The Hamilton-Jacobi Operator with Common Noise and Penalty Drift Has Momentum-Continuous Shifts

In the setting of The Intrinsic Calculus on the Wasserstein Space: Standing Notation, let $(\mathcal{D},\mathcal{D}_{\Sigma},\mathcal{E},\Sigma)$ be a penalty pair on $\mathcal{P}_{2}(\mathbb{R}^{d})$, let $\lambda_{0},\theta\in\mathbb{R}$ be positive, let $p\in\mathbb{N}$, let $\Gamma\in\mathcal{M}_{p\times d}(\mathbb{R})$ be a real $p\times d$ matrix, let $g:\mathcal{P}_{2}(\mathbb{R}^{d})\to\mathbb{R}$, and let $F$ be the Hamilton-Jacobi operator with common noise and penalty drift of the pair with discount $\lambda_{0}$, common-noise matrix $\Gamma$, control cost $\theta$ and running cost $g$, with $\delta$-shifts $F^{-}_{\delta}$ and $F^{+}_{\delta}$ relative to the pair. Momentum-continuous shifts are those of that definition, and $|s|$ is the absolute value of $s\in\mathbb{R}$. In this item the letter $q$ denotes a vector field and the letter $r$ a real number.

1. (Lipschitz bound in the momentum)¶ Let $\delta,R\in\mathbb{R}$ satisfy $0<\delta<1$ and $0<R$, let $\nu\in\mathcal{D}_{\Sigma}$ satisfy $\lVert\Sigma(\nu)\rVert_{\nu}\le R$, let $r\in\mathbb{R}$, let $Y\in\mathcal{S}(d)$, and let $q,q'\in L^{2}(\nu;\mathbb{R}^{d})$ satisfy $\lVert q\rVert_{\nu}\le R$ and $\lVert q'\rVert_{\nu}\le R$. Then

$$\bigl|F^{-}_{\delta}(\nu,r,q,Y)-F^{-}_{\delta}(\nu,r,q',Y)\bigr|\le(2\theta+1)\,R\,\lVert q-q'\rVert_{\nu},\qquad\bigl|F^{+}_{\delta}(\nu,r,q,Y)-F^{+}_{\delta}(\nu,r,q',Y)\bigr|\le(2\theta+1)\,R\,\lVert q-q'\rVert_{\nu}.$$

2. (Momentum-continuous shifts)¶ $F$ has momentum-continuous shifts relative to the pair.