Proof of Plans and Their Velocity Fields: the Marginal, the Composition Isometry, the Pairing, the Velocity Shift and the Lift

Each result cited is universally quantified over the data in its own statement. Throughout, $(\mathbb{R}^{d+d},\mathcal{B}(\mathbb{R}^{d+d}),\pi)$ is a probability space, $\pi$ being a probability measure on $\mathcal{B}(\mathbb{R}^{d+d})$ by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iii), and $L^{2}(\pi;\mathbb{R}^{d})$ and $L^{2}(\nu;\mathbb{R}^{d})$ are the spaces of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (ii) formed over it and over $(\mathbb{R}^{d},\mathcal{B}(\mathbb{R}^{d}),\nu)$.

Claim 1. The projections $\mathrm{pr}_{1}$ and $\mathrm{pr}_{2}$ are Borel and satisfy $\lVert\mathrm{pr}_{i}(z)\rVert\le\lVert z\rVert$ for every $z\in\mathbb{R}^{d+d}$, by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i). Both sides being nonnegative, claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives $\lVert\mathrm{pr}_{i}(z)\rVert^{2}\le\lVert z\rVert^{2}$, so the monotonicity of the integral of nonnegative functions, claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion, gives

$$\int_{\mathbb{R}^{d+d}}\lVert\mathrm{pr}_{i}(z)\rVert^{2}\,\pi(dz)\le\int_{\mathbb{R}^{d+d}}\lVert z\rVert^{2}\,\pi(dz)=M_{2}(\pi)<\infty ,$$

the last inequality because $\pi\in\mathcal{P}_{2}(\mathbb{R}^{d+d})$ by Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (iv). Hence, by the description of $L^{2}(\pi;\mathbb{R}^{d})$ in Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (ii), the classes of $\mathrm{pr}_{1}$ and $\mathrm{pr}_{2}$ belong to it, and their squared norms are the two integrals just bounded, since the nonnegative square root of a nonnegative real number squares back to it by Existence and Uniqueness of the Nonnegative Square Root. The second of them is $\int\lVert p\rVert^{2}\,\pi(dz)$ in the notation of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (iv), and it is finite.

By Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (iii) the first marginal $\nu=(\mathrm{pr}_{1})_{\#}\pi$ belongs to $\mathcal{P}(\mathbb{R}^{d})$, and the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv), applied to the nonnegative Borel function $y\mapsto\lVert y\rVert^{2}$ of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (v), gives

$$\lVert\mathrm{pr}_{1}\rVert_{\pi}^{2}=\int_{\mathbb{R}^{d+d}}\lVert\mathrm{pr}_{1}(z)\rVert^{2}\,\pi(dz)=\int_{\mathbb{R}^{d}}\lVert y\rVert^{2}\,\nu(dy)=M_{2}(\nu),$$

which is therefore finite, so $\nu\in\mathcal{P}_{2}(\mathbb{R}^{d})$ by The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment (ii). This proves claim 1.

Claim 2. By the definition of a random vector the Borel map $\mathrm{pr}_{1}$ is a random vector in $\mathbb{R}^{d}$ on the probability space $(\mathbb{R}^{d+d},\mathcal{B}(\mathbb{R}^{d+d}),\pi)$, and it is square-integrable by claim 1; its law is the push-forward $(\mathrm{pr}_{1})_{\#}\pi=\nu$. Claim 1 of Composition of a Square-Integrable Vector Field with a Random Vector, and the Lifted Score: Isometry, Norm, Weak Identity and Second-Moment Identity, applied with this probability space in place of $(\Omega,\mathcal{F},P)$ and with $\mathrm{pr}_{1}$ in place of the random vector $X$ there, so that the space written $L^{2}(\Omega;\mathbb{R}^{d})$ there is $L^{2}(\pi;\mathbb{R}^{d})$ here and the space written $L^{2}(\mu;\mathbb{R}^{d})$ there is $L^{2}(\nu;\mathbb{R}^{d})$ here, is exactly claim 2, including the assertion $\mathrm{id}\circ\mathrm{pr}_{1}=\mathrm{pr}_{1}$. That a representative of $\eta\circ\mathrm{pr}_{1}$ is Borel follows from that claim, a square-integrable random vector on this probability space being by definition a Borel map $\mathbb{R}^{d+d}\to\mathbb{R}^{d}$.

Claim 3. Let $\eta\in L^{2}(\nu;\mathbb{R}^{d})$ and fix representatives. The function $z\mapsto\eta(x)\cdot p$ is the function $z\mapsto(\eta\circ\mathrm{pr}_{1})(z)\cdot\mathrm{pr}_{2}(z)$, Borel by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (iv) applied to the two Borel maps $\eta\circ\mathrm{pr}_{1}$ and $\mathrm{pr}_{2}$. By The Space of Square-Integrable Random Vectors (iii), read on this probability space as in Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (ii), this function is integrable with respect to $\pi$ and

$$\int\eta(x)\cdot p\,\pi(dz)=\langle\eta\circ\mathrm{pr}_{1},\mathrm{pr}_{2}\rangle_{\pi},$$

a real number depending only on the two classes. The Cauchy-Schwarz inequality The Cauchy–Schwarz Inequality in a Real Inner Product Space in the real inner product space $L^{2}(\pi;\mathbb{R}^{d})$, together with claim 2 and claim 1, gives

$$\Bigl|\int\eta(x)\cdot p\,\pi(dz)\Bigr|\le\lVert\eta\circ\mathrm{pr}_{1}\rVert_{\pi}\,\lVert\mathrm{pr}_{2}\rVert_{\pi}=\lVert\eta\rVert_{\nu}\sqrt{\int\lVert p\rVert^{2}\,\pi(dz)},$$

the last norm being the nonnegative square root of $\int\lVert p\rVert^{2}\,\pi(dz)$ by Existence and Uniqueness of the Nonnegative Square Root. This proves claim 3.

Claim 4. Write $\theta=\mathrm{pr}_{2}+t\,(\eta\circ\mathrm{pr}_{1})$ for the pointwise sum, so that $\Theta^{\eta}_{t}=(\mathrm{pr}_{1},\theta)$. Each component of $\theta$ is a sum of a Borel real-valued function and a real multiple of one, hence Borel by claim 2 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps, so $\theta$ is Borel by claim 2 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets, and $\Theta^{\eta}_{t}$ is Borel by Probability Measures on Euclidean Space and Random Vectors: Standing Notation (v). By The Space of Square-Integrable Random Vectors (ii) the class of $\theta$ is the element $\mathrm{pr}_{2}+t\,(\eta\circ\mathrm{pr}_{1})$ of $L^{2}(\pi;\mathbb{R}^{d})$.

By Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv) the push-forward $(\Theta^{\eta}_{t})_{\#}\pi$ belongs to $\mathcal{P}(\mathbb{R}^{d+d})$, and by the change-of-variables formula there, applied to the nonnegative Borel function $z\mapsto\lVert z\rVert^{2}$, together with claim 3 of Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space, which gives $\lVert\Theta^{\eta}_{t}(z)\rVert^{2}=\lVert\mathrm{pr}_{1}(z)\rVert^{2}+\lVert\theta(z)\rVert^{2}$, and with the additivity of the integral of nonnegative functions in claim 1 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion,

$$M_{2}\bigl((\Theta^{\eta}_{t})_{\#}\pi\bigr)=\int_{\mathbb{R}^{d+d}}\lVert\Theta^{\eta}_{t}(z)\rVert^{2}\,\pi(dz)=\lVert\mathrm{pr}_{1}\rVert_{\pi}^{2}+\lVert\theta\rVert_{\pi}^{2}<\infty ,$$

so $(\Theta^{\eta}_{t})_{\#}\pi$ belongs to $\mathcal{P}_{2}(\mathbb{R}^{d+d})$ by The Second Moment of a Probability Measure on Euclidean Space and the Probability Measures with Finite Second Moment (ii), that is, it is a plan in the sense of Wasserstein Spaces, Random Vectors, Vector Fields and Symmetric Matrices in Every Dimension: Standing Notation (iv). Its first marginal is $\nu$: by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i) one has $\mathrm{pr}_{1}\circ\Theta^{\eta}_{t}=\mathrm{pr}_{1}$, and for $B\in\mathcal{B}(\mathbb{R}^{d})$,

$$\bigl((\mathrm{pr}_{1})_{\#}(\Theta^{\eta}_{t})_{\#}\pi\bigr)(B)=\pi\bigl((\Theta^{\eta}_{t})^{-1}(\mathrm{pr}_{1}^{-1}(B))\bigr)=\pi\bigl((\mathrm{pr}_{1}\circ\Theta^{\eta}_{t})^{-1}(B)\bigr)=\pi\bigl(\mathrm{pr}_{1}^{-1}(B)\bigr)=\nu(B),$$

by the description of the push-forward in Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv) and the identity $(S\circ T)^{-1}(B)=T^{-1}(S^{-1}(B))$ for preimages.

Let $\tilde{\eta}$ be a second representative of the same class. By claim 2 the maps $\eta\circ\mathrm{pr}_{1}$ and $\tilde{\eta}\circ\mathrm{pr}_{1}$ represent the same class of $L^{2}(\pi;\mathbb{R}^{d})$, that is, the set $S$ on which they agree satisfies $\pi(S)=1$; the set $S$ belongs to $\mathcal{B}(\mathbb{R}^{d+d})$, being the preimage of the Borel set $\{0_{\mathbb{R}^{d}}\}$ of Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (v) under the difference of the two maps, which is Borel because each of its components is a difference of Borel real-valued functions by claim 2 of Arithmetic of Measurable Real- and [0,∞]-Valued Maps and claim 2 of The Borel Sigma-Algebra of a Euclidean Space as a Product, and Measurability of Projections, Sequentially Continuous Maps, and Open and Closed Sets. Hence its complement $N$ satisfies $\pi(N)=\pi(\mathbb{R}^{d+d})-\pi(S)=0$ by claim 3 of Basic Properties of a Measure, and $\Theta^{\eta}_{t}$ and $\Theta^{\tilde{\eta}}_{t}$ agree on $S$.

Let $B\in\mathcal{B}(\mathbb{R}^{d+d})$ and put $A=(\Theta^{\eta}_{t})^{-1}(B)$. Since $A\cap N\subseteq N$, the monotonicity of a measure in claim 2 of Basic Properties of a Measure gives $\pi(A\cap N)=0$, the values of a measure being nonnegative; since $A\cap N\subseteq A$ and $\pi$ is finite, claim 3 of Basic Properties of a Measure gives $\pi(A\setminus N)=\pi(A)-\pi(A\cap N)=\pi(A)$, where $A\setminus N=A\setminus(A\cap N)$. On $A\setminus N$ the two maps agree, so $A\setminus N\subseteq(\Theta^{\tilde{\eta}}_{t})^{-1}(B)$, and monotonicity gives $\pi(A)\le\pi((\Theta^{\tilde{\eta}}_{t})^{-1}(B))$; by symmetry the two are equal, so the two push-forwards agree.

For $t=0$ one has $0\,(\eta\circ\mathrm{pr}_{1})=\mathbf{0}$ and $\theta=\mathrm{pr}_{2}$, and $\Theta^{\eta}_{0}(z)=\iota(\mathrm{pr}_{1}(z),\mathrm{pr}_{2}(z))=z$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i), so $\Theta^{\eta}_{0}$ is the identity map and $(\Theta^{\eta}_{0})_{\#}\pi=\pi$ by the description of the push-forward.

Now let $\eta'\in L^{2}(\nu;\mathbb{R}^{d})$. The function $g(z)=\eta'(x)\cdot p$ is Borel by claim 3, and

$$g\bigl(\Theta^{\eta}_{t}(z)\bigr)=\eta'\bigl(\mathrm{pr}_{1}(z)\bigr)\cdot\theta(z)=\eta'(x)\cdot p+t\,\bigl(\eta'\circ\mathrm{pr}_{1}\bigr)(z)\cdot\bigl(\eta\circ\mathrm{pr}_{1}\bigr)(z)$$

by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i) and the bilinearity of the dot product recorded in Difference, Dot Product, and Orthogonality in $\mathbb{R}^n$. Both summands are integrable with respect to $\pi$, the first by claim 3 and the second by claim 3 applied on the probability space $(\mathbb{R}^{d+d},\mathcal{B}(\mathbb{R}^{d+d}),\pi)$ to the two elements $\eta'\circ\mathrm{pr}_{1}$ and $\eta\circ\mathrm{pr}_{1}$ of $L^{2}(\pi;\mathbb{R}^{d})$, whose integral is $\langle\eta'\circ\mathrm{pr}_{1},\eta\circ\mathrm{pr}_{1}\rangle_{\pi}$ by The Space of Square-Integrable Random Vectors (iii); so $g\circ\Theta^{\eta}_{t}$ is integrable and the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv) together with the linearity of the integral in claim 2 of Linearity and Monotonicity of the Lebesgue Integral, and the Integrability Criterion gives

$$\int\eta'(x)\cdot p\,d\bigl((\Theta^{\eta}_{t})_{\#}\pi\bigr)=\int g\circ\Theta^{\eta}_{t}\,d\pi=\int\eta'(x)\cdot p\,\pi(dz)+t\,\langle\eta'\circ\mathrm{pr}_{1},\eta\circ\mathrm{pr}_{1}\rangle_{\pi},$$

and the last inner product is $\langle\eta',\eta\rangle_{\nu}$ by claim 2. Likewise, applying the change-of-variables formula to the nonnegative Borel function $z\mapsto\lVert p\rVert^{2}$ and then Elementary Identities in a Real Inner Product Space (v) in $L^{2}(\pi;\mathbb{R}^{d})$,

$$\int\lVert p\rVert^{2}\,d\bigl((\Theta^{\eta}_{t})_{\#}\pi\bigr)=\lVert\theta\rVert_{\pi}^{2}=\lVert\mathrm{pr}_{2}\rVert_{\pi}^{2}+2t\,\langle\mathrm{pr}_{2},\eta\circ\mathrm{pr}_{1}\rangle_{\pi}+t^{2}\lVert\eta\circ\mathrm{pr}_{1}\rVert_{\pi}^{2},$$

which is the asserted formula by claims 1, 2 and 3, the inner product being symmetric by Real Inner Product Spaces (i). This proves claim 4.

Claim 5. Fix representatives of $Y$ and $Y'$ and write $T=(Y,Y')$ for their pairing, a random vector in $\mathbb{R}^{d+d}$ on $(\Omega,\mathcal{F},P)$ by Random Vectors: Coordinates, Functions, Pairs and Almost-Sure Equality (v), with $\mathcal{L}(T)=T_{\#}P=\pi$ by hypothesis and Random Vector and Its Law (iii). By Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i) one has $\mathrm{pr}_{1}\circ T=Y$, so, exactly as in the computation of the first marginal in claim 4,

$$\mathcal{L}(Y)=(\mathrm{pr}_{1}\circ T)_{\#}P=(\mathrm{pr}_{1})_{\#}\bigl(T_{\#}P\bigr)=(\mathrm{pr}_{1})_{\#}\pi=\nu .$$

Let $\eta\in L^{2}(\nu;\mathbb{R}^{d})$. The class $\eta\circ Y\in L^{2}(\Omega;\mathbb{R}^{d})$ is defined by Composition of a Square-Integrable Vector Field with a Random Vector, and the Lifted Score: Isometry, Norm, Weak Identity and Second-Moment Identity (i), applicable because $\mathcal{L}(Y)=\nu$, and a representative of it is $\eta\circ Y$. The function $g(z)=\eta(x)\cdot p$ of claim 3 satisfies $g\circ T=(\eta\circ Y)\cdot Y'$ by Pairs of Euclidean Points: Coordinate Projections, Pairings, the Product Measure on a Euclidean Space, Borel Norm Functions and Finite Sets (i), and this function is integrable with respect to $P$ by The Space of Square-Integrable Random Vectors (iii), so the change-of-variables formula of Probability Measures on Euclidean Space and Random Vectors: Standing Notation (iv) gives

$$\int\eta(x)\cdot p\,\pi(dz)=\int_{\Omega}(\eta\circ Y)\cdot Y'\,dP=\mathbb{E}\bigl[(\eta\circ Y)\cdot Y'\bigr]=\langle\eta\circ Y,Y'\rangle_{L^{2}},$$

the last equality by The Space of Square-Integrable Random Vectors (iii). Applying the same formula to the nonnegative Borel function $z\mapsto\lVert p\rVert^{2}$, whose composition with $T$ is $\lVert Y'\rVert^{2}$, gives $\int\lVert p\rVert^{2}\,\pi(dz)=\mathbb{E}[\lVert Y'\rVert^{2}]=\lVert Y'\rVert_{L^{2}}^{2}$, again by The Space of Square-Integrable Random Vectors (iii). This proves claim 5.