Proof. By the definition of \(\tilde{W}^{-\delta,k}(\mathbb{R}^d,\mathbb{R}^d)=(\tilde{W}^{-\delta,k})^d\), it suffices to prove that the family \[\{h_n:= 1_{\{|\cdot|\ge n^{-1}\}} h\}_{n\ge 1}\subset {\tilde{W}^{-\delta,k}(\mathbb{R}^d,\mathbb{R}^d)}\] is a Cauchy sequence under \(\|\cdot\|_{\tilde{W}^{-\delta,k}}\), i.e. \[\label{CH} \lim_{n\rightarrow\infty} \sup_{m\ge n} \|h_n-h_m\|_{\tilde{W}^{-\delta,k}}=0,\tag{10}\] so that \(h\) is well-defined in \({\tilde{W}^{-\delta,k}(\mathbb{R}^d,\mathbb{R}^d)}\) as the limit of \(h_n\) when \(n\rightarrow\infty\).

To this end, we use 3 and the formula \[P_t^0(h_n-h_m)(x)= (4\pi t)^{-\frac{d}{2}} \int_{\{m^{-1}\le |z|<n^{-1}\} } h(z) \text{\rm{e}}^{-\frac{|z-x|^2}{4t}}\text{\rm{d}}z,\;\;\; t>0,\;x\in\mathbb{R}^d,\;n\le m.\] By the integral transform \(z\mapsto -z\) and \(h(-z)=-h(z)\), we obtain \[P_t^0(h_n-h_m)(x)= - (4\pi t)^{-\frac{d}{2}} \int_{\{m^{-1}\le |z|<n^{-1}\} } h(z) \text{\rm{e}}^{-\frac{|z+x|^2}{4t}}\text{\rm{d}}z,\;\;\; t>0,\;x\in\mathbb{R}^d.\] So, there exists a constant \(c_1\in (0,\infty)\) such that \[\begin{align} & \big|P_t^0(h_n-h_m)(x)\big| =\frac{1}{2} \bigg|(4\pi t)^{-\frac{d}{2}} \int_{\{m^{-1}\le |z|<n^{-1}\} } h(z) \Big(\text{\rm{e}}^{-\frac{|z-x|^2}{4t}}-\text{\rm{e}}^{-\frac{|z+x|^2}{4t}}\Big)\text{\rm{d}}z\bigg|\\ &\le \frac{1}{2} (4\pi t)^{-\frac{d}{2}} \int_{\{m^{-1}\le |z|<n^{-1}\} } \frac{| h(z)|\cdot |z|\cdot |x|}{t} \Big(\text{\rm{e}}^{-\frac{|z-x|^2}{4t}}+ \text{\rm{e}}^{-\frac{|z+x|^2}{4t}}\Big) \text{\rm{d}}z\\ &= (4\pi t)^{-\frac{d}{2}} \int_{\{m^{-1}\le |z|<n^{-1}\} } \frac{| h(z)|\cdot |z|\cdot |x|}{t} \text{\rm{e}}^{-\frac{|z-x|^2}{4t}} \text{\rm{d}}z\\ &\le c_1 t^{-\frac{d}{2}-1} \int_{B(0,n^{-1})} \frac{ |x|}{|z|^{d+\theta-1}} \text{\rm{e}}^{-\frac{|z-x|^2}{4t}} \text{\rm{d}}z. \end{align}\] Noting that \(|x|\le |z-x|+1\) for \(|z|\le n^{-1}\), by Hölder’s inequality, we find a constant \(c_2\in (0,\infty)\) such that \[\begin{align} & t^{k+\frac{d k}{2}} \big\| P_t^0(h_n-h_m)\big\|_{L^k}^k \le \int_{\mathbb{R}^d} \bigg(c_1 \int_{B(0,n^{-1})} \frac{ |x|}{ |z|^{d+\theta-1}} \text{\rm{e}}^{-\frac{|z-x|^2}{4t}} \text{\rm{d}}z\bigg)^k\text{\rm{d}}x\\ & \le c_1^k \bigg(\int_{B(0,n^{-1})} \frac{ 1}{ |z|^{d+\theta-1}} \text{\rm{d}}z \int_{\mathbb{R}^d} (|z-x|+1)^k \text{\rm{e}}^{-\frac{k|z-x|^2}{4t}} \text{\rm{d}}x \bigg)\\ &\qquad\times \bigg( \int_{B(0,n^{-1})} \frac{ 1}{ |z|^{d+\theta-1}} \text{\rm{d}}z\bigg)^{k-1} \\ &\le c_2^k n^{-k(1-\theta)} \big(1+ t^{\frac{k}{2}}\big) t^{\frac{d }{2}}. \end{align}\] Combining this with the formula 3 and noting that ?? implies \[\frac{\delta}{2}-2 -\frac{d(k-1)}{2k} >-1,\] we find a constant \(c_3 \in (0,\infty)\) such that \[\begin{align} &\sup_{m\ge n} \big\|h_n-h_m\big\|_{\tilde{W}^{-\delta,k}}\le c(\delta)\int_0^\infty t^{\frac{\delta}{2}-1} \text{\rm{e}}^{-t}\big\| P_t^0(h_n-h_m)\big\|_{\tilde{L}^k} \text{\rm{d}}t\\ &\le c_2 n^{-(1- \theta)} \int_0^\infty t^{\frac{\delta}{2}-2} \big(1+ t^{\frac{k}{2}}\big)^{\frac{1}{k}} t^{-\frac{d(k-1)}{2k} } \text{\rm{e}}^{-t}\text{\rm{d}}t\le c_3 n^{-(1-\theta)},\;\;n\ge 1. \end{align}\] This implies 10 since \(\theta\in (0,1)\). ◻

Proof. Noting that \[P_t^0 f(x)= \big(4\pi t\big)^{-\frac{d}{2}} \int_{\mathbb{R}^d} \text{\rm{e}}^{-\frac{|x-y|^2}{4t}} f(y)\text{\rm{d}}y,\] and there exists a constant \(L\in (0,\infty)\) such that \[\Big|\nabla^i \text{\rm{e}}^{-\frac{|\cdot|^2}{4t}}\Big|(x)\le L t^{-\frac{i}{2}} \text{\rm{e}}^{-\frac{|x|^2}{8t}},\;\;\;t>0,\;0\le i\le i_0,\;x\in\mathbb{R}^d,\] we find an increasing function \(C: \mathbb{Z}_+\rightarrow(0,\infty)\) such that \[|(1-\Delta)^n \nabla^i P_t^0f|\le C_{n} t^{-\frac{i}{2} -n} P_{2t}^0 |f|,\;\;t>0,\;0\le i\le i_0,n\in \mathbb{Z}_+, f\in \mathscr B_b(\mathbb{R}^d).\] On the other hand, by [1], there exists a constant \(c_0\in (0,\infty)\) such that \[\|P_{2t}^0\|_{\tilde{L}^k\rightarrow\tilde{L}^p}\le c_0 t^{-\frac{d(p-k)}{2pk}},\;\;t>0,\;1\le k\le p\le \infty.\] So, \[\label{PL42} \|(1-\Delta)^n \nabla^i P_t^0\|_{\tilde{L}^k\rightarrow\tilde{L}^p}\le C_{n} c_0 t^{-\frac{i}{2}-n-\frac{d(p-k)}{2pk}},\;\;t>0,\;0\le i\le i_0,n\in \mathbb{Z}_+.\tag{16}\] Let \(n\in \mathbb{N}\) such that \[\theta_0:=n-\frac{\delta-\varepsilon}{2}\in [1,2).\] By 3 and 16 , we find an increasing function \(\bar{B}: \mathbb{Z}_+\rightarrow(0,\infty)\) such that \[\begin{align} &\|\nabla^i P_t^0\|_{\tilde{W}^{-\delta,k}\rightarrow\tilde{W}^{-\varepsilon,p}}=\|(1-\Delta)^{-\frac{\varepsilon}{2}}\nabla^iP_t^0 (1-\Delta)^{\frac{\delta}{2}}\|_{\tilde{L}^k\rightarrow\tilde{L}^p} \\ &= \|(1-\Delta)^n\nabla^i(1-\Delta)^{-\theta_0}P_t^0\|_{\tilde{L}^k\rightarrow\tilde{L}^p}\\ &=\frac{1}{\Gamma(\theta_0)} \bigg\|\int_0^\infty s^{\theta_0-1}\text{\rm{e}}^{-s} (1-\Delta)^n\nabla^i P_{t+s}^0\text{\rm{d}}s\bigg\|_{\tilde{L}^k\rightarrow\tilde{L}^p}\\ &\le \frac{C_{n}c_0}{\Gamma(\theta_0)} \int_0^\infty s^{\theta_0-1}\text{\rm{e}}^{-s} (t+s)^{-\frac{i}{2}-n-\frac{d(p-k)}{2pk}}\text{\rm{d}}s \\ &\le {\bar{ B}_{n}} t^{-\frac{i+\delta-\varepsilon}{2} - \frac{d(p-k)}{2pk}},\;\;t>0,\;0\le i\le i_0,\;\delta-\varepsilon\in (2(n-2), 2(n-1)],\;n\in\mathbb{N}. \end{align}\] This implies the desired estimate for some increasing function \(B: [0,\infty)\rightarrow(0,\infty).\) ◻

Proof. (1) To prove the completeness of \((\mathscr P_{\delta,k*},\;\|\cdot\|_{\delta,k*})\), let \(\{\mu_n\}_{n\ge 1}\) be a Cauchy sequence in \((\mathscr P_{\delta,k*},\;\|\cdot\|_{\delta,k*})\). Since \((\mathscr P_{\delta,k*},\;\|\cdot\|_{\delta,k*})\) is included by the dual space \((\tilde{W}^{-\delta,k}_*,\;\|\cdot\|_{\delta,k*})\) of the Banach space \((\tilde{W}^{-\delta,k},\|\cdot\|_{\tilde{W}^{-\delta,k}})\), there exists a unique \(\mu\in \tilde{W}^{-\delta,k}_*\) such that \[\lim_{n\rightarrow\infty} \|\mu_n-\mu\|_{\delta,k*}=\sup_{f\in\mathscr B_b(\mathbb{R}^d), \|f\|_{\tilde{W}^{-\delta,k}}\le 1} |\mu_n(f)-\mu(f)|=0.\] Since \(\mathscr P_{\delta,k*}= \tilde{W}_*^{-\delta,k}\cap \mathscr P,\) it remains to show that \(\mu\in \mathscr P\). When \(\delta=0\), we have \[\|f\|_{\tilde{W}^{-0,k}}=\|f\|_{\tilde{L}^k}\le \omega(d)^{\frac{1}{k}} \|f\|_\infty,\] where \(\omega(d)\) is the volume of the unit ball \(B(0,1)\). If \(\delta>0\), by 3 we have \[\begin{align} \|f\|_{\tilde{W}^{-\delta,k}}&=\bigg\|\frac{1}{\Gamma(\delta/2) } \int_0^\infty t^{\frac{\delta}{2}-1} \text{\rm{e}}^{-t} P_t^0f\text{\rm{d}}t\bigg\|_{\tilde{L}^k}\\ &\le \frac{\omega(d)^{\frac{1}{k}}}{\Gamma(\delta/2) } \bigg\| \int_0^\infty t^{\frac{\delta}{2}-1} \text{\rm{e}}^{-t} P_t^0f\text{\rm{d}}t\bigg\|_{\tilde{L}^\infty}\\ &\le \bigg(\frac{\omega(d)^{\frac{1}{k}} }{\Gamma(\delta/2) } \int_0^\infty t^{\frac{\delta}{2}-1} \text{\rm{e}}^{-t} \text{\rm{d}}t \bigg) \|f\|_{\infty}. \end{align}\] In any case, we find a constant \(c\in (0,\infty)\) such that \(\|\cdot\|_{\tilde{W}^{-\delta,k}}\le c \|\cdot\|_\infty\), hence \[\label{Var} \|\mu_n-\mu\|_{var} \le c \|\mu_n-\mu\|_{\delta, k*},\tag{17}\] so that \[\lim_{n\rightarrow\infty} \|\mu_n-\mu\|_{var} \le c \lim_{n\rightarrow\infty}\|\mu_n-\mu\|_{\delta, k*} =0.\] This together with \(\{\mu_n\}_{n\ge 1}\subset \mathscr P\) implies \(\mu\in \mathscr P\).

Next, since \(C_b(\mathbb{R}^d)\) is dense in \(\tilde{W}^{-\delta,k}\), then for any \(f\in \tilde{W}^{-\delta,k}\) and \(\mu\in\mathscr P_{\delta,k*}\), there exists \(\{f_n\}_{n\geq 1}\subset C_b(\mathbb{R}^d)\) satisfying \(\lim_{n\rightarrow\infty}\|f_n-f\|_{\tilde{W}^{-\delta,k}}= 0\), which implies \[\lim_{n\rightarrow\infty}|\mu(f_n)-\mu(f)|\leq \|\mu\|_{\delta,k*}\lim_{n\rightarrow\infty}\|f_n-f\|_{\tilde{W}^{-\delta,k}}= 0.\] Noting that the Borel \(\sigma\)-field in \(\mathscr P_{\delta,k*}\) is induced by \[\big\{\mu\mapsto \mu(f):\;f\in \tilde{W}^{-\delta,k}\big\},\] hence it coincides with the \(\sigma\)-field induced by the weak topology.

(2) It suffices to prove for \(\lambda=0.\) Let \(\{\mu^{(n)}\}_{n\ge 1}\subset \mathscr C_{\varepsilon,p;\delta,k}^{\gamma,T}\) be a Cauchy sequence with respect to \(\rho^{T}_{\varepsilon,p;\delta,k}\). Then for any \(t\in (0,T]\), \(\{\mu_t^{(n)}\}_{n\ge 1}\) is a Cauchy sequence in \(\mathscr P_{\delta,k*}\), so that by (1), there exists a unique \(\mu_t\in \mathscr P_{\delta,k*}\) such that \[\label{IO} \begin{align}&\lim_{n\rightarrow\infty} \rho^{T}_{\varepsilon,p;\delta,k}(\mu^{(n)}, \mu ) = \lim_{n\rightarrow\infty} \sup_{t\in [0,T]} \lim_{l\rightarrow\infty} t^{\frac{\delta-\varepsilon}{2} +\frac{d(p-k)}{2pk} }\|\mu_t^{(n)} -\mu_t^{(l)} \|_{\delta,k*} \\ & \le \lim_{n,l\rightarrow\infty} \rho^{T}_{\varepsilon,p;\delta,k}(\mu^{(n)}, \mu^{(l)})=0. \end{align}\tag{18}\] It remains to show the weak continuity of \((0,T]\ni t \mapsto \mu_t\), which together with 18 implies \(\mu\in \mathscr C_{\varepsilon,p;\delta,k}^{\gamma,T}.\) For any \(f\in C_b(\mathbb{R}^d), s\in (0,T]\) and \(\varepsilon'>0\), by 17 and applying 18 , we find large enough \(n\ge 1\) such that \[\begin{align} &|\mu_t^{(n)}(f)-\mu_t(f)| \le c\|f\|_\infty \|\mu_t^{(n)}- \mu_t\|_{\delta,k*} \\ &\le c\|f\|_\infty t^{-\frac{\delta-\varepsilon}{2}-\frac{d(p-k)}{2pk}} \rho^{T}_{\varepsilon,p;\delta,k}(\mu^{(n)}, \mu ) \le \varepsilon',\;\;t\in [s/2,T]. \end{align}\] Combining this with the weak continuity of \((0,T]\ni t\mapsto \mu_t^{(n)},\) we derive \[\limsup_{t\rightarrow s} |\mu_t (f)-\mu_s(f)| \le \limsup_{t\rightarrow s} \big\{|\mu_t^{(n)} (f)-\mu_s^{(n)} (f)|+2\varepsilon'\big\}=2\varepsilon'.\] Since \(\varepsilon'>0\) is arbitrary, this implies that weak continuity of \((0,T]\ni t \mapsto \mu_t.\) ◻

Proof. By (A) and \(\mu\in \mathscr C_{\varepsilon,p;\delta,k}^{T}\), there exists a constant \(c\in (0,\infty)\) such that \(b_t^{\mu}(x):=b_t(x,\mu_t)\) satisfies \[|b_t^{\mu}(x)|\le c t^{\kappa-\frac{\eta}{2}},\;\;t\in (0,T].\] By this and 6 , we find \(q'>2\) such that \[\label{ZV} \|b^{\mu}\|_{\tilde{L}_{q'}^{\infty}(T)}=\sup_{z\in\mathbb{R}^d}\left(\int_{0}^T\|b_t^\mu 1_{B(z,1)}\|_{\infty}^{q'}\right)^{\frac{1}{q'}}<\infty.\tag{20}\] Then the desired assertion follows from [1]. ◻

Proof. Let \(t\in (0,\infty)\) and \(\mu\in \mathscr C_{\varepsilon,p; \delta,k}^{t}.\) We will complete the proof by the following four steps.

(a) We first observe that when \(p_2>1\), \[\label{42}\sup_{0\le r\le s\le t}(s-r)^{\frac{i}{2}}\|\nabla^i {P}_{r,s}^\mu\|_{\tilde{W}^{-\varepsilon_2,p_2} \rightarrow\tilde{W}^{-\varepsilon_2,p_2}} <\infty,\;\;i=0,1.\tag{22}\] By [1], 20 implies that for \(p_2>1\) \[\label{ES01} \sup_{0\le r\le s\le t} (s-r)^{\frac{i}{2}}\|\nabla^i P_{r,s}^\mu\|_{\tilde{L}^{p_2} \rightarrow\tilde{L}^{p_2}}<\infty, \;\; i=0,1.\tag{23}\] Moreover, by the Duhamel formula, see [1], we have \[\begin{align} \label{DH1} {P}_{s,t}^\mu f= P_{t-s}^0 f+\int_s^t {P}_{s,r}^\mu \langle b_r(\cdot,\mu_r), \nabla P_{t-r}^0 f\rangle\text{\rm{d}}r, \;\; s\in [0,t], \;f\in \mathscr B_b(\mathbb{R}^d). \end{align}\tag{24}\] We now prove 22 by inducing in \(l\in\mathbb{N}\) for \(\varepsilon_2\in [0, lk_0],\) where, due to 6 , \[\label{K0} k_0:= \frac{1}{2} \land [1-(\eta-2\kappa)^+]>0.\tag{25}\]

Let \(\varepsilon_2\in [0, k_0].\) By 23 and \(\|(1-\Delta)^{-\frac{\varepsilon_2}{2}}\|_{\tilde{L}^{p_2}\rightarrow\tilde{L}^{p_2}}<\infty\), we obtain \[\begin{align} & c_1(\mu,t):= \sup_{0\le r\le s\le t} (s-r)^{\frac{i}{2}} \|\nabla^i P_{r,s}^\mu\|_{\tilde{L}^{p_2}\rightarrow\tilde{W}^{-\varepsilon_2, p_2}}\\ &= \sup_{0\le r\le s\le t} (s-r)^{\frac{i}{2}} \|(1-\Delta)^{-\frac{\varepsilon_2}{2}} \nabla^i P_{r,s}^\mu\|_{\tilde{L}^{p_2}\rightarrow\tilde{L}^{ p_2}}\\ &\le \|(1-\Delta)^{-\frac{\varepsilon_2}{2}}\|_{\tilde{L}^{p_2}\rightarrow\tilde{L}^{p_2}} \sup_{0\le r\le s\le t} (s-r)^{\frac{i}{2}} \|\nabla^i P_{r,s}^\mu\|_{\tilde{L}^{p_2}\rightarrow\tilde{L}^{p_2}}<\infty. \end{align}\] Combining this with (A), 4 , 24 and noting that \(s^{\kappa-\frac{\eta}{2}}=s^{-\frac{(\eta-2\kappa)^+}{2}}s^{\frac{(2\kappa-\eta)^+}{2}}\), we find \(c_2(\mu,t)\in (0,\infty)\) increasing in \(t\in (0,\infty)\), such that \[\begin{align} &\|\nabla^i {P}_{r,t}^\mu\|_{\tilde{W}^{-\varepsilon_2,p_2} \rightarrow\tilde{W}^{-\varepsilon_2,p_2}} \le \|\nabla^iP_{t-r}^0\|_{\tilde{W}^{-\varepsilon_2,p_2} \rightarrow\tilde{W}^{-\varepsilon_2,p_2}}\\ &\quad + K_t\int_r^t s^{ \kappa} \|\nabla^i {P}_{r,s}^\mu\|_{\tilde{L}^{p_2} \rightarrow\tilde{W}^{-\varepsilon_2,p_2}} \|\mu_s\|_{\delta,k*}\|\nabla P_{t-s}^0\|_{ \tilde{W}^{-\varepsilon_2,p_2}\rightarrow\tilde{L}^{p_2}}\text{\rm{d}}s\\ &\le {B_{0}} (t-r)^{-\frac{i}{2}} + K_tc_1(\mu,t)\rho^t_{\varepsilon,p;\delta,k} (\mu)\int_r^t s^{\kappa-\frac{\eta}{2}}(s-r)^{-\frac{i}{2}} (t-s)^{-\frac{\varepsilon_2+1}{2} }\text{\rm{d}}s\\ &\leq {B_{0}}(t-r)^{-\frac{i}{2}} + c_2(\mu,t) \int_r^t (s-r)^{-\frac{i+(\eta-2\kappa)^+}{2}} (t-s)^{-\frac{\varepsilon_2+1}{2} } \text{\rm{d}}s. \end{align}\] By 6 , 25 , \(\varepsilon_2\in [0,k_0]\) and \(i\le 1\), we have \[\frac{\varepsilon_2+1}{2}\lor \frac{i+(\eta-2\kappa)^+}{2} <1,\;\;\;\frac{\varepsilon_2+1+(\eta-2\kappa)^+ }{2}\le 1,\] so that 15 with \(\alpha=0\) and \(\lambda=0\) implies \[\int_r^t s^{-\frac{i+(\eta-2\kappa)^+ }{2}} (t-s)^{-\frac{\varepsilon_2+1}{2} } \text{\rm{d}}s\le c (t-r)^{-\frac{i}{2}},\;\;r\in [0,t)\] for some constant \(c\in (0,\infty).\) Therefore, 22 holds for \(\varepsilon_2\in [0,k_0].\)

Assume that for some \(l\in \mathbb{N}\) we have 22 for \(\varepsilon_2\in [0, lk_0]\), then for \(\varepsilon_2\in (lk_0, (l+1)k_0]\), \[\begin{align} & c_l(\mu,t):= \sup_{0\le r\le s\le t} (s-r)^{\frac{i}{2}}\|\nabla^i P_{r,s}^\mu\|_{ \tilde{W}^{-l k_0, p_2}\rightarrow\tilde{W}^{-\varepsilon_2, p_2}}\\ &= \sup_{0\le r\le s\le t} (s-r)^{\frac{i}{2}} \|(1-\Delta)^{-\frac{\varepsilon_2-lk_0}{2}}\nabla^i P_{r,s}^\mu\|_{\tilde{W}^{-l k_0, p_2} \rightarrow\tilde{W}^{-lk_0, p_2}} \\ &\le \|(1-\Delta)^{-\frac{\varepsilon_2-lk_0}{2}}\|_{\tilde{W}^{-l k_0, p_2} \rightarrow\tilde{W}^{-l k_0, p_2}}\sup_{0\le r\le s\le t} (s-r)^{\frac{i}{2}}\|\nabla^i P_{r,s}^\mu\|_{\tilde{W}^{-l k_0, p_2} \rightarrow\tilde{W}^{-l k_0, p_2}}<\infty, \end{align}\] which is increasing in \(t\in (0,\infty)\). By combining this with (A), 4 , 24 , we find \(c_{l+1}(\mu,t)\in (0,\infty)\) increasing in \(t\in (0,\infty)\), such that \[\begin{align} &\|\nabla^i {P}_{r,t}^\mu\|_{\tilde{W}^{-\varepsilon_2,p_2}\rightarrow\tilde{W}^{-\varepsilon_2,p_2}} \le \|\nabla^iP_{t-r}^0\|_{\tilde{W}^{-\varepsilon_2,p_2} \rightarrow\tilde{W}^{-\varepsilon_2,p_2}} \\ &\qquad + K_t\int_r^t s^{\kappa}\|\nabla^i {P}_{r,s}^\mu\|_{\tilde{W}^{-l k_0, p_2} \rightarrow\tilde{W}^{-\varepsilon_2,p_2}} \|\mu_s\|_{\delta,k*}\|\nabla P_{t-s}^0\|_{ \tilde{W}^{-\varepsilon_2,p_2}\rightarrow\tilde{W}^{-lk_0, p_2}}\text{\rm{d}}s\\ &\le {B_{0}} t^{-\frac{i}{2}} + K_tc_l (\mu,t)\rho^t_{\varepsilon,p;\delta,k} (\mu){B_{\varepsilon_2-lk_0}} \int_r^t s^{\kappa-\frac{\eta}{2}} (s-r)^{-\frac{i}{2}} (t-s)^{-\frac{\varepsilon_2-lk_0+1}{2} }\text{\rm{d}}s\\ &\leq {B_{0}}(t-r)^{-\frac{i}{2}} + c_{l+1} (\mu,t) t^{-\frac{i}{2}},\;\;\;r\in [0,t),\; \varepsilon_2\in (l k_0, (l+1)k_0], \end{align}\] where the last step follows from 15 with \(\alpha=0\) and \(\lambda=0\), since \(\frac{\eta+i}{2}-\kappa<1\) for \(i\le 1\), \(\frac{\varepsilon_2-lk_0+1}{2}<1\) for \(\varepsilon_2-lk_0 \in [0, k_0]\), and \(\frac{\varepsilon_2-lk_0 +1}{2}+\frac{(\eta-2\kappa)^{+}}{2}\le 1.\) Hence, 22 holds for all \(\varepsilon_2\in [0,\infty)\).

(b) We intend to find some increasing function \(\beta_0: (0,\infty)\rightarrow(0,\infty)\) such that \[\label{EE} \begin{align}&h_{r,t}:= \|\nabla^i P_{r,t}^\mu\|_{\tilde{W}^{-\varepsilon_2,p_2}\rightarrow\tilde{W}^{-\varepsilon_2,p_2}}\\ &\le \beta_0(t) \exp\big[t\beta_0(t) \rho_{\varepsilon,p;\delta,k}^t(\mu)^{\theta} \big] (t-r)^{-\frac{i}{2}},\; \;\;r\in [0,t), \;i=0,1.\end{align}\tag{26}\] By 4 , 24 and (A), we obtain that \[\begin{align} &h_{r,t } \le \|\nabla^i P_{r,t}^0\|_{\tilde{W}^{-\varepsilon_2,p_2} \rightarrow\tilde{W}^{-\varepsilon_2,p_2}}+ K_t\int_r^t h_{r,s} s^{ \kappa}\|\mu_s\|_{\delta,k*} \|\nabla P_{t-s}^0\|_{\tilde{W}^{-\varepsilon_2,p_2} \rightarrow\tilde{W}^{-\varepsilon_2,p_2}}\text{\rm{d}}s\\ &\le {B_{0}} (t-r)^{-\frac{i}{2}} + K_t\rho_{\varepsilon,p;\delta,k}^t (\mu){B_{0}} t^{(\kappa-\frac{\eta}{2})^+}\int_r^t h_{r,s} (s-r)^{-(\frac{\eta}{2}-\kappa)^+ } (t-s)^{-\frac{1}{2}} \text{\rm{d}}s,\;\;r\in [0,t). \end{align}\] Combining this with 15 for \(\alpha= \frac{1}{\theta}=\frac{1-(\eta-2\kappa)^+}{2}\), we find some constant \(c(t)\in (0,\infty)\) increasing in \(t\in (0,\infty)\), such that \[H_{r,t}(\lambda):= \sup_{s\in (r,t]} (s-r)^{\frac{i}{2}} h_{r,s} \text{\rm{e}}^{-\lambda(s-r)},\;\;\;s\in (r,t]\] satisfies \[\begin{align} H_{r,t}(\lambda)&\le {B_{0}} + K_t\rho_{\varepsilon,p;\delta,k}^t (\mu) {B_{0}} H_{r,t}(\lambda) t^{(\kappa-\frac{\eta}{2})^+} \\ &\quad\times \sup_{s\in (r,t]} (s-r)^{\frac{i}{2}} \int_r^s (u-r)^{-\frac{(\eta-2\kappa)^++i}{2} } (s-u)^{-\frac{1}{2}} \text{\rm{e}}^{-\lambda(s-u)} \text{\rm{d}}u\\ &\le {B_{0}} + c(t) \rho_{\varepsilon,p;\delta,k}^t (\mu) H_{r,t} (\lambda) \lambda^{- \frac{1}{\theta}},\;\;\;r\in [0,t),\;\lambda>0. \end{align}\] Taking \[\lambda:= {\big(\frac{1}{2}c(t) \rho_{\varepsilon,p;\delta,k}^t (\mu) \big)^{\theta},}\] and noting that \(H_{r,t}(\lambda)<\infty\) for \(p_2>1\) due to 22 , we find some increasing \(\beta_0: (0,\infty)\rightarrow(0,\infty)\) such that 26 holds for any \(p_2 \in (1,\infty]\). Since \(\beta_0(t)\) is uniformly in \(p_2>1\), by letting \(p_2\downarrow 1\) the estimate also holds for \(p_2=1.\)

(c) Let \(n\in \mathbb{N}\) such that \[\frac{\xi}{n}<1,\;\;\;\xi:= \varepsilon_1-\varepsilon_2 +\frac{d(p_2-p_1)}{p_1p_2},\] where we take \(n=1\) if \(\xi<1\). Below we prove ?? by inducing in \(0\le l\le n\) such that \[\label{67} \|\nabla^i P_{t}^\mu\|_{\tilde{W}^{-\delta_l,k_l}\rightarrow\tilde{W}^{-\varepsilon_2,p_2}}\le \beta_l(t) \exp\big[t\beta_l(t) \rho_{\varepsilon,p;\delta,k}^t(\mu)^{\theta} \big] t^{-\frac{i}{2}- \frac{l\xi}{2n}}.\tag{27}\] for some increasing \(\beta_l: (0,\infty)\rightarrow(0,\infty)\) and \[\delta_l:=\varepsilon_2 +\frac{l}{n} (\varepsilon_1-\varepsilon_2),\;\;\;k_l:= \frac{n p_1p_2}{(n-l)p_1+lp_2},\;\;\;0\le l\le n.\] In particular, when \(l=n\), 27 reduces to the desired inequality ?? .

By (b), 27 holds for \(l=0\). Assume that 27 holds for some \(0\le l\le n-1\), where \(l=0\) when \(\xi<1\), it suffices to verify it for \(l+1\) in place of \(l\).

To this end, let \[C_l(\mu,t):= \beta_l(t) \exp\big[t \beta_l(t) \rho_{\varepsilon,p;\delta,k}^t(\mu)^{\theta}\big] K_tB_{\varepsilon_1-\varepsilon_2} \rho_{\varepsilon,p;\delta,k}^t(\mu).\] By 4 , 24 , 27 and (A), we obtain \[\begin{align} &\|\nabla^i {P}_{t}^\mu\|_{\tilde{W}^{-\delta_{l+1},k_{l+1}}\rightarrow\tilde{W}^{-\varepsilon_2,p_2}} \le \|\nabla^i P_{t}^0\|_{ \tilde{W}^{-\delta_{l+1},k_{l+1}}\rightarrow\tilde{W}^{-\varepsilon_2,p_2}}\\ &\qquad + K_t\int_0^t s^{ \kappa} \|\nabla^i {P}_{s}^\mu\|_{\tilde{W}^{-\delta_{l},k_{l}}\rightarrow\tilde{W}^{-\varepsilon_2,p_2}} \|\mu_s\|_{\delta,k*}\|\nabla P_{t-s}^0\|_{ \tilde{W}^{-\delta_{l+1},k_{l+1}}\rightarrow\tilde{W}^{-\delta_l,k_l}}\text{\rm{d}}s\\ &\le B_{\varepsilon_1-\varepsilon_2} t^{-\frac{i+(l+1)\xi/n}{2} } + C_l(\mu,t) \int_0^t s^{\kappa- \frac{l\xi}{2n} -\frac{i +\eta}{2}} (t-s)^{ -\frac{1+ \xi/n}{2} }\text{\rm{d}}s. \end{align}\] By \(\xi<n\), 6 and either \(\xi<2-i+2\kappa-\eta\) or \(\xi<1\) with \(l=0\), we have \(\frac{1+\xi/n}{2}<1\) and \[\kappa- \frac{l\xi}{2n} -\frac{i +\eta}{2} > -1.\] Moreover, \(l+1\le n\) together with 6 implies \[1+\kappa-\frac{l\xi}{2n} -\frac{i+\eta}{2} -\frac{1+\xi/n}{2} \ge -\frac{i+\xi}{2} + 1+\kappa-\frac{\eta}{2} >-\frac{i+\xi}{2},\] so by 15 with \(\alpha=0\), we find \(\beta_{l+1}(t)\in (0,\infty)\) which is increasing in \(t>0\) such that 27 holds for \(l+1\) in place of \(l\). Hence, ?? is proved.

(d) Finally, let \(\xi <2-i-(\eta-2\kappa)^+\). For the above defined \((\delta_l,k_l)_{0\le l\le n}\), we intend to find increasing \(\beta_l: (0,\infty)\rightarrow(0,\infty)\) such that \[\label{6739} \|\nabla^i P_{r,t}^\mu\|_{\tilde{W}^{-\delta_l,k_l}\rightarrow\tilde{W}^{-\varepsilon_2,p_2}}\le \beta_l(t) \exp\big[t\beta_l(t) \rho_{\varepsilon,p;\delta,k}^t(\mu)^{\theta} \big] (t-r)^{-\frac{i}{2}- \frac{l\xi}{2n} }\tag{28}\] holds for \(0\le l\le n.\) In particular, when \(l=n\) this inequality reduces to the desired ?? .

By (b), 28 holds for \(l=0\). Assume that 28 holds for some \(0\le l\le n-1\), it suffices to verify it for \(l+1\) in place of \(l\).

By 4 , 24 , 28 and (A), we obtain that for \(r\in [0,t)\), \[\begin{align} &\|\nabla^i {P}_{r,t}^\mu\|_{\tilde{W}^{-\delta_{l+1}, k_{l+1}}\rightarrow\tilde{W}^{-\varepsilon_2,p_2}} \le \|\nabla^i P_{t-r}^0\|_{ \tilde{W}^{-\delta_{l+1},k_{l+1}}\rightarrow\tilde{W}^{-\varepsilon_2,p_2}}\\ &\qquad + K_t\int_r^t s^{ \kappa} \|\nabla^i {P}_{r,s}^\mu\|_{\tilde{W}^{-\delta_l,k_l}\rightarrow\tilde{W}^{-\varepsilon_2,p_2}} \|\mu_s\|_{\delta,k*}\|\nabla P_{t-s}^0\|_{ \tilde{W}^{-\delta_{l+1},k_{l+1}} \rightarrow\tilde{W}^{-\delta_l,k_l}}\text{\rm{d}}s\\ &\le B_{\varepsilon_1-\varepsilon_2} (t-r)^{-\frac{i+(l+1)\xi/n}{2} } + {C_l(\mu,t)} t^{(\kappa-\frac{\eta}{2})^+} \int_r^t (s-r)^{-(\frac{\eta}{2}-\kappa)^+ - \frac{i + l\xi/n}{2}} (t-s)^{ -\frac{1+ \xi/n}{2} }\text{\rm{d}}s. \end{align}\] Since either \(\xi< 2-i-(\eta-2\kappa)^+\) with \(\xi<n\), or \(\xi<1\) with \(l=0\), we have \(\frac{\xi/n +1}{2}<1\) and \[-\Big(\frac{\eta}{2}-\kappa\Big)^+ - \frac{i +l\xi/n}{2}\ge -\Big(\frac{\eta}{2}-\kappa\Big)^+ - \frac{i+ \xi}{2}>-1.\] Moreover, 6 together with \(l+1\le n\) yields \[1- \frac{i+l\xi/n}{2} -\Big(\frac{\eta}{2}- \kappa\Big)^+-\frac{1+\xi/n}{2} >-\frac{i+\xi}{2}.\] So, by 15 with \(\alpha=0\), we find \(\beta_{l+1}(t)\in (0,\infty)\) increasing in \(t>0\) such that 28 holds for \(l+1\) in place of \(l\). Then the proof is finished. ◻

Proof. (1) Let \(T\in (0,\infty)\) and \(\mu\in \mathscr C_{\varepsilon,p;\delta,k}^{\gamma,T}.\) We intend to show \(\Phi\mu\in \mathscr C_{\varepsilon,p;\delta,k}^{\gamma,T}.\) To this end, let \[\label{TA} r=\frac{2}{3+2\kappa},\;\;\; \delta':= r\varepsilon+(1-r)\delta,\;\;\;k':= \frac{pk}{(1-r)p+rk}>1.\tag{31}\] By 6 and \(\eta<\kappa+\frac{3}{2}\) , we have \[\label{T42} \eta<\frac{3+2\kappa}{2}= {\frac{2+2\kappa}{2-r}=\frac{1}{r}},\tag{32}\] \[\label{YY} \delta-\delta' +\frac{d(k'-k)}{k'k}=r\eta,\;\;\;\; \delta'-\varepsilon+\frac{d(p-k')}{pk'}=(1-r)\eta.\tag{33}\] Then \(\delta'-\varepsilon+\frac{d(p-k')}{pk'}=(1-r)\eta<2+2\kappa-\eta\), where \(k'>1\), so that ?? holds for \(i=0\) due to by Lemma 10(1), i.e. \[\label{ES13939} \| {P}_t^\mu\|_{\tilde{W}^{-\delta',k'}\rightarrow\tilde{W}^{-\varepsilon,p}} \le \beta_t \exp\big[t\beta_t \rho_{\varepsilon,p;\delta,k}^t(\mu)^{\theta}\big] t^{-\frac{1-r}{2}\eta},\;\;t\in (0,T],\;\mu\in \mathscr C_{\varepsilon,p;\delta,k}^{\gamma,T}.\tag{34}\] By 24 for \(s=0\), for any \(\mu\in \mathscr C_{\varepsilon,p;\delta,k}^{\gamma,T}\) we have \[\begin{align}&(\Phi_t \mu)(f)=\gamma( {P}_t^\mu f) \\ & = \gamma(P_{t}^0 f) + \int_0^t \gamma\big( {P}_s^\mu \langle b_s(\cdot,\mu_s), \nabla P_{t-s}^0 f\rangle \big)\text{\rm{d}}s,\;\;t\in [0, T],\;f\in\mathscr B_b(\mathbb{R}^d).\end{align}\] Combining this with 4 , 33 , 34 and (A), we find a constant \(c_1 \in (0,\infty)\) depending on \(T\) and \(\mu\) such that \[\label{Y42} \begin{align} &\|\Phi_t\mu\|_{\delta,k*} \le \|\gamma\|_{\varepsilon,p*}\|P_t^0\|_{ \tilde{W}^{-\delta,k}\rightarrow\tilde{W}^{-\varepsilon,p}}\\ &\qquad + K_t\|\gamma\|_{\varepsilon,p*} \int_0^t s^{ \kappa} \| {P}_s^\mu\|_{\tilde{W}^{-\delta',k'}\rightarrow\tilde{W}^{-\varepsilon,p}} \|\mu_s\|_{\delta,k*}\|\nabla P_{t-s}^0\|_{ \tilde{W}^{-\delta,k}\rightarrow\tilde{W}^{-\delta',k'}}\text{\rm{d}}s\\ &\le B_{\delta-\varepsilon}\|\gamma\|_{\varepsilon,p*} t^{-\frac{\eta}{2}} + c_1 \int_0^t s^{ \kappa- \frac{2-r}{2} \eta} (t-s)^{-\frac{1}{2}-\frac{r}{2} \eta}\text{\rm{d}}s,\;\;t\in (0,T].\end{align}\tag{35}\] By 32 , 6 and \(\eta<\kappa+\frac{3}{2}\) , we have \[{\kappa-\frac{2-r}{2}\eta>-1},\;\;-\frac{1}{2} -\frac{r}{2}\eta>-1,\;\; {\kappa- \eta +\frac{1}{2}\ge -\frac{\eta}{2}}.\] By 15 for \(\alpha=0\) and \(\lambda=0\), we find a constant \(c\in (0,\infty)\) such that \[\begin{align} \int_0^t s^{ \kappa- \frac{2-r}{2} \eta} (t-s)^{-\frac{1}{2}-\frac{r}{2} \eta}\text{\rm{d}}s \le c t^{-\frac{\eta}{2}},\;\;t>0. \end{align}\] Therefore, 35 implies \(\Phi\mu\in \mathscr C_{\varepsilon,p;\delta,k}^{\gamma,T}.\)

(2) Let \(\mu\in \tilde{\mathscr} C_{\varepsilon,p; \delta,k}^{\gamma, n}\) where the constant \(A_n\) in \(\tau_n(\gamma)\) is to be determined such that \(\Phi\mu\in \tilde{\mathscr} C_{\varepsilon,p; \delta,k}^{\gamma, n}.\) Then \[\begin{align}& K_nn^{\frac{(2\kappa-\eta)^+}{2}}B_{\delta-\varepsilon}\beta_n\rho^{\tau_n(\gamma)}_{\varepsilon,p;\delta,k} (\mu) \exp\Big[n\beta_n\rho_{\varepsilon,p;\delta,k}^{\tau_n(\gamma)}(\mu)^{\theta}\Big]\\ &\le D_n(\gamma):= 2K_nn^{\frac{(2\kappa-\eta)^+}{2}}B_{\delta-\varepsilon}^2 \|\gamma\|_{\varepsilon,p*} \beta_n\exp\Big[n\beta_n (2B_{\delta-\varepsilon}\|\gamma\|_{\varepsilon,p*})^{\theta}\Big].\end{align}\] By combining this with 24 for \(s=0\), (A), 4 , Lemma 10(1), we obtain \[\label{42D} \begin{align} & \|\Phi_t\mu\|_{\delta,k*} \le \|\gamma\|_{\varepsilon,p*} \|P_t^0\|_{\tilde{W}^{-\delta,k}\rightarrow\tilde{W}^{-\varepsilon,p}}\\ &\qquad + {K_t}\int_0^t s^{ \kappa} \|\gamma\|_{\varepsilon,p*} \|P_s^\mu\|_{\tilde{W}^{-\delta',k'}\rightarrow\tilde{W}^{-\varepsilon,p}} \|\mu_s\|_{\delta,k*} \|\nabla P_{t-s}^0 \|_{\tilde{W}^{-\delta, k}\rightarrow\tilde{W}^{-\delta',k'}}\text{\rm{d}}s\\ & {\le B_{\delta-\varepsilon} \|\gamma\|_{\varepsilon,p*}t^{-\frac{\eta}{2}} + \|\gamma\|_{\varepsilon,p*} 2K_nB_{\delta-\varepsilon}^2 \|\gamma\|_{\varepsilon,p*} \beta_n\exp\Big[n\beta_n (2B_{\delta-\varepsilon}\|\gamma\|_{\varepsilon,p*})^{\theta}\Big]}\\ &\qquad\qquad\qquad\qquad\quad\quad\quad {\times \int_0^t s^{\kappa-\frac{2-r}{2}\eta} (t-s)^{-\frac{1}{2} -\frac{r}{2} \eta}\text{\rm{d}}s}\\ &\le B_{\delta-\varepsilon} \|\gamma\|_{\varepsilon,p*}t^{-\frac{\eta}{2}}+ \|\gamma\|_{\varepsilon,p*} D_n(\gamma) c t^{\frac{1}{\theta}-\frac{\eta}{2}}, \;\;t\in (0,\tau_n(\gamma)],\end{align}\tag{36}\] where \(c\in (0,\infty)\) is a constant due to 15 for \(\alpha=0\) and \(\lambda=0\). Taking \(A_n\in (0,\infty)\) such that \[\label{RB} \Big(\frac{B_{\delta-\varepsilon} }{D_n(\gamma)c}\Big)^{\theta}\ge \Big(A_n \text{\rm{e}}^{A_n \|\gamma\|_{\varepsilon,p*}^{\theta}}\Big)^{-1},\tag{37}\] by the definition of \(\tau_n(\gamma)\) we derive \[\|\Phi_t\mu\|_{\delta,k*}\le 2 B_{\delta-\varepsilon} {t^{-\frac{\eta}{2}}}\|\gamma\|_{\varepsilon,p*},\;\;\;t\le \tau_n(\gamma).\] So, \(\Phi\mu\in\tilde{\mathscr} C_{\varepsilon,p; \delta,k}^{\gamma, n}.\)

If \(\mu\in \mathscr C_{\varepsilon,p; \delta,k}^{\gamma, T}\) is a fixed point of \(\Phi\), then \(\Phi\mu=\mu\). Noting that \(\rho_{\varepsilon,p;\delta,k}^{t}(\mu)\) is non-decreasing in \(t\), \[\rho_{\varepsilon,p;\delta,k}^{t+}(\mu):= \lim_{\varepsilon\downarrow 0} \rho_{\varepsilon,p;\delta,k}^{(t+\varepsilon)\land T}(\mu),\;\; \rho_{\varepsilon,p;\delta,k}^{t-}(\mu):= \lim_{\varepsilon\downarrow 0} \rho_{\varepsilon,p;\delta,k}^{(t-\varepsilon)^+}(\mu)\] exist and are non-decreasing for \(t\in (0,T].\) So, the first inequality in 36 , 4 and Lemma 10(1) yield \[\begin{align} &\|\mu_t\|_{\delta,k*}= \|\Phi_t\mu\|_{\delta,k*}\le B_{\delta-\varepsilon}\|\gamma\|_{\varepsilon,p*}t^{-\frac{\eta}{2}}\\ &\quad + { K_tB_{\delta-\varepsilon}\|\gamma\|_{\varepsilon,p*} \beta_t \exp\big[t\beta_t \rho_{\varepsilon,p;\delta,k}^{t-} (\mu)^{\theta}\big] \rho_{\varepsilon,p;\delta,k}^{t-}(\mu) \int_0^t s^{\kappa-\frac{2-r}{2}\eta}(t-s)^{-\frac{1}{2} -\frac{r}{2} \eta}\text{\rm{d}}s}\\ &\le B_{\delta-\varepsilon}\|\gamma\|_{\varepsilon,p*}t^{-\frac{\eta}{2}}+ { K_tB_{\delta-\varepsilon}\|\gamma\|_{\varepsilon,p*} t^{\frac{(2\kappa-\eta)^+}{2}}\beta_t \exp\big[t\beta_t \rho_{\varepsilon,p;\delta,k}^{t-} (\mu)^{\theta}\big] \rho_{\varepsilon,p;\delta,k}^{t-} (\mu) ct^{\frac{1}{\theta}-\frac{\eta}{2}}}. \end{align}\] Thus, \[\label{PI} \begin{align}&\rho^{t+}_{\varepsilon,p;\delta,k}(\mu)\le B_{\delta-\varepsilon} \|\gamma\|_{\varepsilon,p*} \\ &\quad + { K_tB_{\delta-\varepsilon}\|\gamma\|_{\varepsilon,p*} t^{\frac{(2\kappa-\eta)^+}{2}} \beta_n \exp\big[n\beta_n \rho_{\varepsilon,p;\delta,k}^{t-} (\mu)^{\theta}\big] \rho_{\varepsilon,p;\delta,k}^{t-} (\mu) ct^{\frac{1}{\theta}}},\;\;t\in (0, \tau_n(\gamma)]. \end{align}\tag{38}\] Then \[\rho^{0+}_{\varepsilon,p;\delta,k}(\mu) \le B_{\delta-\varepsilon} \|\gamma\|_{\varepsilon,p*}.\] This and the right continuity of \(\rho_{\varepsilon,p;\delta,k}^{t+}(\mu)\) in \(t\ge 0\) imply \[s_0:= \tau_n(\gamma)\land \inf\big\{t\in (0, \tau_n(\gamma)]:\; \rho^{t+}_{\varepsilon,p;\delta,k}(\mu)\ge 2 B_{\delta-\varepsilon} \|\gamma\|_{\varepsilon,p*}\big\}>0,\] where \(\inf\emptyset:=\infty\) by convention. If \(s_0< \tau_n(\gamma)\), by the non-decreasing of \(\rho^{t}_{\varepsilon,p;\delta,k}\) in \(t\ge 0\), we obtain \[\rho^{s_0+}_{\varepsilon,p;\delta,k}(\mu)\ge 2 B_{\delta-\varepsilon} \|\gamma\|_{\varepsilon,p*}\ge \rho^{s_0-}_{\varepsilon,p;\delta,k}(\mu),\] so that 38 yields \[\begin{align} & 2 B_{\delta-\varepsilon} \|\gamma\|_{\varepsilon,p*} \le \rho^{s_0+}_{\varepsilon,p;\delta,k}(\mu) \leq B_{\delta-\varepsilon} \|\gamma\|_{\varepsilon,p*} + D_n(\gamma)c \|\gamma\|_{\varepsilon,p*} s_0^{\theta}. \end{align}\] So, 37 and the definition of \(\tau_n(\gamma)\) imply \[s_0\ge \Big(\frac{B_{\delta-\varepsilon}}{D_n(\gamma)c}\Big)^{ \theta} \ge \tau_n(\gamma),\] which contradicts to \(s_0< \tau_n(\gamma)\). Hence, \(s_0\ge \tau_n(\gamma)\), which implies \(\mu\in\tilde{\mathscr} C_{\varepsilon,p;\delta,k}^{\gamma, n}.\)

(3) Let \(\mu\in \hat{\mathscr} C_{\delta,k}^{\gamma, n,\lambda}\), where \(\gamma\in \mathscr P, n\in \mathbb{N}\) and \(\lambda\in (0,\infty)\). By 24 , 4 , (A) and noting that \(\|\gamma\|_{0,\infty*}= 1\) for \(\gamma\in \mathscr P\), we obtain \[\begin{align} \|\Phi_t\mu\|_{\delta,k*} &\le B_{\delta}t^{-\frac{\eta}{2}} +K_n B_{0}\int_0^t s^\kappa\|\Phi_s\mu\|_{\delta, k*} \|\mu_s\|_{\delta,k*} (t-s)^{-\frac{1}{2}}\text{\rm{d}}s\\ &\le B_{\delta} t^{-\frac{\eta}{2}} + 2B_{0}B_\delta K_n\int_0^t s^{\kappa-\frac{\eta}{2}} \|\Phi_s\mu\|_{\delta,k*} (t-s)^{-\frac{1}{2}}\text{\rm{d}}s,\;\;t\in (0,n]. \end{align}\] Combining this with 15 for \(\alpha=\frac{1-(\eta-2\kappa)^+}{2}\), when \(\eta<1+\kappa\) such that \(\kappa-\eta >-1,\) we find a constant \(D_n\in (0,\infty)\) such that \[H_{n,\lambda}:= \sup_{t\in (0,n]} t^{\frac{\eta}{2}} \|\Phi_t\mu\|_{\delta, k*} \text{\rm{e}}^{-\lambda t}\] satisfies \[\begin{align} H_{n,\lambda}&\le B_{\delta}+ {2 K_nB_0B_{\delta}} H_{n,\lambda} \sup_{t\in (0,n]} t^{\frac{\eta}{2}} \int_0^t s^{\kappa- \eta } (t-s)^{-\frac{1}{2}} \text{\rm{e}}^{-\lambda(t-s)} \text{\rm{d}}s\\ &\le B_{\delta} + D_n H_{n,\lambda} \lambda^{-\frac{1-(\eta-2\kappa)^+}{2}},\;\;\;\lambda>0. \end{align}\] Noting that ?? for \(\varepsilon=0\) and \(p=\infty\) implies \(H_{n,\lambda}<\infty\), taking \[\lambda_n:= (2D_n)^{-\frac{2}{1-(\eta-2\kappa)^+}},\] we derive \(H_{n,\lambda_n}\le 2 B_{\delta}\), so that \(\Phi: \hat{\mathscr} C_{ \delta,k}^{\gamma, n,\lambda_n}\rightarrow\hat{\mathscr} C_{\delta,k}^{\gamma, n, \lambda_n}.\)

If \(\mu\in \mathscr C_{0,\infty;\delta,k}^{\gamma,n}\) is a fixed point of \(\Phi\), then \(\mu\in \hat{\mathscr} C_{\delta,k}^{\gamma, n,\lambda_n}\) can be proved by the same argument as in step (2) to verify that \[s_0:= n\land \inf\big\{t\in (0, n]:\; \rho^{t+}_{0,\infty;\delta,k}(\mu)\ge {2 B_{\delta}} \big\}=n.\] ◻

Proof. Let \(r,\delta'\) and \(k'\) be in 31 . By 24 for \(s=0\), (A), 4 and 34 , we obtain \[\begin{align} &\|P_t^\mu-P_t^\nu\|_{\tilde{W}^{-\delta',k'}\rightarrow\tilde{W}^{-\varepsilon,p}}=\sup_{f\in\mathscr B_b(\mathbb{R}^d), \|f\|_{\tilde{W}^{-\delta',k'}}\le 1} \|P_t^\mu f-P_t^\nu f\|_{\tilde{W}^{-\varepsilon,p}}\\ &\le \int_0^t \sup_{f\in\mathscr B_b(\mathbb{R}^d), \|f\|_{\tilde{W}^{-\delta',k'}}\le 1}\Big[\big\|(P_s^\mu -P_s^\nu) \langle b_s(\cdot,\mu_s),\nabla P_{t-s}^0f\rangle\big\|_{\tilde{W}^{-\varepsilon,p}}\\ &\qquad \qquad\qquad\qquad \qquad\qquad + \big\|P_s^\nu \langle b_s(\cdot,\mu_s)-b_s(\cdot,\nu_s),\nabla P_{t-s}^0 f\rangle\big\|_{\tilde{W}^{-\varepsilon,p}}\Big]\text{\rm{d}}s\\ &\le B_{0}K_t \rho_{\varepsilon,p;\delta,k}^t(\mu) \int_0^t s^{\kappa-\frac{\eta}{2}} \|P_s^\mu-P_s^\nu\|_{\tilde{W}^{-\delta',k'}\rightarrow\tilde{W}^{-\varepsilon,p}} (t-s)^{-\frac{1}{2} }\text{\rm{d}}s \\ &\quad +B_{0}K_tc_0(t) \exp\big[c_0(t) \rho_{\varepsilon,p;\delta,k}^{t}(\nu)^{\theta}\big] \rho_{\varepsilon,p;\delta,k}^{\lambda,t} (\mu,\nu) \int_0^t s^{\kappa-\frac{2-r}{2} \eta }\text{\rm{e}}^{\lambda s} (t-s)^{-\frac{1}{2}}\text{\rm{d}}s. \end{align}\] Combining this with 15 for \(\alpha= \frac{1}{\theta}\) and \(\alpha=0\) respectively, \(\alpha_1= (\frac{2-r}{2}\eta-\kappa)^+<1\) due to 32 , and \(\alpha_2=\frac{1}{2}\), we find \(c_1(t)\in (0,\infty)\) increasing in \(t>0\) such that \[\begin{align} &H_t(\lambda):=\sup_{u\in (0,t]}\text{\rm{e}}^{-\lambda u}u^{\frac{1-r}{2} \eta} \|P_u^\mu-P_u^\nu\|_{\tilde{W}^{-\delta',k'}\rightarrow\tilde{W}^{-\varepsilon,p}}\\ &\le B_{0}K_t \Big[ \rho_{\varepsilon,p;\delta,k}^t(\mu) H_t(\lambda)+c_0(t) \exp\big[c_0(t) \rho_{\varepsilon,p;\delta,k}^{t}(\nu)^{\theta}\big] \rho_{\varepsilon,p;\delta,k}^{\lambda,t} (\mu,\nu) \Big]\\ &\qquad \times \sup_{u\in (0,t]} u^{\frac{1-r}{2} \eta+(\kappa-\frac{2-r}{2}\eta)^+} \int_0^u s^{-(\frac{2-r}{2} \eta-\kappa)^+} (u-s)^{-\frac{1}{2}} \text{\rm{e}}^{-\lambda(u-s)}\text{\rm{d}}s \\ &\le c_1(t) \rho_{\varepsilon,p;\delta,k}^t(\mu) H_t(\lambda)\lambda^{-\frac{1}{\theta}} + c_1(t) \exp\big[c_0(t) \rho_{\varepsilon,p;\delta,k}^{t}(\nu)^{\theta}\big] \rho_{\varepsilon,p;\delta,k}^{\lambda,t} (\mu,\nu) \end{align}\] holds for all \(\lambda>0\). By ?? for \(i=0\), \((\varepsilon_1,p_1)=(\delta',k')\) and \((\varepsilon_2,p_2)=(\varepsilon,p)\), we have \(H_T(\lambda)<\infty\) for \(\lambda>0\). So, taking \[\label{BLL} \bar{\lambda}(T,\mu)= \big(2c_1(T) \rho_{\varepsilon,p;\delta,k}^T(\mu)\big)^{\theta},\tag{39}\] we find a constant \(c_2(t)\in (0,\infty)\) increasing in \(t\) such that for any \(\lambda\ge \bar{\lambda}(T,\mu)\), \[\label{TL}\begin{align} H_t(\lambda) &\le c_2(t) \exp\Big[c_2(t) \big(\rho_{\varepsilon,p;\delta,k}^{t}(\nu)+\rho_{\varepsilon,p;\delta,k}^t(\mu)\big)^{\theta} \Big] \rho_{\varepsilon,p;\delta,k}^{\lambda,t} (\mu,\nu),\\ &\qquad\qquad \;\;\;t\in (0,T],\;\mu,\nu\in \mathscr C_{\varepsilon,p;\delta,k}^{\gamma,t}.\end{align}\tag{40}\] Let \((r,\delta',k')\) be in 31 . By (A) and 24 for \(s=0\), we have \[\begin{align} &\|\Phi_t\mu-\Phi_t\nu\|_{\delta,k*}= \sup_{f\in \mathscr B_b(\mathbb{R}^d), \|f\|_{\tilde{W}^{-\delta,k} }\le 1} \big|\gamma(P_t^\mu-P_t^\nu)f\big|\\ &\le \|\gamma\|_{\varepsilon,p*} \int_0^t \sup_{f\in \mathscr B_b(\mathbb{R}^d), \|f\|_{\tilde{W}^{-\delta,k}} \le 1} \big\|P_t^\mu\langle b_s(\cdot,\mu_s),\nabla P_{t-s}^0f\rangle - P_t^\nu\langle b_s(\cdot,\nu_s),\nabla P_{t-s}^0f\rangle\big\|_{\tilde{W}^{-\varepsilon,p}} \text{\rm{d}}s\\ &\le \|\gamma\|_{\varepsilon,p*} K_t \int_0^t s^\kappa\|P_s^\mu-P_s^\nu\|_{\tilde{W}^{-\delta',k'}\rightarrow\tilde{W}^{-\varepsilon,p}}\|\mu_s\|_{\delta,k*} \|\nabla P_{t-s}^0\|_{\tilde{W}^{-\delta,k}\rightarrow\tilde{W}^{-\delta',k'}}\text{\rm{d}}s\\ &\quad + \|\gamma\|_{\varepsilon,p*} K_t \int_0^t s^\kappa\| P_s^\nu\|_{\tilde{W}^{-\delta',k'}\rightarrow\tilde{W}^{-\varepsilon,p}}\|\mu_s-\nu_s\|_{\delta,k*} \|\nabla P_{t-s}^0\|_{\tilde{W}^{-\delta,k}\rightarrow\tilde{W}^{-\delta',k'}}\text{\rm{d}}s. \end{align}\] Combining this with 4 , 33 and 34 we find \(c_3(t)\in (0,\infty)\) increasing in \(t\) such that \[\begin{align} & \|\Phi_t\mu-\Phi_t\nu\|_{\delta,k*}\\ &\le c_3(t)\|\gamma\|_{\varepsilon,p*} \rho_{\varepsilon,p;\delta,k}^t(\mu) \int_0^t s^{\kappa-\frac{\eta}{2}} \|P_s^\mu-P_s^\nu\|_{\tilde{W}^{-\delta',k'}\rightarrow\tilde{W}^{-\varepsilon,p}} (t-s)^{-\frac{1}{2}-\frac{r}{2}\eta}\text{\rm{d}}s\\ & + c_3(t)\|\gamma\|_{\varepsilon,p*} \exp\big[c_0(t)\rho_{\varepsilon,p;\delta,k}^t(\mu)^\theta\big] \int_0^t \|\mu_s-\nu_s\|_{\delta,k*}s^{\kappa-\frac{1-r}{2} \eta} (t-s)^{-\frac{1}{2}-\frac{r}{2}\eta}\text{\rm{d}}s.\end{align}\] Thus, by 40 , and noting that \(1+2\kappa-\eta\ge 0\) due to 6 , we find a constant \(c_4(t)\in (0,\infty)\) increasing in \(t\) such that \[\label{YM} \begin{align} &\rho_{\varepsilon,p;\delta,k}^{\lambda,T}(\Phi\mu,\Phi\nu)= \sup_{t\in (0,T]} t^{\frac{\eta}{2}}\text{\rm{e}}^{-\lambda t} \|\Phi_t\mu-\Phi_t\nu\|_{\delta,k*}\\ &\le \|\gamma\|_{\varepsilon,p*} c_3(T) \exp\Big[c_4(T) \big(\rho_{\varepsilon,p;\delta,k}^{T}(\nu)+\rho_{\varepsilon,p;\delta,k}^T(\mu)\big)^{\theta} \Big] \rho_{\varepsilon,p;\delta,k}^{\lambda,T} (\mu,\nu)\\ &\qquad \times\sup_{t\in (0,T]} {t^{\frac{\eta}{2}}} t^{(\kappa-\frac{2-r}{2}\eta)^+} \int_0^t s^{ -(\frac{2-r}{2}\eta-\kappa)^+} (t-s)^{-\frac{1}{2}-\frac{r}{2}\eta}\text{\rm{e}}^{-\lambda(t-s)}\text{\rm{d}}s,\\ & {\qquad \qquad \;\;\;t\in (0,T],\; \mu,\nu\in \mathscr C_{\varepsilon,p;\delta,k}^{\gamma,T},\;\lambda\ge \bar\lambda(T,\mu).} \end{align}\tag{41}\] By 15 for \(\alpha=\theta_1\), \(\alpha_1=(\frac{2-r}{2}\eta-\kappa)^+<1\) and \(\alpha_2= \frac{1}{2}+\frac{r}{2}\eta<1\) due to 32 , we find a constant \(c_5(T)\in (0,\infty)\) increasing in \(T\) such that \[\begin{align} & \sup_{t\in (0,T]} t^{\frac{\eta}{2}+ (\kappa-\frac{2-r}{2}\eta)^+} \int_0^t s^{ -(\frac{2-r}{2}\eta-\kappa)^+} (t-s)^{-\frac{1}{2}-\frac{r}{2}\eta}\text{\rm{e}}^{-\lambda(t-s)}\text{\rm{d}}s \\ & {\le c_5(T) \lambda^{-\theta_1},\;\;\;\lambda\ge \bar{\lambda}(T,\mu),\; t\in (0,T].} \end{align}\] Combining this with 39 and 41 , we derive the desired estimate for some \(C(T)\in (0,\infty)\) increasing in \(T\). ◻

Proof. According to the proof of [1], the first assrtion follows from the second. So, it suffices to show that for any \(n\in \mathbb{N}\) and initial value \(X_0\) with \(\gamma\in \mathscr L_{X_0}\in \mathscr P_{\varepsilon,p*},\) \(\eqref{E0}\) has a unique weak/strong \(\mathscr C_{\varepsilon,p;\delta,k}\)-solution up to \(\tau_n(\gamma)\) such that \[\label{ERR} \mathbb{E}\bigg[\sup_{t\in (0,\tau_n(\gamma)]} |X_t|^q\bigg|\mathscr F_0\bigg] \le c_{n,q}(\gamma) \big(1+|X_0|^q\big),\;\;q\in (0,\infty),\tag{42}\] where \(c_{n,q}(\gamma)\in (0,\infty)\), which does not depend on \(\gamma\) when \(\varepsilon=0\) and \(p=\infty\). Below, we simply denote \(\tau_n=\tau_n(\gamma).\)

(a) By Lemma 9 and ?? , \(\eqref{E0}\) has a unique weak/strong \(\mathscr C_{\varepsilon,p;\delta,k}\)-solution up to \(\tau_n\) provided \(\Phi: \mathscr C_{\varepsilon,p;\delta,k}^{\gamma,\tau_n}\rightarrow\mathscr C_{\varepsilon,p;\delta,k}^{\gamma,\tau_n}\) has a unique fixed point.

By Lemma 11, all fixed points in \(\mathscr C_{\varepsilon,p;\delta,k}^{\gamma, \tau_n}\) of \(\Phi\) are included in \(\tilde{\mathscr} C_{\varepsilon,p;\delta,k}^{\gamma, n}\) when \(\varepsilon>0\) or \(p<\infty\), and in \(\hat{\mathscr} C_{\delta,k}^{\gamma,n,\lambda_n}\) when \(\varepsilon=0\) and \(p=\infty\). So, ?? holds for any (weak) \(\mathscr C_{\varepsilon,p;\delta,k}\)-solutions of \(\eqref{E0}\) with initial distribution \(\gamma\) up to time \(\tau_n\). By Lemma 8 and the contractive fixed point theorem, it suffices to find \(\lambda\in (0,\infty)\) such that the map \(\Phi\) is contractive under the metric \(\rho^{\lambda,\tau_n}_{\varepsilon,p;\delta,k}\) on \(\tilde{\mathscr} C_{\varepsilon,p; \delta,k}^{\gamma, n}\) when \(\varepsilon>0\) or \(p<\infty\), and on \(\hat{\mathscr} C_{\delta,k}^{\gamma, n,\lambda_n}\) otherwise.

Let \(\varepsilon>0\) or \(p<\infty\). By Lemma 12 and 29 , we can find \(C_{n,\gamma}\in (0,\infty)\) such that \[\rho_{\varepsilon,p;\delta,k}^{\lambda,\tau_n}(\Phi\mu,\Phi\nu)\le \lambda^{-\frac{1}{\theta_1}} C_{n,\gamma} \rho_{\varepsilon,p;\delta,k}^{\lambda,\tau_n}(\mu,\nu), \;\;\mu,\nu\in \tilde{\mathscr} C_{\varepsilon,p; \delta,k}^{\gamma, n},\;\;\lambda\ge C_{n,\gamma}.\] Similarly, when \(\varepsilon=0\) and \(p=\infty\), by Lemma 12 and 30 , we find a constant \(C_n\in (0,\infty)\) uniformly in \(\gamma\in \mathscr P\) such that \[\rho_{\varepsilon,p;\delta,k}^{\lambda,n}(\Phi\mu,\Phi\nu)\le \lambda^{-\frac{1}{\theta_1}} C_{n} \rho_{\varepsilon,p;\delta,k}^{\lambda,n}(\mu,\nu),\;\; \mu,\nu\in \hat{\mathscr} C_{\delta,k}^{\gamma, n,\lambda_n},\lambda\ge C_n.\] So, in any case \(\Phi\) is contractive under the metric \(\rho^{\lambda, \tau_n}_{\varepsilon,p;\delta,k}\) when \(\lambda>0\) is large enough. So, \(\eqref{E0}\) has a unique weak/strong \(\mathscr C_{\varepsilon,p;\delta,k}\)-solution up to \(\tau_n\).

(b) Let \(X_t\) be the unique solution up to time \(\tau_n\). Then \[(\mu_t=\mathscr L_{X_t})_{t\in [0,\tau_n]}\in \begin{cases} \tilde{\mathscr} C_{\varepsilon,p; \delta,k}^{\gamma, n}, &\text{if}\;\varepsilon>0\;\text{or}\;p<\infty,\\ \hat{\mathscr} C_{\delta,k}^{\gamma, n,\lambda_n}, &\text{if}\;\varepsilon=0, p=\infty.\end{cases}\] Then by 6 and ?? , there exists \(q'>2\) such that \(\|b^\mu\|_{\tilde{L}_{q'}^\infty(\tau_n)}\) in 20 is bounded above by a constant \(D_{n,\gamma}\in (0,\infty)\) depending on \(n\) and \(\|\gamma\|_{\varepsilon,p*}\). So, 42 follows from [12], where \(c_{n,q}(\gamma)\) is uniform in \(\gamma\in \mathscr P\) when \(\varepsilon=0\) and \(p=\infty\), since \(\|\gamma\|_{0,\infty*}=1\). ◻

Proof of Theorem 3(1). All constants \(\{c_i(t)\}_{i\ge 1}\subset (0,\infty)\) below are increasing in \(t>0\). For any \(\gamma\in \mathscr P_{\varepsilon,p*},\) define \(P_t^{\gamma*}:\mathscr P\rightarrow\mathscr P\) by \[(P_t^{\gamma\ast}\nu)(f):=\int_{\mathbb{R}^d}\big(P_t^\gamma f(x)\big)\nu(\text{\rm{d}}x),\;\;f\in\mathscr B_b(\mathbb{R}^d), \;t\in [0,\tau(\gamma)),\;\nu\in \mathscr P.\] Then \(P_t^*\gamma= P_t^{\gamma*}\gamma\) for \(\gamma\in \mathscr P_{\varepsilon,p*},\) so that \[\label{EQ}\|P_t^*\gamma-P_t^*\tilde{\gamma}\|_{\delta,k*}\le \|P_t^{\tilde{\gamma*}}\gamma-P_t^{\tilde{\gamma*}}\tilde{\gamma}\|_{\delta,k*} + {\|P_t^{\gamma*}\gamma-P_t^{\tilde{\gamma*}}\gamma\|_{\delta,k*}}.\tag{44}\] In the following, we estimate these two terms respectively for fixed \(\gamma,\tilde{\gamma}\in \mathscr P_{\varepsilon,p*}\).

(a) Let \(\pi\in \mathscr C(\gamma,\tilde{\gamma})\) such that \[\label{CPL} \mathbb{W}_q(\gamma,\tilde{\gamma})= \bigg(\int_{\mathbb{R}^d\times\mathbb{R}^d}|x-y|^q\pi(\text{\rm{d}}x,\text{\rm{d}}y)\bigg)^{\frac{1}{q}}.\tag{45}\] By \(\varepsilon\in [0,\frac{d(p-1)}{p}]\), we have \(p_0:= \frac{dp}{d+\varepsilon p}\in [1,p]\) and \[\frac{1}{p_0}= \frac{1}{p}+ \frac{\varepsilon}{d}.\] Then the Sobolev embedding theorem implies \[\|\cdot\|_{\tilde{W}^{-\varepsilon,p}}\le c_0 \|\cdot\|_{\tilde{L}^{p_0}}\] for some constant \(c_0\in (0,\infty),\) so that \[\label{SB} \|\gamma\|_{p_0*}:=\|\gamma\|_{0,p_0*} \le {c_0 \|\gamma\|_{\varepsilon,p*}},\;\;\gamma\in \mathscr P_{\varepsilon,p*}.\tag{46}\] By the upper bound condition in ?? , we have \[k_0:=\frac{p_0q}{q-1}\ge k.\] By the definitions of \(\eta, p_0\) and \(k_0\), we obtain \[\xi(q):=\delta+\frac{dpq-{k(d+\varepsilon p)(q-1)}}{pqk}=\delta+\frac{d(k_0-k)}{k_0k} =\eta +\frac{1}{q} \Big(\varepsilon+ \frac{d}{p}\Big).\] Then ?? implies ?? , so that \[\eta\le \xi(q) <1+(2\kappa-\eta)^+\le 1\lor \big(2-i+2\kappa-\eta\big),\;\;i=0,1.\] Moreover, by the second inequality in ?? , we conclude \(\delta+\frac{d}{k}< 1\vee(2+2\kappa-\eta)\). Thus, by Lemma 10(1), there exists increasing \(\beta: (0,\infty)\rightarrow(0,\infty)\) such that \[\label{X1} \begin{align}&\|\nabla^i P_t^{\tilde{\gamma}}\|_{\tilde{W}^{-\delta,k}\rightarrow\tilde{W}^{0,k_0}} \le \beta_t \exp\big[t\beta_t k_t(\tilde{\gamma})^\theta\big] t^{-\frac{i+\xi(q)}{2}},\;\;\;i=0,1,\;t\in (0,\tau(\tilde{\gamma})),\\ &\|P_t^{\tilde{\gamma}} \|_{\tilde{W}^{-\delta,k}\rightarrow\tilde{W}^{0,\infty}}\leq \beta_t \exp\big[t\beta_t k_t(\tilde{\gamma})^\theta\big] t^{-\frac{\delta+\frac{d}{k}}{2}}, \;\;t\in (0,\tau(\tilde{\gamma})). \end{align}\tag{47}\] Next, consider the maximal functional \[\mathscr M f(x):= \sup_{r\in (0,1)}\frac{1}{|B(x,r)|} \int_{B(x,r)} f(y)\text{\rm{d}}y,\;\;x\in\mathbb{R}^d,\] for a nonnegative measurable function \(f\). By [18] and \(P_t^\gamma f\in C(\mathbb{R}^d)\) for \(f\in \mathscr B_b(\mathbb{R}^d)\) due to 47 , we find a constant \(k_1\in (0,\infty)\) such that \[\begin{align} &|P_t^{\tilde{\gamma}} f(x)-P_t^{\tilde{\gamma}} f(y)|\le k_1 |x-y|\big(\mathscr M|\nabla P_t^{\tilde{\gamma}} f|(x)+\mathscr M|\nabla P_t^{\tilde{\gamma}} f|(y)+ {\|P_t^{\tilde{\gamma}} f\|_\infty}\big),\\ & \big\|\mathscr M|\nabla P_t^{\tilde{\gamma}} f|\big\|_{\tilde{L}^{k_0}}\le k_1 \|\nabla P_t^{\tilde{\gamma}} f\|_{\tilde{L}^{k_0}}, \;\; \;t\in (0,\tau(\gamma)),\;x,y\in\mathbb{R}^d. \end{align}\] Combining this with Hölder’s inequality, 45 , 46 , 47 and \(\|\gamma\|_{\varepsilon,p*}\geq 1\), we find \(k_2, c_1(t)\in (0,\infty)\) such that for any \(f\in \mathscr B_b(\mathbb{R}^d)\) with \(\|f\|_{\tilde{W}^{-\delta,k}}\le 1\) and \(t\in (0,\tau(\gamma)\land \tau(\tilde{\gamma}))\), \[\begin{align} & \big|\gamma(P_t^{\tilde{\gamma}} f)-\tilde{\gamma}(P_t^{\tilde{\gamma}} f)\big| = \bigg|\int_{\mathbb{R}^d\times\mathbb{R}^d} \big(P_t^{\tilde{\gamma}} f(x)- P_t^{\tilde{\gamma}} f(y)\big)\pi(\text{\rm{d}}x,\text{\rm{d}}y)\bigg|\\ &\le k_1 \bigg|\int_{\mathbb{R}^d\times\mathbb{R}^d} |x-y|\big(\mathscr M|\nabla P_t^{\tilde{\gamma}} f|(x)+\mathscr M|\nabla P_t^{\tilde{\gamma}} f|(y)+ {\|P_t^{\tilde{\gamma}} f\|_\infty}\big)\pi(\text{\rm{d}}x,\text{\rm{d}}y)\bigg|\\ &\le k_1 {2^{\frac{1}{q}}} \mathbb{W}_q(\gamma,\tilde{\gamma}) \Big[(\gamma+\tilde{\gamma})\big((\mathscr M|\nabla P_t^{\tilde{\gamma}}f|)^{\frac{q}{q-1}}\big)\Big]^{\frac{q-1}{q}}+ {k_1\mathbb{W}_1(\gamma,\tilde{\gamma})\|P_t^{\tilde{\gamma}} \|_{\tilde{W}^{-\delta,k}\rightarrow\tilde{W}^{0,\infty}}}\\ &\le k_1 {2^{\frac{1}{q}}} \mathbb{W}_q(\gamma,\tilde{\gamma}) (\|\gamma\|_{ {p_0}*}+\|\tilde{\gamma}\|_{ {p_0}*})^{\frac{q-1}{q}} \big\|\mathscr M|\nabla P_t^{\tilde{\gamma}} f|\big\|_{\tilde{L}^{k_0}}+ {k_1\mathbb{W}_1(\gamma,\tilde{\gamma})\|P_t^{\tilde{\gamma}} \|_{\tilde{W}^{-\delta,k}\rightarrow\tilde{W}^{0,\infty}}}\\ &\le k_2 \mathbb{W}_q(\gamma,\tilde{\gamma}) (\|\gamma\|_{\varepsilon,p*}+\|\tilde{\gamma}\|_{\varepsilon,p*})^{\frac{q-1}{q}} \big\| \nabla P_t^{\tilde{\gamma}} \big\|_{\tilde{W}^{-\delta,k}\rightarrow\tilde{W}^{0,k_0}}+ {k_1\mathbb{W}_1(\gamma,\tilde{\gamma})\|P_t^{\tilde{\gamma}} \|_{\tilde{W}^{-\delta,k}\rightarrow\tilde{W}^{0,\infty}}}\\ &\le c_1(t) \mathbb{W}_q(\gamma,\tilde{\gamma}) (\|\gamma\|_{\varepsilon,p*}+\|\tilde{\gamma}\|_{\varepsilon,p*})^{\frac{q-1}{q}} \exp\big[t\beta_t k_t(\tilde{\gamma})^\theta\big] t^{-[(\frac{1+\xi(q)}{2})\vee(\frac{\delta}{2}+\frac{d}{2k})]}. \end{align}\] Therefore, \[\label{X3} \begin{align} &\big\|P_t^{\tilde{\gamma*}}\tilde{\gamma}- P_t^{\tilde{\gamma*}}\gamma\big\|_{\delta,k*} \le D(t,\gamma,\tilde{\gamma}) t^{-[(\frac{1+\xi(q)}{2})\vee(\frac{\delta}{2}+\frac{d}{2k})]},\\ &D(t,\gamma,\tilde{\gamma}):= c_1(t) (\|\gamma\|_{\varepsilon,p*}+\|\tilde{\gamma}\|_{\varepsilon,p*})^{\frac{q-1}{q}} \text{\rm{e}}^{t\beta_t k_t(\tilde{\gamma})^{\theta} }\mathbb{W}_q(\gamma,\tilde{\gamma}),\\ &\qquad \gamma,\tilde{\gamma}\in \mathscr P_{\varepsilon,p*},\;t\in (0,\tau(\gamma)\land \tau(\tilde{\gamma})).\end{align}\tag{48}\]

(b) By Duhamel’s formula [1], we have \[P_t^{\gamma}f-P_t^{\tilde{\gamma}}f= \int_0^t P_s^\gamma\big\langle b_s(\cdot,P_s^*\gamma)- b_s(\cdot,P_s^*\tilde{\gamma}),\;\nabla P_{s,t}^{\tilde{\gamma}}f\big\rangle\text{\rm{d}}s,\;\;f\in C_0^\infty(\mathbb{R}^d),\] and (A) implies \[|b_t(x,P_t^*\gamma)-b_t(x,P_t^*\tilde{\gamma})|\le K_tt^\kappa\|P_t^*\gamma-P_t^*\tilde{\gamma}\|_{\delta,k*},\;\;t\in [0, \tau(\gamma)\land \tau(\tilde{\gamma})).\] Then \[\label{X422} \begin{align} & \big\|P_t^{\gamma*}\gamma - P_t^{\tilde{\gamma*}}\gamma\big\|_{\delta,k*}=\sup_{f\in \mathscr B_b(\mathbb{R}^d), \|f\|_{\tilde{W}^{-\delta,k}}\le 1} \big| \gamma\big(P_t^{\gamma}f-P_t^{\tilde{\gamma}}f\big)\big|\\ &\le K_t \|\gamma\|_{\varepsilon,p*} \int_0^t s^\kappa\big\|P_s^*\gamma-P_s^*\tilde{\gamma}\big\|_{\delta,k*} \|P_s^\gamma\|_{\tilde{W}^{-\delta',k'}\rightarrow\tilde{W}^{-\varepsilon,p}} \|\nabla P_{s,t}^{\tilde{\gamma}}\|_{\tilde{W}^{-\delta,k}\rightarrow\tilde{W}^{-\delta',k'}}\text{\rm{d}}s,\\ &\qquad \;\; t\in (0,\tau(\gamma)\land \tau(\tilde{\gamma})),\end{align}\tag{49}\] where \((\delta',k')\) is in 31 for \(r\in [0,1]\) to be determined.

By ?? , we have either \(\eta<\frac{1}{2} +\kappa\) and \(\kappa\ge \frac{1}{2}\), or \(\eta\in [0,1)\) and \(\kappa<\frac{1}{2}\). Let \[r=\begin{cases} 1- \kappa, &\text{if}\;\kappa<\frac{1}{2}, \eta\in [0,1),\\ \frac{2}{3+2\kappa}, &\text{if} \;\eta<\frac{1}{2} +\kappa, \kappa\ge \frac{1}{2}.\end{cases}\] In each case we have \[\label{YY0} r\eta <1,\;\;\;(2-r)\eta<2+2\kappa,\;\;\;(1-r)\eta\le 2\kappa,\tag{50}\] where \((2-r)\eta<2+2\kappa\) implies \[\delta'-\varepsilon+\frac{d(p-k')}{pk'}=(1-r) \eta<1\lor (2+2\kappa-\eta),\] so that Lemma 10(1) for \(i=0\) gives \[\|P_t^{\gamma}\|_{\tilde{W}^{-\delta',k'}\rightarrow\tilde{W}^{-\varepsilon,p}}\le \beta_t\text{\rm{e}}^{t\beta_t k_t(\gamma)^\theta} t^{-\frac{1-r}{2}\eta},\] and by Lemma 10(1) for \(i=1\), \[\delta-\delta'+\frac{d(k'-k)}{k'k}=r\eta <1\] implies \[\|\nabla P_{s,t}^{\tilde{\gamma}}\|_{\tilde{W}^{-\delta,k}\rightarrow\tilde{W}^{-\delta',k'}}\le \beta_t\text{\rm{e}}^{t\beta_t k_t(\tilde{\gamma})^\theta} (t-s)^{-\frac{1}{2}-\frac{r}{2}\eta}.\] Combining this with 49 , we find \(c_2(t)\in (0,\infty)\) such that \[\begin{align} & {\big\| P_t^{\gamma*}\gamma-P_t^{\tilde{\gamma}*}\gamma\big\|_{\delta,k*}} \le \tilde{D}(t,\gamma,\tilde{\gamma}) \int_0^t s^{\kappa-\frac{1-r}{2}\eta } \big\|P_s^*\gamma-P_s^*\tilde{\gamma}\big\|_{\delta,k*} (t-s)^{-\frac{1}{2}-\frac{r}{2}\eta} \text{\rm{d}}s,\\ & \tilde{D}(t,\gamma,\tilde{\gamma}):= { c_2(t) \|\gamma\|_{\varepsilon,p*} \text{\rm{e}}^{t\beta_t [k_t(\gamma)+k_t(\tilde{\gamma})]^{\theta}}, \; \; t\in [0, \tau(\gamma)\land \tau(\tilde{\gamma}))}. \end{align}\] This together with 44 and 48 yields \[\begin{align} &\big\|P_t^*\gamma-P_t^*\tilde{\gamma}\big\|_{\delta,k*}\le {\big\|P_t^*\gamma- P_t^{\tilde{\gamma} *}\gamma\big\|_{\delta,k*} +\big\| P_t^{\tilde{\gamma} *}\gamma-P_t^*\tilde{\gamma}\big\|_{\delta,k*}}\\ &\le D(t,\gamma,\tilde{\gamma}) t^{-[(\frac{1+\xi(q)}{2})\vee(\frac{\delta}{2}+\frac{d}{2k})]} + \tilde{D}(t,\gamma,\tilde{\gamma}) \int_0^t s^{\kappa-\frac{1-r}{2}\eta} \big\|P_s^*\gamma-P_s^*\tilde{\gamma}\big\|_{\delta,k*} (t-s)^{-\frac{1}{2}-\frac{r}{2}\eta} \text{\rm{d}}s,\\ &\qquad \gamma,\tilde{\gamma}\in \mathscr P_{\varepsilon,p*}, \; t\in [0, \tau(\gamma)\land \tau(\tilde{\gamma})). \end{align}\] Since \((P_t^*\gamma)_{t\in [0,T]}, (P_t^*\tilde{\gamma})_{t\in [0,T]}\in \mathscr C_{\varepsilon,p;\delta,k}^T\) for \(T<\tau(\gamma)\land\tau(\tilde{\gamma})\), and \(1+\xi(q)\ge \eta\) due to ?? , for any constant \(\lambda\in (0,\infty)\) and \(t\in [0, \tau(\gamma)\land \tau(\tilde{\gamma})),\) we have \[I_t:=\sup_{s\in (0, t]} \text{\rm{e}}^{-\lambda s} s^{[(\frac{1+\xi(q)}{2})\vee(\frac{\delta}{2}+\frac{d}{2k})]}\big\|P_s^*\gamma-P_s^*\tilde{\gamma}\big\|_{\delta,k*} <\infty\] and \[\begin{align} I_t \le &\,D(t,\gamma,\tilde{\gamma}) + \tilde{D}(t,\gamma,\tilde{\gamma}) \sup_{s\in (0, t]}s^{[(\frac{1+\xi(q)}{2})\vee(\frac{\delta}{2}+\frac{d}{2k})] + ([(\frac{1+\xi(q)}{2})\vee(\frac{\delta}{2}+\frac{d}{2k})] +\frac{1-r}{2}\eta-\kappa)^-}\\ &\qquad\qquad\qquad\quad\quad\times \int_0^s u^{ -([(\frac{1+\xi(q)}{2})\vee(\frac{\delta}{2}+\frac{d}{2k})] +\frac{1-r}{2}\eta-\kappa)^+} \text{\rm{e}}^{-\lambda(s-u)}(s-u)^{-\frac{1}{2}-\frac{r}{2} \eta}\text{\rm{d}}u. \end{align}\] By 50 , ?? and \(\delta+\frac{d}{k}< 2+(2\kappa-\eta)^+\), we have \[(\frac{1+\xi(q)}{2})\vee(\frac{\delta}{2}+\frac{d}{2k})+ \frac{1-r}{2}\eta-\kappa<1,\;\;\; \frac{1}{2}+\frac{r}{2} \eta<1,\] we may apply 15 to \(\alpha_1 =([(\frac{1+\xi(q)}{2})\vee(\frac{\delta}{2}+\frac{d}{2k})] +\frac{1-r}{2}\eta-\kappa)^+,\alpha_2= \frac{1}{2} +\frac{r}{2} \eta\) and \(\alpha= \frac{1}{\theta},\) such that \[I_t \le D(t,\gamma,\tilde{\gamma}) + c\tilde{D}(t,\gamma,\tilde{\gamma}) \lambda^{-\frac{1}{\theta}}\] holds for some constant \(c\in (0,\infty)\). Taking \[\lambda= \big(2c\tilde{D}(t,\gamma,\tilde{\gamma}) \big)^{\theta}\] we obtain \[I_t\le 2 D(t,\gamma,\tilde{\gamma}),\] which implies ?? for some increasing \(\beta: (0,\infty)\rightarrow(0,\infty).\)

Finally, when \(\varepsilon=0,p=\infty\), \(k_t(\gamma)\) defined in 8 is bounded above by some constant \(c(t)\in (0,\infty)\) uniformly in \(\gamma\in \mathscr P_{\varepsilon,p*}=\mathscr P\). Moreover, in this case ?? implies that \(q=1\) satisfies ?? . So, ?? implies ?? . ◻

Proof of Theorem 3(2). The proof is similar to that of [1], where \(\varepsilon=\delta=0.\) For completeness we figure out it in the present situation.

(a) For \(t\in (0, \tau(\gamma)\land\tau(\tilde{\gamma}))\), we consider the SDEs \[\label{LG} \begin{align} &\text{\rm{d}}X_s= b_s(X_s,P_s^*\gamma)\text{\rm{d}}s + \text{\rm{d}}W_s,\\ &\text{\rm{d}}Y_s= b_s(Y_s,P_s^*\tilde{\gamma})\text{\rm{d}}s + \text{\rm{d}}W_s,\;\;\;s\in [0,t],\end{align}\tag{51}\] where \(X_0,Y_0\) are \(\mathscr F_0\)-measurable such that \[\mathscr L_{X_0}=\gamma,\;\;\; \mathscr L_{Y_0}=\tilde{\gamma},\;\;\;\mathbb{E}|X_0-Y_0|^2= \mathbb{W}_2(\gamma,\tilde{\gamma})^2.\] Then \[\label{GT} P_t^*\gamma=\mathscr L_{X_t},\;\;\; \;P_t^*\tilde{\gamma}=\mathscr L_{Y_t}.\tag{52}\] To estimate \({\rm Ent}(P_t^*\gamma|P_t^*\tilde{\gamma})\), we apply the bi-coupling argument developed in [19].

For fixed \(\theta'\in (\theta,\infty)\), let \[\begin{align} \label{key} t_0=\frac{t}{2}\wedge s_t(\theta',\gamma)\wedge s_t(\theta',\tilde{\gamma}). \end{align}\tag{53}\] According to the bi-coupling method developed in [19], the following SDE will be coupled with those two SDEs in 51 respectively: \[\text{\rm{d}}Z_s = \Big(1_{[0,t_0]}(s) b_s(Z_s, P_s^\ast\tilde{\gamma})+ 1_{(t_0,t]}(s) b_s(Z_s,P_s^\ast\gamma)\Big)\text{\rm{d}}s+ \text{\rm{d}}W_s,\;\;Z_0=Y_0,s\in [0,t].\] By 52 and [19], we have \[\label{RW} \begin{align} &{\rm Ent}(P_t^\ast\gamma| P_t^\ast\tilde{\gamma})={\rm Ent}(\mathscr L_{X_t}|\mathscr L_{Y_t})\\ &\le 2 {\rm Ent}(\mathscr L_{X_t}|\mathscr L_{Z_t}) + \log \int_{\mathbb{R}^d} \Big(\frac{\text{\rm{d}}\mathscr L_{Z_t}}{\text{\rm{d}}\mathscr L_{Y_t}}\Big)^{2}\text{\rm{d}}\mathscr L_{Y_t} =:2 I_1+ I_2.\end{align}\tag{54}\] Below we estimate \(I_1\) and \(I_2\) respectively.

(b) To estimate \(I_1\), we first establish the log-Harnack inequality for \(P_{t}^\gamma:\) for any \(\theta'\in (\theta,\infty)\), there exists \(c_0 (t) \in (0,\infty)\) increasingly in \(t\) such that \[\label{cty}\begin{align} &P_{s,t}^\gamma \log f(x) \leq \log P_{s,t}^\gamma f(y)+\frac{c_0(t) |x-y|^2}{s_t(\theta',\gamma)\wedge (t-s)},\\ &\qquad \;x,y\in\mathbb{R}^d, 0\leq s< t<\tau(\gamma),\; \gamma\in \mathscr P_{\varepsilon,p*},\; f\in \mathscr B_b^+(\mathbb{R}^d) \end{align}\tag{55}\] for \(s_t(\theta',\gamma)\) defined in 9 . By (A), we have \[\|b_t(\cdot,P_t^\ast\gamma)\|_\infty\le K_t k_t(\gamma)t^{\kappa-\frac{\eta}{2}},\;\;\;t\in [0,\tau(\gamma)).\] Since ?? implies 6 so that \((\eta-2\kappa)^+<1\). For any constant \[\theta'> \theta:=\frac{2}{1-(\eta-2\kappa)^+},\] we have \[q':= {\Big(\frac{(\eta-2\kappa)^+}{2} + \frac{1}{\theta'}\Big)^{-1} }\in \Big(2, \frac{2}{(\eta-2\kappa)^+}\Big),\] and for some \(c_1(t)\in(0,\infty)\) increasing in \(t\), \[\label{GM}\|b_\cdot(\cdot,\gamma_\cdot)\|_{\tilde{L}_{q'}^\infty(s,t)}\le c_1(t) k_t(\gamma)(t-s)^{\frac{1}{\theta'}},\;\;0\le s<t<\tau(\gamma).\tag{56}\] By 9 , we find a constant \(k_1\in (0,\infty)\) such that \[k_t(\gamma)(t-s)^{\frac{1}{\theta'}}\le k_1,\;\;0<t-s\le s_t(\theta', \gamma).\] This together with 56 implies that \[\|b_\cdot(\cdot,\gamma_\cdot)\|_{\tilde{L}_{q'}^\infty(s,t)} \le {k_1c_1(t)},\;\;0<t-s\le s_t(\theta',\gamma), \;t\in (0,\tau(\gamma)).\] So, by [1], we derive 55 for \(0<t-s\le s_t(\theta',\gamma), \;t\in (0,\tau(\gamma))\) and some \(c_0 (t) \in (0,\infty)\) increasingly in \(t\). When \(t-s>s_t(\theta',\gamma),\;t\in (0,\tau(\gamma)),\) by the semigroup property and Jensen’s inequality, we deduce \[\begin{align} & P_{s,t}^\gamma\log f(x)=P_{s,s+s_t(\theta',\gamma)}^\gamma P_{s+s_t(\theta',\gamma),t}^\gamma\log f(x)\leq P_{s,s+s_t(\theta',\gamma)}^\gamma\log P_{s+s_t(\theta',\gamma),t}^\gamma f(x)\\ &\leq \log P_{s,s+s_t(\theta',\gamma)}^\gamma P_{s+s_t(\theta',\gamma),t}^\gamma f(y)+\frac{ {c_0(t)|x-y|^2}}{s_t(\theta',\gamma)}\\ &= \log P_{s,t}^\gamma f(y)+ \frac{ c_0(t)|x-y|^2}{s_t(\theta',\gamma)}, \;\;x,y\in\mathbb{R}^d,\;f\in \mathscr B_b^+(\mathbb{R}^d). \end{align}\] So, 55 also holds for \(t-s>s_t(\theta',\gamma)\).

Next, by the Markov property, we have \[\mathbb{E}[f(X_t)]= \mathbb{E}[ (P_{t_0,t}^\gamma f)(X_{t_0})],\;\;\; \mathbb{E}[f(Z_t)] = \mathbb{E}[ (P_{t_0,t}^\gamma f)(Z_{t_0})].\] This together with 55 for \(s=t_0\) and Jensen’s inequality implies \[\label{OG3}\mathbb{E}[\log f(X_t)]\le \log \mathbb{E}[f(Z_t)]+ \frac{2c_0(t)}{ s_t(\theta',\gamma)}\mathbb{E}[|X_{t_0}- Z_{t_0}|^2],\;f\in \mathscr B_b^+(\mathbb{R}^d).\tag{57}\] Moreover, by (A), ?? and 53 , we find \(c_2(t) \in (0,\infty)\) increasing in \(t\) such that \[\begin{align} &\int_0^{t_0} \|b_s(\cdot,P_s^\ast\gamma)-b_s(\cdot,P_s^\ast\tilde{\gamma})\|_\infty\text{\rm{d}}s \le K_t \int_0^{t_0} \|P_s^\ast\gamma-P_s^\ast\tilde{\gamma}\|_{\delta,k} s^\kappa\text{\rm{d}}s \\ &\le K_t \big(\|\gamma\|_{\varepsilon,p*}+\|\tilde{\gamma}\|_{\varepsilon,p*}\big)^{\frac{q-1}{q}} K_{t,\beta}^{(\theta)}(\gamma,\tilde{\gamma}) \mathbb{W}_q(\gamma,\tilde{\gamma})\int_0^{t_0} s^{\kappa- [(\frac{1+\xi(q)}{2})\vee(\frac{\delta}{2}+\frac{d}{2k})]}\text{\rm{d}}s\\ &\le c_2(t) \big(\|\gamma\|_{\varepsilon,p*}+\|\tilde{\gamma}\|_{\varepsilon,p*}\big)^{\frac{q-1}{q}} \mathbb{W}_q(\gamma,\tilde{\gamma}) t_0^{ {(1+\kappa)- [(\frac{1+\xi(q)}{2})\vee(\frac{\delta}{2}+\frac{d}{2k})]}}. \end{align}\] Combining this with [1], we find \(c_2(t) \in (0,\infty)\) increasing in \(t\) such that \[\mathbb{E}[|X_{t_0}- Z_{t_0}|^2] \le c_2(t) \big(\|\gamma\|_{\varepsilon,p*}+\|\tilde{\gamma}\|_{\varepsilon,p*}\big)^{\frac{2(q-1)}{q}} \big(\mathbb{W}_q(\gamma,\tilde{\gamma})^2 {t_0}^{2(1+\kappa)-[(1+\xi(q))\vee(\delta+\frac{d}{k})]}+ \mathbb{W}_2(\gamma,\tilde{\gamma})^2\big).\] This together with the formula \[{\rm Ent}(\mu|\nu)=\sup_{f\in \mathscr B_b^+(\mathbb{R}^d)} \big(\mu(\log f)-\log \nu(f)\big),\;\;\mu,\nu\in \mathscr P\] and 57 yields that for some constant \(c_3(t) \in (0,\infty)\) increasing in \(t\) such that \[\label{I11}\begin{align} &I_1:={\rm Ent}(\mathscr L_{X_t}|\mathscr L_{Z_t})=\sup_{f\in \mathscr B_b^+(\mathbb{R}^d)} \big(\mathbb{E}[\log f(X_t)]-\log \mathbb{E}[ f(Z_t)]\big), \\ & \leq c_3(t) (\|\gamma\|_{\varepsilon,p*}\lor\|\tilde{\gamma}\|_{\varepsilon,p*})^{\frac{2(q-1)}{q}}\bigg( \frac{\mathbb{W}_2(\gamma,\tilde{\gamma})^2}{ s_t(\theta',\gamma)}+ \frac{\mathbb{W}_q(\gamma,\tilde{\gamma})^2}{s_t(\theta',\gamma)^{([(1+\xi(q))\vee(\delta+\frac{d}{k})]-(2\kappa+1) )^+}}\bigg), \end{align}\tag{58}\] where in the last step we have used \(t_0\le s_t(\theta',\gamma)\).

(c) Estimate \(I_2\). By (A) and ?? , we find a constant \(K(t)\in (0,\infty)\) increasing in \(t\) such that \[\xi_s := \big[b_s(Y_s, P_s^\ast\gamma)- b_s(Y_s, P_s^\ast\tilde{\gamma})\big]\] satisfies \[\label{XX}\begin{align}& |\xi_s |^2 \le K_t^2 C(t,\gamma,\tilde{\gamma})s^{2\kappa-[(1+\xi(q))\vee(\delta+\frac{d}{k})]},\;\;\;s\in (0,t],\\ &C(t,\gamma,\tilde{\gamma}):= (\|\gamma\|_{\varepsilon,p*}+\|\tilde{\gamma}\|_{\varepsilon,p*})^{\frac{2(q-1)}{q}} K_{t,\beta}^{(\theta)}(\gamma,\tilde{\gamma})^2 \mathbb{W}_q(\gamma,\tilde{\gamma})^2.\end{align}\tag{59}\] Then \[R_s:=\text{\rm{e}}^{\int_{t_0}^s \langle\xi_r,\text{\rm{d}}W_r\rangle-\frac{1}{2} \int_{t_0}^s|\xi_r|^2\text{\rm{d}}r},\;\;s\in [t_0,t]\] is a martingale, and by Girsanov’s theorem, \[\frac{\text{\rm{d}}\mathscr L_{Z_t}}{\text{\rm{d}}\mathscr L_{Y_t}}(Y_{t})= \mathbb{E}(R_t|Y_{t}).\] Combining this with Jensen’s inequality and 59 , we find \(c_4(t)\in (0,\infty)\) increasing in \(t\) such that \[\begin{align} I_2&:= \log \mathbb{E}\bigg[\Big(\frac{\text{\rm{d}}\mathscr L_{Z_t}}{\text{\rm{d}}\mathscr L_{Y_t}}(Y_{t})\Big)^{2}\bigg] \leq \log\mathbb{E}\Big[R_t^{2}\Big]\\ &\le \log\mathbb{E}\bigg[\text{\rm{e}}^{2\int_{t_0}^t \langle\xi_s,\text{\rm{d}}W_s\rangle- 2 \int_{t_0}^t |\xi_s|^2\text{\rm{d}}s+ C(t,\gamma,\tilde{\gamma}) \int_{t_0}^t s^{2\kappa- 1-\xi(q)}\text{\rm{d}}s }\bigg]\\ &= C(t,\gamma,\tilde{\gamma}) \int_{t_0}^t s^{2\kappa-[(1+\xi(q))\vee(\delta+\frac{d}{k})]}\text{\rm{d}}s \le c_3(t) C(t,\gamma,\tilde{\gamma}) t_0^{-([(1+\xi(q))\vee(\delta+\frac{d}{k})]-(2\kappa+1))^+}. \end{align}\] By combining this with 54 and 58 , we obtain ?? for some \(\beta: (0,\infty)\rightarrow(0,\infty).\)

(c) If \(\varepsilon=0,p=\infty\), we have \(\mathscr P_{\varepsilon,p*}=\mathscr P\), \(\tau(\gamma)=\infty\) and \(\|\gamma\|_{\varepsilon,p*}= 1\) for any \(\gamma\in \mathscr P\), and we may take \(q=1\) so that \(\xi(q)=\eta=\delta+\frac{d}{k}\), and ?? implies \[[(1+\xi(q))\vee(\delta+\frac{d}{k})]-(2\kappa+1) <1.\] Hence ?? implies ?? for some increasing \(\beta: (0,\infty)\rightarrow(0,\infty).\) ◻