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A stochastic evolution equation arising from the fluctuations of a class of interacting particle systems. (English) Zbl 1087.60050

A (finite) measure-valued process \(V\) in \({\mathbb R}^d\) is decribed by a stochastic partial differential equation whose weak (and thus rigorous) form is the following:
\[ \begin{align*}{ \langle\phi,V(t)\rangle- \langle\phi,V(0)\rangle &=\int_0^t\langle\phi d(\cdot, V(s))+ L(V(s))\phi,V(s)\rangle ds\cr &\phantom{=\int_0^t}+\int_{U\times[0,t]}\langle\phi\beta(\cdot,V(s),u)+ \nabla\phi^T\alpha(\cdot,V(s),u),V(s)\rangle W(duds),}\end{align*} \]
where \(L(v)\phi(x)= \frac12 \sum_{i,j} a_{ij}(x,v) \partial_{x_i}\partial_{x_j} \phi(x)+\sum_ib_i(x,v) \partial_{x_i}\phi(x)\) and \(a=(a_{ij})\), \(\alpha=(\alpha_i)\), \(b=(b_i)\), \(a_{ij}, \alpha_i, b_i\) (\(1\leq i,j\leq d\)), \(d,\beta\) are real functions on appropriate spaces, \(U\) is a Polish space, and \(W\) is a space-time Gaussian white noise on \(U\times[0,\infty)\) with covariance measure \(\mu(du)dt\), \(\mu\) being a \(\sigma\)-finite measure on \(U\).
In an earlier paper by the same authors [in: Stochastics in finite and infinite dimensions; 233–258 (2000; Zbl 0991.60053); see also Stochast. Processes Appl. 83, No. 1, 103–126 (1999; Zbl 0996.60071)], it was shown that under suitable assumptions on the coefficients the process \(V\) can be approximated by the weighted empirical measure process
\[ V^n(t)= {1\over n}\sum_{k=1}^nA_k^n(t)\delta_{X_k^n(t)}, \]
of a finite particle system satisfying
\[ \begin{align*}{ X_k^n(t)=&X_k(0)+\int_0^t\sigma(X_k^n(s),V^n(s))dB_k(s)+ \int_0^tc(X_k^n(s),V^n(s))ds\cr &+\int_{U\times[0,t]}\alpha(X_k^n(s),V^n(s),u)W(duds),}\end{align*} \]
\[ \begin{align*}{ A_k^n(t)=&A_k(0)+\int_0^tA_k^n(s)\gamma^T(X_k^n(s),V^n(s))dB_k(s)+ \int_0^tA_k^n(s)d(X_k^n(s),V^n(s))ds\cr &+ \int_{U\times[0,t]}A_k^n(s)\beta(X_k^n(s),V^n(s),u)W(duds),}\end{align*} \]
for \(k=1,2,\dots,n\), where \(B_k\) are independent standard \({\mathbb R}^d\)-valued Brownian motions, independent of \(W\), \((X_k(0),A_k(0))\) are exchangeable random variables in \({\mathbb R}^d\times{\mathbb R}\) independent of \(\{B_k\}\) and \(W\), and \(\sigma, c\) and \(\gamma\) are determined by \(a,b,\alpha,\beta\).
In the present paper a fluctuation result is obtained. It is proved that \(S_n=\sqrt{n}(V^n-V)\) converges in law in the space \(C_{\Phi'}[0,\infty)\), where \(\Phi\) is a Fréchet nuclear (Schwartz modified) space. The limit \(S\) is a distribution-valued process which is the unique solution of a stochastic evolution equation
\[ \langle\phi,S(t)\rangle=\langle\phi,S(0)\rangle +\langle\phi,M(t)\rangle +\int_0^t\langle F_1(V(s))\phi,S(s)\rangle ds +\int_{U\times[0,t]}\langle F_2(V(s),u)\phi,S(s)\rangle W(duds), \]
where \(M\) is a distribution-valued martingale and \(F_1,F_2\) are linear in \(\phi\). This interesting but rather involved theorem is related to old results of M. Hitsuda and I. Mitoma [J. Multivariate Anal. 19, 311–328 (1986; Zbl 0604.60059)], where a much simpler setting was investigated and the fluctuation limit was a Gaussian process.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60F17 Functional limit theorems; invariance principles
60F25 \(L^p\)-limit theorems
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
93E11 Filtering in stochastic control theory
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