On Dean–Kawasaki Dynamics with Smooth Drift Potential

Vitalii Konarovskyi; Tobias Lehmann; Max von Renesse

doi:10.1007/s10955-019-02449-3

Journal of Statistical Physics

pp 1–16 | Cite as

On Dean–Kawasaki Dynamics with Smooth Drift Potential

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Vitalii Konarovskyi
Tobias Lehmann
Max von Renesse

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First Online: 30 November 2019

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Abstract

We consider the Dean–Kawasaki equation with smooth drift interaction potential and show that measure-valued solutions exist only in certain parameter regimes in which case they are given by finite Langevin particle systems with mean field interaction.

Keywords

Dean–Kawasaki equation Langevin dynamics Wasserstein diffusion Itô formula for measure-valued processes

Mathematics Subject Classification

60H15 60K35 82C22 60G57 82C31

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Notes

Appendix A: Approximation of Differentiable Functions on ${\mathcal {M}}_F({\mathbb {R}}^d)$

Appendix A.1: Approximation of Differentiable Functions on ${\mathcal {M}}_F([a,b]^d)$

In this section, we fix $a,b\in {\mathbb {R}}$, $a<b$, and denote $K:=[a,b]^d$ and ${\mathcal {M}}_F:={\mathcal {M}}_F([a,b]^d)$, for convenience of notation. We remark that each function from $\mathrm {C}({\mathcal {M}}_F)$ is bounded on ${\mathcal {N}}_C:={\mathcal {N}}_C(K)=\{\mu \in {\mathcal {M}}_F:\ \mu (K)\le C\}$ for all $C>0$, since ${\mathcal {N}}_C$ is compact in ${\mathcal {M}}_F$.

We are going to introduce an analog of the Weierstrass approximation of functions from $\mathrm {C}^{k,m}({\mathcal {M}}_F)$. For this we use multiplicative Bernstein polynomials on $K=[a,b]^d$ (see e.g. [35]). Let $g\in \mathrm {C}(K)$. We set for $n\ge 1$

$$\begin{aligned} B_n(g)(x)=\sum _{j_1,\ldots j_d=0}^ng\left( a_{j_1,\ldots ,j_d}^n\right) \varphi _{j_1,\ldots ,j_d}^n(x),\quad x\in K, \end{aligned}$$

where

$$\begin{aligned} a_{j_1,\ldots ,j_d}^n=\left( a+\frac{j_1(b-a)}{n},\ldots ,a+\frac{j_d(b-a)}{n}\right) \end{aligned}$$

and

$$\begin{aligned} \varphi _{j_1,\ldots ,j_d}^n(x)=\frac{1}{(b-a)^d}\prod _{k=1}^dC_n^{j_k}(x_k-a)^{j_k}(b-x_k)^{n-j_k},\quad x\in K. \end{aligned}$$

Here $C_n^i=\frac{n!}{i!(n-i)!}$.

We will consider $B_n$, $n\ge 1$, as linear operators from $\mathrm {C}^m(K)$ to $\mathrm {C}^m(K)$.

Proposition 1

Let $m\in {\mathbb {N}}_0$. Then the family of linear operators $B_n:\mathrm {C}^m(K)\rightarrow \mathrm {C}^m(K)$, $n\in {\mathbb {N}}$, satisfies the following properties:

(B1)
$\{B_n\}_{n\ge 1}$ is a family of uniformly bounded linear operators on $\mathrm {C}^m(K)$;
(B2)
For each $g\in \mathrm {C}^m(K)$ and $l\in {\mathbb {N}}_0^d$, $|l|\le m$,
$$\begin{aligned} \frac{\partial ^{|l|}}{\partial x^l}B_n(g)\rightarrow \frac{\partial ^{|l|}}{\partial x^l}g\quad \text{ in }\ \ \mathrm {C}(K),\ \ \text{ as }\ \ n\rightarrow \infty , \end{aligned}$$
that is, $B_n(g)\rightarrow g$ in $\mathrm {C}^m(K)$, as $n\rightarrow \infty $.
(B3)
If $g_k\rightarrow g$ in $\mathrm {C}^m(K)$, $k\rightarrow \infty $, then for each $l\in {\mathbb {N}}_0^d$, $|l|\le m$,
$$\begin{aligned} \frac{\partial ^{|l|}}{\partial x^l}B_n(g_k)\rightarrow \frac{\partial ^{|l|}}{\partial x^l}g\quad \text{ in }\ \ \mathrm {C}(K),\ \ \text{ as }\ \ n,k\rightarrow \infty , \end{aligned}$$
that is, $B_n(g_k)\rightarrow g$ in $\mathrm {C}^m(K)$, as $n,k\rightarrow \infty $.

Proof

For $K=[0,1]^d$ Property (B2) was proved in [35]. The general case can be obtained by the rescaling. Next, for each $g\in \mathrm {C}^m(K)$ Property (B2) implies the boundedness of $\{\Vert B_n(g)\Vert _{\mathrm {C}^m(K)}\}_{n\ge 1}$. By the Banach-Steinhaus theorem, we obtain (B1). Property (B3) easily follows from (B1) and (B2). $\square $

Now, we introduce an analog of Bernstein polynomials on ${\mathcal {M}}_F$. We set for each $\mu \in {\mathcal {M}}_F$

$$\begin{aligned} \chi _n(\mu ):=\sum _{j_1,\ldots ,j_d=0}^n\left\langle \varphi _{j_1,\ldots ,j_d}^n,\mu \right\rangle \delta _{a_{j_1,\ldots ,j_d}^n},\quad n\ge 1, \end{aligned}$$

where $\delta _{c}$ is the point measure at $c\in {\mathbb {R}}^d$, i.e. $\delta _{c}(A)$ equals 1 if $c\in A$ and 0 otherwise. We also define for every $F\in \mathrm {C}({\mathcal {M}}_F)$

$$\begin{aligned} P_n(F)(\mu ):=F\left( \chi _n(\mu )\right) ,\quad \mu \in {\mathcal {M}}_F,\ \ n\ge 1. \end{aligned}$$

(7)

Setting

$$\begin{aligned} u_n^F(z):=F\left( \sum _{j_1,\ldots ,j_d=0}^nz_{j_1,\ldots ,j_d}\delta _{a_{j_1,\ldots ,j_d}^n}\right) ,\quad z\in [0,\infty )^{(n+1)^d}, \end{aligned}$$

it is easy to see that $u_n^F\in \mathrm {C}\left( [0,\infty )^{(n+1)^d}\right) $ and

$$\begin{aligned} P_n(F)(\mu )=u_n^F\left( \left( \left\langle \varphi _{j_1,\ldots ,j_d}^n,\mu \right\rangle \right) _{j_1,\ldots ,j_d=0}^n\right) ,\quad \mu \in {\mathcal {M}}_F,\ \ n\ge 1. \end{aligned}$$

(8)

We will denote by $\mathrm {id}$ the identity map on ${\mathcal {M}}_F$, that is, $\mathrm {id}(\mu )=\mu $, $\mu \in {\mathcal {M}}_F$.

Proposition 2

For each $n\ge 1$ the map $\chi _n:{\mathcal {M}}_F\rightarrow {\mathcal {M}}_F$ is continuous and for each sequence $\{\mu _k\}_{k\ge 1}$ converging $\mu $ in ${\mathcal {M}}_F$ one has $\chi _n(\mu _k)\rightarrow \mu $ in ${\mathcal {M}}_F$ as $n,k\rightarrow \infty $. Moreover, $\chi _n$ maps ${\mathcal {N}}_C$ to ${\mathcal {N}}_C$ for all $C>0$ and $n\ge 1$.

Remark 2

Since the set ${\mathcal {N}}_C=\{\mu \in {\mathcal {M}}_F:\ \mu (K)\le C\}$ is compact in ${\mathcal {M}}_F$, we have that for each $C>0$$\chi _n\rightarrow \mathrm {id}$ uniformly on ${\mathcal {N}}_C$ as $n\rightarrow \infty $, by Proposition 2.

Remark 3

Proposition 2 implies that $P_n$ is a linear map from $\mathrm {C}({\mathcal {M}}_F)$ to $\mathrm {C}({\mathcal {M}}_F)$.

Proof of Proposition 2

The continuity of $\chi _n$ is trivial. We take an arbitrary sequence $\{\mu _k\}_{k\ge 1}$ in ${\mathcal {M}}_F$ which converges to $\mu $ and $g\in \mathrm {C}(K)$. Then by Proposition 1,

$$\begin{aligned} \langle g,\chi _n(\mu _k)\rangle&=\sum _{j_1,\ldots ,j_d=0}^ng\left( a_{j_1,\ldots ,j_d}^n\right) \left\langle \varphi _{j_1,\ldots ,j_d}^n,\mu _k\right\rangle \\&=\left\langle \sum _{j_1,\ldots ,j_d=0}^ng\left( a_{j_1,\ldots ,j_d}^n\right) \varphi _{j_1,\ldots ,j_d}^n,\mu _k\right\rangle \\&=\left\langle B_n(g),\mu _k\right\rangle \rightarrow \langle g,\mu \rangle ,\quad n,k\rightarrow \infty , \end{aligned}$$

since the map $\mathrm {C}(K)\times {\mathcal {M}}_F\ni (g,\mu )\mapsto \langle g,\mu \rangle \in {\mathbb {R}}$ is continuous.

Due to the equality

$$\begin{aligned} \chi _n(\mu )(K)=\sum _{j_1,\ldots ,j_d=0}^n\left\langle \varphi _{j_1,\ldots ,j_d}^n,\mu \right\rangle =\left\langle \sum _{j_1,\ldots ,j_d=0}^n\varphi _{j_1,\ldots ,j_d}^n,\mu \right\rangle =\left\langle 1,\mu \right\rangle =\mu (K), \end{aligned}$$

$\chi _n$ maps ${\mathcal {N}}_C$ to ${\mathcal {N}}_C$. $\square $

Proposition 3

For each $F\in \mathrm {C}({\mathcal {M}}_F)$ and $C>0$ we have that $P_n(F)\rightarrow F$ uniformly on ${\mathcal {N}}_C$ as $n\rightarrow \infty $, that is,

$$\begin{aligned} \sup _{\mu \in {\mathcal {N}}_C}|P_n(F)(\mu )-F(\mu )|\rightarrow 0,\quad n\rightarrow \infty . \end{aligned}$$

Remark 4

Proposition 3 yields that for each $F\in \mathrm {C}({\mathcal {M}}_F)$$P_n(F)\rightarrow F$ in $\mathrm {C}({\mathcal {M}}_F)$ as $n\rightarrow \infty $.

Proof of Proposition 3

We assume that the statement is not true. Then there exist $\varepsilon >0$ and a sequence $\{\mu _n\}_{n\ge 1}$ in ${\mathcal {N}}_C$ such that $|P_n(F)(\mu _n)-F(\mu _n)|\ge \varepsilon $ for all $n\ge 1$. Since ${\mathcal {N}}_C$ is compact, we may assume that $\mu _n\rightarrow \mu $ without loss of generality. But by Proposition 2 and the continuity of F, we have

$$\begin{aligned} P_n(F)(\mu _n)-F(\mu _n)=F(\chi _n(\mu _n))-F(\mu _n)\rightarrow F(\mu )-F(\mu )=0,\quad n\rightarrow \infty , \end{aligned}$$

which contradicts the assumption. $\square $

We note that the space $\mathrm {C}({\mathcal {N}}_C)$ of continuous functions from ${\mathcal {N}}_C$ to ${\mathbb {R}}$ furnished with the uniform norm is a Banach space. It is easy to see that for each $n\ge 1$ the map $P_n$ is a continuous linear operator from $\mathrm {C}({\mathcal {N}}_C)$ to $\mathrm {C}({\mathcal {N}}_C)$. Indeed, the map $\chi _n$ maps ${\mathcal {N}}_C$ to ${\mathcal {N}}_C$, by Proposition 2. The continuity trivially follows from the form of $P_n$ (see (7)).

Corollary 1

The family $\{P_n\}_{n\ge 1}$ of linear operators on $\mathrm {C}({\mathcal {N}}_C)$ is uniformly bounded.

Proof

The corollary immediately follows from Proposition 3 and the Banach–Steinhaus theorem. $\square $

Lemma 2

Let $F\in \mathrm {C}^k({\mathcal {M}}_F)$ for some $k\in \{1,2\}$. Then the function $u_n^F$ belongs to $\mathrm {C}^k\left( [0,\infty )^{(n+1)^d}\right) $. Moreover,

$$\begin{aligned} \frac{\partial }{\partial z_{i_1,\ldots ,i_d}}u_n^F(z)=F'\left( \sum _{j_1,\ldots ,j_d=0}^nz_{j_1,\ldots ,j_d}\delta _{a_{j_1,\ldots ,j_d}^n};a_{i_1,\ldots ,i_d}^n\right) ,\quad z\in [0,\infty )^{(n+1)^d}, \end{aligned}$$

for all $i_1,\ldots ,i_d$ and

$$\begin{aligned} \frac{\partial ^2}{\partial z_{j_1,\ldots ,j_d}\partial z_{i_1,\ldots ,i_d}}u_n^F(z){=}F''\left( \sum _{l_1,\ldots ,l_d=0}^nz_{l_1,\ldots ,l_d}\delta _{a_{l_1,\ldots ,l_d}^n};a_{j_1,\ldots ,j_d}^n,a_{i_1,\ldots ,i_d}^n\right) ,\quad z{\in }[0,\infty )^{(n+1)^d}, \end{aligned}$$

for all $j_1,\ldots ,j_d$, $i_1,\ldots ,i_d$, if $k=2$.

Proof

The proof easily follows from the definition of $F'$ and $F''$. $\square $

Proposition 4

Let $F\in \mathrm {C}^{k,m}({\mathcal {M}}_F)$ for some $k\in \{1,2\}$ and $m\ge 0$. Then for every $n\ge 1$$P_n(F)\in \mathrm {C}^{k,\infty }$ and for each $\mu \in {\mathcal {M}}_F$, $x,y\in K$

$$\begin{aligned} P_n'(F)(\mu ;x)&=\sum _{j_1,\ldots ,j_d=0}^nF'\left( \chi _n(\mu );a_{j_1,\ldots ,j_d}^n\right) \varphi _{j_1,\ldots ,j_d}^n(x)\\&=\sum _{j_1,\ldots ,j_d=0}^nP_n\left( F'\left( \cdot ;a_{j_1,\ldots ,j_d}^n\right) \right) (\mu )\varphi _{j_1,\ldots ,j_d}^n(x). \end{aligned}$$

and if $k=2$

$$\begin{aligned} P_n''(F)(\mu ;x,y)&=\sum _{j_1,\ldots ,j_d=0}^n\sum _{i_1,\ldots ,i_d=0}^nF''\left( \chi _n(\mu );a_{j_1,\ldots ,j_d}^n,a_{i_1,\ldots ,i_d}^n\right) \varphi _{j_1,\ldots ,j_d}^n(x)\varphi _{i_1,\ldots ,i_d}^n(y)\\&=\sum _{j_1,\ldots ,j_d=0}^n\sum _{i_1,\ldots ,i_d=0}^nP_n\left( F''\left( \cdot ;a_{j_1,\ldots ,j_d}^n,a_{i_1,\ldots ,i_d}^n\right) \right) (\mu )\varphi _{j_1,\ldots ,j_d}^n(x)\varphi _{i_1,\ldots ,i_d}^n(y). \end{aligned}$$

Proof

The proposition follows from the definition of the derivatives $F'$, $F''$, equality (8) and Lemma 2. Indeed,

$$\begin{aligned} P_n'(F)(\mu ;x)&=\sum _{j_1,\ldots ,j_d=0}^n\frac{\partial }{\partial z_{j_1,\ldots ,j_d}}u_n^F\left( \left( \left\langle \varphi _{j_1,\ldots ,j_d}^n,\mu \right\rangle \right) _{j_1,\ldots ,j_d=0}^n\right) \varphi _{j_1,\ldots ,j_d}^n(x)\\&=\sum _{j_1,\ldots ,j_d=0}^nP_n\left( F'\left( \cdot ;a_{j_1,\ldots ,j_d}^n\right) \right) (\mu )\varphi _{j_1,\ldots ,j_d}^n(x). \end{aligned}$$

Similarly, one can obtain the equality for $P_n''(F)(\mu ;x,y)$. $\square $

Theorem 4

Let $F\in \mathrm {C}^{k,m}({\mathcal {M}}_F)$ for some $k\in \{1,2\}$ and $m\ge 0$. Then for each $l,{\tilde{l}}\in {\mathbb {N}}_0^d$, $|l|+|{\tilde{l}}|\le m$ and $C>0$ one has

$$\begin{aligned} \sup _{x\in K,\ \mu \in {\mathcal {N}}_C}\left| \frac{\partial ^{|l|}}{\partial x^l} P_n'(F)(\mu ;x)-\frac{\partial ^{|l|}}{\partial x^l}F'(\mu ;x)\right| \rightarrow 0,\quad n\rightarrow \infty . \end{aligned}$$

(9)

and if $k=2$

$$\begin{aligned} \sup _{x,y\in K,\ \mu \in {\mathcal {N}}_C}\left| \frac{\partial ^{|l|}}{\partial x^l}\frac{\partial ^{|{\tilde{l}}|}}{\partial y^{{\tilde{l}}}} P_n''(F)(\mu ;x,y)-\frac{\partial ^{|l|}}{\partial x^l}\frac{\partial ^{|{\tilde{l}}|}}{\partial y^{{\tilde{l}}}}F''(\mu ;x,y)\right| \rightarrow 0,\quad n\rightarrow \infty . \end{aligned}$$

(10)

Proof

We will prove the theorem similarly as Proposition 3. We start with (9). If (9) does not hold, then there exist $\varepsilon >0$ and sequences $\{\mu _n\}_{n\ge 1}\subset {\mathcal {N}}_C$, $\{x_n\}_{n\ge 1}\in K$ such that

$$\begin{aligned} \left| \frac{\partial ^{|l|}}{\partial x^l} P_n'(F)(\mu _n;x_n)-\frac{\partial ^{|l|}}{\partial x^l}F'(\mu _n;x_n)\right| \ge \varepsilon \end{aligned}$$

(11)

for all $n\ge 1$. Since ${\mathcal {N}}_C$ and K are compact sets, we may assume that $\mu _n\rightarrow \mu _0$ and $x_n\rightarrow x_0$ as $n\rightarrow \infty $, without loss of generality. So, we compute

$$\begin{aligned} \frac{\partial ^{|l|}}{\partial x^l} P_n'(F)(\mu _n;x_n)&=\frac{\partial ^{|l|}}{\partial x^l}\sum _{j_1,\ldots ,j_d=0}^nF'\left( \chi _n(\mu _n);a_{j_1,\ldots ,j_d}^n\right) \varphi _{j_1,\ldots ,j_d}^n(x_n)\\&=\frac{\partial ^{|l|}}{\partial x^l}B_n\left( F'\left( \chi _n(\mu _n);\cdot \right) \right) (x_n). \end{aligned}$$

Since $F'$ is continuous on ${\mathcal {M}}_F\times K$ and K is compact, it is easy to see that $F'(\chi _n(\mu _n);\cdot )\rightarrow F'(\mu _0;\cdot )$ in $\mathrm {C}(K)$ as $n\rightarrow \infty $, using Proposition 2. Thus, by Proposition 1 (B3),

$$\begin{aligned} \frac{\partial ^{|l|}}{\partial x^l} P_n'(F)(\mu _n;x_n)=\frac{\partial ^{|l|}}{\partial x^l}B_n\left( F'\left( \chi _n(\mu _n);\cdot \right) \right) (x_n)\rightarrow \frac{\partial ^{|l|}}{\partial x^l}F'\left( \mu _0;x_0\right) ,\quad n\rightarrow \infty , \end{aligned}$$

that contradicts (11).

The uniform convergence (10) can be proved by the same argument taking into an account that

$$\begin{aligned}&\frac{\partial ^{|l|}}{\partial x^l} \frac{\partial ^{|{\tilde{l}}|}}{\partial y^{{\tilde{l}}}}P_n''(F)(\mu ;x,y)\\&\quad =\frac{\partial ^{|l|}}{\partial x^l}\frac{\partial ^{|{\tilde{l}}|}}{\partial y^{{\tilde{l}}}}\sum _{j_1,\ldots ,j_d=0}^n\sum _{i_1,\ldots ,i_d=0}^nF''\left( \chi _n(\mu _n);a_{j_1,\ldots ,j_d}^n,a_{i_1,\ldots ,i_d}^n\right) \varphi _{j_1,\ldots ,j_d}^n(x)\varphi _{i_1,\ldots ,i_d}^n(y)\\&\quad =\frac{\partial ^{|l|}}{\partial x^l}\frac{\partial ^{|{\tilde{l}}|}}{\partial y^{{\tilde{l}}}}{\tilde{B}}_n\left( F''\left( \chi _n(\mu _n);\cdot ,\cdot \right) \right) (x,y), \end{aligned}$$

where ${\tilde{B}}_n$, $n\ge 1$, are the Bernstein polynomials defined for functions from $\mathrm {C}(K^2)$. $\square $

Appendix A.2: Approximation of Differentiable Functions on ${\mathcal {M}}_F({\mathbb {R}}^d)$

We fix a smooth bounded function $\psi :{\mathbb {R}}^d\rightarrow {\mathbb {R}}$ and define a map from ${\mathcal {M}}_F({\mathbb {R}}^d)$ to ${\mathcal {M}}_F({\mathbb {R}}^d)$ as follows

$$\begin{aligned} \vartheta _{\psi }(\mu )(dx):=\psi (x)\mu (dx). \end{aligned}$$

We also assume that $\psi $ has a compact support. Let $K=[a,b]^d$ such that ${{\,\mathrm{supp}\,}}\psi \subset K$. Then the measure $\vartheta _{\psi }(\mu )$ is supported on K and, consequently, we can consider $\vartheta _{\psi }$ as a map from ${\mathcal {M}}_F({\mathbb {R}}^d)$ to ${\mathcal {M}}_F(K)\subset {\mathcal {M}}_F({\mathbb {R}}^d)$.

Lemma 3

The map $\vartheta _{\psi }:{\mathcal {M}}_F({\mathbb {R}}^d)\rightarrow {\mathcal {M}}_F(K)$ is continuous.

Proof

The proof trivially follows from the definition of $\vartheta _{\psi }$. $\square $

We define for each $F\in \mathrm {C}({\mathcal {M}}_F(K))$ a new function as follows

$$\begin{aligned} \varGamma _{\psi }(F)(\mu ):=\varGamma _{\psi ,K}(F)(\mu ):=F(\vartheta _{\psi }(\mu )),\quad \mu \in {\mathcal {M}}_F({\mathbb {R}}^d). \end{aligned}$$

Lemma 4

If $F\in \mathrm {C}^{k,m}({\mathcal {M}}_F(K))$ for some $k\in \{0,1,2\}$ and $m\ge 0$, then $\varGamma _{\psi }(F)\in \mathrm {C}^{k,m}({\mathcal {M}}_F({\mathbb {R}}^d))$. Moreover,

$$\begin{aligned} \varGamma _{\psi }'(F)(\mu ;x)=\varGamma _{\psi }(F'(\cdot ;x))(\mu )\psi (x)=F'(\vartheta _{\psi }(\mu );x)\psi (x),\quad \mu \in {\mathcal {M}}_F({\mathbb {R}}^d),\ \ x\in {\mathbb {R}}^d, \end{aligned}$$

and

$$\begin{aligned}&\varGamma _{\psi }''(F)(\mu ;x)=\varGamma _{\psi }(F''(\cdot ;x,y))(\mu )\psi (x)\psi (y)\\&\quad =F''(\vartheta _{\psi }(\mu );x,y)\psi (x)\psi (y),\quad \mu \in {\mathcal {M}}_F({\mathbb {R}}^d),\ \ x\in {\mathbb {R}}^d. \end{aligned}$$

Remark 5

We remark that $\psi (x)=0$ for all $x\in K^c:={\mathbb {R}}^d\setminus K$, thus, we assume that the multiplication $f(x)\psi (x)=0$, even if f is not defined for such x.

Proof of Lemma 4

The continuity of $\varGamma _{\psi }(F)$ immediately follows from Lemma 3. The derivatives of $\varGamma _{\psi }(F)$ can be computed using the following observation

$$\begin{aligned} \varGamma _{\psi }(F)(\mu +\varepsilon \delta _x)=F(\vartheta _{\psi }(\mu )+\varepsilon \psi (x)\delta _x),\quad \mu \in {\mathcal {M}}_F({\mathbb {R}}^d),\ \ x\in {\mathbb {R}}^d,\ \ \varepsilon >0. \end{aligned}$$

$\square $

Lemma 5

Let $\{\psi _n\}_{n\ge 1}$ be a sequence of uniformly bounded continuous functions on ${\mathbb {R}}^d$ which pointwise converges to $\psi \in \mathrm {C}_b({\mathbb {R}})$, then $\vartheta _{\psi _n}(\mu )\rightarrow \vartheta _{\psi }(\mu )$, $n\rightarrow \infty $, for each $\mu \in {\mathcal {M}}_F$.

Proof

The lemma easily follows from the dominated convergence theorem. $\square $

Proposition 5

Let $F\in \mathrm {C}^{k,m}({\mathcal {M}}_F({\mathbb {R}}^d))$ for some $k\in \{0,1,2\}$ and $m\ge 0$. Let $\{\psi _n\}_{n\ge 1}$ be a sequence of smooth bounded functions on ${\mathbb {R}}^d$ such that $\psi _n\rightarrow \psi $ in $\mathrm {C}^m({\mathbb {R}}^d)$, $n\rightarrow \infty $, and $\{\psi _n\}_{n\ge 1}$ is uniformly bounded. Then for each $\mu \in {\mathcal {M}}_F({\mathbb {R}}^d)$

$$\begin{aligned}&\varGamma _{\psi _n}(F)(\mu )\rightarrow \varGamma _{\psi }(F)(\mu ),\quad n\rightarrow \infty , \\&\varGamma _{\psi _n}'(F)(\mu ;\cdot )\rightarrow \varGamma _{\psi }'(F)(\mu ;\cdot )\ \ \text{ in }\ \ \mathrm {C}^m({\mathbb {R}}^d),\quad n\rightarrow \infty ,\quad \text{ if }\ \ k\ge 1, \end{aligned}$$

and

$$\begin{aligned} \varGamma _{\psi _n}''(F)(\mu ;\cdot )\rightarrow \varGamma _{\psi }''(F)(\mu ;\cdot )\ \ \text{ in }\ \ \mathrm {C}^m({\mathbb {R}}^{2d}),\quad n\rightarrow \infty ,\quad \text{ if }\ \ k=2. \end{aligned}$$

Proof

We first note that $F'(\mu _n;\cdot )\rightarrow F'(\mu ;\cdot )$ in $\mathrm {C}^m({\mathbb {R}}^d)$ and $F''(\mu _n;\cdot )\rightarrow F''(\mu ;\cdot )$ in $\mathrm {C}^m({\mathbb {R}}^{2d})$ as $\mu _n\rightarrow \mu $, if $F\in \mathrm {C}^{2,m}({\mathcal {M}}_F({\mathbb {R}}^d))$. Thus, the statement immediately follows from lemmas 4 and 5. $\square $

We denote by $\mathrm {C}_P^k({\mathcal {M}}_F({\mathbb {R}}^d))$ the set of functions on ${\mathcal {M}}_F({\mathbb {R}}^d)$ of the form

$$\begin{aligned} G(\mu )=u(\langle \varphi _1,\mu \rangle ,\ldots ,\langle \varphi _p,\mu \rangle ),\quad \mu \in {\mathcal {M}}_F({\mathbb {R}}^d), \end{aligned}$$

where $\varphi _i$, $i\in [p]=\{1,\ldots ,p\}$, are positive smooth functions with compact supports, ${u\in C^k([0,+\infty )^p)}$ and $p\in {\mathbb {N}}$.

Remark 6

We remark that a function belongs to $\mathrm {C}^k([0,+\infty )^p)$ if and only if it can be extended to a function from $\mathrm {C}^k({\mathbb {R}}^p)$.

Let $F_K$ denote the restriction of a function F from $\mathrm {C}({\mathcal {M}}_F({\mathbb {R}}^d))$ to ${\mathcal {M}}_F(K)$.

Lemma 6

For each $F\in \mathrm {C}^{k,m}({\mathcal {M}}_F({\mathbb {R}}^d))$ the function $F_K$ belongs to $\mathrm {C}^{k,m}({\mathcal {M}}_F(K))$ and

$$\begin{aligned} F_K'(\mu ;x)= & {} F'(\mu ;x),\quad \mu \in {\mathcal {M}}_F(K),\ \ x\in K, \\ F_K''(\mu ;x,y)= & {} F'(\mu ;x,y),\quad \mu \in {\mathcal {M}}_F(K),\ \ x,y\in K. \end{aligned}$$

Proof

The proof of the lemma is trivial. $\square $

Theorem 5

Let $F\in \mathrm {C}^{k,m}({\mathcal {M}}_F({\mathbb {R}}^d))$ for some $k\in \{0,1,2\}$ and $m\ge 0$. Then there exists a sequence $\{F_n\}_{n\ge 1}$ from $\mathrm {C}_P^k({\mathcal {M}}_F({\mathbb {R}}^d))$ such that for all $\mu \in {\mathcal {M}}_F({\mathbb {R}}^d)$

$$\begin{aligned}&F_n(\mu )\rightarrow F(\mu ),\quad n\rightarrow \infty , \\&F_n'(\mu ;\cdot )\rightarrow F'(\mu ;\cdot )\ \ \text{ in }\ \ \mathrm {C}^m({\mathbb {R}}^d),\quad n\rightarrow \infty ,\quad \text{ if }\ \ k\ge 1, \end{aligned}$$

and

$$\begin{aligned} F''_n(\mu ;\cdot )\rightarrow F''(\mu ;\cdot )\ \ \text{ in }\ \ \mathrm {C}^m({\mathbb {R}}^{2d}),\quad n\rightarrow \infty ,\quad \text{ if }\ \ k=2. \end{aligned}$$

Moreover, if for some $C>0$ the functions F, $F'$ and $F''$ and their derivatives are bounded on sets ${\mathcal {N}}_C({\mathcal {M}}_F({\mathbb {R}}^d))$, ${\mathcal {N}}_C({\mathcal {M}}_F({\mathbb {R}}^d))\times {\mathbb {R}}^d$ and ${\mathcal {N}}_C({\mathcal {M}}_F({\mathbb {R}}^d))\times {\mathbb {R}}^{2d}$, respectively, then the sequence $\{F_n\}_{n\ge 1}$ can be chosen with $\{F_n\}_{n\ge 1}$, $\{F_n'\}_{n\ge 1}$, $\{F_n''\}_{n\ge 1}$ and their derivatives uniformly bounded in n on ${\mathcal {N}}_C({\mathcal {M}}_F({\mathbb {R}}^d))$, ${\mathcal {N}}_C({\mathcal {M}}_F({\mathbb {R}}^d))\times {\mathbb {R}}^d$ and ${{\mathcal {N}}_C({\mathcal {M}}_F({\mathbb {R}}^d))\times {\mathbb {R}}^{2d}}$, respectively.

Proof

We assume that $k=2$. Let $\psi _n$ be a sequence of smooth functions on ${\mathbb {R}}^d$ such that they take values from [0, 1], ${{\,\mathrm{supp}\,}}\psi _n\subset K_n:=[-n,n]^d$, $\psi _n(x)=1$, $x\in [-n+1,n-1]^d$, and all derivatives are uniformly bounded in x and n, i.e. for each $l\in {\mathbb {N}}_0^d$, the set $\left\{ \frac{\partial ^{|l|}}{\partial x^l}\psi _n(x),\ \ x\in {\mathbb {R}}^d,\ \ n\ge 1\right\} $ is bounded. Let us fix a function $F\in \mathrm {C}^{k,m}({\mathcal {M}}_F({\mathbb {R}}^d))$. We are going to approximate $F_{K_n}$ by polynomials introduced in the previous section. So, by Proposition 3 and Theorem 4, for every $n\ge 1$ there exists a number $N_n\in {\mathbb {N}}$ such that

$$\begin{aligned}&\sup _{\mu \in {\mathcal {N}}_n({K_n})}\left| F_{K_n}(\mu )-P_{N_n}(F_{K_n})(\mu )\right| \le \frac{1}{n}, \\&\sup _{\mu \in {\mathcal {N}}_n({K_n})}\left\| F_{K_n}'(\mu ;\cdot )-P_{N_n}'(F_{K_n})(\mu ;\cdot )\right\| _{\mathrm {C}^m(K_n)}\le \frac{1}{n} \end{aligned}$$

and

$$\begin{aligned} \sup _{\mu \in {\mathcal {N}}_n({K_n})}\left\| F_{K_n}''(\mu ;\cdot )-P_{N_n}''(F_{K_n})(\mu ;\cdot )\right\| _{\mathrm {C}^m(K_n^2)}\le \frac{1}{n}, \end{aligned}$$

where ${\mathcal {N}}_n({K_n})$ is defined in Lemma 1 with $C=n$ and $K=K_n$, and $P_{N_n}$ is defined by (7) for $K=K_n$.

We set $F_n(\mu ):=\varGamma _{\psi _n}\left( P_{N_n}(F_{K_n})\right) (\mu )=P_{N_n}(F_{K_n})\left( \vartheta _{\psi _n}(\mu )\right) $, $\mu \in {\mathcal {M}}_F({\mathbb {R}}^d)$. By Lemma 4, $F_n\in \mathrm {C}^{k,m}({\mathcal {M}}_F({\mathbb {R}}^d))$. Moreover, it is easy to see that $F_n\in \mathrm {C}_P^k({\mathcal {M}}_F({\mathbb {R}}^d))$, by the definition of $P_{N_n}$ and $\varGamma _{\psi _n}$.

Next, we are going to show that $\{F_n\}_{n\ge 1}$ is the sequence which approximates F. We fix $\varepsilon >0$, $\mu \in {\mathcal {M}}_F({\mathbb {R}}^d)$ and a compact set $K\subset {\mathbb {R}}^d$. We choose ${\tilde{n}}\in {\mathbb {N}}$ such that $\frac{1}{{\tilde{n}}}<\frac{\varepsilon }{2}$, $K\subset K_{{\tilde{n}}}$, $\mu ({\mathbb {R}}^d)\le {\tilde{n}}$ and for all $n\ge {\tilde{n}}$

$$\begin{aligned}&\left| F(\mu )-\varGamma _{\psi _n}(F)(\mu )\right|<\frac{\varepsilon }{2}, \\&\left\| F'(\mu ;\cdot )-\varGamma _{\psi _n}'(F)(\mu ;\cdot )\right\| _{C^m(K)}<\frac{1}{\varepsilon } \end{aligned}$$

and

$$\begin{aligned} \left\| F''(\mu ;\cdot )-\varGamma _{\psi _n}''(F)(\mu ;\cdot )\right\| _{C^m(K^2)}<\frac{1}{\varepsilon }. \end{aligned}$$

Such ${\tilde{n}}$ exists due to Proposition 5, since $\{\psi _n\}_{n\ge 1}$ converges to the function $\psi =1$ in $\mathrm {C}^m({\mathbb {R}}^d)$. Let us remark that $\varGamma _{\psi _n}(F)=\varGamma _{\psi _n}(F_{K_n})$, $\varGamma _{\psi _n}'(F)=\varGamma _{\psi _n}'(F_K)$ and $\varGamma _{\psi _n}''(F)=\varGamma _{\psi _n}''(F_K)$, by Lemma 6. So, now we can estimate for each $n\ge {\tilde{n}}$

$$\begin{aligned} \left| F(\mu )-F_n(\mu )\right|&\le \left| F(\mu )-\varGamma _{\psi _n}(F)(\mu )\right| +\left| \varGamma _{\psi _n}(F_{K_n})(\mu )-\varGamma _{\psi _n}\left( P_{N_n}(F_{K_n})\right) (\mu )\right| \\&\le \frac{\varepsilon }{2}+\left| F_{K_n}(\vartheta _{\psi _n}(\mu ))-P_{N_n}(F_{K_n})(\vartheta _{\psi _n}(\mu ))\right| \le \frac{\varepsilon }{2}+\frac{1}{n}\le \varepsilon , \end{aligned}$$

since $\vartheta _{\psi _n}(\mu )\in {\mathcal {N}}_n({K_n})$. Similarly, for each $n\ge {\tilde{n}}+1$, we have

$$\begin{aligned} \left\| F'(\mu ;\cdot )-F_n'(\mu ;\cdot )\right\| _{C^m(K)}&\le \left\| F'(\mu ;\cdot )-\varGamma _{\psi _n}'(F)(\mu ;\cdot )\right\| _{C^m(K)}\\&\quad +\left\| \varGamma _{\psi _n}'(F_{K_n})(\mu ;\cdot )-\varGamma _{\psi _n}'\left( P_{N_n}(F_{K_n})\right) (\mu ;\cdot )\right\| _{C^m(K)}\\&\le \frac{\varepsilon }{2}+\left| F_{K_n}'(\vartheta _{\psi _n}(\mu );\cdot )\psi _n-P_{N_n}'(F_{K_n})(\vartheta _{\psi _n}(\mu ;\cdot ))\psi _n\right\| _{C^m(K)}\\&\le \frac{\varepsilon }{2}+\frac{1}{n}\le \varepsilon , \end{aligned}$$

since $\psi _n(x)=1$ on K for all $n\ge {\tilde{n}}+1$. Analogously, $\left\| F''(\mu ;\cdot )-F_n''(\mu ;\cdot )\right\| _{C^m(K^2)}<\varepsilon $ for all $n\ge {\tilde{n}}+1$. The theorem is proved. $\square $

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Vitalii Konarovskyi
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Tobias Lehmann
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Max von Renesse
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1.Fakultät für Mathematik und InformatikUniversität LeipzigLeipzigGermany

On Dean–Kawasaki Dynamics with Smooth Drift Potential

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