Interacting particle mean field and vortex blob error

Let $\omega ,\nu$ be finite signed Borel measures on ${\mathbb{R}}^d$ such that $\mathrm{\mid }\omega -\nu \mathrm{\mid }({\mathbb{R}}^d)<\mathrm{\infty }$. For each bounded Lipschitz test function, integration against $\omega -\nu$ is defined by Integration against a signed or complex measure, and the class L^1(nu) = L^1(|nu|); boundedness and finite total variation make every test in the following set integrable. Set $\mathrm{\parallel }\varphi {\mathrm{\parallel }}_{\mathrm{\infty }}:={\sup }_x\mathrm{\mid }\varphi (x)\mathrm{\mid }$ and $\mathrm{Lip}(\varphi ):={\sup }_{x\mathrm{\ne }y}\mathrm{\mid }\varphi (x)-\varphi (y)\mathrm{\mid }\mathrm{/}\mathrm{\mid }x-y\mathrm{\mid }$, with $\mathrm{Lip}(\varphi )=0$ when $d=0$, using the Euclidean norm from The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$. Define the bounded-Lipschitz dual discrepancy $d_{BL}(\omega ,\nu ):=\sup \{\mid \int \varphi d(\omega -\nu )\mid :\mathrm{\parallel }\varphi {\mathrm{\parallel }}_{\mathrm{\infty }}\le 1,\mathrm{Lip}(\varphi )\le 1\}\mathrm{.}$ The integral bound by total variation shows this supremum is finite. This page uses only the displayed dual quantity and makes no separate metric-separation claim. No Jordan decomposition, positivity, or probability normalization is needed.

Let $d\ge 1$, with the Euclidean norm from The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$. An interaction kernel is a continuous map $K:{\mathbb{R}}^d\times {\mathbb{R}}^d\to {\mathbb{R}}^d$. It is globally Lipschitz with constant $L\ge 0$ in the convention used on this page if $\mathrm{\mid }K(x,y)-K(x^{\mathrm{\prime }},y^{\mathrm{\prime }})\mathrm{\mid }\le L(\mathrm{\mid }x-x^{\mathrm{\prime }}\mathrm{\mid }+\mathrm{\mid }y-y^{\mathrm{\prime }}\mathrm{\mid })$ for all $x,y,x^{\mathrm{\prime }},y^{\mathrm{\prime }}\in {\mathbb{R}}^d$. This implies the linear-growth estimate $\mathrm{\mid }K(x,y)\mathrm{\mid }\le \mathrm{\mid }K(0,0)\mathrm{\mid }+L(\mathrm{\mid }x\mathrm{\mid }+\mathrm{\mid }y\mathrm{\mid })\mathrm{.}$ The notation $L$ always refers to this two-variable convention; the stability factor below is therefore $2L$.

In this definition only, use the local one-based coordinate names $z_1:=z(0)$ and $z_2:=z(1)$ for $z\in {\mathbb{R}}^2$, whose coordinate domain is $\{0,1\}$ as in The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$. Let $z^{\perp }=(-z_2,z_1)$ and $G(z)=\frac{1}{2\pi }\log \mathrm{\mid }z\mathrm{\mid },K(z)={\mathrm{\nabla }}^{\perp }G(z)=\frac{z^{\perp }}{2\pi \mathrm{\mid }z{\mathrm{\mid }}^2}(z\mathrm{\ne }0)\mathrm{.}$ The locally integrable value assigned to $K(0)$ is immaterial. Fix a radial, nonnegative smooth bump $\rho$ supported in the unit ball with $\int \rho =1$; its scaled family is the mollifier family of The mollifier family generated by a unit-mass smooth bump. Put ${\rho }_{\epsilon }(z)={\epsilon }^{-2}\rho (z\mathrm{/}\epsilon )$, $G_{\epsilon }=G\ast {\rho }_{\epsilon }$, and $K_{\epsilon }=K\ast {\rho }_{\epsilon }\mathrm{.}$ The kernel is a regularized interaction; its smoothness and global Lipschitz property are proved in a separate lemma. It does not define the singular physical interaction at a collision.

Let $d\ge 1$. For Borel probability measures $\mu ,\nu$ on ${\mathbb{R}}^d$ with finite first moments $\int \mathrm{\mid }x\mathrm{\mid }d\mu (x)<\mathrm{\infty }$ and $\int \mathrm{\mid }y\mathrm{\mid }d\nu (y)<\mathrm{\infty }$, let $\mathrm{\Pi }(\mu ,\nu )$ be the set of Borel probability measures on ${\mathbb{R}}^d\times {\mathbb{R}}^d$ whose first and second marginals are $\mu$ and $\nu$. Define $W_1(\mu ,\nu ):={\inf }_{\pi \in \mathrm{\Pi }(\mu ,\nu )}{\int }_{{\mathbb{R}}^d\times {\mathbb{R}}^d}\mathrm{\mid }x-y\mathrm{\mid }d\pi (x,y)\mathrm{.}$ This definition makes no assertion that the infimum is attained. It applies only to probability measures with finite first moment; it is not a distance assigned here to a general signed measure.

Here $\mathrm{\mid }\cdot \mathrm{\mid }$ is the Euclidean norm from The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$, and probability measure has the meaning of Probability measures and probability spaces.

Assume $d\ge 1$ and the Axiom of Choice when invoking Jordan decomposition of a signed measure into unique mutually singular positive parts. Let $\omega ,\nu$ be finite signed Borel measures on ${\mathbb{R}}^d$ whose total variations have finite first moments, and use the Euclidean norm from The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$. Write their Jordan parts as ${\omega }^{+},{\omega }^{-}$ and ${\nu }^{+},{\nu }^{-}$. If the positive-part masses agree and the negative-part masses agree, define using finite nonnegative Borel measures ${\pi }^{\pm }$ on ${\mathbb{R}}^d\times {\mathbb{R}}^d$ whose first and second marginals are respectively the corresponding positive or negative Jordan parts: $C_{\pm }(\omega ,\nu ):={\inf }_{{\pi }^{+},{\pi }^{-}}(\int \mathrm{\mid }x-y\mathrm{\mid }d{\pi }^{+}(x,y)+\int \mathrm{\mid }x-y\mathrm{\mid }d{\pi }^{-}(x,y)),$ where ${\pi }^{+}$ ranges over finite nonnegative couplings of ${\omega }^{+}$ and ${\nu }^{+}$, and ${\pi }^{-}$ over finite nonnegative couplings of ${\omega }^{-}$ and ${\nu }^{-}$. The infimum is over both sign couplings independently. This cost is not the probability $W_1$.

Assume Countable Choice. Let $(E,\mathcal{A},\mu )$ be a probability space, $T>0$, and $d\ge 1$. Use the Euclidean norm of The Euclidean inner product $⟨x,y⟩={\sum }_{k<n}{x}_k{y}_k$ on ${\mathbb{R}}^n$. Let $\mathcal{X}$ be the space, modulo equality $\mu$-almost everywhere, of jointly measurable maps $X:[0,T]\times E\to {\mathbb{R}}^d$ whose paths are continuous for almost every label and whose time-supremum function is measurable, and for which $\mathrm{\parallel }X{\mathrm{\parallel }}_{\mathcal{X}}:={\int }_E{\sup }_{0\le t\le T}\mathrm{\mid }{X}_t(z)\mathrm{\mid }d\mu (z)<\mathrm{\infty }\mathrm{.}$ In this formula and in $\mathrm{\parallel }X-Y{\mathrm{\parallel }}_{\mathcal{X}}$, take the supremum on a measurable conull set where the relevant paths are continuous and extend it by zero off that set; equality almost everywhere means equality of whole paths outside one measurable null set. The seminorm identifies maps equal almost everywhere, and the induced distance $d_{\mathcal{X}}(X,Y)=\mathrm{\parallel }X-Y{\mathrm{\parallel }}_{\mathcal{X}}$ is a metric on the quotient; it is complete.

Input real endpoints $a,b,c,d$, integer panel counts $n_x,n_y$, and an exact evaluation oracle that, when queried at a point, returns either a value in ${\mathbb{R}}^2$ or evaluation-failure. First, if $a\ge b$ or $c\ge d$, return invalid-domain without querying the oracle. Next, if either panel count is odd or is less than two, return invalid-panel-count without querying the oracle. For the remaining inputs set $R=[a,b]\times [c,d]$, $h_x=(b-a)\mathrm{/}{n}_x$, $h_y=(d-c)\mathrm{/}{n}_y$, $x_i=a+i{h}_x$ for $0\le i\le n_x$, and $y_j=c+j{h}_y$ for $0\le j\le n_y$. Query the oracle at every grid point in lexicographic order with $i$ first and $j$ second, storing the returned values. If a query fails, stop at the first such point and return evaluation-failure with that point. Otherwise, for each $j$ form the continuous piecewise-linear interpolant through the stored values $F(x_i,y_j)$ and apply Composite Simpson rule on an even uniform grid componentwise in $x$ to this interpolant, obtaining $G_j=Q_{x}F(\cdot ,y_j)$. Form the continuous piecewise-linear interpolant through the stored $G_j$ and apply the same valid Simpson procedure componentwise in $y$ to obtain $Q_{y}G=Q_y{Q}_{x}F$. Return this vector $Q_y{Q}_{x}F$. Each interpolant is queried only at its stored grid nodes, so no new oracle call is needed. Both calls to the supplier use strictly ordered endpoints and even panel counts at least two. This is an exact-real procedure; termination and force-accuracy bounds are established in Tensor composite Simpson error for a two-dimensional force integral ↗, and floating-point roundoff is outside its scope.

Fix $T>0$, a globally Lipschitz kernel $K$ as in Globally Lipschitz mean-field interaction kernel, and a Borel probability measure ${\mu }_0$ on ${\mathbb{R}}^d$ with finite first moment as in The coupling definition of the 1-Wasserstein distance. A characteristic solution on $[0,T]$ consists of a jointly Borel map $X:[0,T]\times {\mathbb{R}}^d\to {\mathbb{R}}^d$, defined up to a ${\mu }_0$-null set of labels, such that $t\mapsto X_t(x)$ is absolutely continuous for ${\mu }_0$-almost every $x$, $X_0(x)=x$, and $X_t(x)=x+{\int }_0^t{\int }_{{\mathbb{R}}^d}K(X_s(x),X_s(y))d{\mu }_0(y)ds$ for ${\mu }_0$-almost every $x$. It transports ${\mu }_0$ to ${\mu }_t=(X_t{)}_{\mathrm{\#}}{\mu }_0$, where $(X_t{)}_{\mathrm{\#}}{\mu }_0(A):={\mu }_0(X_t^{-1}(A))$. The weighted path integrability convention is ${\int }_{{\mathbb{R}}^d}{\sup }_{0\le t\le T}\mathrm{\mid }{X}_t(x)\mathrm{\mid }d{\mu }_0(x)<\mathrm{\infty }$; in particular every time slice has finite first moment.

For a globally Lipschitz interaction kernel $K$ and $N\ge 1$, the equal-probability mean-field particle system is ${\dot{x}}_i(t)=\frac{1}{N}{\sum }_{j=1}^{N}K(x_i(t),x_j(t)),i=1,\dots ,N\mathrm{.}$ Its empirical probability measure is ${\mu }_N(t)=N^{-1}{\sum }_i{\delta }_{x_i(t)}$.

For a mollified two-dimensional Biot-Savart kernel $K_{\epsilon }$ from Smooth mollified Biot-Savart kernel and real weights ${\mathrm{\Gamma }}_1,\dots ,{\mathrm{\Gamma }}_N$, the weighted signed vortex particle system is ${\dot{x}}_i(t)={\sum }_{j=1}^N{\mathrm{\Gamma }}_j{K}_{\epsilon }(x_i(t)-x_j(t)),i=1,\dots ,N,$ with vorticity measure ${\omega }_N(t)={\sum }_i{\mathrm{\Gamma }}_i{\delta }_{x_i(t)}$. If ${\mathrm{\Gamma }}_i=1\mathrm{/}N$ and $K(x,y)=K_{\epsilon }(x-y)$, the first system is the probability special case. The weighted signed system is a first-order Biot-Savart model.

Assume AC. Let ${\omega }_0$ be a finite signed Borel measure on ${\mathbb{R}}^2$, set $M=\mathrm{\mid }{\omega }_0\mathrm{\mid }({\mathbb{R}}^2)$, and let ${\omega }_0={\omega }_0^{+}-{\omega }_0^{-}$ be its Jordan decomposition. The Jordan total-variation formula gives $M={\omega }_0^{+}({\mathbb{R}}^2)+{\omega }_0^{-}({\mathbb{R}}^2)$. If $M=0$, its characteristic solution is the constant zero measure curve. If $M>0$, set $E={\mathbb{R}}^2\times \{+1,-1\}$ with sigma-algebra $\mathcal{B}({\mathbb{R}}^2)\otimes 2^{\{+1,-1\}}$, and define the probability measure ${\lambda }_0$ by ${\lambda }_0(A\times \{s\})={\omega }_0^s(A)\mathrm{/}M$ for Borel $A$ and $s\in \{+1,-1\}$; its total mass is one by the displayed formula. A signed characteristic solution for the regularized kernel $K_{\epsilon }$ on $[0,T]$ is a jointly measurable path map $Z:[0,T]\times E\to {\mathbb{R}}^2$, modulo ${\lambda }_0$-almost everywhere equality, with absolutely continuous paths for almost every label, $Z_0(x,s)=x$, and

$Z_t(x,s)=x+M{\int }_0^t{\int }_{E}r{K}_{\epsilon }(Z_u(x,s)-Z_u(y,r))d{\lambda }_0(y,r)du,$

such that its time-supremum function is measurable and ${\int }_E{\sup }_{0\le t\le T}\mathrm{\mid }{Z}_t(x,s)\mathrm{\mid }d{\lambda }_0(x,s)<\mathrm{\infty }$. It represents the signed measure curve

${\omega }_t=M{\int }_{E}s{\delta }_{Z_t(x,s)}d{\lambda }_0(x,s)\mathrm{.}$

Here ${\omega }_0^s$ denotes ${\omega }_0^{+}$ when $s=+1$ and ${\omega }_0^{-}$ when $s=-1$. The total variation mass is $M$ at time zero, and the sign coordinate records each Jordan part separately.

Assume Countable Choice and the mollifier of Smooth mollified Biot-Savart kernel. For every $\epsilon >0$, $G_{\epsilon }$ and $K_{\epsilon }$ are $C^{\mathrm{\infty }}$ on ${\mathbb{R}}^2$, and there is $C_{\rho }<\mathrm{\infty }$ independent of $\epsilon$ such that ${\sup }_{x\in {\mathbb{R}}^2}\mathrm{\parallel }D{K}_{\epsilon }(x){\mathrm{\parallel }}_{\mathrm{o}\mathrm{p}}\le C_{\rho }{\epsilon }^{-2}\mathrm{.}$ Consequently $K_{\epsilon }$ is globally Lipschitz with constant at most $C_{\rho }{\epsilon }^{-2}$.

Assume AC for the Jordan decompositions in Sign-preserving transport cost for finite signed measures. Let $d\ge 1$. If finite signed Borel measures $\omega ,\nu$ on ${\mathbb{R}}^d$ have the same positive-part mass and the same negative-part mass, and their total variations have finite first moments, then the couplings in $C_{\pm }(\omega ,\nu )$ exist, $C_{\pm }(\omega ,\nu )<\mathrm{\infty }$, and $d_{BL}(\omega ,\nu )\le C_{\pm }(\omega ,\nu )\mathrm{.}$

If $\mu ,\nu$ are Borel probability measures on ${\mathbb{R}}^d$ with finite first moments, then $\mathrm{\Pi }(\mu ,\nu )$ is nonempty and $W_1(\mu ,\nu )<\mathrm{\infty }$ for the definition in The coupling definition of the 1-Wasserstein distance.

For $N\ge 2$, real weights ${\mathrm{\Gamma }}_i$, and $\epsilon >0$, define the open collision-free configuration set $Q_{cf}:=\{q=(q_1,\dots ,q_N)\in ({\mathbb{R}}^2{)}^N:q_i\mathrm{\ne }{q}_j\text{ for }i\mathrm{\ne }j\}$. Let $q\in Q_{cf}$ and $p=(p_1,\dots ,p_N)\in ({\mathbb{R}}^2{)}^N$. Let $H_{\epsilon }(q,p)=\frac{1}{2}{\sum }_{i=1}^N\mathrm{\mid }{p}_i{\mathrm{\mid }}^2+V_{\epsilon }(q),V_{\epsilon }(q)={\sum }_{i<j}{\mathrm{\Gamma }}_i{\mathrm{\Gamma }}_j{G}_{\epsilon }(q_i-q_j),$ where $G_{\epsilon }$ is from Smooth mollified Biot-Savart kernel. This is a separable canonical Hamiltonian on $Q_{cf}$; its local leapfrog map is the partial composition One-step leapfrog map as force and drift shears restricted as in Local domain restriction for kick, drift, and leapfrog maps. It is a distinct second-order canonical comparison model, not the first-order Biot-Savart vortex system of Equal-probability mean-field and weighted signed vortex particle systems.

Let $K$ have two-variable Lipschitz constant $L$ as in Globally Lipschitz mean-field interaction kernel, and let ${\mu }_t,{\nu }_t$ be characteristic solutions from probability measures with finite first moments. For every $t\in [0,T]$, $W_1({\mu }_t,{\nu }_t)\le e^{2Lt}{W}_1({\mu }_0,{\nu }_0)\mathrm{.}$ The estimate is proved for every prescribed initial coupling before taking the infimum; existence of an optimal coupling is not assumed.

Let $x_1,\dots ,x_N$ solve the equal-probability system of Equal-probability mean-field and weighted signed vortex particle systems on $[0,T]$, and put ${\mu }_N(t)=N^{-1}{\sum }_i{\delta }_{x_i(t)}$. Then for every $\phi \in C_c^1({\mathbb{R}}^d)$, $\int \phi d{\mu }_N(t)-\int \phi d{\mu }_N(0)={\int }_0^t\int \mathrm{\nabla }\phi (x)\cdot (\int K(x,y)d{\mu }_N(s)(y))d{\mu }_N(s)(x)ds\mathrm{.}$ Thus the empirical measure satisfies the same weak transport equation exactly, with no particle quadrature error in this identity.

Assume Countable Choice. Let $K$ be globally Lipschitz with constant $L$ in the convention of Globally Lipschitz mean-field interaction kernel, and let ${\mu }_0$ be a Borel probability measure on ${\mathbb{R}}^d$ with finite first moment. For every finite $T>0$ there exists a characteristic solution on $[0,T]$ in the sense of Characteristic solution of a probability mean-field equation. It is unique up to equality of trajectories for ${\mu }_0$-almost every label. No compact support assumption is needed.

Let ${\mu }_t=(X_t{)}_{\mathrm{\#}}{\mu }_0$ be the characteristic solution for a globally Lipschitz kernel on $[0,T]$. For every $\phi \in C_c^1({\mathbb{R}}^d)$ and $t\in [0,T]$, $\int \phi d{\mu }_t-\int \phi d{\mu }_0={\int }_0^t{\int }_{{\mathbb{R}}^d}\mathrm{\nabla }\phi (x)\cdot ({\int }_{{\mathbb{R}}^d}K(x,y)d{\mu }_s(y))d{\mu }_s(x)ds\mathrm{.}$

Assume Countable Choice and the radial compactly supported mollifier of Smooth mollified Biot-Savart kernel. Write $\mathrm{\parallel }F{\mathrm{\parallel }}_{L^1}:={\int }_{{\mathbb{R}}^2}\mathrm{\mid }F(x)\mathrm{\mid }dx$ and let $\mathrm{\parallel }g{\mathrm{\parallel }}_{L^{\mathrm{\infty }}}$ be the essential supremum of $\mathrm{\mid }g\mathrm{\mid }$. There is a constant $C_{\rho }<\mathrm{\infty }$, independent of $\epsilon >0$, such that $\mathrm{\parallel }{K}_{\epsilon }-K{\mathrm{\parallel }}_{L^1({\mathbb{R}}^2)}\le C_{\rho }\epsilon \mathrm{.}$ Consequently, for every bounded measurable vorticity density $\omega$, $\mathrm{\parallel }(K_{\epsilon }-K)\ast \omega {\mathrm{\parallel }}_{L^{\mathrm{\infty }}}\le C_{\rho }\epsilon \mathrm{\parallel }\omega {\mathrm{\parallel }}_{L^{\mathrm{\infty }}}\mathrm{.}$ This is an error estimate at fixed $\epsilon$; it does not prove a singular-limit theorem for the particle dynamics.