Large and moderate deviations and exponential convergence for stochastic damping Hamiltonian systems.
The paper is devoted to an in-depth study of long-time behaviour of solutions to a stochastic Hamiltonian system with damping \[ dx_ {t} = y_ {t}\, dt, \quad dy_ {t} = -c(x_ {t},y_ {t})y_ {t} \,dt -\nabla V(x_ {t})\, dt + \Sigma (x_ {t},y_ {t})\,dW_ {t} \tag{1} \] in \(\mathbb R^ {2d}\), where \(W\) is a standard \(d\)-dimensional Wiener process. It is supposed that the potential \(V\in C^ 1(\mathbb R^ {d})\) is lower bounded, the damping coefficient \(c\) is continuous and bounded on every set \(\{\| x\| \leq N\}\times \mathbb R^ {d}\), \(N>0\), and the matrix-valued function \(\Sigma \) is \(C^ \infty \)-smooth and \(0<\Sigma \leq \sigma I\) on \(\mathbb R^ {2d}\) for some \(\sigma >0\). Moreover, it it assumed that there exist \(c,L>0\) such that \(c^ {s}(x,y)\geq cI\) for all \(\| x\| \geq L\), \(y\in \mathbb R^ {d}\), \(c^ {s} = \frac 12(c+c^ {T})\) denoting the symmetrization of the matrix \(c\). First, it is shown that there exists a unique weak solution to (1) for every initial datum \((x_ 0,y_ 0)\in \mathbb R^ {2d}\). A Girsanov formula is established and used to prove that the transition semigroup \((P_ {t})\) defined by (1) is strong Feller in the strict sense. Further, several general theorems on sufficient conditions for (empirical measures of) strong Feller topologically transitive Markov processes on Polish spaces to obey large or moderate deviation principles are proven. Finally, these results are applied to solutions of the equation (1). In particular, assume that there exist a function \(G\in C^ 1 _ {b}(\mathbb R^ {d};\mathbb R^ {d})\), \(\| G\| \leq 1\) on \(\mathbb R^ {d}\), and a lower bounded function \(U\in C^ 1(\mathbb R^ {d})\) satisfying \[ \begin{aligned} & \liminf _ {\| x\| \to \infty } \nabla V(x)\cdot G(x)>0,\;\lim _ {\| x\| \to \infty } \| (\partial _ {x_ {i}} G_ {j}(x))\| = 0,\\ & \lim _ {\| x\| \to \infty } \sup _ {y\in \mathbb R^ {d}} \| c(x,y)^ {T}G(x) - \nabla U(x) \| = 0.\end{aligned} \] Then the transition semigroup \((P_t)\) is exponentially ergodic: There exist a unique invariant probability measure \(\mu \) for \((P_ {t})\), a continuous function \(\Psi \geq 1\), \(\Psi \in L^ 1(\mu )\), and constants \(D>0\), \(\varrho \in ]0,1[\) such that \[ \sup _ {| f| \leq \Psi } \bigl | P_tf(z) - \mu (f)\bigr | \leq D\Psi (z)\varrho ^ {t} \] for all \(t\geq 0\) and \(z\in \mathbb R^ {2d}\).
scientific article; zbMATH DE number 2102367
scientific article