Transportation cost inequalities for SDEs with irregular drifts (Q2066969)

Let \(\mu\) and \(\nu\) be probability measures on a metric space and denote by \(\mathbb W_p(\mu,\nu)\) and \(H(\nu|\mu)\) the \(L^p\)-Wasserstein distance between \(\mu\) and \(\nu\) and the relative entropy of \(\nu\) with respect \(\mu\), respectively. The authors consider an SDE \[ dX_t=\{B_t(X_t)+b_t(X_t)\}\,dt+\sigma_t(X_t)\,dW_t, \] where \(W\) is an \(d\)-dimensional Wiener process, \(B,b:[0,T]\times\mathbb R^d\to\mathbb R^d\) and \(\sigma:[0,T]\times\mathbb R^d\to\mathbb R^d\times\mathbb R^d\) are measurable, \[ |b_t(x)-b_t(y)|\le\phi(|x-y|) \] for \(\phi:[0,\infty)\to[0,\infty)\) increasing such that \(\phi^2\) is concave and \[ \int_0^1\frac{\phi(s)}{s}\,ds<\infty, \] \(B\) is Lipschitz continuous in the spatial variable, \(t\mapsto B_t(0)\) is bounded, \(\sigma\) takes values in the space of invertible matrices, \(\sigma_t\) is \(C^2\)-smooth and \(b\), \(\nabla B\), \(\sigma\), \(\nabla\sigma\), \(\nabla^2\sigma\) and \((\sigma\sigma^*)^{-1}\) are uniformly bounded. Denote by \(\mathbb P^\mu\) the law of the solution of the above equation with the initial distribution \(\mu\). Then, for every \(x\in\mathbb R^d\), there exists a constant \(C\) such that \[ \mathbb W_2(\mathbb Q,\mathbb P^x)\le\sqrt{CH(\mathbb Q|\mathbb P^x)} \] holds for every probability law \(\mathbb Q\) on \(C([0,T];\mathbb R^d)\), and \[ \mathbb W_2(\mathbb Q,\mathbb P^\mu)\le\sqrt{CH(\mathbb Q|\mathbb P^\mu)} \] holds for every probability law \(\mathbb Q\) on \(C([0,T];\mathbb R^d)\) if and only if there exists a constant \(C_2\) such that \[ \mathbb W_2(\nu,\mu)\le\sqrt{C_2H(\nu|\mu)} \] holds for every probability law \(\nu\) on \(\mathbb R^d\). Secondly, a particular case of the above SDE is considered, i.e., \[ dX_t=b(X_t),dt+\sigma(X_t)\,dW_t \] but now it is assumed that \(\sigma\) is uniformly continuous, \(\nabla\sigma\in L^p(\mathbb R^d)\), \[ c_1|\xi|\le|\sigma^*(x)\xi|\le c_2|\xi|, \] \(b=b_1+b_2\), \(b_1\in L^p(\mathbb R^d)\) and \(b_2\) satisfies either \[ \langle x,b_2(x)\rangle\le-\kappa_1|x|^{2+r}+\kappa_2\quad\text{and}\quad|b_2(x)|\le\kappa_2(1+|x|^{1+r}), \] or \[ |b_2(x)|\le\kappa_4(1+|x|) \] for some positive constants \(c_1\), \(c_2\), \(\kappa_1\), \(\kappa_2\), \(\kappa_3\), \(\kappa_4\) and some \(r>-1\) and \(p>d\). Let \(\mu\) satisfy \[ \mu\left(e^{\delta_0|\cdot|^{2+r^+}}\right)<\infty \] for some \(\delta_0>0\). Then there exists a constant \(C\) such that \[\mathbb W_1(\mathbb Q,\mathbb P^\mu)\le\sqrt{CH(\mathbb Q|\mathbb P^\mu)} \] holds for every probability law \(\mathbb Q\) on \(C([0,T];\mathbb R^d)\).