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PROBABILISTIC REPRESENTATIONS OF SOLUTIONS

OF THE FORWARD EQUATIONS

S. THANGAVELU And

B. RAJEEV

Department of Mathematics Indian Institute of Science

Bangalore

Technical Report no. 2007/32

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Probabilistic Representations of Solutions of the Forward Equations

By

B. Rajeev, S. Thangavelu,

Statistics and Mathematics Unit, Department of Mathematics, Indian Statistical Institute, Indian Institute of Science, Bangalore - 560 059. Bangalore - 560 012.

brajeev@isibang.ac.in veluma@math.iisc.ernet.in Abstract: In this paper we prove a stochastic representation for solutions of the evolution equation

tψt = 1 2Lψt

where L is the formal adjoint of a second order elliptic differential operator L, with smooth coefficients, corresponding to the infinitesimal generator of a finite dimensional diffusion (Xt). Givenψ0 =ψ, a distribution with compact support, this representation has the form ψt = E(Yt(ψ)) where the process (Yt(ψ)) is the solution of a stochastic partial differential equation connected with the stochastic differential equation for (Xt) via Ito’s formula.

Key words : Stochastic differential equation, Stochastic partial differen- tial equation, evolution equation, stochastic flows, Ito’s formula, stochastic representation, adjoints, diffusion processes, second order elliptic partial dif- ferential equation, monotonicity inequality .

1 Introduction

The first motivation for the results of this paper is that they extend the results in [17] for Brownian motion, to more general diffusions. To recall, there we had recast the classical relationship between Brownian motion and the heat equation in the language of distribution theory. To be more precise,

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if S0 denotes the space of tempered distributions onIRd, the solutions of the initial value problem,

tψt = 1 2∆ψt, ψ0 = ψ

for anyψ ∈ S0, were represented in terms of a standard d-dimensional Brow- nian motion (Xt)t≥0, asψt =Xt(ψ). Hereτx :IRd →IRd is the translation operator and the expectation is taken in a Hilbert space Sp ⊂ S0 in which the process takes values.

The main result of this paper is that the above result extends to the solutions of the initial value problem

tψt = 1 2Lψt,

ψ0 = ψ ( 1.1 )

where ψ ∈ E0, i.e it is a distribution on IRd with compact support and L is the formal adjoint of L, a second order elliptic differential operator with smooth coefficients given as the first component of the pair (L, A), where

L = 1

2

X

i,j

(σσt)ij(x) ij2 +X

i

bi(x) i, A = (A1,· · ·, Ad),

Ak =

Xd

i=1

σij(x)∂i.

To describe the stochastic representation of solutions of (1.1), let (X(t, x)) denote the solutions of the stochastic differential equation

dXt = σ(Xt)·dBt+b(Xt) dt, X0 = x.

Then it is well known that a.s, x X(t, x) is smooth and induces a map Xt:C→C, namely Xt(φ)(x) =φ(X(t, x)) for φ ∈C. Here and in the rest of the paperCdenotes the space of all infinitely differentiable functions

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on IRd. Let Yt := Xt : E0 → E0 be the adjoint of the map Xt : C C. If ψ ∈ E0 ⊂ S0 , then we can show that the process (Yt(ψ)) takes values in one of the Hilbert spaces S−p,p >0 that define the countable Hilbertian structure of S0 (Proposition 3.1). Our stochastic representation now reads, ψt = E(Yt(ψ))(Theorem 4.3). These results extend the well known results connecting diffusion processes and partial differential equations(see [1], [5], [19], [6], [3]). Moreover, they establish a natural link with the subject of stochastic partial differential equations (see [13],[9], [12],[11] [22],[4]), viz. the process (Yt(ψ)) for ψ ∈ E0, is the solution of a stochastic partial differential equation (eqn (3.7) below) associated naturally with the equation for (Xt) via Ito’s formula. This stochastic partial differential equation is different from the one satisfied by the process (δXt) in [16],[7],[8] - the former is associated with the operators (L, A) as above, the latter being associated with the random operators (L(t, ω), A(t, ω)),A(t, ω) := (A1(t, ω),· · ·Ar(t, ω)),

Ak(t, ω) =

Xd

i=1

σik(Xt(ω)∂i, L(t, ω) = 1

2

Xd

i,j=1

(σσt)ij(Xt(ω)) ij2

Xd

i=1

bi(Xt(ω)) i

(see [16],[8]). However it is easily seen that whenψ =δx, the process (Yt(ψ)) is the same as the process (δXt). We also note that solutions (ψt) of equation (1.1) are obtained by averaging out the diffusion term in the stochastic partial differential equation satisfied by (Yt(ψ)) (see Theorem 4.3), a result that corresponds quite well with the original motivation for studying stochastic partial differential equations, viz. ‘Stochastic PDE = PDE + noise’ (see [22]

for example).

The definition of Yt as the adjoint of the map Xt : C C induced by the flow (X(t, x, ω)) does not automatically lead to good path properties for the process (Yt(ψ)), ψ ∈ E0. To get these we generalise the representation

Yt(ψ)(ω) =

Z

δX(t,x,ω) dψ(x)

which is easily verified when ψ is a measure with compact support. Indeed, hYt(ψ)(ω), ϕi = hψ, Xt(ω)◦ϕi

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=

Z

Xt(ω)◦ϕ(x) dψ(x)

=

Z

ϕ(X(t, x, ω)) dψ(x)

=

Z

X(t,x,ω), ϕi dψ(x)

= h

Z

δX(t,x,ω) dψ(x), ϕi.

Here the integral R δX(t,x,ω)dψ(x) is understood in the sense of Bochner and takes values in a suitable Hilbert spaceS−p ⊆ S0. In Section 3, we define the process (Yt(ψ)), for ψ ∈ E0, via a representation such as the one above and verify that indeed Yt(ψ) = Xt(ψ). In Theorem (3.3) we show that (Yt(ψ)) satisfies a stochastic partial differential equation.

In Section 2, we state some well known results from the theory of stochastic flows [13] in a form convenient for our purposes. These are used in Section 3 for constructing the process (Yt(ψ)) and proving its properties. In Section 4, we prove the representation result for solutions of equation (1.1). Our results amount to a proof of existence of solutions for equation (1.1). Our proofs require that the coefficients be smooth. However, we do not require that the diffusion matrix be non degenerate. The uniqueness of solutions of (1.1) can be shown to hold if the so called ‘monotonicity inequality’ (see (4.2) below) for the pair of operators (L, A) is satisfied (Theorem 4.4). It may be mentioned here that the ‘monotonicity inequality’ is known to hold even when the diffusion matrix is degenerate (see [8]). As mentioned above, when ψ = δx, x IRd, Yt(ψ) = δXt where X0 = x. As discussed above, for ψ ∈ E0, the solutions (Yt(ψ)) of the stochastic partial differential equation (3.7) can be constructed out of the particular solutions (δXt) corresponding to ψ = δx. In other words the processes (δXt), X0 = x, can be regarded as the ‘fundamental solution’ of the stochastic partial differential equation (3.7) that (Yt(ψ)) satisfies. This property is preserved on taking expecta- tions: in other wordsXt, X0 =xis the fundamental solution of the partial differential equation (1.1) satisfied by ψt = EYt(ψ) - a well known result for probabilists, if one notes thatXt =P(t, x,·) the transition probability measure of the diffusion (Xt) starting at x (Theorem 4.5). Assuming that this diffusion has a density p(t, x, y) satisfying some mild integrability con- ditions, we deduce some well known results. If L = L then the density is symmetric (Theorem (4.6)), and in the constant coefficient case we further

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have p(t, x, y) = p(t,0, y−x). Finally if Tt : C C denotes the semi- group corresponding to the diffusion (Xt), and St : E0 → E0 is the adjoint, then St, given by St(ψ) = EYt(ψ), is a uniformly bounded (in t) operator when restricted to the Hilbert spaces S−p (Theorem (4.8)).

2 Stochastic Flows

Let Ω = C([0,∞), IRr) be the set of continuous functions on [0,∞) with values in IRr. Let F denote the Borel σ-field on Ω and let P denote the Wiener measure. We denote Bt(ω) := ω(t), ω Ω, t 0 and recall that under P, (Bt) is a standard r dimensional Brownian motion. Let (Xt)t≥0 be a strong solution on (Ω,F, P) of the stochastic differential equation

dXt = σ(Xt)·dBt+b(Xt)dt X0 = x

)

( 2.1 ) with σ = (σji),i = 1. . . d, j = 1. . . r and b = (b1, . . . bd), where σji and bi are given C functions on IRd with bounded derivatives satisfying

kσ(x)k+kb(x)k=

Xd

i=1

Xr

j=1

ji(x)|2

1/2

+

à d X

i=1

|bi(x)|2

!1/2

≤K(1 +|x|) for some K > 0. Under the above assumptions on σ and b, it is well known that a unique, non-explosive strong solution (X(t, x, ω))t≥0,x∈IRd exists on (Ω,F, P) (see [10]). We also have the following theorem (see [2], [13] and [10], p.251 ).

Theorem 2.1 Forx∈IRdandt≥0, let(X(t, x, ω))be the unique strong so- lution of equation (2.1) above. Then there exists a process( ˜X(t, x, ω))t≥0,x∈IRd such that

(1) For all x∈IRd, P{X(t, x, ω) =˜ X(t, x, ω), t 0}= 1.

(2) For a.e. ω(P), x→X(t, x, ω)˜ is a diffeomorphism for all t≥0.

(3) Let θt : Ω be the shift operator i.e. θtω(s) = ω(s +t); then for s, t≥0, we have

X(t˜ +s, x, ω) = ˜X(s,X(t, x, ω), θ˜ tω)

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for all x∈IRd, a.e. ω (P).

Denote by ∂X(t, x, ω) the d×d matrix valued process, given for a.e. ω by (∂X)ij(t, ω, x) = ∂X∂xji(t, x, ω) for allt 0 andx∈IRd. Denote by ∂σα(x) the d×dmatrix (∂σα(x))ij = ∂σ∂xαi(x)j and by∂b(x) thed×dmatrix (∂b(x))ij = ∂b∂xi(x)j . Then it is well known (see [13]) that almost surely, ∂X(t, ω, x) is invertible for all t and xand the inverse satisfies the SDE

dJt =

Xr

α=1

Jt·∂σα(Xt)dBtα

−Jt·

"

∂b(Xt)

Xr

α=1

(∂σα)·(∂σα)(Xt)

#

dt, J0 = I

whereI is the identity matrix andJt·∂σα(Xt) etc. denote the product ofd×d matrices. In proving our results, we will need to show that sup

x∈K|∂rX(t, x)|q and sup

x∈K|(∂X)−1(t, x)| (here r := 1r1. . . ∂drd and | · | denotes the Euclidean norm on IRd in the first case and on IRd2 in the second case) have finite expectation forq 1 andK ⊆IRd a compact set. To do this we will use the results of section 4.6 of [13], as also the notation there. First, we note that the stochastic differential equations for (Xt) and (∂X(t)) can be combined into a single stochastic differential equation inIRd+d2, which in the language of [13], can be based on a spatial semi-martingaleF(x, t) = (F1(x, t), . . . Fd+d2(x, t)).

Having done this and having verified the regularity hypothesis on the local characteristics ofF(x, t) we can apply Corollary 4.6.7 of [13] to get our results.

To fix notation we note that the set {k :d < k ≤d2+d}is in 1-1 correspon- dence with {(i, j) : 1 i, j d}. We fix such a correspondence and write k (i, j) if k corresponds to (i, j). Ifx IRd+d2 we will write x = (x1, x2) where x1 ∈IRd and x2 ∈IRd2. For 1≤k≤d, let

Fk(x, t) =

Xr

α=1

σkα(x1)Btα+bk(x1)t.

Ford+ 1≤k ≤d+d2, let Fk(x, t) =

Xr

α=1

(x2·∂σα(x1))ij Btα

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t

Ã

x2·

"

∂b(x1)

Xr

α=1

(∂σα)·(∂σα)(x1)

#!i

j

wherek↔(i, j), and the notationx2·∂σα(x1) stands for the product ofd×d matrices x2 and ∂σα(x1) . The local characteristics of F(x, t) are then given by (α(x, y, t), β(x, t), t) where:

βk(x, t) = bk(x1), 1≤k≤d

= x2·∂b(x1)

Xr

α=1

(∂σα)·(∂σα)(x1), d+ 1 ≤k≤d+d2 β(x, t) = (β1(x, t), . . . βd+d2(x, t)), x= (x1, x2)∈IRd+d2.

Further, for x = (x1, x2), y = (y1, y2) ,α(x, y, t) = αk`(x, y, t), 1 k, ` d+d2 where

αk`(x, y, t) = (σ(x1)·σt(y1))k`, 1≤k, `≤d

=

Xr

α=1

(x2·∂σα(x1))ij(y2·∂σα(y1))ij00

d+ 1 ≤k, `≤d+d2, k (i, j), `(i0, j0)

=

Xr

α=1

σαk(x1)(y2·∂σα(y1))ij

1≤k≤d, d+ 1≤`≤d+d2 and `↔(i, j)

=

Xr

α=1

(x2·∂σα(x1))ijσ`α(y1)

1≤`≤d, d+ 1 ≤k ≤d+d2 and k (i, j).

Let n=d+d2 and f :IRn→IRn, g :IRn×IRn →IRn2. Consider for m≥1 and δ >0, the following semi-norms (see [13]),

kfkm,δ = sup

x∈IRn

|f(x)|

(1 +|x|)+ X

1≤|α|≤m

sup

x∈IRn

|∂αf(x)|

+ X

|α|=m

sup

x,y∈IRn

x6=y

|∂αf(x)−∂αf(y)|

|x−y|δ

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kgkm,δ = sup

x∈IRn

|g(x, y)|

(1 +|x|)(1 +|y|) + X

1≤|α|≤m

sup

x,y∈IRn

|∂xαyαg(x, y)|

+ X

|α|=m

sup

x,x0,y,y0∈IRn

x6=x0,y6=y0

|∂xαyαg(x, y)−∂xαyαg(x0, y)−∂xαyαg(x, y0) +xαyαg(x0, y0)|

|x−x0|δ|y−y0|δ . Let for 0 < s < T, s,t(x), s t T} be the solution of Ito’s stochastic differential equation based on the semi-martingale F(x, t), x∈IRd+d2, i.e.

ϕs,t(x) =x+

Z t

s

Fs,r(x), dr).

For x = (x1, x2), where x1 IRd and x2 IRd2 corresponding to the identity matrix I, let ϕk0,t(x), 1 k d + d2 denote the components of the vector ϕ0,t(x). Note that ϕk0,t(x) = Xk(t, x1) for 1 k d and ϕk0,t(x) = (∂X−1)ij(t, x1), for d+ 1 k d+d2 and k (i, j). We then have the following theorem.

Theorem 2.2 Given 0 s T, α= (α1. . . αd) a multi-index, N > 0, and q≥1, there exists C =C(s, T, α, N, q)>0 such that

E sup

|x|≤N

|∂αϕs,t(x)|q< C

for any t satisfying s ≤t ≤T. In particular, for any compact K ⊆IRd, q 1, α a multi-index,and 1≤i, j ≤d, there exists C > 0

Esup

x∈K|∂αX(t, x)|q < C and

Esup

x∈K|(∂X−1(t, x))ij|q < C for 0≤t≤T.

Proof: It is easily verified that the local characteristics (α, β, t) verify for m≥1, δ = 1:

sup

t≤T kα(t)km,1 <∞ and sup

t≤T kβ(t)km,1 <∞.

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In the language of [13], the local characteristics (α, β, t) belong to the class Bubm,1 for allm≥1. Thus the hypothesis of Corollary 4.6.7 in [13] is satisfied.

Hence for p > 1, α a multi-index, N > 0 and 0 < s < T, there exists C =C(p, α, N, s, T) such that

E sup

|x|≤N

|∂αϕs,t(x)|2p < C.

The result for q= 2p >2 and hence for q 1 follows. 2

3 The Induced Flow on Distributions with compact support

We will denote the modification obtained in Theorem 2.1 again by (X(t, x, ω)).

Forωoutside a null set ˜N, the flow of diffeomorphisms induces, for eacht≥0 a continuous linear map, denoted by Xt(ω) on C. Xt(ω) : C C is given by (Xt(ω)(ϕ)(x) = ϕ(X(t, x, ω)). This map is linear and continuous with respect to the topology on C given by the following family of semi- norms: For K ⊆IRd a compact set, let kϕkn,K = max

|α|≤n sup

x∈K|Dαf(x)| where ϕ∈ C and n 1 an integer and α = (α1, . . . αd) and |α| =α1 +. . .+αd. Let Kt,ω denote the image of K under the map x→ X(t, x, ω). Then using the ‘chain rule’ it is easy to see that there exists a constant C(t, ω)>0 such that

kXt(ω)(ϕ)kn,K ≤C(t, ω)kϕkn,Kt,ω.

Let Xt(ω) denote the transpose of the map Xt(ω) : C →C. Then if E0 denotes the space of distributions with compact support we have Xt(ω) : E0 → E0 is given by

hXt(ω)ψ, ϕi=hψ, Xt(ω)ϕi

for all ϕ C and ψ ∈ E0. Let ψ ∈ E0. Let supp ψ K and let N = order(ψ) + 2d. Then there exist continuous functions gα,|α| ≤N,supp gα V where V is an open set having compact closure, containingK, such that

ψ = X

|α|≤N

αgα. ( 3.1 )

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See [21]. Let ϕ C. Let fi C and f = (f1, . . . fd). Let α be a multi index. We now describe each of the numbers α◦f)(x), x IRd as the result of a distribution (depending on x IRd) acting on the test function φ. Let βi, i = 1,· · ·, d be multi indices each with d components. Using the chain rule for differentiation , it is easy to see that for each multi index γ with |γ| ≤ |α|, there exist polynomials Pγ, in a finite number of variables ,with deg Pγ =|γ|, such that

α◦f)(x) = X

|γ|≤|α|

(−1)|γ| ( 3.2 )

Pγ((∂β1f1, ∂β2f2, ..., ∂βdfd)i|≤|α|)(x)hϕ, ∂γδf(x)i.

Forω 6∈N˜, defineYt(ω) :E0 → E0 by Yt(ω)(ψ) = X

|α|≤N

(−1)|α| X

|γ|≤|α|

(−1)|γ|

Z

V

gα(x)

Pγ((∂β1X1. . . ∂βdXd)i|≤|α|)(t, x, ω) γδX(t,x,ω) dx ( 3.3 ) TakeYt(ω) = 0 if ω∈N˜.

Let S be the space of smooth rapidly decreasing functions on IRd with dual S0, the space of tempered distributions. It is well known ([9]), that S is a nuclear space, and that S = \

p>0

(Sp,k · kp), where the Hilbert spaces Sp are equipped with increasing norms k · kp, defined by the inner products

hf, gip =

X

|k|=0

(2|k|+d)2phf, hki hg, hki, f, g∈ S

Above, {hk}|k|=0 is an orthonormal basis for L2³IRd, dx´ given by Hermite functions (for d = 1, hk(t) = (2kk!√

π)−1/2exp{−t2/2}Hk(t), with Hk(t), a Hermite polynomial. see [9]), and h·,·i is the usual inner product in L2³ Rd, dx´. We also have S0 = [

p>0

(S−p,k · k−p). Note that for all −∞ <

p <∞, Sp is the completion ofS ink · kp and Sp0 is isometrically isomorphic with S−p, p >0. We then have the following proposition.

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Proposition 3.1 Let ψ be a distribution with compact support having rep- resentation (3.1). Let p > 0 be such that αδx S−p for |α| ≤ N. Then (Yt(ψ))t≥0 is an S−p valued continuous adapted process such that for allt 0,

Yt(ψ) =Xt(ψ) a.s. P.

Proof: From Theorem 4.6.5 of [13], it follows that for any multi indexγ and compact set K ⊆IRd, sup

s≤T sup

x∈K|∂γXi(s, x, ω)|<∞a.s. for all T >0.

From this result, the fact that i : S−p−1

2 S−p is bounded, and Theorem 2.1 of [17], it follows that for all T >0 a.s.

Z

V

|gα(x)||Pγ((∂β1X1. . . ∂βdXd)i|≤|α|)(t, x, ω)| k∂γδX(t,x,ω)k−pdx < C(ω)( 3.4 ) for some finite positive constantC(ω) that does not depend ontfor 0≤t≤T .It follows that (Yt(ψ)) is a well defined S−p-valued process. Since for all x IRd,(∂γXi(t, x, ω)) is an adapted process, and jointly measurable in (t, x, ω) it follows that (Yt(ψ)) is anS−p-valued adapted process.

To show that the mapt→Yt(ψ) is almost surely continuous in S−p, we first observe that for any p IR and any φ Sp, the map x τxφ : IRd Sp is continuous, where τx : S0 S0 is translation by x IRd. To see this, let xn x IRd. Note that from Theorem 2 of [17], given ² > 0, there exists ψ ∈S such that

xφ−τxnφkp < xψ−τxnψkp+ ² 2. From the definition of k.kp we have

xψ−τxnψk2p = X

k

(2|k|+d)2pxψ−τxnψ, hki2.

Sinceψ ∈Sq for everyq and since the result is true forp= 0, the right hand side above tends to zero by dominated convergence theorem and

xφ−τxnφkp < ²

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for largenthus proving the continuity of the map x→τxφ. In particular ifp andαare as in the statement of the theorem, the mapx→∂γδxis continuous in S−p for |γ| ≤ |α| . Now the continuity of t Yt(ψ)(ω) follows from the continuity in the t variable of the processes (∂γXi(t, x, ω)) and (∂γδX(t,x,ω)) and the dominated convergence theorem .

We now verify that Yt(ψ) = Xt(ψ). Let ϕ∈C. From (3.3), hYt(ω)ψ, φi = X

|α|≤N

(−1)|α| X

|γ|≤|α|

(−1)|γ|

Z

V

gα(x)Pγ((∂β1X1. . . ∂βdXd)i|≤|α|)(t, x, ω) h∂γδX(t,x,ω), φi dx

= X

|α|≤N

(−1)|α|

Z

V

gα(x)∂α◦X(t, x, ω))dx (by (3.2))

= X

|α|≤N

h∂αgα, φ◦X(t,·, ω)i (by (3.1))

= X

|α|≤N

h∂αgα, Xt(ω)φi

= hψ, Xt(ω)φi.

2 We now define the operators A :C→L(IRr, C) and L:C →C(IRd) as follows: For ϕ∈C, x∈IRd,

= (A1ϕ,· · ·, Arϕ), Aiϕ(x) =

Xd

k=1

σik(x)∂kϕ(x), Lϕ(x) = 1

2

Xd

i,j=1

(σσt)ij(x) ij2ϕ(x) +

Xd

i=1

bi(x)iϕ(x).

We define the adjoint operators A : E0 L(IRr,E0) and L : E0 → E0 as follows:

Aψ = (A1ψ,· · ·, Arψ), Aiψ =

Xd

k=1

kkiψ),

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Lψ = 1 2

Xd

i,j=1

ij2((σσt)ijψ)−

Xd

i=1

i(biψ).

The following proposition gives the boundedness properties of L and A. ForK ⊂IRd, let E0(K)⊆ E0 be the subspace of distributions whose support is contained in K.

Proposition 3.2 Let p > 0 and q > [p] + 2, where [p] denotes the largest integer less than or equal to p. Then, A :S−p∩E0(K)→L(IRr, S−q∩E0(K)) and L : S−p ∩ E0(K) S−q ∩ E0(K). Moreover, there exists constants C1(p)>0, C2(p)>0 such that

kAψkHS(−q)≤C1(q) kψk−p, kLψk−q ≤C2(q) kψk−p where

kAψk2HS(−q)=

Xr

i=1

°°

°°

° Xd

k=1

kkiψ)

°°

°°

°

2

−q

=

Xr

i=1

kAiψk2−q.

Proof Clearly A : E0(K) L(IRr,E0(K)) and L : E0(K) → E0(K). We prove the bounds for A. The bounds for L follow in a similar fashion. By definition, if q≥p+ 1/2,

kAψk2HS(−q) =

Xr

i=1

°°

°°

° Xd

k=1

kkiψ)

°°

°°

°

2

−q

C

Xr

i=1

Xd

k=1

ikψk2−(q−1

2). ( 3.5 ) Let σ denote a C function. We first show that the map ψ σψ : Sn E0(K)→ Sn∩ E0(K) satisfies

kσψkn≤C kψkn

where the constant C depends on σ, K and n. First assume that ψ S ∩ E0(K). We then have (see [17], Proposition 3.3b )

kσψk2n≤C1 X

|α|+|β|≤2n

kxαβ(σψ)k20.

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Clearly,

kxαβ(σψ)k20 ≤C2 X

|γ|≤|β|

kxαγψk20.

It follows that

kσψk2n C3 X

|α|+|γ|≤2n

kxαγψk20

Ckψk2n.

We now extend this to ψ ∈ Sn∩ E0(K): Since S is dense in Sn, we can get ψm ∈ S, ψm ψ in Sn. By multiplying by an appropriate C function with compact support can assume, ψm ∈ S ∩ E0(K). By the above inequality applied to ψm, it follows that σψm converges in Sn, and hence converges weakly to a limit ϕin S0. Hence if f isC(IRd) with compact support,

hϕ, fi = lim

m→∞hσψm, fi

= lim

m→∞m, σfi

= hψ, σfi=hσψ, fi.

Hence σψm σψ in Sn and the required inequality follows for ψ ∈ Sn E0(K).

Now suppose ψ ∈ S ∩ E0(K).

kσψk−n = sup

kϕkn≤1

ϕ∈S

|hσψ, ϕi|

= sup

kϕkn≤1

ϕ∈S

|hgσψ, ϕi|

where g C, g = 1 on K, supp(g) K², an ²-neighbourhood of K.

Therefore,

kσψk−n ≤ kψk−n sup

kϕkn≤1

ϕ∈S

kσgϕkn

Ckψk−n. ( 3.6 )

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In the same way as for n 0, we can extend the above inequality to ψ S−n∩ E0(K). Now the proof can be completed using (3.5) and (3.6) and by choosing a q IR such that q m+ 12 > m p for some integer m. In

particular, we may take q >[p] + 2. 2

Theorem 3.3 Let ψ ∈ E0 have the representation (3.1). Let p > 0 be such that γδx ∈S−p,|γ| ≤N. Let q > p be as in Proposition 3.2. Then the S−p- valued continuous, adapted process (Yt(ψ))t≥0 satisfies the following equation in S−q: a.s.,

Yt(ψ) =ψ+

Z t

0

A(Ys(ψ))·dBs+

Z t

0

L(Ys(ψ)) ds ( 3.7 ) for all t≥0.

Proof: From Proposition 3.2 and the estimate (3.4), it follows that fort≥0,

a.s. Z t

0

kAYs(ψ)k2HS(−q) ds+

Z t

0

kLYs(ψ)k−qds <∞.

Hence the right hand side of (3.7) is well defined. Let ϕ C. Then by Ito’s formula

Xt(ϕ) = ϕ+

Z t

0

Xs(Aϕ)·dBs+

Z t

0

Xs(Lϕ)ds

where the integrals on the right hand side are understood asC valued pro- cesses given by (t, x, ω) Rt

0 Aϕ(X(s, x, ω)·dBsand (t, s, ω) Rt

0 Lϕ(X(s, x, ω))ds.

By multiplying by a smooth function with support contained in the support of ψ, we may assume that these processes have their supports contained in a fixed compact set not depending on t and ω viz. the support of ψ. In particular, they belong to S ⊂Sp. Using Proposition 3.1, we get, for ϕ∈S,

hYt(ψ), ϕi = hψ, Xt(ϕ)i

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= hψ, ϕi+

Z t

0

hψ, Xs(Aϕ)i ·dBs+

Z t

0

hψ, Xs(Lϕ)ids

= hψ, ϕi+

Z t

0

hAYs(ψ), ϕi ·dBs+

Z t

0

hLYs(ψ), ϕi ds

= +

Z t

0

AYs(ψ)·dBs+

Z t

0

LYs(ψ) ds, ϕi

and the result follows. In the above calculations we have used the fact that T Rt

0 AYs(ψ)·dBs = Rt

0 T A(Ys(ψ))·dBs for any bounded linear functional

T :S−q IR. 2

4 Probabilistic representations

In this section we prove the probabilistic representations of solutions to the initial value problem for the parabolic operatort−L. We also show unique- ness for the solutions of the initial value problem under the ‘Monotonicity conditions’. We first prove some estimates on xk−p and k∂γδxk−p that are required later.

Theorem 4.1 a) δx ∈S−p iff p > d4. Further if p > d4, then

|x|→∞lim xk−p = 0.

b) Further if γ ZZd+ is a multi index and p > d4 + |γ|2 , then sup

x∈IRd

k∂γδxk−p <∞.

Proof: We first prove b). Since γ :S−q →S−q−|γ|

2 is a continuous operator we have for p > d4 +|γ|2 ,

k∂γδxk−p ≤Ckδxk(−p−|γ|

2 ).

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Further from Theorem (2.1) of [17], it follows that for any compact set K contained in IRd, sup

x∈Kxk−(p+|γ|

2 ) < ∞. Since p+ |γ|2 > d4, the statement in part b) of the theorem now follows from part a).

The proof of part a) of the theorem uses the generating function for Hermite functions given by Mehler’s formula (see [20], page 2). First we note that

(2n+d)−2p = 1 (2p1)!

Z

0

t2p−1 e−(2n+d)tdt.

Hence,

xk2−p =

X

n=0

(2n+d)−2p X

|k|=n

|hk(x)|2

= 1

(2p1)!

Z

0

t2p−1 g(t, x)dt where

g(t, x) :=

X

n=0

e−(2n+d)t X

|k|=n

|hk(x)|2. Using Mehler’s formula ,

g(t, x) = e−dtπd2(1−e−4t)d2

× e12(1+e

−4t

1−e−4t)2|x|2+(1−ee−2t−4t)2|x|2

= e−dtπd2(1−e−4t)d2

× e−(tanh t)|x|2. It is easy to see that for allx

g(t, x)∼(1−e−4t)d2, t→0 and g(t, x)∼e−dte−(tanh t)|x|2, t→ ∞

where a(t) b(t) if and only if a(t)b(t) converges to a positive number c. It follows that for all ² >0 andx∈IRd,

Z ²

0

t2p−1g(t, x)dt <∞

References

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