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A Hamilton–Jacobi PDE Associated with Hydrodynamic Fluctuations from a Nonlinear Diffusion Equation

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Abstract

We study a class of Hamilton–Jacobi partial differential equations in the space of probability measures. In the first part of this paper, we prove comparison principles (implying uniqueness) for this class. In the second part, we establish the existence of a solution and give a representation using a family of partial differential equations with control. A large part of our analysis exploits special structures of the Hamiltonian, which might look mysterious at first sight. However, we show that this Hamiltonian structure arises naturally as limit of Hamiltonians of microscopical models. Indeed, in the third part of this paper, we informally derive the Hamiltonian studied before, in a context of fluctuation theory on the hydrodynamic scale. The analysis is carried out for a specific model of stochastic interacting particles in gas kinetics, namely a version of the Carleman model. We use a two-scale averaging method on Hamiltonians defined in the space of probability measures to derive the limiting Hamiltonian.

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Correspondence to Jin Feng.

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Communicated by M. Hairer.

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Jin Feng’s work was supported in part by US NSF Grant No. DMS-1440140 while he was in residence at MSRI - Berkeley, California, during fall 2018; and in part by a Simons Visiting Professorship to Mathematisches Forschungsinstitut Oberwolfach, Germany and to Technical University of Eindhoven, the Netherlands in November 2017, and in part by LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by French National Research Agency (ANR). He thanks Albert Fathi for helpful discussions on weak KAM theory on numerous occasions. Toshio Mikami’s work was supported by JSPS KAKENHI Grant Numbers JP26400136 and JP16H03948 from Japan. Johannes Zimmer’s work was supported through a Royal Society Wolfson Research Merit Award.

Appendix A. Some Properties of \({\mathcal {P}}({\mathcal {O}})\)

Appendix A. Some Properties of \({\mathcal {P}}({\mathcal {O}})\)

1.1 A.1 Quotients and Coverings

1.1.1 A.1.1 Projections and lifts

As defined in Sect. 1.3, we view \({\mathcal {O}}:= {{\mathbb {R}}}/{{\mathbb {Z}}}\) as a quotient space with corresponding quotient metric r. We define a projection \({{\mathsf {p}}}:{{\mathbb {R}}}\mapsto {\mathcal {O}}\) by

$$\begin{aligned} x := {{\mathsf {p}}}({\hat{x}}) = {\hat{x}} \pmod {1}, \quad \forall {\hat{x}} \in {{\mathbb {R}}}. \end{aligned}$$
(A.1)

Let \({\mathcal {P}}_2({{\mathbb {R}}})\) be the Wasserstein order-2 metric space [2]. For every \({\hat{\mu }} \in {\mathcal {P}}_2({{\mathbb {R}}})\), we define push forward of \({\hat{\mu }}\) by \(\mu := {{\mathsf {p}}}_\# {\hat{\mu }} \in {\mathcal {P}}({\mathcal {O}})\). That is, we project \({\hat{\mu }}\) to \(\mu \) in the following way

$$\begin{aligned} \mu (A) := {\hat{\mu }} \big ({{\mathsf {p}}}^{-1}(A)\big ):= \sum _{k \in {{\mathbb {Z}}}} {\hat{\mu }}(A + k), \quad \forall A \in {{\mathcal {B}}}({\mathcal O}). \end{aligned}$$
(A.2)

Here, we use the set \(A+k:= \{ x +k : x \in A\}\), and we write \({{\mathcal {B}}}({\mathcal {O}})\) to denote the collection of Borel sets in \({\mathcal {O}}\).

There are many ways to lift a probability measure \(\mu \in \mathcal P({\mathcal {O}})\) to a probability measure \({\hat{\mu }} \in \mathcal P_2({{\mathbb {R}}})\) such that \({{\mathsf {p}}}_\# {\hat{\mu }} = \mu \). Following Galaz-García, Kell, Mondino and Sosa [27], we now describe one class of such lifts. Given a family of weights

$$\begin{aligned} \alpha := \{ \alpha _k \in [0,1] : \sum _{k \in {{\mathbb {Z}}}} \alpha _k =1 , \sum _{k \in {{\mathbb {Z}}}} k^2 \alpha _k <\infty \}_{k \in {{\mathbb {Z}}}}, \end{aligned}$$
(A.3)

we introduce a probability measure on \({{\mathbb {R}}}\) by

$$\begin{aligned} \nu _x(d{\hat{x}}) :=\nu _x^\alpha (d {\hat{x}}):= \sum _{k \in {{\mathbb {Z}}}} \alpha _k \delta _{x+k}(d{\hat{x}}), \quad \forall x \in {{\mathcal {O}}}. \end{aligned}$$

Second, using the family of measures \(\{\nu _x \}_{x \in \mathcal O}\), we define a lift operator \(\Lambda : = \Lambda ^\alpha : \mathcal P({\mathcal {O}}) \mapsto {\mathcal {P}}_2({{\mathbb {R}}})\) as follows

$$\begin{aligned} \mu \mapsto {\hat{\mu }} := \Lambda (\mu ) := \int _{x \in {\mathcal {O}}} \nu _x \mu (dx). \end{aligned}$$
(A.4)

We note that \({{\mathsf {p}}}_\# {\hat{\mu }} = \mu \).

Let \(C_{\mathrm{per}}({{\mathbb {R}}})\) be the collection of continuous functions which are 1-periodic on \({{\mathbb {R}}}\). Similarly, we define \(C_{\mathrm{per}}^p({{\mathbb {R}}})\) for \(p =1,2,\ldots , \infty \). For each \({\hat{\varphi }} \in C_{\mathrm{per}}^p({{\mathbb {R}}})\), we have translation invariance

$$\begin{aligned} {\hat{\varphi }}({\hat{x}}) = {\hat{\varphi }}({\hat{x}}+k), \quad \forall k \in Z. \end{aligned}$$

Hence such a function \({\hat{\varphi }}\) projects to an element \(\varphi \in C^p({\mathcal {O}})\) as follows

$$\begin{aligned} \varphi (x):= {\hat{\varphi }} ({\hat{x}}), \quad \forall {\hat{x}} \in {{\mathsf {p}}}^{-1}(x). \end{aligned}$$
(A.5)

On the other hand, each \(\varphi \in C^p({\mathcal {O}})\) has a lift to \({\hat{\varphi }} \in C^p_{\mathrm{per}}({{\mathbb {R}}})\) defined by

$$\begin{aligned} {\hat{\varphi }}({\hat{x}}):= \varphi \big ({{\mathsf {p}}}({\hat{x}})\big ). \end{aligned}$$
(A.6)

Such a lift is translation invariant in \({{\mathbb {Z}}}\) and its projection (as defined in (A.1)) gives \(\varphi \).

Given \(\rho , \gamma \in {\mathcal {P}}({\mathcal {O}})\) and \(\varphi \in C^p({\mathcal {O}})\), for a fixed \(\alpha \), let \({\hat{\rho }}, {\hat{\gamma }} \in {\mathcal {P}}_2({{\mathbb {R}}})\) and \({\hat{\varphi }} \in C^p_{\mathrm{per}}({{\mathbb {R}}})\) be lifts as just defined. Then

$$\begin{aligned} \langle \rho - \gamma , \varphi \rangle = \langle {\hat{\rho }} - {\hat{\gamma }}, {\hat{\varphi }} \rangle . \end{aligned}$$

In particular, this implies that

$$\begin{aligned} \Vert \rho - \gamma \Vert _{-1} = \sup \Big ( \langle {\hat{\rho }} - {\hat{\gamma }}, {\hat{\varphi }} \rangle : \varphi \in C^\infty (\mathcal O), \int _{{\mathcal {O}}} |\partial _x \varphi |^2 dx \le 1 \Big ) . \end{aligned}$$
(A.7)

1.1.2 A.1.2 A random variable description

The constructions of projection and lifts of the previous subsection A.1.1 can be described using the language of random variables. In certain situations, this can be more intuitive.

Let \((\Omega , {\mathcal {F}}, {\mathbb {P}})\) be a probability space and let \((X,K):\Omega \mapsto {\mathcal {O}} \times {{\mathbb {Z}}}\) be a pair of random variables. We define the \({{\mathbb {R}}}\)-valued random variable

$$\begin{aligned} {\hat{X}}:=X+K. \end{aligned}$$

Then

$$\begin{aligned} X= {\hat{X}} \pmod {1}, \quad K:=\lfloor {X}\rfloor . \end{aligned}$$

If \({\hat{X}}\) has the probability law \({\hat{\rho }}\), then

$$\begin{aligned} \rho (dx):={\mathbb {P}}(X \in dx) = \sum _{k \in {{\mathbb {Z}}}} {\mathbb {P}}({\hat{X}} \in dx+k; K=k) = \sum _{k\in {{\mathbb {Z}}}} {\hat{\rho }}(dx+k). \end{aligned}$$

On the other hand, if X has the probability law \(\rho \), depending on the conditional probability law of the K,

$$\begin{aligned} \alpha _k:=\alpha _k(x):={{\mathbb {P}}}(K =k | X=x), \end{aligned}$$

or equivalently for the conditional law of \({\hat{X}}\)

$$\begin{aligned} \nu _x(d{\hat{x}}) := {{\mathbb {P}}}({\hat{X}} \in d {\hat{x}}|X=x) = {\mathbb P}(X+K \in d{\hat{x}} | X=x) =\sum _{k\in {{\mathbb {Z}}}} \alpha _k\delta _{x+k}(d{\hat{x}}), \end{aligned}$$

the lift defined in (A.4) becomes

$$\begin{aligned} {\hat{\rho }}(d{\hat{x}})&= {{\mathbb {P}}}({\hat{X}} \in d{\hat{x}}) = \int _{x \in {\mathcal {O}}} {{\mathbb {P}}}( X+K \in d {\hat{x}} |X=x) {{\mathbb {P}}}(dx) =\int _{x \in {\mathcal {O}}} \nu _x(d{\hat{x}}) \rho (dx). \end{aligned}$$

1.2 A.2 Equivalence of metric topologies

We recall inequality (1.18)

$$\begin{aligned} W_1(\rho ,\gamma ) \le \Vert \rho -\gamma \Vert _{-1}. \end{aligned}$$

Next, we establish a converse of sorts. The proof below is an adaptation of Lemma 4.1 in Mischler–Mouhot [39].

Lemma A.1

For every \(\rho , \gamma \in {\mathcal {P}}({\mathcal {O}})\), we have

$$\begin{aligned} \Vert \rho - \gamma \Vert _{-1} \le \frac{2}{\sqrt{\pi }} \sqrt{W_1(\rho ,\gamma )}. \end{aligned}$$

Proof

We can construct a probability space \((\Omega , {\mathcal {F}}, {\mathbb P})\) with two pairs of \({\mathcal {O}} \times {{\mathbb {Z}}}\)-valued random variables \((X,K_1), (Y,K_2)\) such that

$$\begin{aligned} \rho (dx)= {{\mathbb {P}}}(X \in dx), \quad \gamma (dy)={{\mathbb {P}}}(Y \in dy). \end{aligned}$$

We introduce

$$\begin{aligned} {\hat{X}}:=X+K_1 \text { and } {\hat{Y}}:=Y+K_2 \end{aligned}$$

and denote \({\hat{\rho }}(d{\hat{x}}) := {{\mathbb {P}}}({\hat{X}} \in d {\hat{x}})\) and \({\hat{\gamma }}(d{\hat{y}}):={{\mathbb {P}}} ({\hat{Y}} \in d {\hat{y}})\). We use the Fourier transform

$$\begin{aligned} {{\mathcal {F}}}[{\hat{\rho }}](\xi )&:=\frac{1}{\sqrt{2\pi }} \int _{{{\mathbb {R}}}} e^{-i z \cdot \xi } {\hat{\rho }}(dz ) = \frac{1}{\sqrt{2\pi }} {\mathbb E}[ e^{-i {\hat{X}} \cdot \xi } ], \end{aligned}$$

and

$$\begin{aligned} {\mathcal F}[{\hat{\gamma }}](\xi )&:= \frac{1}{\sqrt{2\pi }} \int _{{{\mathbb {R}}}} e^{-i z \cdot \xi } {\hat{\gamma }}(dz ) = \frac{1}{\sqrt{2\pi }} {{\mathbb {E}}}[ e^{-i {\hat{Y}} \cdot \xi }]. \end{aligned}$$

Then

$$\begin{aligned} |{{\mathcal {F}}}[{\hat{\rho }}](\xi ) -{{\mathcal {F}}}[{\hat{\gamma }}](\xi )| \le \frac{1}{\sqrt{2\pi }} \big | {{\mathbb {E}}} [e^{-i {\hat{X}} \cdot \xi } - e^{-i {\hat{Y}} \cdot \xi }) ] \big | \le \frac{|\xi |}{\sqrt{2\pi }} {{\mathbb {E}}} [|X-Y-K|], \end{aligned}$$

where \(K:=K_1-K_2\).

On the other hand, it holds that

$$\begin{aligned} \Vert \rho - \gamma \Vert _{-1}^2&= \sup \Big ( \langle {\hat{\rho }} - {\hat{\gamma }}, {\hat{\varphi }} \rangle : \varphi \in C^\infty (\mathcal O), \int _{{\mathcal {O}}} |\partial _x \varphi |^2 dx \le 1, \\&\quad {\hat{\varphi }} \text { is defined from} \varphi \text { as in }~(A.6), {\hat{\rho }}, {\hat{\gamma }} \text { are lifts of }\rho ,\gamma \text { as in }~(A.4) \Big )^2 \\&\le \sup \Big ( \langle {\hat{\rho }} - {\hat{\gamma }}, {\hat{\varphi }} \rangle : {\hat{\varphi }} \in C_c^\infty ({{\mathbb {R}}}), \int _{{{\mathbb {R}}}} |\partial _x {\hat{\varphi }}|^2 dx \le 1 \Big )^2 \\&= \int _{{{\mathbb {R}}}} \frac{|{{\mathcal {F}}}[{\hat{\rho }}] (\xi )- {\mathcal F}[{\hat{\gamma }}](\xi )|^2}{|\xi |^2} d \xi \\&\le \inf _{R>0} \Big ( \sup _{\xi \in {{\mathbb {R}}}}\frac{|{\mathcal F}[{\hat{\rho }}] (\xi )- {{\mathcal {F}}}[{\hat{\gamma }}](\xi )|^2}{|\xi |^2} \int _{|\xi |\le R} d \xi +\frac{ 4}{2 \pi } \int _{|\xi |>R} \frac{1}{|\xi |^2} d \xi \Big ). \end{aligned}$$

Therefore

$$\begin{aligned} \Vert \rho - \gamma \Vert _{-1}^2&\le \frac{1}{2 \pi } \inf _{R>0} \Big ( (2R) {{\mathbb {E}}}^2[| X-Y-K|] + \frac{8}{R} \Big ) = \frac{4}{\pi } {{\mathbb {E}}}[|X-Y-K|]. \end{aligned}$$

Next, we note that

$$\begin{aligned}&\big \{ \varphi (X,Y) \quad \big | \quad \varphi : {\mathcal {O}} \times {\mathcal {O}} \mapsto {{\mathbb {Z}}}\text { is measurable} \big \} \\&\quad \subset \big \{ K:=K_1-K_2 \quad \big | \quad K_1, K_2 \text { are }{{\mathbb {Z}}}\text {-valued random variables} \big \}. \end{aligned}$$

Therefore, with the quotient metric r defined in (1.15), we have

$$\begin{aligned}&\inf _K {{\mathbb {E}}}[|X-Y-K|] \\&\le \inf \Big \{ {{\mathbb {E}}}[ |X-Y-\varphi (X,Y)| ] : \text{ where } \varphi \text { is measurable function from } {\mathcal {O}} \times {\mathcal {O}} \text { to } {{\mathbb {Z}}}\Big \} \\&= {{\mathbb {E}}}[ \inf _{k \in {{\mathbb {Z}}}}|X-Y-k|] = {{\mathbb {E}}}[ r(X,Y)] = \int _{{\mathcal {O}} \times {\mathcal {O}}} r(x,y) {\varvec{\nu }}(dx, dy), \quad \forall {\varvec{\nu }} \in \Pi (\rho , \gamma ) \end{aligned}$$

(see (1.16) for the definition of \(\Pi (\rho ,\gamma )\)). This leads to

$$\begin{aligned} \Vert \rho - \gamma \Vert _{-1}^2 \le \frac{4}{\pi } W_1(\rho ,\gamma ). \end{aligned}$$

\(\square \)

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Feng, J., Mikami, T. & Zimmer, J. A Hamilton–Jacobi PDE Associated with Hydrodynamic Fluctuations from a Nonlinear Diffusion Equation. Commun. Math. Phys. 385, 1–54 (2021). https://doi.org/10.1007/s00220-021-04110-1

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