Convergence rate of some semi-groups to their invariant probability
Let \(W\) be a \(d\)-dimensional Wiener process and \(b:\mathbb R^{d}\to \mathbb R^{d}\) a locally Lipschitz function such that \(\langle b(y),y\rangle \geq 0\) for all \(y\in \mathbb R^{d}\). It exists a unique solution \(X^{x}\) to the stochastic differential equation \[ X_{t} = x + W_{t} -\frac 12\int ^{t}_{0} b(X_{s}) ds \tag{1} \] for every initial value \(x\in \mathbb R^{d}\). Denote by \((P_{t})\) the transition semigroup associated to (1), \(P_{t}\varphi (x) = E\varphi (X^{x}_{t})\) for any bounded Borel function \(\varphi \). Assume that \(b\) is a gradient field, that is, \(b = \nabla V\) for a function \(V:\mathbb R^{d}\to \mathbb R\), and that \(\int _{\mathbb R^{d}} \exp (-V(x)) dx = 1\). Then the probability measure \(\mu \) defined by \( d\mu = \exp (-V(x)) dx\) is invariant for the semigroup \((P_{t})\) and it is known that \(\|P_{t}f -\mu (f)\|_2 \to 0\) as \(t\to \infty \) for every function \(f\in L^{2}(\mu)\), where we denote by \(\|\cdot \|_{p}\) the norm of the space \(L^{p}(\mu)\). The authors aim at obtaining several estimates of the rate of convergence of \(P_{t}\) to \(\mu \), these estimates being dependent on the behaviour of the function \(b\) at infinity. For example, if \(|b(x)|\sim C|x|^\alpha \) as \(|x|\to \infty \) for an \(\alpha > 1\) and \(b\) is a radial function, \(b(x) = k(|x|)|x|^{-1}x\), then \(\|P_{t}f -\mu (f)\|_\infty \leq K\exp (-\sigma t) \|f\|_{1}\) for some \(K>0\), \(t_0 >0\), \(\sigma >0\) and any \(t\geq t_{0}\), \(f\in L^{2}(\mu)\). Related results are proved also in the case \(d=1\) and \(|b(x)|\sim C|x|^\alpha \), \(\alpha \in [-1,1]\).
scientific article; zbMATH DE number 1554284