Small time chaos approximations for heat kernels of multidimensional diffusions

Abstract

In this article, we consider the problem of obtaining general Hermite type approximations for densities of uniformly elliptic multi-dimensional diffusions with smooth coefficients in small time. These approximations resemble the Itô chaos expansions for square integrable random variables on Wiener space. The proofs are constructive and the approximations are explicit enough so that they can be used for constructing algorithms of approximation in the total variation sense. As examples, we provide some density expansions on some multi-dimensional examples.

Appendices

Algorithm for calculating the terms of the chaos decomposition

1.1 Description of the algorithm

The algorithm consists of finding an expansion of (4.1) of a certain time order. As our proof is constructive this expansion can be obtained from the proof. Suppose that we want an expansion with t-order N. By induction, we suppose that we have the chaos monomial expansion of t-order $N-1$, denoted by $p^{(N-1)}$ in Proposition 4.2. The algorithm has two parts:

Part I: Chaos monomial decomposition of t-order N/2 for $\varXi _t(x,y)$ (The first term in (4.1))

Step I.1: Collect all terms in Taylor’s formula for $\varPhi _t(\xi ;x,y)$ w.r.t. $\xi$ around x as in Corollary 4.12. That is, the terms where $l+m/2\le N/2$.

Step I.2: The Taylor formula for increments of a, b. Although this is mentioned in Corollary 4.2, the polynomials P and Q are not given explicitly. One has to recall that each polynomial of order k in the Taylor expansion is equivalent to a t-order of k/2 and therefore one needs to perform this expansion as long as the t-order of that term does not exceed N/2.

Step I.3: Transformation of the products into chaos monomials. This implies that each difference $(y-x)$ has to be corrected using the mean $\mp a(x)t$. Then use formula B.6 in order to convert the product of the polynomial and the derivative of $\varPhi _t(x,y)$ into sums of chaos monomials. This procedure is also used repeatedly in Part II.

Part II: Chaos decomposition of $p^{(N-1)}\circledast \Upsilon$ (the second term in (4.1)).

Step II.1: elimination of negligible terms: Note that the kernel $\Upsilon _t(z,y)=\Upsilon ^{\textrm{drift}}_t(z,y)+\Upsilon ^{\textrm{diffusion}}_t(z,y)$ where each kernel has a different t-time order according to Lemma 4.1. Therefore the order of the required expansions will be different for each term. Still the procedure is similar as in Step I and one uses the Taylor formula for $\varPhi _t^{(i)}(\xi ;z,y)$,$\varPhi _t^{(i,j)}(\xi ;z,y)$ w.r.t. $\xi$ around x.

Step II.2: Apply the Taylor formula for increments of a, b as in Step I.2 above.

Step II.3: Transformation of the products into 3-point monomials: Now we will have polynomials in the variables $(z-x)$ and $(z-y)$ due to the Step II.2. As in Step I.3, we add and subtract the mean term a(x)t and use B.6.

Step II.4: Integration of the 3-point monomials into chaos monomials using (B.10) and computing the integrals with respect to time in $p^{(N-1)}\circledast \Upsilon$ .

Hermite functions and polynomials

In this section, we gather a series of identities and properties of general Hermite functions and polynomials as introduced in Section 2. These functions/polynomials do not appear often in the literature, which typically focuses on standardized Gaussian measures $\mathscr {N}(0, \textrm{Id}_{{\mathbb {R}}^d})$ or $\mathscr {N}(0, t\textrm{Id}_{{\mathbb {R}}^d})$; thus their properties can be hardly found explicitly written⁵ in the present generality.

To provide a comprehensive discussion, in addition to the Hermite functions introduced in Sect. 2, we will also consider their standardized analogues

$$\begin{aligned} \phi _t(0,w)=(2\pi t)^{-d/2}\exp \left( -{1\over 2t}|w|^2\right) , \\ \phi ^{(i_1, \dots , i_n)}_t(0,w)=(-1)^n \partial _{w_{i_1}}\dots \partial _{w_{i_n}}\phi _t(0,w), \quad w\in {\mathbb {R}}^d \end{aligned}$$

and corresponding Hermite polynomials defined by the equation

$$\begin{aligned} \phi ^{(i_1, \dots , i_n)}_t(0,w)=\mathscr {H}^{(i_1, \dots , i_n)}_t(w)\phi _t(0,w). \end{aligned}$$

Note that the Hermite functions/polynomials introduced in Section 2 reduce to the above definition in the particular case that $\textbf{a}=0$ and $\textbf{b}=I_{d}$. Furthermore, let $\textbf{a}\in {\mathbb {R}}^d, \textbf{b}\in {\mathbb {R}}^{d\times d}_\textrm{sym}$ be given. As $\textbf{b}$ is assumed to be a strictly positive definite matrix then there exists a $d\times d$ matrix $\sigma$ such that $\textbf{b}=\sigma \sigma ^*$. From now on, $\sigma$ is considered to be fixed and it is invertible since $\textbf{b}\in {\mathbb {R}}^{d\times d}_\textrm{sym}$. It follows simply by the definition of $\varPhi$ that if we let $w=\sigma ^{-1}(y-x-t\textbf{a})$ then

$$\begin{aligned} \varPhi _t (\textbf{a}, \textbf{b}; x,y)=|\det \sigma |^{-1}\phi _t(0,w). \end{aligned}$$

Thus using the chain rule for derivatives we have the following relation:

$$\begin{aligned} \begin{aligned} H^{(i_1, \dots , i_n)}_t (\textbf{a}, \textbf{b}; x,y)=&\sum _{j_1, \dots , j_n=1}^d\mathscr {H}^{(j_1, \dots , j_n)}_t(w)(\sigma ^{-1})_{j_1 i_1}\dots (\sigma ^{-1})_{j_n i_n} \\=&\sum _{j_1, \dots , j_n=1}^d ((\sigma ^*)^{-1})_{i_1 j_1}\dots ((\sigma ^*)^{-1})_{i_n j_n}\mathscr {H}^{(j_1, \dots , j_n)}_t(w). \end{aligned}\nonumber \\ \end{aligned}$$

(B.1)

Furthermore, the above formulas show that any property on the standardized Hermite polynomials $\mathscr {H}$ can be transferred to the general Hermite polynomials H.

For example,

$$\begin{aligned} \mathscr {H}^{(i)}_t(w)={1\over t}w_i, \quad H^{(i)}_t(\textbf{a}, \textbf{b}; x,y)={1\over t}\Big (\textbf{b}^{-1}(y-x-\textbf{a}t)\Big )_i, \end{aligned}$$

the second identity can be either derived directly or obtained from the first one using (B.1) and the fact that $\textbf{b}^{-1}= (\sigma ^*)^{-1}\sigma ^{-1}$. Similarly,

$$\begin{aligned} \mathscr {H}^{(i,j)}_t(w)= & {} {1\over t^2}w_iw_j-{1\over t}1_{i=j}, \\ H^{(i,j)}_t(\textbf{a}, \textbf{b}; x,y)= & {} {1\over t^2}\Big (\textbf{b}^{-1}(y-x-\textbf{a}t)\Big )_i\Big (\textbf{b}^{-1}(y-x-\textbf{a}t)\Big )_j-{1\over t}\textbf{b}^{-1}_{ij} \end{aligned}$$

and

$$\begin{aligned} \mathscr {H}^{(i,j,k)}_t(w)= & {} {1\over t^3}w_iw_jw_k-{1\over t^2}w_i1_{j=k}-{1\over t^2}w_j1_{i=k}-{1\over t^2}w_k1_{i=j},\\ H^{(i,j,k)}_t(\textbf{a}, \textbf{b}; x,y)= & {} {1\over t^3}\Big (\textbf{b}^{-1}(y-x-\textbf{a}t)\Big )_i\Big (\textbf{b}^{-1}(y-x-\textbf{a}t)\Big )_j\Big (\textbf{b}^{-1}(y-x-\textbf{a}t)\Big )_k\\{} & {} -{1\over t^2}\textbf{b}^{-1}_{ij}\Big (\textbf{b}^{-1}(y-x-\textbf{a}t)\Big )_k-{1\over t^2}\textbf{b}^{-1}_{ik}\Big (\textbf{b}^{-1}(y-x-\textbf{a}t)\Big )_j\\{} & {} -{1\over t^2}\textbf{b}^{-1}_{jk}\Big (\textbf{b}^{-1}(y-x-\textbf{a}t)\Big )_i. \end{aligned}$$

Next, we have the identities

$$\begin{aligned}{} & {} H_t^{(i_1, \dots ,i_{m-1},i, i_{m+1}, \dots , i_n)}(\textbf{a}, \textbf{b};x,y)\nonumber \\{} & {} \quad = {1\over t}\Big [\textbf{b}^{-1}(y-x-t\textbf{a})\Big ]_{i} H_t^{(i_1, \dots ,i_{m-1}, i_{m+1}, \dots , i_n)}(\textbf{a}, \textbf{b};x,y)\nonumber \\{} & {} \quad -\partial _{y_{i}} H_t^{(i_1, \dots ,i_{m-1}, i_{m+1}, \dots , i_n)}(\textbf{a}, \textbf{b};x,y), \end{aligned}$$

(B.2)

$$\begin{aligned}{} & {} \partial _{y_i} H_t^{(i_1, \dots ,i_{n})}(\textbf{a}, \textbf{b};x,y)\nonumber \\{} & {} \quad = {1\over t}\sum _{m=1}^n \textbf{b}_{i_mi}^{-1}H_t^{(i_1, \dots ,i_{m-1}, i_{m+1},\dots , i_{n})}(\textbf{a}, \textbf{b};x,y), \end{aligned}$$

(B.3)

which can be derived either directly or using (B.1) and the well known identities for canonical Hermite polynomials:

$$\begin{aligned} \mathscr {H}_t^{(i_1, \dots ,i_{m-1},i, i_{m+1}, \dots , i_n)}(w)= & {} {1\over t}w_i \mathscr {H}_t^{(i_1, \dots ,i_{m-1}, i_{m+1}, \dots , i_n)}(w)-\partial _{w_i} \mathscr {H}_t^{(i_1, \dots ,i_{m-1}, i_{m+1}, \dots , i_n)}(w),\\ \partial _{w_i} \mathscr {H}_t^{(i_1, \dots ,i_{n})}(w)= & {} {1\over t}\sum _{m=1}^n 1_{i_m=i}\mathscr {H}_t^{(i_1, \dots ,i_{m-1}, i_{m+1},\dots , i_{n})}(x,y). \end{aligned}$$

Combining (B.2) and (B.3) and re-arranging the notation for the multi-indices we get

$$\begin{aligned}{} & {} \Big [\textbf{b}^{-1}(y-x-t\textbf{a})\Big ]_{i} H_t^{(i_1, \dots , i_n)}(\textbf{a}, \textbf{b};x,y)\nonumber \\{} & {} \quad = t H_t^{(i,i_1, \dots , i_n)}(\textbf{a}, \textbf{b};x,y)+\sum _{m=1}^n \textbf{b}_{i_mi}^{-1}H_t^{(i_1, \dots ,i_{m-1}, i_{m+1},\dots , i_{n})}(\textbf{a}, \textbf{b};x,y). \end{aligned}$$

(B.4)

Multiplying (B.4) by $\textbf{b}_{j i}$ and taking the sum over i, we obtain for $n\ge 0$

$$\begin{aligned} \begin{aligned} (y-x-t\textbf{a})_j H_t^{(i_1, \dots , i_n)}(\textbf{a}, \textbf{b};x,y)&= t\sum _{i=1}^d \textbf{b}_{ji} H_t^{(i,i_1, \dots , i_n)}(\textbf{a}, \textbf{b};x,y) \\ {}&\quad +1_{n\ge 1}\sum _{m=1}^n 1_{i_m=j}H_t^{(i_1, \dots ,i_{m-1}, i_{m+1},\dots , i_{n})}(\textbf{a}, \textbf{b};x,y). \end{aligned}\nonumber \\ \end{aligned}$$

(B.5)

In calculations, it is better to simplify the notation as follows (here we use the summation over double indexes notation)

$$\begin{aligned} \begin{aligned} (y-x-t\textbf{a})_j \varPhi _t^{(i_1, \dots , i_n)}(\textbf{a}, \textbf{b};x,y)&= t \textbf{b}_{ji} \varPhi _t^{(i,i_1, \dots , i_n)}(\textbf{a}, \textbf{b};x,y)\\&\quad +1_{n\ge 1}\varPhi _t^{(i_1, \dots , i_{n})(j)}(\textbf{a}, \textbf{b};x,y). \end{aligned}\nonumber \\ \end{aligned}$$

(B.6)

In the above equation, we have also used the following index deleting operation:

$$\begin{aligned} \varPhi _t^{(i_1, \dots , i_{n})(j)}(\textbf{a}, \textbf{b};x,y)=&\sum _{m=1}^n 1_{i_m=j}\varPhi _t^{(i_1, \dots ,i_{m-1}, i_{m+1},\dots , i_{n})}(\textbf{a}, \textbf{b};x,y). \end{aligned}$$

Extensions of this operation are defined by iteration. For example,

$$\begin{aligned} \varPhi _t^{(i_1, \dots , i_n)(j,j')}(\textbf{a}, \textbf{b};x,y)=&\sum _{m=1}^n 1_{i_m=j}\varPhi _t^{(i_1, \dots ,i_{m-1}, i_{m+1},\dots , i_{n})(j')}(\textbf{a}, \textbf{b};x,y). \end{aligned}$$

Also, we will need to use the following index addition operation

$$\begin{aligned} \varPhi _t^{i(i_1, \dots , i_n)(j)}(\textbf{a}, \textbf{b};x,y)=&\sum _{m=1}^n 1_{i_m=j}\varPhi _t^{(i,i_1, \dots ,i_{m-1}, i_{m+1},\dots , i_{n})}(\textbf{a}, \textbf{b};x,y). \end{aligned}$$

We remark that the above operations with indexes have to be carried out in an order from right to left.

The Hermite functions $\varPhi _t^{(i_1, \dots , i_n)}(\textbf{a}, \textbf{b};x,y)$ also have nice properties as functions of the parameters $\textbf{a}\in {\mathbb {R}}^d, \textbf{b}\in {\mathbb {R}}^{d\times d}_\textrm{sym}$:

$$\begin{aligned} \partial _{\textbf{a}_j}\varPhi _t^{(i_1, \dots , i_n)}(\textbf{a}, \textbf{b};x,y)= & {} t\varPhi _t^{(i_1, \dots , i_n,j)}(\textbf{a}, \textbf{b};x,y), \end{aligned}$$

(B.7)

$$\begin{aligned} \partial _{\textbf{b}_{jk}}\varPhi _t^{(i_1, \dots , i_n)}(\textbf{a}, \textbf{b};x,y)= & {} {t\over 2}\varPhi _t^{(i_1, \dots , i_n,j,k)}(\textbf{a}, \textbf{b};x,y). \end{aligned}$$

(B.8)

These properties are difficult to prove using the canonical Hermite functions, thus we provide the direct proof based on the Fourier transform. Namely, let

$$\begin{aligned} {{\widehat{\varPhi }}}_t^{(i_1, \dots , i_n)}(\textbf{a}, \textbf{b};x,\lambda )=\int _{{\mathbb {R}}^d} e^{\textbf{i} y\cdot \lambda } \varPhi _t^{(i_1, \dots , i_n)}(\textbf{a}, \textbf{b};x,y)\, dy,\quad \lambda \in {\mathbb {R}}^d \end{aligned}$$

then

$$\begin{aligned} {{\widehat{\varPhi }}}_t(\textbf{a}, \textbf{b};x,\lambda )=e^{t\textbf{i}\textbf{a}\cdot \lambda -\frac{t}{2}(\textbf{b}\lambda , \lambda )} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} {{\widehat{\varPhi }}}_t^{(i_1, \dots , i_n)}(\textbf{a}, \textbf{b};x,\lambda )&=\int _{{\mathbb {R}}^d} e^{\textbf{i} y\cdot \lambda } (-\partial _{y_{i_1}})\dots (-\partial _{y_{i_n}})\varPhi _t(\textbf{a}, \textbf{b};x,y)\, dy \\ {}&=\int _{{\mathbb {R}}^d} \partial _{y_{i_1}}\dots \partial _{y_{i_n}} e^{\textbf{i} y\cdot \lambda } \varPhi _t(\textbf{a}, \textbf{b};x,y)\, dy \\ {}&=(\textbf{i}\lambda _{i_1})\dots (\textbf{i}\lambda _{i_n}){{\widehat{\varPhi }}}_t(\textbf{a}, \textbf{b};x,\lambda )=(\textbf{i}\lambda _{i_1})\dots (\textbf{i}\lambda _{i_n})e^{\textbf{i}\textbf{a}\cdot \lambda -\frac{t}{2}(\textbf{b}\lambda , \lambda )},\quad \lambda \in {\mathbb {R}}^d. \end{aligned}\nonumber \\ \end{aligned}$$

(B.9)

Then, we conclude that

$$\begin{aligned} \partial _{\textbf{b}_{jk}} {{\widehat{\varPhi }}}_t^{(i_1, \dots , i_n)}(\textbf{a}, \textbf{b};x,\lambda )= & {} -\frac{t}{2}(\textbf{i}\lambda _{i_1})\dots (\textbf{i}\lambda _{i_n})\lambda _j\lambda _k e^{\textbf{i}\textbf{a}\cdot \lambda -\frac{1}{2}(\textbf{b}\lambda , \lambda )}\\= & {} \frac{t}{2}\widehat{\varPhi }_t^{(i_1, \dots , i_n, j,k)}(\textbf{a}, \textbf{b};x,\lambda ), \end{aligned}$$

which proves (B.8). The proof of (B.7) is similar and omitted.

Finally, we mention the following convolution property of the Hermite functions: for any $t,s>0$ and $m,n\ge 0$, $i_1, \dots , i_n, j_1, \dots , j_m$

$$\begin{aligned} \Big (\varPhi _t^{(i_1, \dots , i_n)}*\varPhi _s^{(j_1, \dots , j_m)} \Big )(\textbf{a}, \textbf{b};x,y)= \varPhi _{t+s}^{(i_1, \dots , i_n,j_1, \dots , j_m)} (\textbf{a}, \textbf{b};x,y). \end{aligned}$$

(B.10)

Identity (B.10) easily follows from (B.9) and the fact that the Fourier transform of a convolution is a product.

Carrying out the multi-dimensional algorithm

Here we will give the steps carried out in order to obtain Example 3.2. In fact, recall that the existence of the residue $\varDelta _t^2(x,y)$ is of order $1+\rho /2$ is already assured by Theorem 3.1. Therefore, we will only find expansions up to order 1 for all terms. For the one dimensional case, we refer the reader to [5].

In order to simplify the expressions, we use the summation over double indexes convention.

Step 1: Consider Proposition 4.2 with $N=2$. Therefore we now need to find $\varGamma ^n_t(x,y)$ for $n=0,1,2$. Instead of this approach, one uses an iterative approach where one computes $p^{(j)}_t(x,y)$ for $j=0,1$.

Step 1.0: From the statement of Theorem 3.1 in the case $n=0$, we have $\varGamma ^0_t(x,y)=\varPhi _t(x,y)$ and $p^{(0)}_t(x,y)=\varPhi _t(x,y)$. Therefore we need to find an expansion for $\varXi _t(x,y)= \varPhi _t(y;x,y)$ around $\varPhi _t(x;x,y)\equiv \varPhi _t(x,y)$.

In order to understand how this step should be carried out it is important to note the following two rules:

• Each derivative of $\varPhi _t(\textbf{a}, \textbf{b}; x,y)$ with respect to an element of Åleads to a term which increases its t-order in 1/2 while a derivative with respect to an element of $\textbf{b}$ increases the t-order in 1.

• Then the expansion is obtained from the equality $\varPhi _t(\xi ;x,y)\!=\!\varPhi _t(a(\xi ),b(\xi ) ; x,y)$.

Carrying out these steps one obtains

$$\begin{aligned} \begin{aligned} \varPhi _t(y;x,y)-\varPhi _t(x,y)&\approx _{O(t^{3/2})} \frac{t}{2}(b_{jk}(y)-b_{jk}(x))\varPhi _t^{(j,k)}(x,y) +t(a_j(y)-a_j(x))\varPhi _t^{(j)}(x,y)\\ {}&\quad +\frac{t^2}{8}(b_{jk}(y)-b_{jk}(x))(b_{j'k'}(y)-b_{j'k'}(x))\varPhi _t^{(j,k,j',k')}(x,y)=:(*). \end{aligned}\nonumber \\ \end{aligned}$$

(C.1)

This finishes Step I.1. Now we proceed with Step I.2, which is the Taylor expansions of the coefficients. That is,

$$\begin{aligned} b_{jk}(y)-b_{jk}(x)&\approx \partial _{x_l}b_{jk}(x)(y-x)_l+\frac{1}{2}\partial _{x_lx_{l'}}b_{jk}(x)(y-x)_l(y-x)_{l'},\nonumber \\ a_j(y)-a_j(x)&\approx \partial _{x_l}a_{j}(x)(y-x)_l. \end{aligned}$$

(C.2)

From now on, in order to shorten expressions we will use the usual convention for products of functions: $(fg)(x)\equiv f(x)g(x)$

For Step I.3 we now add and subtract $-ta(x)$. Putting these results together and using (B.6), we obtain:

$$\begin{aligned}&t \partial _{x_l}a_j(x)\left( tb_{li}(x)\varPhi _t^{(i,j)}(x,y)+\varPhi _t^{(j)(l)}(x,y)\right) + \left( \partial _{x_l}a_ja_l\right) (x)t^2\varPhi _t^{(j)}(x,y)\\ {}&\quad +\partial _{x_l}b_{jk}(x)\frac{t}{2}\left( tb_{li}(x)\varPhi _t^{(i,j,k)}(x,y)+\varPhi _t^{(j,k)(l)}(x,y)+a_l(x)t\varPhi _t^{(j,k)}(x,y)\right) \\ {}&\quad +\frac{t^2}{8}\left( \partial _{x_l}b_{jk}\partial _{x_l'}b_{j'k'}\right) (x)\\ {}&\quad \times \left( t^2\left( b_{li}b_{l'i'}\right) (x)\varPhi _t^{(i,i',j,k,j',k')}(x,y)+tb_{li}(x)\varPhi _t^{(i,j,k,j',k')(l')}(x,y)\right. \\ {}&\quad \left. +tb_{l'i'}(x)\varPhi _t^{i'(j,k,j',k')(l)}(x,y)+\varPhi _t^{(j,k,j',k')(l,l')}(x,y)\right) \\ {}&\quad +\frac{t}{4} \partial _{x_l}\partial _{x_l'}b_{jk}(x) \left( t^2\left( b_{li}b_{l'i'}\right) (x)\varPhi _t^{(i,i',j,k)}(x,y)\right. \\ {}&\quad \left. +tb_{li}(x)\varPhi _t^{(i,j,k)(l')}(x,y)+tb_{l'i'}(x)\varPhi _t^{i'(j,k)(l)}(x,y)+\varPhi _t^{(j,k)(l,l')}(x,y)\right) . \end{aligned}$$

Deleting the terms of high time order, we obtain

$$\begin{aligned}&\varPhi _t(y;x,y)-\varPhi _t(x,y)\approx _{O(t^{1+\rho /2})}t \partial _{x_l}a_j(x)\left( { tb_{li}(x)\varPhi _t^{(i,j)}(x,y)+ }\varPhi _t^{(j)(l)}(x,y)\right) +\\ {}&\quad {\left( \partial _{x_l}a_ja_l\right) (x)t^2\varPhi _t^{(j)}(x,y) }\\ {}&\quad +\partial _{x_l}b_{jk}(x)\frac{t}{2}\left( tb_{li}(x)\varPhi _t^{(i,j,k)}(x,y)+\varPhi _t^{(j,k)(l)}(x,y)+a_l(x)t\varPhi _t^{(j,k)}(x,y)\right) \\ {}&\quad +\frac{t^2}{8}\left( \partial _{x_l}b_{jk}\partial _{x_l'}b_{j'k'}\right) (x)\\ {}&\quad \times \left( t^2\left( b_{li}b_{l'i'}\right) (x)\varPhi _t^{(i,i',j,k,j',k')}(x,y)+tb_{li}(x)\varPhi _t^{(i,j,k,j',k')(l')}(x,y)\right. \\ {}&\quad \left. +tb_{l'i'}(x)\varPhi _t^{i'(j,k,j',k')(l)}(x,y)+\varPhi _t^{(j,k,j',k')(l,l')}(x,y)\right) \\ {}&\quad +\frac{t}{4} \partial ^2_{x_lx_l'}b_{jk}(x) \left( t^2\left( b_{li}b_{l'i'}\right) (x)\varPhi _t^{(i,i',j,k)}(x,y)\right. \\ {}&\quad \left. +tb_{li}(x)\varPhi _t^{(i,j,k)(l')}(x,y)+tb_{l'i'}(x)\varPhi _t^{i'(j,k)(l)}(x,y)+\varPhi _t^{(j,k)(l,l')}(x,y)\right) . \end{aligned}$$

Further algebraic simplifications and arranging the terms give:

$$\begin{aligned}&\frac{t}{2}\partial _{x_l}b_{jk}(x)\varPhi _t^{(j,k)(l)}(x,y) +\frac{t^2}{2}\left( \partial _{x_l}b_{jk}b_{li}\right) (x)\varPhi _t^{(i,j,k)}(x,y)+\frac{t}{4} \partial ^2_{x_lx_l'}b_{jk}(x) \varPhi _t^{(j,k)(l,l')}(x,y)\\ {}&+t \partial _{x_l}a_j(x)\varPhi _t^{(j)(l)}(x,y)+\frac{t^2}{2} \left( \partial _{x_l}b_{jk} a_l\right) (x)\varPhi _t^{(j,k)}(x,y)+t^2 \left( \partial _{x_l}a_j b_{li}\right) (x)\varPhi _t^{(i,j)}(x,y) \\ {}&+\frac{t^2}{4} \partial ^2_{x_lx_l'}b_{jk}(x) \left( b_{li}(x)\varPhi _t^{(i,j,k)(l')}(x,y)+b_{l'i'}(x)\varPhi _t^{i'(j,k)(l)}(x,y)\right) \\ {}&+\frac{t^2}{8} \left( \partial _{x_l}b_{jk}\partial _{x_l'}b_{j'k'}\right) (x) \varPhi _t^{(j,k,j',k')(l,l')}(x,y)+\frac{t^3}{4} \left( \partial ^2_{x_lx_l'}b_{jk} b_{li}b_{l'i'}\right) (x)\varPhi _t^{(i,i',j,k)}(x,y)\\ {}&+\frac{t^2}{8} \left( \partial _{x_l}b_{jk}\partial _{x_l'}b_{j'k'}\right) (x)\left( tb_{li}(x)\varPhi _t^{(i,j,k,j',k')(l')}(x,y)+tb_{l'i'}(x)\varPhi _t^{i'(j,k,j',k')(l)}(x,y)\right) \\ {}&+\frac{t^4}{8} \left( \partial _{x_l}b_{jk}\partial _{x_l'}b_{j'k'}b_{li}b_{l'i'}\right) (x)\varPhi _t^{(i,i',j,k,j',k')}(x,y). \end{aligned}$$

Now, we need to find the expansion of t-order 1/2 for $\left( \varXi \circledast \Upsilon \right) _t(x,y)$. For this, given Lemma 4.1 and Proposition 4.1, we just need to compute the terms of this order in

$$\begin{aligned} (\varPhi \circledast \Upsilon ^{\textrm{diffusion}})_t(x,y). \end{aligned}$$

We start with the expansion of $\Upsilon ^{\textrm{diffusion}}$. Applying Taylor’s expansion of b at y, B.6 and (C.2), we have

$$\begin{aligned}&\frac{1}{2}\left( b_{jk}(z)-b_{jk}(y)\right) \varPhi _s^{(j,k)}(y;z,y)\\&\quad \approx _{O(t^{\rho /2})}\frac{1}{2}\partial _{x_l} b_{jk}(y)\left( b_{li}(y)s\varPhi _s^{(i,j,k)}(x;z,y)+\varPhi _s^{(j,k)(l)}(x;z,y)\right) . \end{aligned}$$

After integration in $\circledast$ and we obtain that

$$\begin{aligned} (\varPhi \circledast \Upsilon ^{\textrm{diffusion}})_t(x,y)\approx _{O(t^{1/2+\rho /2})}-\frac{1}{2}\partial _{x_l} b_{jk}(x)\left( b_{li}(x)\frac{t^2}{2}\varPhi _t^{(i,j,k)}(x,y)+t\varPhi _t^{(j,k)(l)}(x,y)\right) \end{aligned}$$

Putting this together with the result of Step I.3, we have

$$\begin{aligned} p^{(1)}_t(x,y)=&p^{(0)}_t(x,y)+\frac{t^2}{4}\partial _{x_l}b_{jk}b_{li}(x)\varPhi _t^{(i,j,k)}(x,y). \end{aligned}$$

Step 1.1: We need to find the expansions described in Part II. We start with Step II.1:

$$\begin{aligned} \begin{aligned} p^{(1)}_{t-s}(x,z)\Upsilon _s(z,y)&\approx _{O(t^{\rho /2})}p^{(0)}_{t-s}(x,z)\Upsilon ^{\textrm{drift}}_s(z,y)+p^{(1)}_{t-s}(x,z)\Upsilon ^{\textrm{diffusion}}_s(z,y) \\ {}&=\varPhi _{t-s}(x,z)(a_j(z)-a_j(y))\varPhi _s^{(j)}(y; z,y) \\ {}&\quad +\frac{1}{2}\varPhi _{t-s}(x,z)(b_{jk}(z)-b_{jk}(y))\varPhi _s^{(j,k)}(y; z,y) \\ {}&\quad +\frac{(t-s)^2}{8} \left( \partial _{x_l}b_{jk}b_{li}\right) (x)\varPhi _t^{(i,j,k)}(x,z)(b_{j'k'}(z)\\&\quad -b_{j'k'}(y))\varPhi _s^{(j',k')}(y; z,y)=:(\circledast ). \end{aligned} \end{aligned}$$

Repeating similar steps as in the one dimensional case, one obtains at the end of Step II.2:

$$\begin{aligned} (\circledast )\approx _{O(t^{\rho /2})}&-\frac{1}{2}\partial _{x_l}b_{jk}(x)\varPhi _{t-s}(x,z)(y-z)_l\varPhi _s^{(j,k)}(x; z,y) \\ {}&-\frac{1}{2}\partial ^2_{x_lx_{l'}}b_{jk}(x)\varPhi _{t-s}(x,z)(z-x)_{l'}(y-z)_l\varPhi _s^{(j,k)}(x; z,y) \\ {}&-\frac{1}{4}\partial ^2_{x_lx_{l'}}b_{jk}(x)\varPhi _{t-s}(x,z)(y-z)_l(y-z)_{l'}\varPhi _s^{(j,k)}(x; z,y) \\ {}&-\partial _{x_l}a(x)\varPhi _{t-s}(x,z)(y-z)_l\varPhi _s^{(j)}(x; z,y) \\ {}&-\frac{s}{4} \left( \partial _{x_l}b_{jk}\partial _{x_{l'}}b_{j'k'}\right) (x)\varPhi _{t-s}(x,z)(y-z)_l(z-x)_{l'}\varPhi _s^{(j,k,j',k')}(x; z,y) \\ {}&-\frac{s}{4} \left( \partial _{x_l}b_{jk}\partial _{x_{l'}}b_{j'k'}\right) (x)\varPhi _{t-s}(x,z)(y-z)_l(y-z)_{l'}\varPhi _s^{(j,k,j',k')}(x; z,y) \\ {}&-\frac{(t-s)^2}{8} \left( b_{li}\partial _{x_l}b_{jk}\right) (x)\partial _{x_{l'}}b_{j'k'}(x)\varPhi _{t-s}^{(i,j,k)}(x,z)(y-z)\varPhi _s^{(j',k')}(x; z,y)=:(\circledast \circledast ). \end{aligned}$$

Step II.3. gives

$$\begin{aligned}&(\circledast \circledast )\approx _{O(t^{1/2})} -\frac{1}{2} \left( b_{li}\partial _{x_l}b_{jk}\right) (x)\varPhi _{t-s}(x,z)\left( s\varPhi _s^{(i,j,k)}(x; z,y)+\varPhi _s^{(j,k)(l)}(x; z,y)\right) \\ {}&\quad -\frac{s}{2} \left( \partial _{x_l}b_{jk}a_l\right) (x)\varPhi _{t-s}(x,z)\varPhi _s^{(j,k)}(x; z,y) \\ {}&\quad -\frac{(t-s)}{2} \left( b_{l'i'}\partial ^2_{x_lx_{l'}}b_{jk}\right) (x)\varPhi _{t-s}^{(i')}(x,z)\left( sb_{li}(x)\varPhi _s^{(i,j,k)}(x; z,y)+\varPhi _s^{(j,k)(l)}(x; z,y)\right) \\ {}&\quad -\frac{1}{4}\partial ^2_{x_lx_{l'}}b_{jk}(x)\varPhi _{t-s}(x,z)\\ {}&\quad \times \left( s^2 \left( b_{li}b_{l'i'}\right) (x)\varPhi _s^{(i,i',j,k)}(x; z,y)+sb_{li}(x)\varPhi _s^{(i,j,k)(l')}(x; z,y)\right. \\ {}&\quad \left. +sb_{l'i'}(x)\varPhi _s^{i'(j,k)(l)}(x; z,y)+\varPhi _s^{(j,k)(l,l')}(x; z,y)\right) \\ {}&\quad -\partial _{x_l}a_j(x)\varPhi _{t-s}(x,z)\left( sb_{li}(x)\varPhi _s^{(i,j)}(x; z,y)+\varPhi _s^{(j)(l)}(x; z,y)\right) \\ {}&\quad -\frac{(t-s)s}{4} \left( b_{l'i'} \partial _{x_l}b_{jk}\partial _{x_{l'}}b_{j'k'}\right) (x)\varPhi _{t-s}^{(i')}(x,z) \\ {}&\quad \left( sb_{li}(x)\varPhi _s^{(i,j,k,j',k')}(x; z,y)+\varPhi _s^{(j,k,j',k')(l)}(x; z,y)\right) \\ {}&\quad -\frac{s}{4} \left( \partial _{x_l}b_{jk}\partial _{x_{l'}}b_{j'k'}\right) (x)\varPhi _{t-s}(x,z)\Big (s^2 \left( b_{li}b_{l'i'}\right) (x)\varPhi _s^{(i,i',j,k,j',k')}(x; z,y)\\ {}&\quad +sb_{li}(x)\varPhi _s^{(i,j,k,j',k')(l')}(x; z,y)\\ {}&\quad +sb_{l'i'}(x) \varPhi _s^{i'(j,k,j',k')(l)}(x; z,y)+\varPhi _s^{(j,k,j',k')(l,l')}(x; z,y) \Big ) \\ {}&\quad - \frac{(t-s)^2}{8} \left( b_{li} \partial _{x_l}b_{jk}\partial _{x_{l'}}b_{j'k'}\right) (x)\varPhi _{t-s}^{(i,j,k)}(x,z)\\ {}&\quad \left( sb_{l'i'}(x)\varPhi _s^{(i',j',k')}(x; z,y)+\varPhi _s^{(j',k')(l')}(x; z,y)\right) =:(\circledast \circledast \circledast ). \end{aligned}$$

Now we integrate in the time and space variables (z) to obtain that

$$\begin{aligned}&(p^{(1)}\circledast \Upsilon )_t(x,y)\approx _{O(t^{1+\rho /2})} -\frac{1}{2} \left( b_{li}\partial _{x_l}b_{jk}\right) (x)\left( \frac{t^2}{2}\varPhi _t^{(i,j,k)}(x,y)+t\varPhi _t^{(j,k)(l)}(x,y)\right) \\ {}&\quad -\frac{t^2}{4} \left( \partial _{x_l}b_{jk}a_l\right) (x)\varPhi _t^{(j,k)}(x,y) \\ {}&\quad -\frac{1}{2} \left( b_{l'i'}\partial ^2_{x_lx_{l'}}b_{jk}\right) (x)\left( \frac{t^3}{6}b_{li}(x)\varPhi _t^{(i',i,j,k)}(x,y)+\frac{t^2}{2}\varPhi _t^{i'(j,k)(l)}(x,y)\right) \\ {}&\quad -\frac{1}{4}\partial ^2_{x_lx_{l'}}b_{jk}(x)\\ {}&\quad \times \left( \frac{t^3}{3} \left( b_{li}b_{l'i'}\right) (x)\varPhi _t^{(i,i',j,k)}(x,y)+\frac{t^2}{2}b_{li}(x)\varPhi _t^{(i,j,k)(l')}(x,y)\right. \\ {}&\quad \left. +\frac{t^2}{2}b_{l'i'}(x)\varPhi _t^{i'(j,k)(l)}(x,y)+t\varPhi _t^{(j,k)(l,l')}(x,y)\right) \\ {}&\quad -\partial _{x_l}a_j(x)\left( \frac{t^2}{2}b_{li}(x)\varPhi _t^{(i,j)}(x,y)+t\varPhi _t^{(j)(l)}(x,y)\right) \\ {}&\quad -\frac{1}{4} \left( b_{l'i'} \partial _{x_l}b_{jk}\partial _{x_{l'}}b_{j'k'}\right) (x) \left( \frac{t^4}{12}b_{li}(x)\varPhi _t^{(i',i,j,k,j',k')}(x,y)+\frac{t^3}{6}\varPhi _s^{i'(j,k,j',k')(l)}(x,y)\right) \\ {}&\quad -\frac{1}{4} \left( \partial _{x_l}b_{jk}\partial _{x_{l'}}b_{j'k'}\right) (x)\left( \frac{t^4}{4} \left( b_{li}b_{l'i'}\right) (x)\varPhi _t^{(i,i',j,k,j',k')}(x,y)+\frac{t^3}{3}b_{li}(x)\varPhi _t^{(i,j,k,j',k')(l')}(x,y) \right. \\ {}&\quad \left. \quad +\frac{t^3}{3}b_{l'i'}(x) \varPhi _t^{i'(j,k,j',k')(l)}(x,y)+\frac{t^2}{2}\varPhi _t^{(j,k,j',k')(l,l')}(x,y) \right) \\ {}&\quad - \frac{1}{8} \left( b_{li} \partial _{x_l}b_{jk}\partial _{x_{l'}}b_{j'k'}\right) (x) \left( \frac{t^4}{12}b_{l'i'}(x)\varPhi _t^{(i,j,k,i',j',k')}(x,y)+\frac{t^3}{3}\varPhi _t^{i,j,k(j',k')(l')}(x,y)\right) . \end{aligned}$$

Further algebraic simplifications give:

$$\begin{aligned}&-\frac{t}{2}b_{li}(x)\varPhi _t^{(j,k)(l)}(x,y) -\frac{t^2}{4} \left( b_{li}\partial _{x_l}b_{jk}\right) (x)\varPhi _t^{(i,j,k)}(x,y)-t\partial _{x_l}a_j(x)\varPhi _t^{(j)(l)}(x,y)\\ {}&\quad -\frac{t}{4}\partial ^2_{x_lx_{l'}}b_{jk}(x) \varPhi _t^{(j,k)(l,l')}(x,y)-\frac{t^2}{4} \left( \partial _{x_l}b_{jk}a_l\right) (x)\varPhi _t^{(j,k)}(x,y)\\ {}&\quad -\frac{t^2}{2} \left( \partial _{x_l}a_jb_{li}\right) (x)\varPhi _t^{(i,j)}(x,y) -\frac{t^2}{8} \left( \partial _{x_l}b_{jk}\partial _{x_{l'}}b_{j'k'}\right) (x)\varPhi _t^{(j,k,j',k')(l,l')}(x,y) \\ {}&\quad -\frac{t^2}{8} \partial ^2_{x_lx_{l'}}b_{jk}(x)\left( b_{li}(x)\varPhi _t^{(i,j,k)(l')}(x,y)+3b_{l'i'}(x)\varPhi _t^{i'(j,k)(l)}(x,y)\right) \\ {}&\quad -\frac{t^3}{6}\left( \partial ^2_{x_lx_{l'}}b_{jk}b_{li}b_{l'i'}\right) (x)\varPhi _t^{(i,i',j,k)}(x,y)-\frac{t^3}{24} \left( \partial _{x_l}b_{jk}\partial _{x_{l'}}b_{j'k'}\right) (x)\\ {}&\quad \times \left( 2b_{li}(x)\varPhi _t^{(i,j,k,j',k')(l')}(x,y)+ 2b_{l'i'}(x)\varPhi _t^{i'(j,k,j',k')(l)}(x,y)\right. \\ {}&\quad \left. +b_{l'i'}(x)\varPhi _t^{i(j,k,j',k')(l)}(x,y) +b_{li}(x)\varPhi _t^{i,j,k(j',k')(l')}(x,y)\right) \\ {}&\quad -\frac{3t^4}{32} \left( \partial _{x_l}b_{jk}\partial _{x_{l'}}b_{j'k'}b_{li}b_{l'i'}\right) (x)\varPhi _t^{(i,i',j,k,j',k')}(x,y). \end{aligned}$$

As in the one dimensional case, adding the above result to the expansion of $\varXi _t(x,y)= \varPhi _t(y;x,y)$ around $\varPhi _t(x;x,y)\equiv \varPhi _t(x,y)$ obtained in $\mathbf{Step\ I.3}$, we have

$$\begin{aligned} p_t(x,y)&\approx _{O(t^{1+\rho /2})}\varPhi _t(x,y)+\frac{t^2}{4} \left( \partial _{x_l}b_{jk}b_{li}\right) (x)\varPhi _t^{(i,j,k)}(x,y)\\ {}&\quad + \frac{t^2}{4} \left( \partial _{x_l}b_{jk} a_l\right) (x)\varPhi _t^{(j,k)}(x,y)\\ {}&\quad +\frac{t^2}{2} \left( \partial _{x_l}a_j b_{li}\right) (x)\varPhi _t^{(i,j)}(x,y) +\frac{t^2}{8} \partial _{x_l}\partial _{x_l'}b_{jk}(x)\\ {}&\quad \left( b_{li}(x)\varPhi _t^{(i,j,k)(l')}(x,y)-b_{l'i'}(x)\varPhi _t^{i'(j,k)(l)}(x,y)\right) \\ {}&\quad +\frac{t^3}{12} \left( \partial ^2_{x_lx_l'}b_{jk} b_{li}b_{l'i'}\right) (x)\varPhi _t^{(i,i',j,k)}(x,y) +\frac{t^3}{24} \left( \partial _{x_l}b_{jk}\partial _{x_{l'}}b_{j'k'}\right) (x)\\ {}&\quad \times \left( b_{li}(x)\varPhi _t^{(i,j,k,j',k')(l')}(x,y)+ b_{l'i'}(x)\varPhi _t^{i'(j,k,j',k')(l)}(x,y)\right. \\ {}&\quad \left. -b_{l'i'}(x)\varPhi _t^{i(j,k,j',k')(l)}(x,y) -b_{li}(x)\varPhi _t^{i,j,k(j',k')(l')}(x,y)\right) \\ {}&\quad +\frac{t^4}{32}\left( \partial _{x_l}b_{jk}\partial _{x_l'}b_{j'k'}b_{li}b_{l'i'}\right) (x)\varPhi _t^{(i,i',j,k,j',k')}(x,y). \end{aligned}$$

Remark C.1

Note that a natural approach with lower orders of expansion may be obtained as follows. In order to write an expansion of order $1+\rho /2$, we need to use Lemma 4.1 and Proposition 4.1 to see that

• 1. The t-order of $\Upsilon$ is $-1/2$ and $\Upsilon ^{\circledast {2}}$ is 0.

• 2. Therefore in order to obtain $\left( \varXi \circledast \Upsilon ^{\circledast {k}}\right) _t(x,y)$ of t-order $1+\rho /2$, we need to expand $\varXi$ up to t-order 1/2 in the case $k=1$ and t-order 0 in the case $k=2$.

This proposal can be cumbersome when general orders are required. For this reason, we prefer the iterative scheme.