TY - JOUR A1 - Lie, Han Cheng A1 - Sullivan, T. J. T1 - Cameron--Martin theorems for sequences of Cauchy-distributed random variables JF - arXiv N2 - Given a sequence of Cauchy-distributed random variables defined by a sequence of location parameters and a sequence of scale parameters, we consider another sequence of random variables that is obtained by perturbing the location or scale parameter sequences. Using a result of Kakutani on equivalence of infinite product measures, we provide sufficient conditions for the equivalence of laws of the two sequences. Y1 - 2016 SP - 1608.03784 ER - TY - JOUR A1 - Lie, Han Cheng A1 - Sullivan, T. J. A1 - Stuart, Andrew T1 - Strong convergence rates of probabilistic integrators for ordinary differential equations JF - Statistics and Computing N2 - Probabilistic integration of a continuous dynamical system is a way of systematically introducing model error, at scales no larger than errors inroduced by standard numerical discretisation, in order to enable thorough exploration of possible responses of the system to inputs. It is thus a potentially useful approach in a number of applications such as forward uncertainty quantification, inverse problems, and data assimilation. We extend the convergence analysis of probabilistic integrators for deterministic ordinary differential equations, as proposed by Conrad et al.\ (\textit{Stat.\ Comput.}, 2016), to establish mean-square convergence in the uniform norm on discrete- or continuous-time solutions under relaxed regularity assumptions on the driving vector fields and their induced flows. Specifically, we show that randomised high-order integrators for globally Lipschitz flows and randomised Euler integrators for dissipative vector fields with polynomially-bounded local Lipschitz constants all have the same mean-square convergence rate as their deterministic counterparts, provided that the variance of the integration noise is not of higher order than the corresponding deterministic integrator. Y1 - 2019 U6 - https://doi.org/10.1007/s11222-019-09898-6 VL - 29 IS - 6 SP - 1265 EP - 1283 ER - TY - JOUR A1 - Lie, Han Cheng A1 - Sullivan, T. J. T1 - Equivalence of weak and strong modes of measures on topological vector spaces JF - Inverse Problems N2 - Modes of a probability measure on an infinite-dimensional Banach space X are often defined by maximising the small-radius limit of the ratio of measures of norm balls. Helin and Burger weakened the definition of such modes by considering only balls with centres in proper subspaces of X, and posed the question of when this restricted notion coincides with the unrestricted one. We generalise these definitions to modes of arbitrary measures on topological vector spaces, defined by arbitrary bounded, convex, neighbourhoods of the origin. We show that a coincident limiting ratios condition is a necessary and sufficient condition for the equivalence of these two types of modes, and show that the coincident limiting ratios condition is satisfied in a wide range of topological vector spaces. Y1 - 2018 U6 - https://doi.org/10.1088/1361-6420/aadef2 VL - 34 IS - 11 SP - 115013 ER - TY - JOUR A1 - Lie, Han Cheng A1 - Sullivan, T. J. T1 - Quasi-invariance of countable products of Cauchy measures under non-unitary dilations JF - Electronic Communications in Probability Y1 - 2018 U6 - https://doi.org/10.1214/18-ECP113 VL - 23 IS - 8 SP - 1 EP - 6 ER - TY - JOUR A1 - Lie, Han Cheng A1 - Sullivan, T. J. T1 - Erratum: Equivalence of weak and strong modes of measures on topological vector spaces (2018 Inverse Problems 34 115013) JF - Inverse Problems Y1 - 2018 U6 - https://doi.org/10.1088/1361-6420/aae55b VL - 34 IS - 12 SP - 129601 ER - TY - JOUR A1 - Lie, Han Cheng A1 - Sullivan, T. J. A1 - Teckentrup, Aretha T1 - Error bounds for some approximate posterior measures in Bayesian inference JF - Numerical Mathematics and Advanced Applications ENUMATH 2019 Y1 - 2020 U6 - https://doi.org/10.1007/978-3-030-55874-1_26 SP - 275 EP - 283 PB - Springer ER - TY - JOUR A1 - Ayanbayev, Birzhan A1 - Klebanov, Ilja A1 - Lie, Han Cheng A1 - Sullivan, T. J. T1 - Γ-convergence of Onsager–Machlup functionals: I. With applications to maximum a posteriori estimation in Bayesian inverse problems JF - Inverse Problems Y1 - 2022 U6 - https://doi.org/10.1088/1361-6420/ac3f81 VL - 38 IS - 2 ER - TY - JOUR A1 - Ayanbayev, Birzhan A1 - Klebanov, Ilja A1 - Lie, Han Cheng A1 - Sullivan, T. J. T1 - Γ-convergence of Onsager–Machlup functionals: II. Infinite product measures on Banach spaces JF - Inverse Problems Y1 - 2022 U6 - https://doi.org/10.1088/1361-6420/ac3f82 VL - 38 IS - 2 ER - TY - JOUR A1 - Koltai, Péter A1 - Lie, Han Cheng A1 - Plonka, Martin T1 - Fréchet differentiable drift dependence of Perron–Frobenius and Koopman operators for non-deterministic dynamics JF - Nonlinearity N2 - We prove the Fréchet differentiability with respect to the drift of Perron–Frobenius and Koopman operators associated to time-inhomogeneous ordinary stochastic differential equations. This result relies on a similar differentiability result for pathwise expectations of path functionals of the solution of the stochastic differential equation, which we establish using Girsanov's formula. We demonstrate the significance of our result in the context of dynamical systems and operator theory, by proving continuously differentiable drift dependence of the simple eigen- and singular values and the corresponding eigen- and singular functions of the stochastic Perron–Frobenius and Koopman operators. Y1 - 2019 U6 - https://doi.org/10.1088/1361-6544/ab1f2a VL - 32 IS - 11 SP - 4232 EP - 4257 ER - TY - JOUR A1 - Lie, Han Cheng A1 - Fackeldey, Konstantin A1 - Weber, Marcus T1 - A Square Root Approximation of Transition Rates for a Markov State Model JF - SIAM. J. Matrix Anal. Appl. Y1 - 2013 U6 - https://doi.org/10.1137/120899959 VL - 34 IS - 2 SP - 738 EP - 756 ER -