TY  - JOUR
A1  - Cattiaux, Patrick
A1  - Fradon, Myriam
A1  - Kulik, Alexei M.
A1  - Roelly, Sylvie
T1  - Long time behavior of stochastic hard ball systems
JF  - Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability
N2  - We study the long time behavior of a system of n = 2, 3 Brownian hard balls, living in R-d for d >= 2, submitted to a mutual attraction and to elastic collisions.
KW  - hard core interaction
KW  - local time
KW  - Lyapunov function
KW  - normal reflection
KW  - Poincare inequality
KW  - reversible measure
KW  - stochastic differential equations
Y1  - 2016
U6  - https://doi.org/10.3150/14-BEJ672
SN  - 1350-7265
SN  - 1573-9759
VL  - 22
SP  - 681
EP  - 710
PB  - International Statistical Institute
CY  - Voorburg
ER  - 
TY  - JOUR
A1  - Maoutsa, Dimitra
A1  - Reich, Sebastian
A1  - Opper, Manfred
T1  - Interacting particle solutions of Fokker–Planck equations through gradient–log–density estimation
JF  - Entropy
N2  - Fokker-Planck equations are extensively employed in various scientific fields as they characterise the behaviour of stochastic systems at the level of probability density functions. Although broadly used, they allow for analytical treatment only in limited settings, and often it is inevitable to resort to numerical solutions. Here, we develop a computational approach for simulating the time evolution of Fokker-Planck solutions in terms of a mean field limit of an interacting particle system. The interactions between particles are determined by the gradient of the logarithm of the particle density, approximated here by a novel statistical estimator. The performance of our method shows promising results, with more accurate and less fluctuating statistics compared to direct stochastic simulations of comparable particle number. Taken together, our framework allows for effortless and reliable particle-based simulations of Fokker-Planck equations in low and moderate dimensions. The proposed gradient-log-density estimator is also of independent interest, for example, in the context of optimal control.
KW  - stochastic systems
KW  - Fokker-Planck equation
KW  - interacting particles
KW  - multiplicative noise
KW  - gradient flow
KW  - stochastic differential equations
Y1  - 2020
U6  - https://doi.org/10.3390/e22080802
SN  - 1099-4300
VL  - 22
IS  - 8
PB  - MDPI
CY  - Basel
ER  - 
TY  - JOUR
A1  - Koltai, Peter
A1  - Lie, Han Cheng
A1  - Plonka, Martin
T1  - Frechet 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.
KW  - stochastic differential equations
KW  - transfer operator
KW  - Koopman operator
KW  - Perron-Frobenius operator
KW  - smooth drift dependence
KW  - linear response
KW  - pathwise expectations
Y1  - 2019
U6  - https://doi.org/10.1088/1361-6544/ab1f2a
SN  - 0951-7715
SN  - 1361-6544
VL  - 32
IS  - 11
SP  - 4232
EP  - 4257
PB  - IOP Publ. Ltd.
CY  - Bristol
ER  -