Infinite Particle Systems with Hard-Core and Long-Range Interaction

Abstract

A system of Brownian hard balls is regarded as a reflecting Brownian motion in the configuration space and can be represented by a solution to a Skorohod-type equation. In this article, we consider the case that there are an infinite number of balls, and the interaction between balls is given by the long-range pair interaction. We discuss the existence and uniqueness of strong solutions to the infinite-dimensional Skorohod equation.

Appendix

1.1 A.1  Solutions of (ISKE)

We give precise definitions of solutions of the (ISKE).

Let $\overline{\mathcal {B}}$ and $\overline{\mathcal {B}}_t$ be the completions of $\mathcal {B}(W((\mathbb {R}^d)^\mathbb {N})$ and $\mathcal {B}_t (W((\mathbb {R}^d)^\mathbb {N})$ with respect to $P_{Br}^{\infty }= P({\boldsymbol{B}}\in \cdot )$, respectively.

Definition A.1

(Strong Solutions Starting at ${\boldsymbol{x}}$) A weak solution $\boldsymbol{X}$ of (ISKE) with (3.10) and (3.11) and an $(\mathbb {R}^d)^{\mathbb {N}}$-valued $\{\mathcal {F}_t\}_{t\ge 0}$-Brownian motion ${\boldsymbol{B}}$ on $({\varOmega }, \mathcal {F}, P, \{ \mathcal {F}\}_{t\ge 0})$ is called a strong solution starting at ${\boldsymbol{x}}$ if $\boldsymbol{X}_0={\boldsymbol{x}}$ a.s. and if there exists a function $F_{\boldsymbol{x}}: W_0((\mathbb {R}^d)^\mathbb {N}) \rightarrow W((\mathbb {R}^d)^\mathbb {N})$ such that $F_{\boldsymbol{x}}$ is $\overline{\mathcal {B}}/ \mathcal {B}(W((\mathbb {R}^d)^\mathbb {N})$-measurable, and $\overline{\mathcal {B}}_t/ \mathcal {B}_t(W((\mathbb {R}^d)^\mathbb {N})$-measurable for each t and that $F_{\boldsymbol{x}}$ satisfies $\boldsymbol{X} = F_{\boldsymbol{x}}({\boldsymbol{B}})$ a.s. We also call $\boldsymbol{X}= F_{\boldsymbol{x}}({\boldsymbol{B}})$ a strong solution starting at ${\boldsymbol{x}}$. Additionally, we call $F_{\boldsymbol{x}}$ itself a strong solution starting at ${\boldsymbol{x}}$.

Definition A.2

(Unique Strong Solution Starting at ${\boldsymbol{x}}$) We say (ISKE) with (3.10) and (3.11) has a unique strong solution starting at ${\boldsymbol{x}}$ if there exists a function $F_{\boldsymbol{x}}: W_0((\mathbb {R}^d)^\mathbb {N}) \rightarrow W((\mathbb {R}^d)^\mathbb {N})$ such that, for any weak solution $(\hat{\boldsymbol{X}}, \hat{{\boldsymbol{L}}}, \hat{{\boldsymbol{B}}})$ of (ISKE) with (3.10) and (3.11) $\hat{\boldsymbol{X}} = F_{\boldsymbol{x}}(\hat{{\boldsymbol{B}}})$ a.s. and if, for any $(\mathbb {R}^d)^\mathbb {N}$-valued $\{\mathcal {F}_t\}_{t\ge 0}$-Brownian motion ${\boldsymbol{B}}$ defined on $({\varOmega }, \mathcal {F}, P, \{\mathcal {F}\}_{t\ge 0})$ with ${\boldsymbol{B}}_0=0$, the process $\boldsymbol{X}=F_{\boldsymbol{x}}({\boldsymbol{B}})$ is a strong solution of (ISKE) with (3.10) and (3.11) starting at ${\boldsymbol{x}}$. We also call $F_{\boldsymbol{x}}$ a unique strong solution starting at ${\boldsymbol{x}}$.

We next present a variant of the notion of a unique strong solution.

Definition A.3

(A Unique Strong Solution Under a Constraint) For a condition (Cond), we say (ISKE) with (3.10) and (3.11) has a unique strong solution starting at ${\boldsymbol{x}}$ under the constraint of (Cond) if there exists a function $F_{\boldsymbol{x}}: W_0((\mathbb {R}^d)^\mathbb {N}) \rightarrow W((\mathbb {R}^d)^\mathbb {N})$ such that for any weak solution $(\hat{\boldsymbol{X}}, \hat{{\boldsymbol{B}}})$ of (ISKE) with (3.10) and (3.11) starting at ${\boldsymbol{x}}$ satisfying (Cond), it holds that $\hat{\boldsymbol{X}} = F_{\boldsymbol{x}}(\hat{{\boldsymbol{B}}})$ a.s. and if, for any $(\mathbb {R}^d)^\mathbb {N}$-valued $\{\mathcal {F}_t\}_{t\ge 0}$-Brownian motion ${\boldsymbol{B}}$ on $({\varOmega }, \mathcal {F}, P, \{\mathcal {F}\}_{t\ge 0})$ with ${\boldsymbol{B}}_0=0$, the process $\boldsymbol{X}=F_{\boldsymbol{x}}({\boldsymbol{B}})$ is a strong solution of (ISKE) with (3.10) and (3.11) starting at ${\boldsymbol{x}}$ satisfying (Cond). We also call $F_{\boldsymbol{x}}$ a strong solution starting at ${\boldsymbol{x}}$ under the constraint of (Cond).

For a family of strong solutions $\{F_{\boldsymbol{x}}\}$ satisfying (MF), we put

$$ P_{\{F_{\boldsymbol{x}}\}} = \int P(F_{\boldsymbol{x}}({\boldsymbol{B}})\in \cdot ) P \circ \boldsymbol{X}_0^{-1}(d{\boldsymbol{x}}). $$

Let $(\boldsymbol{X}, {\boldsymbol{L}}, {\boldsymbol{B}})$ be a solution of (ISKE) with (3.10) and (3.11) under P. Suppose that $(\boldsymbol{X}, {\boldsymbol{B}})$ is a unique strong solution under $P_{\boldsymbol{x}}$ for $P\circ \boldsymbol{X}_0^{-1}$-a.s. ${\boldsymbol{x}}$. Let $\{F_{\boldsymbol{x}}\}$ be a family of the unique strong solution given by $(\boldsymbol{X}, {\boldsymbol{B}})$ under $P_{\boldsymbol{x}}$. Then, (MF) is automatically satisfied and $P_{\{F_{\boldsymbol{x}}\}}= P\circ \boldsymbol{X}^{-1}$.

Definition A.4

(A Family of Unique Strong Solutions Under Constraints) For a condition (Cond), we say (ISKE) with (3.10) and (3.11) has a family of unique strong solutions $\{F_{\boldsymbol{x}}\}$ starting at ${\boldsymbol{x}}$ for $P\circ \boldsymbol{X}_0^{-1}$-a.s. ${\boldsymbol{x}}$ under the constraints of (MF) and (Cond) if $\{F_{\boldsymbol{x}}\}$ satisfies (MF) and $P_{\{F_{\boldsymbol{x}}\}}$ satisfies (Cond). Furthermore, (i) and (ii) are satisfied.

• (i) For any weak solution $(\hat{\boldsymbol{X}}, \hat{{\boldsymbol{B}}})$ under $\hat{P}$ of (ISKE) with (3.10) and (3.11) with $\hat{P}\circ \boldsymbol{X}_0^{-1} \prec P\circ \boldsymbol{X}_0^{-1}$ satisfying (Cond), it holds that, for $\hat{P}\circ \boldsymbol{X}_0^{-1}$-a.s. ${\boldsymbol{x}}$, $\hat{\boldsymbol{X}}=F_{\boldsymbol{x}}(\hat{{\boldsymbol{B}}})$ $\hat{P}_{\boldsymbol{x}}$-a.s., where $\hat{P}_{\boldsymbol{x}}= \hat{P}(\cdot | \hat{\boldsymbol{X}}_0 ={\boldsymbol{x}})$.

• (ii) For an arbitrary $(\mathbb {R}^d)^\mathbb {N}$-valued $\{\mathcal {F}_t\}$-Brownian motion ${\boldsymbol{B}}$ on $({\varOmega }, \mathcal {F}, P, \{\mathcal {F}\}_{t\ge 0})$ with ${\boldsymbol{B}}_0=\textbf{0}$, $F_{\boldsymbol{x}}({\boldsymbol{B}})$ is a strong solution of (ISKE) with (3.10) and (3.11) starting at ${\boldsymbol{x}}$ for $P\circ \boldsymbol{X}_0^{-1}$-a.s. ${\boldsymbol{x}}$.

1.2 A.2  Definition of (IFC)

In this subsection, we introduce (IFC) for our situation. Let $\mathbb {I}$ be a finite subset of $\mathbb {N}$. Put $\mathbb {I}^c = \mathbb {N} \setminus \mathbb {I}$. For ${\boldsymbol{y}}= (y^1, y^2, \ldots ) \in \boldsymbol{S}_\textrm{hard}$, we put ${\boldsymbol{y}}^{\mathbb {I}} = (y^j)_{j\in \mathbb {I}}$ and ${\boldsymbol{y}}^{\mathbb {I}^c} = (y^j)_{j\in \mathbb {I}^c}$ Let $(\boldsymbol{X}, {\boldsymbol{B}})= ((X^j)_{j\in \mathbb {N}}, (B^j)_{j\in \mathbb {N}})$ be a weak solution of (ISKE) starting at ${\boldsymbol{x}}=(x^j)_{j\in \mathbb {N}}$ defined on $({\varOmega }, \mathcal {F}, P, \{\mathcal {F}_t\})$. We consider the SDE

$$\begin{aligned} dY_t^{\mathbb {I},j}&= dB_t^j + b^{\mathbb {I},j}_{\boldsymbol{X}} (t,\textbf{Y}_t^{\mathbb {I}})dt +\sum \limits _{k\in \mathbb {I}\setminus \{j\}}(Y_t^{\mathbb {I},j} - Y_t^{\mathbb {I},k}) dL_t^{\mathbb {I},jk} \nonumber \\&\qquad +\sum \limits _{k\in \mathbb {I}^c}^\infty (Y_t^{\mathbb {I},j} - X_t^{k}) dL_t^{\mathbb {I},jk}, \quad j\in \mathbb {I},\qquad \qquad \qquad \qquad (\text {SKE}_{{\boldsymbol{X}}}(\mathbb {I}))\nonumber \\ Y_0^{\mathbb {I},j}&= X_0^j=x^j, \quad j\in \mathbb {I},\nonumber \end{aligned}$$

where $L_t^{\mathbb {I},jk}$, $j\in \mathbb {I}$, $k\in \mathbb {N}$ are increasing functions satisfying

$$\begin{aligned}&L_t^{\mathbb {I},jk}=\int \limits _0^t \textbf{1}(|Y_s^{\mathbb {I},j}-Y_s^{\mathbb {I},k}|=r)dL_s^{\mathbb {I},jk}, \quad j,k \in \mathbb {I}, \\&L_t^{\mathbb {I},jk}=\int \limits _0^t \textbf{1}(|Y_s^{\mathbb {I},j}-X_s^k|=r)dL_s^{\mathbb {I},jk}, \quad j\in \mathbb {I}, \ k\in \mathbb {I}^c. \end{aligned}$$

We denote by $\mathcal {C}^\mathbb {I}$ the completion of $\mathcal {B}(W_\textbf{0}((\mathbb {R}^{d})^{\mathbb {I}})\times W((\mathbb {R}^d)^{\mathbb {N}}))$ with respect to $P_{{\boldsymbol{x}}}\circ ({\boldsymbol{B}}^\mathbb {I}, \boldsymbol{X}^{\mathbb {I}^c})^{-1}$. Let $(\textbf{v}, {\boldsymbol{w}})\in W_\textbf{0}((\mathbb {R}^{d})^\mathbb {I})\times W((\mathbb {R}^d)^{\mathbb {N}})$. We denote by $\mathcal {C}_t^\mathbb {I}$ the completion of $\sigma [(\textbf{v}_s, {\boldsymbol{w}}_s) : 0\le s \le t]$ with respect to $P_{{\boldsymbol{x}}}\circ ({\boldsymbol{B}}^\mathbb {I}, \boldsymbol{X}^{\mathbb {I}^c})^{-1}$.

Definition A.5

(Strong Solution for $(\boldsymbol{X},{\boldsymbol{B}})$ Starting at ${\boldsymbol{x}}^\mathbb {I}$) $\textbf{Y}^\mathbb {I}$ is called a strong solution of $(\textrm{SKE}_{\boldsymbol{X}}(\mathbb {I}))$ for $(\boldsymbol{X},{\boldsymbol{B}})$ under $P_{{\boldsymbol{x}}}$ if $(\textbf{Y}^\mathbb {I}, {\boldsymbol{B}}^{\mathbb {I}}, \boldsymbol{X}^{\mathbb {I}^c})$ satisfies $(\textrm{SKE}_{\boldsymbol{X}}(\mathbb {I}))$ and there exists a $\mathcal {C}^\mathbb {I}$-measurable function

$$ F_{{\boldsymbol{x}}}^\mathbb {I}:W_\textbf{0}((\mathbb {R}^d)^\mathbb {I}) \times W((\mathbb {R}^d)^{\mathbb {I}^c}) \rightarrow W((\mathbb {R}^d)^{\mathbb {I}}) $$

such that $F_\textbf{x}^\mathbb {I}$ is $\mathcal {C}_t^\mathbb {I}/ \mathcal {B}_t(W(\mathbb {R}^d)^\mathbb {I}$-measurable for each t, and $F_{{\boldsymbol{x}}}^\mathbb {I}$ satisfies $\textbf{Y}^I =F_{{\boldsymbol{x}}}^\mathbb {I}({\boldsymbol{B}}^\mathbb {I}, \boldsymbol{X}^{\mathbb {I}^c})$, $P_{\boldsymbol{x}}$-a.s.

Definition A.6

(A Unique Strong Solution for $(\boldsymbol{X},{\boldsymbol{B}})$ Starting at ${\boldsymbol{x}}_m$) The SDE $(\textrm{SKE}_{\boldsymbol{X}}(\mathbb {I}))$ is said to have a unique strong solution for ($\boldsymbol{X}, {\boldsymbol{B}}$) under $P_{\boldsymbol{x}}$ if there exists a strong solution $F_{{\boldsymbol{x}}}^\mathbb {I}$ such that for any solution $(\hat{\textbf{Y}}^\mathbb {I}, {\boldsymbol{B}}^\mathbb {I}, \boldsymbol{X}^{\mathbb {I}^c})$ of $(\textrm{SKE}_{\boldsymbol{X}}(\mathbb {I}))$ under $P_{\boldsymbol{x}}$, $\hat{\textbf{Y}}^\mathbb {I}= F_{{\boldsymbol{x}}}^\mathbb {I}({\boldsymbol{B}}^\mathbb {I}, \boldsymbol{X}^{\mathbb {I}^c})\quad \text {for}\,P_{{\boldsymbol{x}}}\text {-a.s.}$.

We can then give the definition of (IFC).

• (IFC) For each finite subset $\mathbb {I}\subset \mathbb {N}$, $(\textrm{SKE}_{\boldsymbol{X}}(\mathbb {I}))$ has a unique strong solution under $P_{{\boldsymbol{x}}}:= P(\cdot | \boldsymbol{X}_0 ={\boldsymbol{x}})$ for $P \circ \boldsymbol{X}_0^{-1}$-a.s. ${\boldsymbol{x}}$.