On one-dimensional Riccati diffusions

Bernoulli, 2005

We establish a law of large numbers and a central limit theorem for a class of additive functionals related to the solution of a one-dimensional stochastic differential equation perturbed by a large noise.

Lecture Notes in Mathematics, 2010

We define horizontal diffusion in C 1 path space over a Riemannian manifold and prove its existence. If the metric on the manifold is developing under the forward Ricci flow, horizontal diffusion along Brownian motion turns out to be length preserving. As application, we prove contraction properties in the Monge-Kantorovich minimization problem for probability measures evolving along the heat flow. For constant rank diffusions, differentiating a family of coupled diffusions gives a derivative process with a covariant derivative of finite variation. This construction provides an alternative method to filtering out redundant noise.

Communications in Mathematical Physics, 1985

The [α, β, y]-Langevin equation describes the time evolution of a real stationary process with T-positivity (reflection positivity) originating in the axiomatic quantum field theory. For this [α,/?,y]-Langevin equation a generalized fluctuation-dissipation theorem is proved. We shall obtain, as its application, a generalized fluctuation-dissipation theorem for the onedimensional non-linear diffusion process, which presents one solution of Ryogo Kubo's problem in physics.

2020

We are interested in the Euler-Maruyama discretization of a stochastic differential equation in dimension d with constant diffusion coefficient and bounded measurable drift coefficient. In the scheme, a randomization of the time variable is used to get rid of any regularity assumption of the drift in this variable. We prove weak convergence with order 1/2 in total variation distance. When the drift has a spatial divergence in the sense of distributions with ρ-th power integrable with respect to the Lebesgue measure in space uniformly in time for some ρ> d, the order of convergence at the terminal time improves to 1 up to some logarithmic factor. In dimension d=1, this result is preserved when the spatial derivative of the drift is a measure in space with total mass bounded uniformly in time. We confirm our theoretical analysis by numerical experiments.

2015

In this dissertation, we investigate various problems in the analysis of stochastic (partial) differential equations. A part of the dissertation introduces several notions of nonlinear integrations. Some differential equations associated with nonlinear integrations are investigated. Examples include transport differential equations in space-time random fields and parabolic equations with potentials of the type ∂tW , where W is continuous in time variable and smooth in the spatial variables. Another part of the dissertation studies nonlinear stochastic convolution equations driven by a multiplicative Gaussian noise which is white in time and which has the covariance of a fractional Brownian motion with Hurst parameter H ∈ (1/4, 1/2) in the spatial variable. The other part of the dissertation gives rigorous meaning to the Brox differential equation X(t) B(t) − 2 t 0 Ẇ (X(s))ds where B and W are independent Brownian motions. Furthermore, it is shown that the Brox differential equation ...

Stochastic Processes and their Applications, 2009

We consider the long term behaviour of a one-dimensional mixed effects diffusion process (X(t)) with a multivariate random effect φ in the drift coefficient. We first study the estimation of the random variable φ based on the observation of one sample path on the time interval [0, T ] as T tends to infinity. The process (X(t)) is not Markov and we characterize its invariant distributions. We build moment and maximum likelihood-type estimators of the random variable φ which are consistent and asymptotically mixed normal with rate √ T. Moreover, we obtain non asymptotic bounds for the moments of these estimators. Examples with a bivariate random effect are detailed. Afterwards, the estimation of parameters in the distribution of the random effect from N i.i.d. processes (Xj (t), t ∈ [0, T ]), j = 1,. .. , N is investigated. Estimators are built and studied as both N and T = T (N) tend to infinity. We prove that the convergence rate of estimators differs when deterministic components are present in the random effects. For true random effects, the rate of convergence is √ N whereas for deterministic components, the rate is √ N T. Illustrative examples are given.

Probability Theory and Related Fields, 2002

This paper is concerned with a general class of self-interacting diffusions {X t } t≥0 living on a compact Riemannian manifold M. These are solutions to stochastic differential equations of the form : dX t = Brownian increments + drift term depending on X t and µ t , the normalized occupation measure of the process. It is proved that the asymptotic behavior of {µ t } can be precisely related to the asymptotic behavior of a deterministic dynamical semi-flow = { t } t≥0 defined on the space of the Borel probability measures on M. In particular, the limit sets of {µ t } are proved to be almost surely attractor free sets for . These results are applied to several examples of self-attracting/repelling diffusions on the n-sphere. For instance, in the case of self-attracting diffusions, our results apply to prove that {µ t } can either converge toward the normalized Riemannian measure, or to a gaussian measure, depending on the value of a parameter measuring the strength of the attraction.

SIAM Journal on Applied Dynamical Systems, 2020

Solving inverse problems without the use of derivatives or adjoints of the forward model is highly desirable in many applications arising in science and engineering. In this paper we propose a new version of such a methodology, a framework for its analysis, and numerical evidence of the practicality of the method proposed. Our starting point is an ensemble of over-damped Langevin diffusions which interact through a single preconditioner computed as the empirical ensemble covariance. We demonstrate that the nonlinear Fokker-Planck equation arising from the mean-field limit of the associated stochastic differential equation (SDE) has a novel gradient flow structure, built on the Wasserstein metric and the covariance matrix of the noisy flow. Using this structure, we investigate large time properties of the Fokker-Planck equation, showing that its invariant measure coincides with that of a single Langevin diffusion, and demonstrating exponential convergence to the invariant measure in a number of settings. We introduce a new noisy variant on ensemble Kalman inversion (EKI) algorithms found from the original SDE by replacing exact gradients with ensemble differences; this defines the ensemble Kalman sampler (EKS). Numerical results are presented which demonstrate its efficacy as a derivative-free approximate sampler for the Bayesian posterior arising from inverse problems.

Probability Theory and Related Fields, 1996

A general comparison argument for expectations of certain multitime functionals of infinite systems of linearly interacting diffusions differing in the diffusion coefficient is derived. As an application we prove clustering occurs in the case when the symmetrized interaction kernel is recurrent, and the components take values in an interval bounded on one side. The technique also gives an alternative proof of clustering in the case of compact intervals. Classification (1991): 60K35, 60J60, 60J15 is one-sided bounded. We show that clustering is universal in the diffusion coefficient. This had been conjectured in Cox, Greven and Shiga [CGS95a] (see also Shiga [Shi92]). On the way, we obtain a new proof, in the case where the state space of a component is compact, based on the interacting Fisher-Wright diffusion where a well-known duality is available.

Annales Polonici Mathematici, 1993

Let X(t) be a diffusion process satisfying the stochastic differential equation dX(t) = a(X(t)) dW (t) + b(X(t)) dt. We analyse the asymptotic behaviour of p(t) = Prob{X(t) ≥ 0} as t → ∞ and construct an equation such that lim sup t→∞ t −1 t 0 p(s) ds = 1 and lim inf t→∞ t −1 t 0 p(s) ds = 0.