Derivative formula for singular McKean-Vlasov SDEs (Q6166271)

Let \(k\in\mathbb N\), let \(\mathcal P_k\) denote the Polish space of Borel probability measures on \(\mathbb R^d\) with the finite \(k\)-th moment equipped with the \(L^k\)-Wasserstein metric \(\mathbb W_k\) and consider spaces \(\tilde L^p_q\) consisting of measurable functions \(f:[0,T]\times\mathbb R^d\) such that \[ \sup_{z\in\mathbb R^d}\|\mathbf 1_{z+B}f\|_{L^q(0,T;L^p(\mathbb R^d))}<\infty, \] where \(B\) is the unit ball in \(\mathbb R^d\) and \(p,q\in(2,\infty)\) satisfy \[ \frac{d}{p}+\frac{2}{q}<\infty. \] A stochastic differential equation \[dX_t=b(t,X_t,\operatorname{Law}X_t)\,dt+\sigma(t,X_t)\,dW_t, \tag{1}\] where \(b:[0,\infty)\times\mathbb R^d\times\mathcal P_k\to\mathbb R^d\) and \(\sigma:[0,\infty)\times\mathbb R^d\to\mathbb R^d\otimes\mathbb R^m\) are measurable functions and \(W\) is a Wiener process in \(\mathbb R^m\). It is assumed that \(\sigma\) is differentiable in the space variable, \[ \|\nabla\sigma\|\le\sum_{i=1}^\ell f_i \] for some non-negative \(f_i\in\tilde L^{p_i}_{q_i}\), \(a=\sigma\sigma^*\) is invertible, \(a\) and \(a^{-1}\) are uniformly bounded, \(a\) is uniformly continuous in the space variable, uniformly in \(t\), \[ b(t,x,\mu)=b^{(0)}(t,x)+b^{(1)}(t,x,\mu), \] where \(b^{(0)}\in\tilde L^{p_0}_{q_0}\), \(b^{(1)}\) is differentiable in \(x\), intrinsically \(L\)-differentiable in \(\mu\), \(D^Lb^{(1)}\) has a jointly continuous version such that \[ |D^Lb^{(1)}|\le c(x,\mu)(1+|y|^{k-1}) \] and \[ \sup_{(t,x,\mu)\in[0,T]\times\mathbb R^d\times\mathcal P_k}\left\{|b^{(1)}(t,0,\delta_0)|+\|\nabla b^{(1)}(t,x,\mu)\|+\|D^Lb^{(1)}(t,x,\mu)\|_{L^{k^*}}\right\}<\infty. \] It is proved that, under the above assumptions, the equation (1) is well-posed both in the strong and the weak sense, estimations on moments of the solutions are derived, if \((P_t)\) denotes the Markov semigroup then \[ \|P^*_t\mu-P^*_t\nu\|_{\operatorname{TV}}\le\frac{c}{\sqrt t}\mathbb W_k(\mu,\nu),\qquad t\ge0,\,\mu,\nu\in\mathcal P_k, \] \(P_tf\) is intrinsically differentiable on \(\mathcal P_k\) for every bounded measurable function \(f\), a Bismut formula for the intrinsic derivative \(D^IP_tf\) is derived, as well as an estimation (given \(p>1\)) \[ \|D^IP_tf(\mu)\|_{L^{k^*}(\mu)}\le\frac{c}{\sqrt t}\|\left(\mathbb E\,\left[|f|^p(X^\mu_t)|\mathcal F_0\right]\right)^\frac{1}{p}\|_{L^{k^*}(\mathbb P)},\qquad t\in(0,T] \] which holds for every bounded measurable function \(f\) and every \(\mu\in\mathcal P_k\). If \(k>1\) and \(f\), \(\sigma\), \(\nabla b^{(1)}\) and \(D^Lb^{(1)}\) are uniformly continuous in the spatial variables, uniformly in \(t\) then \(P_tf\) is intrinsically \(L\)-differentiable on \(\mathcal P_k\) for every \(t\in(0,T]\) and every bounded measurable function \(f\). Analogous results are then derived for singular drifts \(b\) independent of the \(\mu\)-variable, among them, a Bismut formula for bounded measurable functions \(f\).