Higher order regularity of nonlinear Fokker-Planck PDEs with respect to the measure component

TL;DR

This paper derives a general formula for higher order derivatives of functionals on the Wasserstein space, applied to solutions of nonlinear Fokker-Planck PDEs, with implications for mean-field games and propagation of chaos.

Contribution

It introduces a new formula for higher order derivatives of functionals composed with Fokker-Planck PDE solutions, advancing the understanding of measure-dependent PDEs.

Findings

Derived a general formula for higher order linear functional derivatives.

Established connections with propagation of chaos and mean-field game theory.

Provides tools for analyzing regularity of nonlinear measure-valued PDEs.

Abstract

In this article, we establish a general formula for higher order linear functional derivatives for the composition of an arbitrary smooth functional on the 1-Wasserstein space with the solution of a Fokker-Planck PDE. This formula has important links with the theory of propagation of chaos and mean-field games.

Equations

$$\{\begin{array}{l}{\partial }_{t}m+\text{div}(b(\cdot ,m)m)-\Delta m=0,t\in [0,T],\\ m(0,\mu )=\mu ,\end{array}$$

$$\mathcal{U}(t,\mu ):=\Phi (m(t,\mu )),$$

$$\mathcal{V}(m^{\prime })-\mathcal{V}(m)={\int }_0^1{\int }_{{\mathbb{T}}^d}\frac{\delta \mathcal{V}}{\delta m}((1-s)m+s{m}^{\prime },y)(m^{\prime }-m)(dy)ds.$$

$$\frac{{\delta }^{p-1}\mathcal{V}}{\delta m^{p-1}}(m^{\prime },y)-\frac{{\delta }^{p-1}\mathcal{V}}{\delta m^{p-1}}(m,y)={\int }_0^1{\int }_{{\mathbb{T}}^d}\frac{{\delta }^p\mathcal{V}}{\delta m^p}((1-s)m+s{m}^{\prime },y,y^{\prime })(m^{\prime }-m)(d{y}^{\prime })ds,$$

$${\int }_{{\mathbb{T}}^d}\frac{{\delta }^p\mathcal{V}}{\delta m^p}(m,y_1,\dots ,y_p)m(d{y}_i)=0,i\in \{1,\dots ,p\}.$$

$$\frac{{\delta }^k\mathcal{U}}{\delta m^k}(t,\mu )(z_1,\dots ,z_k)$$

$$\sup_{z_{1},\ldots,z_{k}\in\mathbb{T}^{d}}\sup_{\mu\in\mathcal{P}(\mathbb{T}^{d})}\sup_{t\in[0,T]}\bigg{|}\frac{\delta^{k}\mathcal{U}}{\delta m^{k}}(t,\mu)(z_{1},\ldots,z_{k})\bigg{|}<+\infty.$$

$$\{\begin{array}{l}{X}_t^{0,\eta }=\eta +{\int }_0^{t}b(X_s^{0,\eta },\mathcal{L}(X_s^{0,\eta }))ds+\sqrt{2}{W}_t,\\ \mathcal{L}(\eta )=\mu .\end{array}$$

$$m(s,\mu )=\mathcal{L}(X_s^{0,\eta }).$$

$$\begin{cases}Y^{i,N}_{t}=\eta_{i}+\int_{0}^{t}b\big{(}Y^{i,N}_{s},\mu^{N}_{s}\big{)}\,ds+\sqrt{2}W^{i}_{t},\quad 1\leq i\leq N,\quad t\in[0,T],\\ \mu^{N}_{s}:=\frac{1}{N}\sum_{i=1}^{N}\delta_{Y^{i,N}_{s}},\end{cases}$$

$$\mathbb{E}[\Phi(\mu^{N}_{T})]-\Phi({\mathscr{L}}{{(X^{0,\eta}_{T})}})=\frac{1}{N}\int_{0}^{T}\mathbb{E}\bigg{[}\int_{\mathbb{T}^{d}}\text{Tr}\bigg{(}\partial_{y_{1}}\partial_{y_{2}}\frac{\delta^{2}\mathcal{U}}{\delta m^{2}}(T-s,\mu^{N}_{s})(z,z)\bigg{)}\,\mu^{N}_{s}(dz)\bigg{]}\,ds.$$

$$\mathbb{E}[\Phi ({\mu }_T^N)]-\Phi (\mathcal{L}(X_T^{0,\eta }))=\sum _{j=1}^{k-1}\frac{C_j}{N^j}+O(\frac{1}{N^k}),$$

$${\partial }_{z_1}\dots {\partial }_{z_k}\frac{{\delta }^k\mathcal{U}}{\delta m^k}(t,\mu )(z_1,\dots ,z_k)$$

$$\displaystyle\int_{\mathbb{T}^{d}}\phi(t,x)\big{(}m(t,(1-\epsilon)\mu+\epsilon\hat{\mu})\big{)}(dx)-\int_{\mathbb{T}^{d}}\phi(0,x)\big{(}(1-\epsilon)\mu+\epsilon\hat{\mu}\big{)}(dx)$$

$$m^{(1)}(s,\mu,\hat{\mu}):=\frac{d}{d\epsilon}\bigg{|}_{\epsilon=0^{+}}m(s,(1-\epsilon)\mu+\epsilon\hat{\mu})$$

$$\frac{d}{d\epsilon}\bigg{|}_{\epsilon=0^{+}}\Phi\big{(}m(s,(1-\epsilon)\mu+\epsilon\hat{\mu})\big{)}=\int_{\mathbb{T}^{d}}\frac{\delta\Phi}{\delta m}(m(s,\mu))(y)\,m^{(1)}(s,\mu,\hat{\mu})(dy).$$

$$\frac{d}{d\epsilon}\bigg{|}_{\epsilon=0^{+}}\Phi\big{(}(1-\epsilon)\mu+\epsilon\hat{\mu}\big{)}=\int_{\mathbb{T}^{d}}\frac{\delta\Phi}{\delta m}(\mu)(y)\,(\hat{\mu}-\mu)(dy),$$

$${\int }_{{\mathbb{T}}^d}\varphi (t,y)m^{(1)}(t,\mu ,\hat{\mu })(dy)-{\int }_{{\mathbb{T}}^d}\varphi (0,y)m^{(1)}(0,\mu ,\hat{\mu })(dy)$$

$$\left\{\begin{array}[]{rrl}{}\hfil&\partial_{t}m^{(1)}(t,\mu,\hat{\mu})+\text{div}(b(\cdot,m(t,\mu))m^{(1)}(t,\mu,\hat{\mu}))&\\ &+\text{div}\big{(}m(t,\mu)\frac{\delta b}{\delta m}(\cdot,m(t,\mu))(m^{(1)}(t,\mu,\hat{\mu}))\big{)}-\Delta m^{(1)}(t,\mu,\hat{\mu})&=0,\\ &&\\ {}\hfil&m^{(1)}(0,\mu,\hat{\mu})&=\hat{\mu}-\mu.\\ \end{array}\right.$$

$$\underset{t\in [0,T]}{\sup }\Vert m(t,\hat{\mu })-m(t,\mu )-m^{(1)}(t,\mu ,\hat{\mu }){\Vert }_{-(n,\infty )}\le C{W}_1(\mu ,\hat{\mu }{)}^2,$$

$$\frac{d}{d\epsilon}\bigg{|}_{\epsilon=0^{+}}\Phi\big{(}m(t,(1-\epsilon)\mu+\epsilon\hat{\mu})\big{)}=\int_{\mathbb{T}^{d}}\frac{\delta\Phi}{\delta m}(m(t,\mu))(y)\,m^{(1)}(t,\mu,\hat{\mu})(dy).$$

$$X_s^{0,x,\mu }=x+{\int }_0^{s}b(X_r^{0,x,\mu },m(r,\mu ))dr+\sqrt{2}{W}_s,0\le s\le t.$$

$$v(s,x,\mu;\xi,t):=\mathbb{E}\big{[}\xi(X^{0,x,\mu}_{t})\big{|}X^{0,x,\mu}_{s}=x\big{]},$$

$$\{\begin{array}{l}{\partial }_{s}v(s,x,\mu )+b(x,m(s,\mu ))\cdot \nabla v(s,x,\mu )+\Delta v(s,x,\mu )=0,\\ v(t,x,\mu )=\xi (x).\end{array}$$

$$v(0,x,\mu;\xi,t)=\mathbb{E}\big{[}\xi(X^{0,x,\mu}_{t})\big{]}$$

$${\int }_{{\mathbb{T}}^d}\xi (x)m(t,\mu )(dx)=\mathbb{E}[\xi (X_t^{0,\mu })]={\int }_{{\mathbb{T}}^d}v(0,x,\mu ;\xi ,t)\mu (dx).$$

$$\int_{\mathbb{T}^{d}}\xi(x)\,m^{(1)}(t,\mu,\hat{\mu})(dx)=\int_{\mathbb{T}^{d}}\bigg{[}v(0,x,\mu;\xi,t)+\int_{\mathbb{T}^{d}}\frac{\delta v}{\delta m}(0,z,\mu,x;\xi,t)\,\mu(dz)\bigg{]}(\hat{\mu}-\mu)(dx).$$

$$\Phi (\mu )={\int }_{{\mathbb{R}}^d}\zeta (y)\mu (dy),$$

$$b(x,\mu)=\varphi_{2}\bigg{(}x,\int_{\mathbb{R}^{d}}\varphi_{1}(y)\mu(dy)\bigg{)},\quad\quad\Phi(\mu)=\int_{\mathbb{R}^{d}}\zeta(y)\,\mu(dy),$$

$$W_1(\mu ,\nu ):=\underset{\pi \in \Pi (\mu ,\nu )}{\inf }{\int }_{{\mathbb{T}}^d\times {\mathbb{T}}^d}|x-y|\pi (dx,dy),$$

Full text

Résumé

Dans cet article, nous établissons une formule générale pour les dérivées fonctionnelles linéaires d’ordre supérieur pour la composition d’une fonctionnelle régulière arbitraire sur l’espace 1-Wasserstein avec la solution d’une EDP de Fokker-Planck. Cette formule a des liens importants avec la théorie de la propagation du chaos et des jeux à champ moyen.

**Keywords: Fokker-Planck PDEs, Linear functional derivatives, Propagation of chaos

**

2010 AMS subject classifications: 35R06, 60H30, 65C35

1 Introduction

Let $\mathcal{P}(\mathbb{T}^{d})$ denote the 1-Wasserstein space of probability measures on $\mathbb{T}^{d}$, where $\mathbb{T}^{d}:=\mathbb{R}^{d}/\mathbb{Z}^{d}$ denotes the $d$-dimensional torus. In this paper, we consider nonlinear Fokker-Planck PDEs of the form

[TABLE]

for some function $b:\mathbb{T}^{d}\times\mathcal{P}(\mathbb{T}^{d})\to\mathbb{R}^{d}$ and probability measure $\mu\in\mathcal{P}(\mathbb{T}^{d})$. This type of equations has been a rich area of research in the last decades. The case in which $b$ does not depend on $m$ has been treated in most classical works, such as Chapter 6 of [3]. In [1], this type of equations is considered to construct weak solutions to a class of distribution-dependent SDEs. The case corresponding to probability measures on the path space is considered in [19].

Let $\Phi:\mathcal{P}(\mathbb{T}^{d})\to\mathbb{R}$ be a continuous function (w.r.t. the topology of $\mathcal{P}(\mathbb{T}^{d})$). This paper explores the smoothness w.r.t. the measure component for function $\mathcal{U}:[0,T]\times\mathcal{P}(\mathbb{T}^{d})\to\mathbb{R}$ defined by

[TABLE]

under sufficient regularity of $b$ and $\Phi$. The notion of smoothness that we consider, i.e. the linear functional derivative, is widely adopted in the literature of McKean-Vlasov equations and mean-field games, such as [8], [9] and [12]. A continuous function (w.r.t. the product topology of $\mathcal{P}(\mathbb{T}^{d})\times\mathbb{T}^{d}$) $\frac{\delta\mathcal{V}}{\delta m}:\mathcal{P}(\mathbb{T}^{d})\times\mathbb{T}^{d}\to\mathbb{R}$ is said to be the linear functional derivative of $\mathcal{V}:\mathcal{P}(\mathbb{T}^{d})\to\mathbb{R}$, if for any $m,m^{\prime}\in\mathcal{P}(\mathbb{T}^{d})$,

[TABLE]

We then introduce higher-order derivatives through iteration: for any $m,m^{\prime}\in\mathcal{P}(\mathbb{T}^{d})$ and $y\in(\mathbb{T}^{d})^{p-1}$,

[TABLE]

provided that the $(p-1)$-th order derivative is well defined. These derivatives are defined up to an additive constant via (1.3) and (1.4). They are normalised by the convention

[TABLE]

The main result of this paper is Theorem 4.5. The definitions of the assumptions are found in Section 1.4.2. The definitions of the higher-order Kolmogorov equations $m^{(\beta)}$ and the multi-indices $\Lambda\in e(\Lambda_{k})$ can be found in (3.4) and Definitions 4.1- 4.3 respectively.

Theorem (Main result)****.

Let $k\in\mathbb{N}$. Assume (Int-$b$-(${k+2,k}$)), (Lip-$b$-(${k+1,k}$)), (TLip-$\Phi$-(${k}$)) and (TReg-$\Phi$-(${k+2,k}$)). Then $\frac{\delta^{k}\mathcal{U}}{\delta m^{k}}$ exists and is given by

[TABLE]

In particular, if we also assume (TInt-$\Phi$-(${k+1,k}$)), then

[TABLE]

1.1 Links of the main result with the theory of quantitative propagation of chaos

This result has intricate links with the theory of McKean-Vlasov stochastic differential equations (MVSDEs) and mean-field optimal control. Let us consider a probability space $(\Omega,\mathcal{F},\mathbb{P})$ equipped with a $d$-dimensional Brownian motion $W$. Denoting the law of random variable $\eta$ by ${\mathscr{L}}{{(\eta)}}$, we consider a $d$-dimensional MVSDE given by

[TABLE]

Lipschitz condition on $b$ ensures uniqueness of the solution to (1.7) ([30]) and it can be easily checked that in this case

[TABLE]

MVSDEs provide a probabilistic representation to the solutions of a class of nonlinear PDEs. A particular example of such nonlinear PDEs was first studied by McKean ([27]). These equations describe the limiting behaviour of an individual particle evolving within a large system of particles undergoing diffusive motion and interacting in a ‘mean-field’ sense, as the population size grows to infinity. More precisely, we consider the following system of particles,

[TABLE]

where $W^{i},$ $1\leq i\leq N$, are independent $d$-dimensional Brownian motions and $\eta_{i},$ $1\leq i\leq N$, are i.i.d. random variables with the same distribution as $\eta$. A particular characteristic of the limiting behaviour of the system, is that any finite subset of particles becomes asymptotically independent of each other. This phenomenon is known as propagation of chaos. We refer the reader to [17, 28, 30] for the classical results in this direction and to [6, 15, 21, 24, 29] for an account (non-exhaustive) of recent results. Nonetheless, most results are only qualitative and do not give us a rate of convergence.

For deterministic $\eta=c\in\mathbb{R}^{d}$, it is shown in [11] that under sufficient regularity of $b$ and $\Phi$, the weak error between the particle system (1.8) and its mean-field limit (1.7) is given by

[TABLE]

(A more complicated formula is also given in [11] for non-deterministic initial conditions.) To obtain a full expansion of the form

[TABLE]

for some positive constants $C_{1},\ldots,C_{k-1}$ that do not depend on $N$, one would even need to consider higher order linear derivatives $\frac{\delta^{k}\mathcal{U}}{\delta m^{k}}$ (see [11]).

Note that in most practical applications, the test function $\Phi$ being considered is linear, therefore its linear derivatives have simple closed-form formulae. In this case, the advantage of (LABEL:k_order_full_formula) is that it expresses $\frac{\delta^{k}\mathcal{U}}{\delta m^{k}}$ completely in terms of higher order Kolmogorov equations $m^{(\beta)}$, which are intrinsically Cauchy problems.

Despite being out of the scope of this paper, we remark that it is not difficult to compute the expression for

[TABLE]

by perturbing each of the measures $\mu_{1},\ldots,\mu_{\beta}$ in $m^{(\beta)}(t,\mu,\mu_{1},\ldots,\mu_{\beta})$. This is much simpler than the linearisation procedure performed in this paper, where we perturb measure $\mu$, which is more cumbersome and technical. Through more sophisticated techniques of global Schauder estimates, it should even be possible to obtain a control of (1.10) that decays over time $t$, which allows us to obtain a uniform estimate of propagation of chaos in $T$, by (1.9). This is a closely related research direction.

1.2 Main method of proof in this paper

The main idea of proof comes from [8], based on their idea of ‘linearising’ a forward-backward mean-field game system by perturbating the measure component. Our strategy follows a similar argument as Proposition 3.4.3 and Corollary 3.4.4 in [8].

To explore regularity of (1.1) along the measure component, we perturb probability measure $\mu\in\mathcal{P}(\mathbb{T}^{d})$ along direction $\hat{\mu}\in\mathcal{P}(\mathbb{T}^{d})$. Take any smooth test function $\phi:[0,T]\times\mathbb{T}^{d}\to\mathbb{R}$. We have

[TABLE]

We define

[TABLE]

in the sense of distributions. Then one should expect that

[TABLE]

(In particular, for the linear case when $m(s,\mu)=\mu$, we have

[TABLE]

which is a consequence of the definition of the linear functional derivative.) Applying (1.12) to (1.11), by differentiating (1.11) w.r.t. $\epsilon$ at [math], we have

[TABLE]

Note that, in the distribution sense, (1.13) can be rewritten as the linearised forward Kolmogorov equation

[TABLE]

This is what we expect by differentiating (1.1) formally in $m$. To show that this is indeed the case, we consider the difference $m(t,\hat{\mu})-m(t,\mu)-m^{(1)}(t,\mu,\hat{\mu})$ to prove differentiability of $m$ with respect to the measure.

We adopt the approach of Schauder theory and most of the results follow from Theorem 2.2, which is a fundamental result of Schauder estimates on the viscous transport equation. Based on Schauder theory, it is shown in Theorem 2.6 that there exists some constant $C>0$ such that

[TABLE]

under the assumptions (Int-$b$-(${n,1}$)), (Lip-$b$-(${0,1}$)), (TLip-$\Phi$-(${1}$)) and (TReg-$\Phi$-(${n,1}$)), where $n\geq 2$. Therefore, we can show that

[TABLE]

Nonetheless, to show that $\mathcal{U}$ indeed has a linear functional derivative, we need to express the integral on the right hand side in terms of the signed measure $\hat{\mu}-\mu$. Here is where probability theory comes into action. For every $t\in[0,T]$ and $x\in\mathbb{R}^{d}$, we consider the decoupled process $\{X^{0,x,\mu}_{u}\}_{u\in[0,t]}$ defined by

[TABLE]

For every $\xi:\mathbb{T}^{d}\to\mathbb{R}$ and $t\in[0,T]$, we define a function $v(\cdot,\cdot,\cdot;\xi,t):[0,t]\times\mathbb{T}^{d}\times\mathcal{P}(\mathbb{T}^{d})\to\mathbb{R}$ such that

[TABLE]

which satisfies the backward Kolmogorov equation

[TABLE]

Note that

[TABLE]

and therefore 111Note that if the law of $\eta_{1}$ is equal to the law of $\eta_{2}$, then the law of $X^{0,\eta_{1}}_{t}$ is also equal to the law of $X^{0,\eta_{2}}_{t}$. Therefore, if we are only interested in the law of the process $X^{0,\eta}_{t}$, where $\eta$ is distributed as $\mu$, then it is proper to adopt the notation $X^{0,\mu}_{t}$.

[TABLE]

By linearisating with respect to $\mu$ in the same way as (1.11) and (1.12), we obtain that

[TABLE]

Consequently, by replacing $\xi$ by $\frac{\delta{\Phi}}{\delta m}(m(s,\mu))(\cdot)$, we can deduce from (1.12) the existence of the first order linear derivative of $\mathcal{U}$. We repeat the same procedure for higher order linear derivatives of $\mathcal{U}$. It is precisely this combination of forward and backward equations that allows us to prove existence of the linear derivatives of $\mathcal{U}$.

1.3 Comparison with other approaches in the literature

There are various alternative methods for establishing smoothness of functions of the form (1.2) in the literature, all of which are probabilistic.

The method of Malliavin calculus is adopted in [12]. That paper proves smoothness of $\mathcal{U}$, for $\Phi$ being in the form

[TABLE]

where $\zeta:\mathbb{R}^{d}\to\mathbb{R}$ is infinitely differentiable with bounded partial derivatives.

The method of parametrix is considered in [13]. We represent $\mathcal{U}$ in terms of the transition density $p(s,\mu;t^{\prime},y^{\prime};t,y)$ of $X^{s,x,\mu}_{t}$ (defined above in (1.15)). This method is applied to the case in which $b$ and $\Phi$ are of the form

[TABLE]

for some functions $\varphi_{1}:\mathbb{R}^{d}\to\mathbb{R}$, $\varphi_{2}:\mathbb{R}^{d}\times\mathbb{R}\to\mathbb{R}^{d}$ and $\zeta:\mathbb{R}^{d}\to\mathbb{R}$. Nonetheless, it is not clear whether this method can be applied to $b$ and $\Phi$ with more general forms.

Finally, a ‘variational’ approach is adopted in [7]. The core idea is to prove smoothness of $\mathcal{U}$ by viewing the lift of $\mathcal{U}$ (i.e. the map $Y\mapsto\mathcal{U}({\mathscr{L}}{{(Y)}})$) as a composition of the map $\eta\mapsto X^{0,\eta}_{t}$ and the lift of $\Phi$ (i.e. the map $Y\mapsto\Phi({\mathscr{L}}{{(Y)}})$). In [7], the smoothness of $\mathcal{U}$ is proven up to the second order, under fairly general conditions on $b$ and $\Phi$.

1.4 Notations and main assumptions

1.4.1 Notations

The scalar product between two vectors $a,b\in\mathbb{R}^{d}$ is denoted by $a\cdot b$. $\mathcal{P}(\mathbb{T}^{d})$ denotes the space of integrable probability measures and $W_{1}$ denotes the 1-Wasserstein distance, defined by

[TABLE]

where $\Pi(\mu,\nu)$ denotes the set of couplings between $\mu$ and $\nu$, i.e. all measures on $\mathscr{B}(\mathbb{T}^{d}\times\mathbb{T}^{d})$ such that $\pi(B\times\mathbb{T}^{d})=\mu(B)$ and $\pi(\mathbb{T}^{d}\times B)=\nu(B)$ for every $B\in\mathscr{B}(\mathbb{T}^{d})$.

To write the norms of a Sobolev space $W^{n,\infty}(\mathbb{T}^{d})$ and its dual, we use the notations

[TABLE]

Moreover, for dual elements with their arguments, we use the notation

[TABLE]

Denoting $W^{0,\infty}(\mathbb{T}^{d}):=L^{\infty}(\mathbb{T}^{d})$, for any $f\in W^{n-1,\infty}(\mathbb{T}^{d},\mathbb{R}^{d})$ and $\eta\in L^{\infty}([0,T],(W^{n-1,\infty}(\mathbb{T}^{d}))^{\prime})$, we use the notation

[TABLE]

$W^{0,n,\infty}([0,T]\times{\mathbb{T}}^{d})$ denotes, for $n\geq 1$, the space of measurable functions $f:[0,T]\times{\mathbb{T}}^{d}\rightarrow{\mathbb{R}}$ with spatial generalized derivatives up to order $n$ that all belong to $L^{\infty}([0,T]\times{\mathbb{T}}^{d})$. We define

[TABLE]

For functions $f=(f_{1},\ldots,f_{d}):[0,T]\times\mathbb{T}^{d}\to\mathbb{R}^{d}$ such that each component function $f_{i}$ belongs to $W^{0,n,\infty}([0,T]\times{\mathbb{T}}^{d})$, we write $f\in W^{0,n,\infty}([0,T]\times{\mathbb{T}}^{d},\mathbb{R}^{d})$ with

[TABLE]

For any signed measures $\mu_{1},\ldots,\mu_{n}$, we write $\frac{\delta^{n}\Phi}{\delta m^{n}}(\mu)(\mu_{1},\ldots,\mu_{n})$ to denote

[TABLE]

if this iterated integral is well-defined.

Unless otherwise specified, $C$ is a constant that only depends on $n$, $k$, $T$, $b$ and $\Phi$, whose value varies from line to line.

1.4.2 Main assumptions

Throughout this work, we work with the following assumptions on $b=(b_{i})_{1\leq i\leq d}$ and $\Phi$. (Int-$b$-(${n,k}$)) denotes the condition that, for each $i\in\{1,\ldots,d\}$, $\ell\in\{1,\ldots,k\}$,

[TABLE]

(Lip-$b$-(${n,k}$)) denotes the condition that, for each $i\in\{1,\ldots,d\}$, $\ell\in\{1,\ldots,k\}$,

[TABLE]

where

[TABLE]

For the test function $\Phi:\mathcal{P}(\mathbb{T}^{d})\to\mathbb{R}$, we shall impose the following assumptions. (TLip-$\Phi$-(${k}$)) denotes the condition that, for each $\ell\in\{1,\ldots,k\}$,

[TABLE]

where

[TABLE]

(TReg-$\Phi$-(${n,k}$)) denotes the condition that, for each $\ell\in\{1,\ldots,k\}$ and $i\in\{1,\ldots,\ell\}$,

[TABLE]

Finally, (TInt-$\Phi$-(${n,k}$)) denotes the integrability condition that, for each $\ell\in\{1,\ldots,k\}$,

[TABLE]

1.5 Practical examples of our model

We now give a result of a class of drift terms $b$ and test functions $\Phi$ that satisfies the above assumptions, followed by practical examples of our model.

Theorem 1.1**.**

Let $n\in\mathbb{N}$. Suppose that for each $i\in\{1,\ldots,d\}$, $F_{i}:\mathbb{T}^{d}\times\mathbb{T}^{d}\to\mathbb{R}$ belongs to $W^{n,\infty}(\mathbb{T}^{d}\times\mathbb{T}^{d})$ and that $G:\mathbb{T}^{d}\to\mathbb{R}$ belongs to $W^{n,\infty}(\mathbb{T}^{d})$. We then define functions $b_{i}:\mathbb{T}^{d}\times\mathcal{P}(\mathbb{T}^{d})\to\mathbb{R}$ and $\Phi:\mathcal{P}(\mathbb{T}^{d})\to\mathbb{R}$ by

[TABLE]

Then $b$ satisfies (Int-$b$-(${n,k}$)) and (Lip-$b$-(${n,k}$)). Moreover, $\Phi$ satisfies (TLip-$\Phi$-(${k}$)), (TReg-$\Phi$-(${n,k}$)) and (TInt-$\Phi$-(${n,k}$)).

Proof.

Let $k\in\mathbb{N}$ be arbitrary. Let

[TABLE]

It can be shown easily by the definition of linear functional derivatives (along with the condition of normalisation) that

[TABLE]

It can be easily checked that

[TABLE]

[TABLE]

and

[TABLE]

Moreover, by the Kantorovich Rubinstein duality (see Remark 6.5 in [31]),

[TABLE]

Similarly,

[TABLE]

This allows us to show that

[TABLE]

[TABLE]

[TABLE]

These calculations show that $b$ and $\Phi$ satisfy the aforementioned regularity properties in the theorem. Note that $k$ is arbitrary in $\mathbb{N}$, since the dependence on measure is linear for functions $b$ and $\sigma$. ∎

Example 1.2** (Kuramoto model).**

The Kuramoto model is used to describe the behaviour of synchronization for a large set of coupled oscillators and is defined in dimension $d=1$ (see, e.g., [2]):

[TABLE]

Example 1.3** (Aggregation models).**

Aggregation models are commonly used in the analysis of mean-field models in biology, ecology, for space homogeneous granular media (see [4, 5, 10, 20, 26]). In such models, the drift term $b$ typically takes the form

[TABLE]

for some smooth functions $V,W:\mathbb{T}^{d}\to\mathbb{R}$. According to Theorem 1.1, our analysis would be applicable to functions $V,W\in W^{n,\infty}$, where $n\geq 2$.

2 Regularity of first order linear derivative in measure of $\mathcal{U}$

2.1 Analysis of the forward Kolmogorov equation

The first step in the analysis of PDEs is the regularity of $m$. The following result concerns regularity of (1.1) and is standard in the literature.

Lemma 2.1**.**

Suppose that $b$ is jointly Lipschitz continuous in the space and measure variables w.r.t. the Euclidean and $W_{1}$ metrics. Then (1.1) has a unique solution and satisfies

[TABLE]

for some constant $C>0$.

Proof.

The fact that (1.1) has a unique solution follows from the strong uniqueness of (1.7), by Theorem 1.1 of [30]. The estimate follows from the proof of Lemma 3.1 in [7]. ∎

The following result is a modified version of Proposition 3.4.3 in [8] from Hölder spaces to Sobolev spaces.

Proposition 2.2**.**

Let $n\geq 1$, $f\in W^{0,n-1,\infty}([0,T]\times{\mathbb{T}}^{d})$ and $g\in W^{0,n-1,\infty}([0,T]\times{\mathbb{T}}^{d},\mathbb{R}^{d})$. Then, for any $z_{T}\in W^{n,\infty}({\mathbb{T}}^{d})$, the Cauchy problem

[TABLE]

has a unique solution in the following space:

[TABLE]

where ${\mathcal{C}}^{0,1}([0,T)\times{\mathbb{T}}^{d},{\mathbb{R}})$ is the space of real-valued functions $z$ (on $[0,T)\times{\mathbb{T}}^{d}$) that are continuous in time and space, differentiable in space, and the derivative of which is continuous in time and space, and where $W^{1,2,d+1}_{\textrm{\rm loc}}([0,T)\times{\mathbb{T}}^{d})$ is the space of functions $z$ such that $|z|$, $|\nabla_{x}z|$, $|\nabla_{x}^{2}z|$ and $|\partial_{t}z|$ belong to $L^{d+1}_{\textrm{\rm loc}}([0,T)\times{\mathbb{T}}^{d})$. The unique solution satisfies

[TABLE]

where $C$ only depends on $\|g\|_{n-1,\infty}$.

Proof.

First Step. We start with uniqueness. Uniqueness of a solution (in ${\mathcal{V}}$) is a trivial consequence of the solvability of the SDE:

[TABLE]

and, then, of Itô-Krylov’s formula (see for instance [22]), which guarantees that

[TABLE]

Notice that Itô’s formula does not suffice since the solution may just have first order $t$-derivative and second order $x$-derivatives in $L^{d+1}$. Obviously, Itô-Krylov’s formula here applies because of the non-degeneracy of the noise.

Second Step. Existence of a solution with generalized second order derivatives is a well known fact in the literature. The main reference is the monograph of Ladyzenskaja et al., [25]; a more precise application of the results of [25] to our setting may be found in [14], see Theorem 2.1 therein. The latter says that existence of a solution hold in the space ${\mathcal{V}}$ defined in the statement.

Third Step. Now, the main point is to prove that the solution satisfies the required bounds. By mollifying the coefficients $f$ and $g$ in space (using a standard convolution argument), we may easily assume that the coefficients $f$ and $g$ are smooth in space, and that their derivatives up the order $n-1$ satisfy the same Lipschitz bounds as the original coefficients. If we can prove that the solution associated with the equation with mollified coefficients satisfies the inequality announced in the statement, with a constant $C$ therein that remains uniform along the mollification, then we are done: it suffices to observe that the solution associated with the mollified equation converges (as the mollification parameter tends to [math]) to the original $z$ by passing to the limit along the stochastic representation (based upon (2.3)–(2.4)).

So, from now on, we assume that the coefficients $f$ and $g$ are smooth in space, and that their derivatives up the order $n-1$ satisfy the same Lipschitz bounds as the original coefficients. The key point is then to observe that we can differentiate with respect to $x$ in the representation formula (2.4), since the solution to (2.3) generates a smooth flow (see for instance [23]). As a by-product, we deduce that, for any $k\geq 1$,

[TABLE]

Obviously, the bound of the above left-hand side depends on the (additional) smoothness of $f$ and $g$. Now, by expanding $(z(T-s,x+B_{s-t}))_{t\leq s\leq T}$ by means of Itô’s formula, we get that

[TABLE]

where $p$ is the standard heat kernel. We know from Theorem 11 in [16, Chapter 1] that there exists a bounded density $g$ on the torus such that, for any $0\leq s<s^{\prime}$ with $s^{\prime}-s\leq 1$,

[TABLE]

for a constant $C$ only depending on the bound of $V$. Taking the derivative with respect to $x$,

[TABLE]

where the constant $C$ depends on $\|g\|_{\infty}$ (and is allowed to vary from line to line). By a standard variant of Gronwall’s lemma (see for instance [18, Lemma 7.1.1 and Exercise 1]), we get

[TABLE]

which is exactly the announced result when $n=1$. Differentiating twice (2.5) (hence differentiating twice the heat kernel in the right-hand side), performing an integration by parts in the resulting second and third terms in the right-hand side of (2.5) and eventually plugging the above bound in the resulting formula, we then get

[TABLE]

where $C$ now depends on $\|g\|_{1,\infty}$, and in turn

[TABLE]

which yields, by the same variant of Gronwall’s lemma,

[TABLE]

This is exactly the desired result when $n=2$.

Now, we can iterate by induction, assuming that the result holds true for a given $k\in\{2,\cdots,n-1\}$. It suffices to take $k+1$ derivatives (in $x$) in the left-hand side of (2.5) and then to use an integration by parts to pass $k$ derivatives from the heat kernel onto $f$ and $g\cdot\nabla z$ in the resulting second and third terms in the right-hand side of (2.5). We then get

[TABLE]

where $C$ now depends on $\|g\|_{k,\infty}$. Plugging the bound, we have, for $\|z(s,\cdot)\|_{k,\infty}$ (as given by the induction assumption), we get

[TABLE]

and, then, refined Gronwall’s lemma applies as before. ∎

The core analysis of forward Kolmogorov equations depends heavily on the following fact. The main ideas of the proof follow from the proof of Lemma 3.3.1 in [8].

Theorem 2.3** (Bound for forward Kolmogorov equations).**

Let $n\geq 1$ and $q_{0}\in(W^{n,\infty}(\mathbb{T}^{d}))^{\prime}$. Assume (Int-$b$-(${n,1}$)). Let r\in L^{\infty}\big{(}[0,T],(W^{n,\infty}(\mathbb{T}^{d}))^{\prime}\big{)}. Then the Cauchy problem defined by

[TABLE]

interpreted as

[TABLE]

for each $\phi\in C^{\infty}([0,T]\times\mathbb{T}^{d})$, has a unique solution in L^{\infty}\big{(}[0,T],(W^{n,\infty}(\mathbb{T}^{d}))^{\prime}\big{)} such that

[TABLE]

for some constant $C>0$.

Proof.

We consider the space $X:=C^{\beta}([0,T],(W^{n,\infty}(\mathbb{T}^{d}))^{\prime})$, where $\beta\in(0,\frac{1}{2})$. We recall that the norm of $X$ is given by

[TABLE]

For $q\in X$, we consider the Cauchy problem

[TABLE]

By Schauder estimates, setting $T(q):=\tilde{q}$ defines a continuous and compact map $T:X\to X$. (See Step 1 in the proof of Lemma 3.3.1 in [8]). We show the existence of solution to (2.7) by applying the Leray-Schauder theorem, i.e. by showing that the set

[TABLE]

is bounded. To this end, we pick an arbitrary $q\in X_{0}$, which satisfies the Cauchy problem

[TABLE]

The estimates rely on the classical argument of duality pairing. Fix $t\in[0,T]$ and $\xi\in W^{n,\infty}(\mathbb{T}^{d})$. Let $w$ be the solution to the Cauchy problem

[TABLE]

By Theorem 2.2, $w$ satisfies

[TABLE]

By the definition of (2.9), we have

[TABLE]

Therefore, by (2.10),

[TABLE]

We now estimate each of the three terms on the right hand side by (2.11). Firstly,

[TABLE]

By (2.11) and (Int-$b$-(${n,1}$)), we obtain the estimate

[TABLE]

Finally, by (2.11),

[TABLE]

By (2.12), along with estimates (2.13), (2.14) and (2.15), we have

[TABLE]

which concludes by Gronwall’s inequality that

[TABLE]

Now we pick $t,t^{\prime}\in[0,T]$. Then (2.12) becomes

[TABLE]

By combining (2.16) with the argument of (2.14),

[TABLE]

Similarly,

[TABLE]

Therefore, by combining (2.17), (2.18) and (2.19), we have

[TABLE]

Combining with (2.16) gives

[TABLE]

Consequently, by the Leray-Schauder theorem, the map $T$ admits a fixed point. This shows the existence of solution to (2.7). For uniqueness, one simply has to apply a Gronwall argument to (2.12). Finally, the estimate for the solution follows by repeating the proof up to (2.16), but with $\sigma=1$. ∎

Lemma 2.4**.**

Assume (Int-$b$-(${n,1}$)), where $n\geq 1$. Then the Cauchy problem $m^{(1)}$ defined in (1.14) has a unique solution in L^{\infty}\big{(}[0,T],(W^{n,\infty}(\mathbb{T}^{d}))^{\prime}\big{)}.

Proof.

This is immediate from Theorem 2.3. ∎

For every $t\in[0,T]$, $\mu,\hat{\mu}\in\mathcal{P}(\mathbb{T}^{d})$, let

[TABLE]

Let $\phi\in C^{\infty}([0,T]\times\mathbb{T}^{d})$. By (1.1) and (1.14), we have

[TABLE]

We rewrite the final two terms as

[TABLE]

Therefore, we obtain that

[TABLE]

In distributional sense, we write

[TABLE]

where

[TABLE]

We first establish the regularity of $c$.

Lemma 2.5**.**

Assume (Int-$b$-(${n-1,1}$)) and (Lip-$b$-(${0,1}$)), where $n\geq 2$. Then c(\cdot,\mu,\hat{\mu})\in L^{\infty}\big{(}[0,T],(W^{n,\infty}(\mathbb{T}^{d}))^{\prime}\big{)}.

Proof.

For any $\xi\in W^{n,\infty}(\mathbb{T}^{d}),$

[TABLE]

Next, we estimate each of the two terms. By (2.1) and (Int-$b$-(${n-1,1}$)), since $n\geq 2$,

[TABLE]

Similarly, by (Lip-$b$-(${0,1}$)),

[TABLE]

Combining (2.24) and (2.25), we have

[TABLE]

which implies that $c(t,\mu,\hat{\mu})$ is a bounded operator with its operator norm given by

[TABLE]

∎

The following theorem is a straightforward consequence of the above results.

Theorem 2.6**.**

Assume (Int-$b$-(${n,1}$)), (Lip-$b$-(${0,1}$)), (TLip-$\Phi$-(${1}$)) and (TReg-$\Phi$-(${n,1}$)), where $n\geq 2$. Then the following statements hold.

1. (i)

There exists some constant $C>0$ such that

[TABLE] 2. (ii)

For $\mathcal{U}$ defined by (1.2),

[TABLE] 3. (iii)

[TABLE]

Proof.

1. (i)

This follows from (2.22), estimate (2.26) and Theorem 2.3. 2. (ii)

Let $\pi$ be the optimal transport plan from $m(t,\mu)$ to $m(t,\hat{\mu})$. The computation from the proof of Proposition 5.44 from [9] shows that

[TABLE]

By (TLip-$\Phi$-(${1}$)), (2.1) and the fact that

[TABLE]

there exists some constant $C>0$ such that

[TABLE]

By assumption (TReg-$\Phi$-(${n,1}$)) and part (i), there exists some constant $C^{\prime}>0$ such that

[TABLE]

which completes the proof. 3. (iii)

[TABLE]

by (2.30) and the fact that $m^{(1)}(t,\mu,(1-\epsilon)\mu+\epsilon\hat{\mu})=\epsilon m^{(1)}(t,\mu,\hat{\mu})$.

∎

2.2 Analysis of the backward Kolmogorov equation

We observe that, in (2.27), the integral is with respect to the signed measure $m^{(1)}(t,\mu,\hat{\mu})$. To show that $\mathcal{U}$ indeed has a linear functional derivative, we need to express the integral in terms of the signed measure $\hat{\mu}-\mu$. To this end, we fix $t\in[0,T]$ and $x\in\mathbb{R}^{d}$ and introduce the decoupled process $\{X^{0,x,\mu}_{u}\}_{u\in[0,t]}$ by

[TABLE]

For every $\xi:\mathbb{T}^{d}\to\mathbb{R}$ and $t\in[0,T]$, we define a function $v(\cdot,\cdot,\cdot;\xi,t):[0,t]\times\mathbb{T}^{d}\times\mathcal{P}(\mathbb{T}^{d})\to\mathbb{R}$ such that

[TABLE]

Note that

[TABLE]

It is well-known that (see, for example, equation (3.4) in [7])

[TABLE]

Therefore,

[TABLE]

for any $\mu,\hat{\mu}\in\mathcal{P}(\mathbb{T}^{d})$. If $\frac{\delta v}{\delta m}$ exists, taking derivative w.r.t. $\epsilon$ at [math] gives

[TABLE]

Hence, it suffices to study the regularity of $v$. In most of the analysis for $v$, we suppress the parameters $\xi$ and $t$, for simplicity of notations. By the standard Feynman-Kac equation (Kolmogorov backward equation), $v$ satisfies the PDE

[TABLE]

Lemma 2.7**.**

Assume (Int-$b$-(${n,1}$)), where $n\geq 2$. Suppose that $\xi\in W^{n+1,\infty}$. Then the Cauchy problem $v$ defined in (2.34) has a unique solution in L^{\infty}\big{(}[0,t],W^{n+1,\infty}(\mathbb{T}^{d})\big{)}. Moreover, there exists a constant $C>0$ (depending on $\xi$) such that for any $\mu,\hat{\mu}\in\mathcal{P}(\mathbb{T}^{d})$,

[TABLE]

Proof.

The fact that v\in L^{\infty}\big{(}[0,t],W^{n+1,\infty}(\mathbb{T}^{d})\big{)} follows from Proposition 2.2. For the second part, take any $\mu,\hat{\mu}\in\mathcal{P}(\mathbb{T}^{d})$. Let

[TABLE]

Then $z$ satisfies the Cauchy problem

[TABLE]

Using the same argument as (2.24), by (2.1), (Int-$b$-(${n,1}$)) and Proposition 2.2, there exists a constant $C>0$ such that

[TABLE]

∎

The core analysis of backward Kolmogorov equations depends on the following fact.

Theorem 2.8** (Bound for backward Kolmogorov equations).**

Assume (Int-$b$-(${n,1}$)), where $n\geq 2$. Suppose that $\xi\in W^{n+1,\infty}$. Let q\in L^{\infty}\big{(}[0,t],(W^{n,\infty}(\mathbb{T}^{d}))^{\prime}\big{)} and \gamma\in L^{\infty}\big{(}[0,t],W^{n,\infty}(\mathbb{T}^{d})\big{)}. Then the Cauchy problem $h$

[TABLE]

has a unique solution in L^{\infty}\big{(}[0,t],W^{n+1,\infty}(\mathbb{T}^{d})\big{)} such that

[TABLE]

for some constant $C>0$ depending on $\xi$.

Proof.

By (Int-$b$-(${n,1}$)) and Proposition 2.2,

[TABLE]

∎

Formal differentiation of (2.34) w.r.t. the measure component gives

[TABLE]

We now study the regularity of $v^{(1)}$.

Lemma 2.9**.**

Assume (Int-$b$-(${n,1}$)), where $n\geq 2$. Suppose that $\xi\in W^{n+1,\infty}$. Then the Cauchy problem $v^{(1)}$ defined in (2.37) has a unique solution in L^{\infty}\big{(}[0,t],W^{n+1,\infty}(\mathbb{T}^{d})\big{)}. Moreover, $v^{(1)}$ satisfies the relation

[TABLE]

Proof.

The first part of the lemma follows directly from Theorem 2.8. For the second part, we note that $v^{(1)}(0,x,\mu,\delta_{z})$ satisfies

[TABLE]

where the final term uses the normalisation condition of $\frac{\delta{b}}{\delta m}$. Integrating both sides w.r.t. $z$ with measure $\hat{\mu}-\mu$, we have

[TABLE]

By comparing (2.37) and (2.38), the result follows by an argument of stability similar to Corollary 3.4.2 of [8]. ∎

As before, we consider the difference

[TABLE]

Then $\Gamma$ satisfies the Cauchy problem

[TABLE]

where

[TABLE]

The following result is immediate.

Theorem 2.10**.**

Assume (Int-$b$-(${n,1}$)) and (Lip-$b$-(${0,1}$)), where $n\geq 2$. Suppose that $\xi\in W^{n+1,\infty}$. Then $\frac{\delta{v}}{\delta m}(0,x,\mu,y)$ exists and is given by

[TABLE]

Proof.

We proceed in the same way as in the proof of Lemma 2.5. By (Int-$b$-(${n,1}$)), (Lip-$b$-(${0,1}$)), (2.1), (2.30) and Lemma 2.7, we deduce from (2.40) that

[TABLE]

for some constant $C>0$ depending on $\xi$. Therefore, by Theorem 2.8,

[TABLE]

Therefore, by Lemma 2.9,

[TABLE]

We conclude the result by the characterisation of linear functional derivatives in Remark 5.47 of [9]. ∎

Corollary 2.11** (Existence of the first order linear derivative).**

Assume (Int-$b$-(${n,1}$)), (Lip-$b$-(${0,1}$)), (TLip-$\Phi$-(${1}$)) and (TReg-$\Phi$-(${n+1,1}$)), where $n\geq 2$. Then $\frac{\delta{\mathcal{U}}}{\delta m}$ exists and is given by

[TABLE]

for every $\mu\in\mathcal{P}(\mathbb{T}^{d})$.

Proof.

Fix $\mu_{0}\in\mathcal{P}(\mathbb{T}^{d})$. Firstly, we recall from (2.32) that

[TABLE]

Since $\Phi$ satisfies (TLip-$\Phi$-(${1}$)) and (TReg-$\Phi$-(${n+1,1}$)), the function

[TABLE]

satisfies (TLip-${\widetilde{\Phi}-(1)}$) and (TReg-$\widetilde{\Phi}$-(${n+1},1$)). Moreover,

[TABLE]

lies in $W^{n+1,\infty}$. Therefore, by part (iii) of Theorem 2.6 and Theorem 2.10, we differentiate (2.41) w.r.t. $\epsilon$ at [math], which gives (by (LABEL:v_phi_connection_2))

[TABLE]

Putting $\mu_{0}=\mu$, we have

[TABLE]

Finally, by part (iii) of Theorem 2.6, we conclude that $\frac{\delta{\mathcal{U}}}{\delta m}$ exists and is given by

[TABLE]

∎

3 Higher order forward and backward Kolmogorov equations

In this section, we repeat the same procedure in the previous section to establish regularity of higher order Kolmogorov equations. In order to proceed with an iteration argument, we first introduce the following class of multi-indices in the class $\tau_{k}$.

3.1 Definitions and notations for iteration in multi-indices in the class $\tau_{k}$

Definition 3.1** (Class $\tau_{k}$ of multi-indices).**

For any $k\in\mathbb{N}$, the class $\tau_{k}$ contains all multi-indices of the form

[TABLE]

where ${\hat{n}}$, $\beta_{j}$ and $\hat{\beta}$ are non-negative integers and $\alpha_{i,j}$, $\hat{\alpha}_{\ell}$, $1\leq i\leq{\hat{n}}$, $1\leq j\leq\beta_{i}$, $1\leq\ell\leq\hat{\beta}$, are positive integers satisfying

1. (i)

${\hat{n}}\leq k,\quad\quad 1\leq\alpha_{i,1}<\ldots<\alpha_{i,\beta_{i}}\leq k,\quad\quad 1\leq\hat{\alpha}_{1}<\ldots<\hat{\alpha}_{\hat{\beta}}\leq k,$ 2. (ii)

$\beta_{1},\ldots,\beta_{\hat{n}},\hat{\beta}<k,$ 3. (iii)

exactly one of $\alpha_{i,j}$ and $\hat{\alpha}_{\ell}$ is equal to $k$, 4. (iv)

[TABLE] 5. (v)

for any $i,i^{\prime}\in\{1,\ldots,\hat{n}\},$

[TABLE]

In particular, $o(\lambda)$ is called the order of $\lambda$ defined by

[TABLE]

Moreover, for any $(\lambda^{(1)},\ldots,\lambda^{(q)})\in(\tau_{k})^{q}$, we define the magnitude of $(\lambda^{(1)},\ldots,\lambda^{(q)})$ by

[TABLE]

If $\lambda=\lambda^{(i)}$, for some $i\in\{1,\ldots,q\}$, we write

[TABLE]

Remark 3.2**.**

This definition is modified accordingly when one of ${\hat{n}}$, $\beta_{j}$ and $\hat{\beta}$ is zero. When ${\hat{n}}=0$, we set \lambda:=\big{(}0,\hat{\beta},(\hat{\alpha}_{\ell})_{1\leq\ell\leq\hat{\beta}}\big{)}. On the other hand, when $\hat{\beta}=0$, we set \lambda:=\Big{(}{\hat{n}},(\beta_{j})_{j=1}^{{\hat{n}}},({\alpha_{i,j}})_{\begin{subarray}{c}1\leq i\leq{\hat{n}}\\ 1\leq j\leq\beta_{i}\end{subarray}},0\Big{)}. Finally, when $\beta_{j_{0}}=0$, for some $j_{0}\in\{1,\ldots,{\hat{n}}\}$, the column entry of $j_{0}$ disappears in the array $({\alpha_{i,j}})$.

Next, we introduce the recurrence map $T_{k}$ for multi-indices, followed by the sequence of multi-dimensional vectors $\lambda_{k}$ of elements in $\tau_{k}$.

Definition 3.3** (Recurrence map $T_{k}$).**

Let $\lambda\in\tau_{k}$ be given by the form (3.1). We define a recurrence map $T_{k}$ by

[TABLE]

Definition 3.4** (Multi-dimensional vectors $\lambda_{k}$ of elements in $\tau_{k}$).**

We first define

[TABLE]

For every $k\geq 2$, we define a multi-dimensional vector $\lambda_{k+1}$ of elements in $\tau_{k+1}$ by the recurrence relation

[TABLE]

for \lambda_{k}=\big{(}\lambda^{(1)}_{k},\ldots,\lambda^{(m(\lambda_{k}))}_{k}\big{)}.

3.2 Analysis of higher order forward Kolmogorov equations

In this subsection, we consider the following Cauchy problem (defined recursively by (3.5), (3.7), Definition 3.3 and Definition 3.4):

[TABLE]

where, for $k=1$, $F_{\lambda_{1}}(t,\mu,\mu_{1}):=0.$ For $\lambda\in\tau_{k}$ given by (3.1), we define

[TABLE]

Note that $F_{\lambda}(t,\mu,\mu_{1},\ldots,\mu_{k})$ can be interpreted as an element in the dual space $(W^{n+k-1,\infty}(\mathbb{T}^{d}))^{\prime}$ (under the assumption (Int-$b$-(${n+k-1,k}$))):

[TABLE]

For any $(\lambda^{(1)},\ldots,\lambda^{(q)})\in(\tau_{k})^{q}$, we define

[TABLE]

Theorem 3.5**.**

Let $k\in\mathbb{N}$. Assume (Int-$b$-(${n+k-1,k}$)), where $n\geq 2$. Then (3.6) is well-defined and the Cauchy problem defined by (3.4) has a unique solution in L^{\infty}\big{(}[0,T],(W^{n+k-1,\infty}(\mathbb{T}^{d}))^{\prime}\big{)} and satisfies

[TABLE]

Also, if we assume (Int-$b$-(${n+k,k+1}$)), then

[TABLE]

for any $\mu,\mu_{1},\ldots,\mu_{k+1}\in\mathcal{P}(\mathbb{T}^{d})$, for some constant $C>0$.

Proof.

We proceed by strong induction for (3.8). The base step follows clearly from (1.14) and Theorem 2.3, since

[TABLE]

Suppose that (3.8) holds for $\{1,\ldots,k-1\}$. Take any $\xi\in W^{{{n}}+k-1,\infty}(\mathbb{T}^{d})$ and $k\geq 2$. We first show that (3.6) is well-defined, i.e. $F_{\lambda}(t,\mu,\mu_{1},\ldots,\mu_{k})$ is indeed in $(W^{{{n}}+k-1,\infty}(\mathbb{T}^{d}))^{\prime}$, for any $\lambda\in\tau_{k}$. Note that $\beta_{1},\ldots,\beta_{\hat{n}},\hat{\beta}\leq k-1$, which implies by (3.2) and (3.3) that

[TABLE]

where the final step follows from (Int-$b$-(${n+k-1,k}$)). Therefore, the first statement that the Cauchy problem has a unique solution in L^{\infty}\big{(}[0,T],(W^{n+k-1,\infty}(\mathbb{T}^{d}))^{\prime}\big{)} and (3.8) both follow directly from Theorem 2.3, by the assumption of (Int-$b$-(${n+k-1,k}$)) and the fact that $F_{\lambda}(\cdot,\mu,\mu_{1},\ldots,\mu_{k})$ is in $L^{\infty}([0,T],(W^{{{n}}+k-1,\infty}(\mathbb{T}^{d}))^{\prime})$.

It remains to prove (3.9) under the stronger assumption (Int-$b$-(${n+k,k+1}$)). Let ${\xi}\in W^{n+k,\infty}(\mathbb{T}^{d})$. Again, we proceed by strong induction. The base step is omitted as it is a special case of the procedure of the induction step. Suppose that (3.9) holds for $\{1,\ldots,k\}$. Replacing $\mu$ by $\mu_{k+1}$ in (3.4), we have

[TABLE]

On the other hand, we have

[TABLE]

Next, we compute that

[TABLE]

Note that the first term in (3.13) can be rewritten as

[TABLE]

by which we can estimate by the assumption (Int-$b$-(${n+k,k+1}$)). For every $\lambda\in\tau_{k}$, we know that $\beta_{1},\ldots,\beta_{n},\hat{\beta}<k$ by definition. For $i\in\{1,\ldots,\hat{n}\}$ and $\hat{\mu}\in\{\mu,\mu_{k+1}\},$

[TABLE]

By the induction hypothesis, for every $\beta_{\ell}<k$,

[TABLE]

Similarly, by the induction hypothesis, for $\hat{\beta}<k$,

[TABLE]

Hence, by (3.13), (LABEL:eq:_linear_derivative_higher_order_trick), (3.15), (3.16), (3.17) and the assumption of (Int-$b$-(${n+k,k+1}$)), we obtain that

[TABLE]

Let

[TABLE]

Subtracting (3.11) by (3.12) gives

[TABLE]

Let $\eta^{(k+1)}(t,\mu,\mu_{1},\ldots,\mu_{k},\mu_{k+1})$ be an element in the dual space $(W^{n+k,\infty}(\mathbb{T}^{d}))^{\prime}$ defined by

[TABLE]

Clearly, by (Int-$b$-(${n+k,k+1}$)) and (3.18), it follows from the same argument as Lemma 2.5 to deduce that

[TABLE]

By (LABEL:dk+1_formula) (and replacing $\xi$ by arbitrary test functions $\phi\in C^{\infty}([0,T]\times\mathbb{T}^{d})$) we note that $d^{(k+1)}$ satisfies the Cauchy problem

[TABLE]

Therefore, by Theorem 2.3 and (Int-$b$-(${n+k,1}$)),

[TABLE]

This completes the proof by (3.21). ∎

Theorem 3.6**.**

Let $k\in\mathbb{N}\cup\{0\}$. Assume (Int-$b$-(${n+k+1,k+1}$)) and (Lip-$b$-(${n+k,k+1}$)), where $n\geq 2$. Then

[TABLE]

for any $\mu,\mu_{1},\ldots,\mu_{k+1}\in\mathcal{P}(\mathbb{T}^{d})$, for some constant $C>0$.

Proof.

We proceed by strong induction. The base case is done in Theorem 2.6. Assume that the theorem holds for $\{1,\ldots,k-1\}$, where $k\geq 2$. Then

[TABLE]

Take $\xi\in W^{n+k+1,\infty}(\mathbb{T}^{d})$. We first recall from the definition of $\lambda_{k+1}$ (given in Definition 3.4) that the PDE for $m^{(k+1)}$ is given by

[TABLE]

Recalling the definition of $d^{(k+1)}$ in (3.19), we define

[TABLE]

Subtracting (LABEL:dk+1_formula) by (3.25) (and replacing $\xi$ by arbitrary test functions $\phi\in C^{\infty}([0,T]\times\mathbb{T}^{d})$), we observe that ${\rho}^{(k+1)}$ satisfies the Cauchy problem

[TABLE]

where

[TABLE]

and $c_{i}^{(k+1)}(t,\mu,\mu_{1},\ldots,\mu_{k},\mu_{k+1}),i\in\{1,\ldots,4\}$, are elements in the dual space $(W^{n+k+1,\infty}(\mathbb{T}^{d}))^{\prime}$ defined by

[TABLE]

and, by (3.13),

[TABLE]

Note that the term {\Big{\langle}\xi,c_{1}^{(k+1)}(t,\mu,\mu_{1},\ldots,\mu_{k},\mu_{k+1})\Big{\rangle}}_{n+k+1,\infty} can be rewritten as

[TABLE]

By Theorem 3.5, the first term of (3.27) is controlled by

[TABLE]

By the same argument as (LABEL:eq:_linear_derivative_higher_order_trick) and (3.15), the second and third terms of (3.27) are controlled by

[TABLE]

where the estimate for the first term follows from (Lip-$b$-(${n+k,k+1}$)) with the same argument as (2.25). This shows that

[TABLE]

Similarly, by (Int-$b$-(${n+k,k+1}$)), (Lip-$b$-(${n+k,k+1}$)) and Theorem 3.5, along with a similar argument applied to the induction hypothesis (3.24) (as in estimates (3.15), (3.16) and (3.17)), we can show that, for $i\in\{2,3,4\}$,

[TABLE]

Therefore,

[TABLE]

Finally, by (Int-$b$-(${n+k+1,1}$)), (3.26) and Theorem 2.3, we conclude that

[TABLE]

∎

3.3 Analysis of higher order backward Kolmogorov equations

In this subsection, we fix $t\in[0,T]$ and consider the following Cauchy problem (defined recursively by (3.29), Definition 3.3 and Definition 3.4):

[TABLE]

where

[TABLE]

The following theorem gives the regularity of $v^{(k)}$ by Schauder estimates.

Theorem 3.7**.**

Let $k\in\mathbb{N}$. Assume (Int-$b$-(${n+k-1,k}$)), where $n\geq 2$. Suppose that $\xi\in W^{n+1,\infty}$. Then the Cauchy problem $v^{(k)}$ defined by (3.28) has a unique solution in L^{\infty}\big{(}[0,t],W^{n+1,\infty}(\mathbb{T}^{d})\big{)}.

Proof.

We proceed by strong induction. The base step is proven in Lemma 2.9. For the induction step, we assume that the statement is true for $1,\ldots,k-1$, where $k\geq 2$. For each $\lambda\in e(\lambda_{k})$, by (Int-$b$-(${n+k-1,k}$)),

[TABLE]

which implies that $G_{\lambda_{k}}(\cdot,\cdot,\mu,\mu_{1},\ldots,\mu_{k})\in L^{\infty}([0,t],W^{n,\infty}(\mathbb{T}^{d}))$. This completes the induction step by repeating the same argument as in the proof of Theorem 2.8. ∎

The following theorem is an analogue of Theorem 3.6 for backward Kolmogorov equations. The computations in the proof follow the same ideas as those in the previous subsection, i.e. Theorem 3.5 and Theorem 3.6. Consequently, the proof is omitted for brevity.

Theorem 3.8**.**

Let $k\in\mathbb{N}$. Assume (Int-$b$-(${n+k+1,k+1}$)) and (Lip-$b$-(${n+k,k+1}$)), where $n\geq 2$. Suppose that $\xi\in W^{n+1,\infty}$. Then

[TABLE]

for any $\mu,\mu_{1},\ldots,\mu_{k+1}\in\mathcal{P}(\mathbb{T}^{d})$, for some constant $C>0$.

We now establish the $k$th order linear derivative of $v$ in terms of $v^{(k)}$.

Theorem 3.9**.**

Let $k\in\mathbb{N}$. Assume (Int-$b$-(${n+k,k}$)) and (Lip-$b$-(${n+k-1,k}$)), where $n\geq 2$. Suppose that $\xi\in W^{n+1,\infty}$. Then

[TABLE]

where the linear derivative $\frac{\delta{v^{(k-1)}}}{\delta m}$ is taken with respect to $\mu$. Consequently, $\frac{\delta^{k}v}{\delta m^{k}}(0,x,\mu,y_{1},\ldots,y_{k})$ exists and is given by

[TABLE]

Proof.

Replacing $k$ by $k-1$ in Theorem 3.8 gives

[TABLE]

It follows from a similar argument as Lemma 2.9 to show that

[TABLE]

This proves the first equality. For the second equality, an inductive argument gives

[TABLE]

∎

3.4 Connection between higher order forward and backward equations

In this section, we follow the same approach as Section 2.2 to show that integrals with respect to the signed measure $m^{(k)}(t,\mu,\mu_{1},\ldots,\mu_{k})$ can be re-expressed in terms of the signed measure $\mu_{k}-\mu$.

Theorem 3.10**.**

Let $k\in\mathbb{N}$. Assume (Int-$b$-(${n+k,k}$)) and (Lip-$b$-(${n+k-1,k}$)), where $n\geq 2$. Suppose that $\xi\in W^{n+k,\infty}$. We define a sequence of functions $I^{(j)}(x,\mu,\mu_{1},\ldots,\mu_{j-1};\xi,t)$, $j\in\{1,\ldots,k\}$, by the following iteration:

[TABLE]

[TABLE]

for $j\in\{2,\ldots,k\}$, where $\frac{\delta I^{(j-1)}}{\delta m}$ is taken with respect to $\mu$. Then the sequence is well-defined and

[TABLE]

Proof.

By Theorem 3.9, the sequence $I^{(j)}$ is well-defined. To prove the equality, we proceed via an induction argument. The base step is established in (LABEL:v_phi_connection_2). For the inductive step, we assume that

[TABLE]

By replacing $k$ by $k-1$ in Theorem 3.6, we have

[TABLE]

for any $\mu,\mu_{1},\ldots,\mu_{k}\in\mathcal{P}(\mathbb{T}^{d})$, for some constant $C>0$. Since $\xi\in W^{n+k,\infty}$, it follows from the proof of Theorem 2.6 to observe that

[TABLE]

On the other hand, by the chain rule of differentiation,

[TABLE]

The proof is complete by combining (3.33) and (3.34). ∎

4 Regularity of higher order derivatives in measure of $\mathcal{U}$

4.1 Definitions and notations for iteration in multi-indices in the class $\Delta_{k}$

In order to obtain a general formula for the $k$th order linear derivative of $\Phi$, we proceed with another iteration argument. Therefore, we need to introduce another class $\Delta_{k}$ of multi-indices.

Definition 4.1** (Class $\Delta_{k}$ of multi-indices).**

For any $k\in\mathbb{N}$, the class $\Delta_{k}$ contains all multi-indices of the form

[TABLE]

where ${\hat{n}}$ and $\beta_{j}$ are non-negative integers and $\alpha_{i,j}$, $1\leq i\leq{\hat{n}}$, $1\leq j\leq\beta_{i}$, are positive integers satisfying

1. (i)

[TABLE] 2. (ii)

[TABLE] 3. (iii)

for any $i,i^{\prime}\in\{1,\ldots,\hat{n}\},$

[TABLE]

In particular, $o(\Lambda)$ is called the order of $\Lambda$ defined by

[TABLE]

Moreover, for any $(\Lambda^{(1)},\ldots,\Lambda^{(q)})\in(\Delta_{k})^{q}$, we define the magnitude of $(\Lambda^{(1)},\ldots,\Lambda^{(q)})$ by

[TABLE]

If $\Lambda=\Lambda^{(i)}$, for some $i\in\{1,\ldots,q\}$, we write

[TABLE]

Next, we introduce the recurrence map $Q_{k}$ for multi-indices in $\Delta_{k}$, followed by the sequence of multi-dimensional vectors $\Lambda_{k}$ of elements in $\Delta_{k}$.

Definition 4.2** (Recurrence map $Q_{k}$).**

Let $\Lambda\in\Delta_{k}$ be given by the form (4.1). We define a recurrence map $Q_{k}$ by

[TABLE]

Definition 4.3** (Multi-dimensional vectors $\Lambda_{k}$ of elements in $\Delta_{k}$).**

We first define

[TABLE]

For every $k\geq 2$, we define a multi-dimensional vector $\Lambda_{k+1}$ of elements in $\Delta_{k+1}$ by the recurrence relation

[TABLE]

for \Lambda_{k}=\big{(}\Lambda^{(1)}_{k},\ldots,\Lambda^{(m(\Lambda_{k}))}_{k}\big{)}.

4.2 Analysis of higher order linear derivatives of $\mathcal{U}$

We begin by establishing a higher-order analogue of Theorem 2.6.

Lemma 4.4**.**

*Let $k\in\mathbb{N}\setminus\{1\}$. Assume (Int-$b$-(${n+k,k}$)), (Lip-$b$-(${n+k-1,k}$)) and

(TReg-$\Phi$-(${n+k,k-1}$)), where $n\geq 2$. Then, for $\hat{n},\beta\leq k-1$ and $i\in\{1,\ldots,\hat{n}\}$,*

[TABLE]

for every $m,\mu,\mu_{1},\ldots,\mu_{\beta},\mu_{k}\in\mathcal{P}(\mathbb{T}^{d}).$

Proof.

Since $\beta\leq k-1$, the condition (Int-$b$-(${n+k,k}$)) implies (Int-$b$-(${n+\beta+1,\beta+1}$)). Similarly, the condition (Lip-$b$-(${n+k-1,k}$)) implies (Lip-$b$-(${n+\beta,\beta+1}$)). By Theorem 3.6, we have

[TABLE]

for any $\mu,\mu_{1},\ldots,\mu_{\beta},\mu_{k}\in\mathcal{P}(\mathbb{T}^{d})$, for some constant $C>0$. On the other hand, the condition (TReg-$\Phi$-(${n+k,k-1}$)) implies (TReg-$\Phi$-(${n+\beta+1,k-1}$)). The rest of the proof is identical to the proof of Theorem 2.6. ∎

We are now in a position to prove the main result of the paper. Clearly, one can obtain the minimal condition by setting $n=2$ (as in the introduction).

Theorem 4.5**.**

Let $k\in\mathbb{N}$ and $n\geq 2$. Assume (Int-$b$-(${n+k,k}$)), (Lip-$b$-(${n+k-1,k}$)), (TLip-$\Phi$-(${k}$)) and (TReg-$\Phi$-(${n+k,k}$)). Then $\frac{\delta^{k}\mathcal{U}}{\delta m^{k}}$ exists and is given by

[TABLE]

In particular, if we also assume (TInt-$\Phi$-(${n+k-1,k}$)), then

[TABLE]

Proof.

We proceed by induction on $k$. We first prove the statement for $k=1$. By Corollary 2.11, we know that $\frac{\delta{\mathcal{U}}}{\delta m}$ exists. Therefore, by (2.28),

[TABLE]

By the normalisation convention of $\frac{\delta{\mathcal{U}}}{\delta m}$,

[TABLE]

Therefore, putting $\hat{\mu}:=\delta_{z_{1}}$ gives

[TABLE]

We now assume that this statement holds for $k-1$. Therefore, for any $\epsilon>0$, we have

[TABLE]

By the chain rule of differentiation,

[TABLE]

Therefore, the right-hand derivative at $\epsilon=0$ exists and is given by

[TABLE]

By the assumptions (TReg-$\Phi$-(${n+k,k}$)) (which implies (TReg-$\Phi$-(${n,k-1}$))) and (TLip-$\Phi$-(${k}$)), we can repeat the same argument as in Theorem 2.6. For any signed measures $m_{1},\ldots,m_{\hat{n}}$ and $\hat{n}\leq k-1$,

[TABLE]

The second part of (LABEL:right_hand_derivative_long_expression) can be computed by Lemma 4.4. Therefore, by (LABEL:right_hand_derivative_long_expression), (4.9) and Lemma 4.4,

[TABLE]

For each $\Lambda\in e(\Lambda_{k-1})$ and $i\in\{1,\ldots,\hat{n}\}$, we define functions $\Theta^{(1)},\Theta^{(2)}_{\Lambda,i}:\mathbb{T}^{d}\to\mathbb{R}$ by

[TABLE]

By the assumption (TReg-$\Phi$-(${n+k,k}$)), it is clear that $\Theta^{(1)},\Theta^{(2)}_{\Lambda,i}\in W^{n+k,\infty}.$ Therefore, using the notations (3.31) and (3.32), Theorem 3.10 implies that

[TABLE]

which shows that $\frac{\delta^{{k}}\mathcal{U}}{\delta m^{k}}$ exists and is given by

[TABLE]

By adopting the same normalisation argument as (4.5) and (4.6), formula (4.10) gives

[TABLE]

Finally, if we also assume (TInt-$\Phi$-(${n+k-1,k}$)), then by Theorem 3.5, for any $\Lambda\in e(\Lambda_{k})$,

[TABLE]

∎

Acknowledgements

The author is indebted to Prof. Pierre Cardaliaguet and Dr. Łukasz Szpruch for useful suggestions in various occasions, and to Prof. François Delarue for the help in developing the proof of Proposition 2.2.