests that if we expand sGLt (x, x+ s(y−x))

with respect to s, we should be able to recover equation (6.17).

Notice that limt→0α(t)t = a(0), limt→0

β(t)t = b(0), thus

sGLt (x, x+ s(y − x))∣∣s=0

=s

2√πα(s2)

e− (β(s2)+s(x−y))2

4α(s2)

∣∣∣∣∣s=0

=1

2√πa(0)

e−(x−y)24a(0)

This is exactly eL0 .

Next, we shall expand sGLs2(x, x+s(y−x)) = s

2√πα(s2)

e− (β(s2)+s(x−y))2

4α(s2) in powers

of s and compare it with (6.17) term by term. To carry out the comparison, the

scheme we shall apply is to expand s

2√πα(s2)

and e− (β(s2)+s(x−y))2

4α(s2) with respect to

s respectively and then take the product. For the later exponential, in order to

obtain its power expansion, we expand its power term, namely (β(s2)+s(x−y))2

4α(s2) first.

More precisely, notice that

α(s2) = a(0)s2 +1

2a′(0)s4 +

1

6a′′(0)s6 +O(s8)

β(s2) = b(0)s2 +1

2b′(0)s4 +

1

6b′′(0)s6 +O(s8)

then plugging in the expansion of α(s) and β(s) to the two functions as we men-tioned above, we have

s√α(s2)

=1√

a(0)√

1 + a′(0)2a(0)s

2 + a′′(0)6a(0) s

4 +O(s6)

=1√a(0)

(1− a′(0)

4a(0)s2 − a′′(0)

12a(0)s4 +

3a′(0)2

32a(0)2s4 +O(s6)

)(6.19)

40

(β(s2) + s(x− y)

)24α(s2)

=1

4

(b(0)s2 + 1

2b′(0)s4 + 1

6b′′(0)s6 +O(s8) + s(x− y)

)2a(0)s2 + 1

2a′(0)s4 + 1

6a′′(0)s6 +O(s8)

=1

4a(0)

(b(0)s+

1

2b′(0)s3 +

1

6b′′(0)s5 +O(s7) + (x− y)

)21

1 + a′(0)2a(0)s

2 + a′(0)6a(0)s

4 +O(s6)

=1

4a(0)

(b(0)s+

1

2b′(0)s3 +

1

6b′′(0)s5 +O(s7) + (x− y)

)2

(1− a′(0)

2a(0)s2 − a′(0)

6a(0)s4 +

a′(0)2

4a(0)2s4 +O(s6)

)=

1

4a(0)

((x− y)2 + 2b(0)(x− y)s+ (b(0)2 − a′(0)

2a(0)(x− y)2)s2

+a(0)b′(0)− a′(0)b(0)

a(0)(x− y)s3 +O(s4)

)Therefore,

e− (β(s2)+s(x−y))

2

4α(s2) = e−(x−y)24a(0)

(1− b(0)

2a(0)(x− y)s−

(b(0)2

4a(0)− a′(0)

8a(0)2(x− y)2

)s2

− a(0)b′(0)− a′(0)b(0)

4a(0)2(x− y)s3 +

b(0)2

8a(0)2(x− y)2s2

+b(0)

2a(0)

(b(0)2

4a(0)− a′(0)

8a(0)2(x− y)2

)(x− y)s3 − b(0)3

48a(0)3(x− y)3s3 +O(s4)

)= e−

(x−y)24a(0)

(1− b(0)

2a(0)(x− y)s−

(b(0)2

4a(0)− b(0)2 + a′(0)

8a(0)2(x− y)2

)s2

+

[b(0)3 − 2a(0)b′(0) + 2a′(0)b(0)

8a(0)2(x− y)− b(0)(b(0)2 + 3a′(0))

48a(0)3(x− y)3

]s3 +O(s4)

).

(6.20)

Taking the product of (6.19) and (6.20), we obtain

s

2√πα(s2)

e− (β(s2)+s(x−y))2

4α(s2) =1

2√πa(0)

e−(x−y)24a(0)

[1− b(0)

2a(0)(x− y)s

+

((x− y)2

8a(0)2− 1

4a(0)

)(b(0)2 + a′(0)

)s2+(

b(0)3 − 2a(0)b′(0) + 3a′(0)b(0)

8a(0)2(x− y)− b(0)(b(0)2 + 3a′(0))

48a(0)3(x− y)3

)s3 +O(s4)

]

(6.21)

A routine check shows that (6.21) is exactly the same as (6.17), which confirmsour Dyson-Taylor Commutator method.

6.4. Applications to Stochastic Volatility models. In this subsection we willapply our Dyson-Taylor commutator method to stochastic volatility models thatappear in option pricing theory. Though our method is applicable to very generaltypes of financial derivatives, for simplicity we shall only consider European styleoptions. A European call option is a financial contract that gives the option holder

41

the right (not obligation) to buy the underlying asset, which we assume to be astock throughout this section, at a predetermined future date T (the maturity orexpiry date) for a predetermined price K (the strike). At time T, if the stock priceXT is greater than K, then the option will be exercised with realized payoff XT −K,because the option holder buy the stock with price K and he or she can immediatelysell it in the market with price XT . But at time T if XT < K then nothing willhappen and the option will just go expired, so the payoff will be zero. As a result,the general payoff function is (XT −K)+ := max{XT −K, 0} for a European call.

To price such a call option, stochastic volatility models have been intensivelystudied in the literature ([7, 13]). One major difficulty in such models is thatwe do not have tractable or explicitly computable solutions for the opton prices(for instance, the SABR model ([12, 10]) we will discuss shortly), thus numericalmethods need to be applied. An alternative is to asymptotically approximate theoption prices as in many papers ([12, 8, 4, 11]). In this section we shall carry out thisidea and show how our method can be used to price European style options understochastic volatility models. We shall first take the Heston model for example,the reason is that this model admits a semi-closed form formula for European calloptions ,so we are able to compare our results with the exact solutions numerically.Again for simplicity we assume the interest rate r = 0, then the stock price Xt

satisfies the following stochastic process

dXt =√YtXtdW

1t

and the volatility term Yt itself satisfies another CIR process given by

dYt = (a− bYt)dt+ σ√YtdW

2t ,

whereW 1t andW 2

t are standard Brownian motions with correlation dW 1t dW

2t = ρdt.

Using the same notations as we elaborated above, a call option price u(t, x, y) isthe solution to a parabolic PDE

ut(t, x, y) =

1

2yx2uxx(t, x, y) + ρσyxuxy(t, x, y) +

1

2yσ2uyy(t, x, y) + (a− by)uy(t, x, y)

(6.22)

with initial condition u(0, x, y) = (x−K)+ by Feymann-Kac formula [25]. In thisPDE we abused the notations and actually did the change of variable t→ T − t.

Then by exactly the same procedure as we did in previous sections, we are ableto obtain the zeroth order approximated Green’s function of the Heston PDE

H[0]t (x, y, w, z) =

1

2πtσxy√

1− ρ2exp

(−σ

2(x− w)2 − 2ρσx(x− w)(y − z) + x2(y − z)2

2tσ2x2y(1− ρ2)

)(6.23)

Next we compute the first order approximation of the Green’s function for theHeston model. By our main result, the first order approximation is

H[1]t = (1 + sQ1) eL0 ,

where

Q1 = (L1 +1

2[L0, L1]).

42

We can compute it explicitly as follows:

Q1eL0(x, y) = C3,0

(∂3

∂x3eL0(x, y)

)+ C2,1

(∂3

∂x2∂yeL0(x, y)

)+ C1,2

(∂3

∂x∂y2eL0(x, y)

)+ C0,3

(∂3

∂y3eL0(x, y)

)+ C0,1

(∂

∂yeL0(x, y)

),

(6.24)

where

C3,0(x, y) =

(1

2y +

1

4ρσ

)x3y

C2,1(x, y) =

(ρσy +

1

2ρ2σ2 +

1

4σ2

)x2y

C1,2(x, y) =1

2ρ2σ2xy2 +

1

4ρσ3xy +

1

2ρσ3xy

C0,3(x, y) =1

4σ4y

C0,1(x, y) = a− by.

Denote

xt =x− w√

t, yt =

y − z√t

eL0(x, y) := const(x0, y0) · eαx2+βxy+γy2 ,

Di,j(x, y)eL0(x, y) =∂i+j

∂xiyjeL0(x, y),

where

const(x0, y0) =1

2πσx0y0

√1− ρ2

.

Then

D3,0(x, y) = 12α2x+ 6αβy + 8α3x3 + 12α2βx2y + 6αβ2xy2 + β3y3

D2,1(x, y) = 6αβx+ (4αγ + 2β2)y + 4α2βx3 + (8α2γ + 4αβ2)x2y

+(8αβγ + β3)xy2 + 2β2γy3

D1,2(x, y) = (2β2 + 4αγ)x+ 6βγy + 2αβ2x3 + (8αβγ + β3)x2y

+(8αγ2 + 4β2γ)xy2 + 4βγ2y3

D0,3(x, y) = 6βγx+ 12γ2y + β3x3 + 6β2γx2y + 12βγ2xy2 + 8γ3y3

D0,1(x, y) = βx+ 2γy.

In the Heston case, we have

α(x0, y0) = − 1

2(1− ρ2)x20y0

β(x0, y0) =ρ

σ(1− ρ2)x0y0

γ(x0, y0) = − 1

2σ2(1− ρ2)y0.

43

Note that in all the Ds, α, β and γ are evaluated at (x, y). Finally, the first orderapproximation of the Green function for the Heston model reads

H[1]t (x, y, w, z) =

H[0]t (x, y, w, z)

(1 +√t

3∑j=0

(C3−j,j(x, y) ·D3−j,j(xt, yt) + C0,1(x, y) ·D0,1(xt, yt))).

(6.25)

Once we have the Green’s functions of the Heston PDE, we can obtain the calloption price by integrating the Green’s function against the payoff function (initialdata). In in a work in progress, we plan to compare numerically our approximationwith Heston’s formula, and we will show that our approximations are extremelyaccurate. We plan to actually give general formulas of the first and second orderapproximations of the Green’s function of a general parabolic PDE. In the aboveHeston example, the final formula we obtained is just a special case of our generalfirst order approximation. If we want to get more accurate results, then the secondorder approximation should be caught out. It is worth indicating that our generalformulas actually can be applied to any other option pricing models. As anotherexample, we consider a min/max options under the Bivariate Black-Scholes-Merton(BSM) model. In this type of options there are two underlying assets, and underthe BSM framework the prices of two assets St and S2 satisfies

dS1 = rS1dt+ σ1S1dW1t

dS2 = rS2dt+ σ2S2dW2t ,

where again W 1t and W 2

t are Brownian motions with correlation ρ. For the Euro-pean min option written on these two assets with strike K, the payoff at maturityis max(min(S1(T ), S2(T ))−K, 0). A routine argument show that when time to ma-turity is t, the min option price u(t, S1, S2) is the solution to the following Cauchyproblem

(6.26)

ut = 1

2σ21S

21∂

2S1u+ ρσ1σ2S1S2∂S1

∂S2u+ 1

2σ22S

22∂

2S2u

+rσ1S1∂S1u+ rσ2S2∂S2

u− ruu(0, S1, S2) = max(min(S1, S2)−K, 0).

This is also a second order parabolic PDE and it fits into our setting. Therefore,the same approximation procedure as we did in the Heston case applies to Euro-pean min/man options as well. Similarly, bivariate BSM models are also tractable.Johnson (1987) and Stulz (1982) show that the min option price has a closed formrepresentation. So we are also able to compare our approximations of the Europeanmin option prices with the exact prices.

So far we have considered two tractable option pricing models. Though closedform exact formulas are available in these models, they can not accurately capturethe features of option prices. To overcome this difficulty, complicated models havebeen developed. But the tradeoff is that these models do not admit closed formsolutions. Take the SABR model ([10]) for example, like the Heston model, it isalso a stochastic volatility model, and the two processes that drive the stock price

44

and the volatility are

dXt = YtXβt dW

1t

dYt = σYtdW2t ,

(6.27)

where σ, β are constant and 0 ≤ β ≤ 1. This model has been intensively usedin fixed income and commodities modeling. If we want, we can also add mean-reverting feature in the volatility pricess. Unless β = 0, 1, no explicit pricingformulas for this model are available in the literature to the authors’ knowledge.Therefore, numerical methods like finite difference or finite element have to be used.But these numerical methods are in general very slow and there are also other issues.However, our approximation methods are still applicable to this case, and we areable to obtain accurate first order and second order formulas for the SABR model.More detailed numerical tests and discussions will be included in work in progress.

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E-mail address: cheng@math.psu.edu

E-mail address: mazzucat@math.psu.edu

All authors, Pennsylvania State University, Math. Dept., University Park, PA

16802, USA,, V.N. also Inst. Math. Romanian Acad. PO BOX 1-764, 014700 BucharestRomania

E-mail address: nistor@math.psu.edu

46