Dynamical systems generated by Sobolev class vector fields in finite and infinite dimensions
The authors consider the fundamental problems of dynamical systems theory on existence and uniqueness of a global flow \(\{U_t\}\) generated by a vector field \(A\). The authors introduce the following definition: If \(X\) is a topological vector space equipped with a positive Radon measure \(\mu\) on its Borel \(\sigma\)-field and a measurable vector field \(A: X\to X\), a mapping \((t,x)\in \mathbb{R}\times X\to U^A_t(x)\in X\) is said to be a solution to the equation \[ U_t(x)=x+ \int^t_0 A\bigl(U_s(x)\bigr) ds\tag{*} \] if (a) for \(\mu\)-almost every \(x\), (*) holds with \(U=U^A\) for all \(t\in \mathbb{R}\) (its right-hand side being implicitly well defined), and (b) for every \(t\in\mathbb{R}\), the measure \(\mu\circ U_t^{-1}\) is absolutely continuous with respect to \(\mu\). The Radon-Nikodym derivative \(d(\mu\circ U^{-1}_t)/d\mu\) will always be denoted by \(r_t\). Moreover, \(U=U^A\) is a flow solution if, in addition, \(\mu\) -- a.e., \(\bigwedge_{s,t\in\mathbb{R}} U_{t+s} =U_tU_s\). The authors prove the following main theorem: Theorem 1. Let \(\mu\) be a measure on \(\mathbb{R}^n\) possessing a density \(\rho\in W^{1,1}_{\text{loc}} (\mathbb{R}^n)\), which is locally uniformly positive, and let \(A\in W^{1,1}_{\text{loc}} (\mathbb{R}^n, \mathbb{R}^n)\) be a vector field such that either \(e^{\|\nabla A\|}\in \bigcap_{p>1} L^p(\mu)\) or \(e^{\|\nabla A\|}\in \bigcap_{p>1} L^p_{\text{loc}} (\mu)\) and \(\|A\|\in L^{1+\varepsilon} (\mathbb{R}^n)\), and such that \(e^{|\delta_\mu A|} \in\bigcap_{p>1} L^p(\mu)\), where \(\delta_\mu A=\text{div} A+ (A,{\nabla\rho \over\rho})\). Then equation (*) possesses a flow solution. Assume that the measure \(\mu\) on a locally convex space \(X\) is such that there is a separable Hilbert space \(H\) continuously embedded into \(X\). This gives rise to the embedding \(j:X^*\to H\). The measure \(\mu\) is assumed to be Fomin differentiable along all vectors in \(j(X^*)\) such that the corresponding logarithmic derivatives have all exponential moments. Then one can define the Sobolev-type classes \(G_{\rho,1}(H,\mu,H)\) of \(H\)-valued vector fields on \(X\) differentiable along \(H\) as well as the divergence \(\delta_\mu A\) with respect to \(\mu\) of a vector field \(A\). Let \(\{l_n\}^\infty_{n=1} \subset X^*\) be the sequence, which separates the points in \(X\), and assume that the sequence \(\{e_n:=j_H (l_n)\}^\infty_{n=1}\) is an orthogonormal basis of \(H\); moreover \(\{j_H(l_n)\}^\infty_{n=1}\) is dense in \(H\). For each \(n\in\mathbb{N}\), let \(L_n\) be the linear span of \(\{l_1, \dots, l_n\}\subset X^*\), and \(H_n\) the linear span of \(\{e_1,\dots, e_n\}\subset H\) \((H_n=j(L_n))\); denote \({\mathcal F}_n= {\mathcal F}_{L_n}\) and define \(E_n:L^1(\mu)\to L^1_{{\mathcal F}_n} (\mu)\) to be the conditional expectation \(E_n\varphi= E(\varphi \mid{\mathcal F}_n)\) and \(P_{H_n}\) to be the orthogonal projection of \(H\) onto \(H_n\), which can be extended to the continuous linear operator \(P_n:X\to H_n\subset X\) by setting \(P_nx= \sum^n_{i=1} l_i(x)e_i\), \(x\in X\). Finally, \(\mu_n=\mu\circ P_n^{-1}\) is the marginal measure of \(\mu\) on \(H_n\) and \(\beta^{(n)}= \sum^n_{k=1} \beta_{e_k} e_k\in L^1(\mu,H)\). Assume that the probability measure \(\mu\) and the vector field \(A:H\to H\) satisfy the following conditions: (A1) \(\mu\) is differentiable along all directions in \(j_H(X^*)\). In addition, for every \(h\in j_H(X^*)\) there exists \(c=c(h)>0\) such that \[ Ee^{c|\beta_n |}=\int_X e^{c|\beta_n(x)|} \mu(dx)<\infty. \] (A2) \(A\in G_{\alpha,1} (\mu,H)\) for some \(\alpha>1\), and for every \(\lambda>0\), one has \[ \int_X e^{2\|\nabla_H A\|_{L(H)}} \mu(dx)<\infty. \] (A3) For every \(\lambda>0\), one has \[ \sup_n Ee^{\lambda|\delta_\mu (P_nA)|} =\sup_n \int_Xe^{\lambda|\delta_\mu P_nA(x)|} \mu(dx)<\infty. \] (A3\('\)) There exist two increasing sequences \(\{k_n\}\) and \(\{m_n\}\) with \(k_n\geq m_n\) such that for every \(\lambda>0\), one has \[ \sup_n \int_Xe^{\lambda |\delta_\mu (E_kP_mA)|} \mu(dx)<\infty, \quad \sup_n\int_Xe^{\lambda \|\nabla_H (E_{k_n} P_{m_n}A) \|_{L(H)}} \mu(dx)< \infty. \] Theorem 2. Let \(\mu\) be a probability measure on a locally convex space \(X\) described above, and let \(A:X\to H\) be a measurable vector field. Assume that (A1), (A2) and (A3) (or (A3\('\))) are satisfied. Then (*) has a flow solution, and the Radon-Nikodym process \(r_t\) has a measurable modification, all of whose sample paths are continuous; in addition, for every \(t\in\mathbb{R}^n\) and \(p>1\), one has \[ \|r_t\|_{L^p(\mu)}\leq 4e^{\Lambda(p,t)|t|}, \] where \[ \Lambda(p,t)= \biggl(\int_Xe^{(|t|\vee 1)pq|\delta_\mu A(x)|} \mu(dx)\biggr)^{1/q}, \qquad 1/p+1/q=1. \] Moreover, \(V_t:=U_t-I\in G_{\theta,1} (H,\mu,H)\) for each \(\theta< \alpha\), and for every \(p\geq 1\), letting \(\nabla_HU_t: =\nabla_H V_t+I_H\), one has \[ \|\nabla_H U_t\|_{L^p (\mu)}\leq 4\Bigl(\int_X e^{2\bigl(|t |+1\bigr) p\bigl\|\nabla_H A(x)\bigr\|} \mu(dx)\Bigr)^{1/2p} e^{( \Lambda(2,t)+ 1)|t|/p}. \] Finally, \(\psi\circ U_t\in G_{\alpha, 1}(\mu)\) for every \(\psi\in S(X)\), and \[ {d\over dt}\int_X\psi\circ U_t(x)\mu (dx)=- \int_X\psi \bigl(U_t(x)\bigr) \delta_\mu A(x)\mu(dx). \] (If we assume that \(A\in W^{\alpha,1}(H,\mu,H)\), then \(\psi\circ U_t\in W^{\alpha,1} (H,\mu)\) for every \(\psi\in S(X))\).
scientific article; zbMATH DE number 1348492
scientific article