arXiv subjects

Linan Chen

Publications and source records attributed to Linan Chen.

15 recordsLinked to original sources

arXiv2025

A Regime-Switching Approach to the Unbalanced Schr\"odinger Bridge Problem

The unbalanced Schr\"odinger bridge problem (uSBP) seeks to interpolate between a probability measure $\rho_0$ and a sub-probability measure $\rho_T$ while minimizing KL divergence to a reference measure $\mathbf{R}$ on a path space. In this work, we investigate the case where $\mathbf{R}$ is the path measure of a diffusion process with killing, which we interpret as a regime-switching diffusion. In addition to matching the initial and terminal distributions of trajectories that survive up to time $T$, we consider a general constraint $\psi(t,x)$ on the distribution of killing times and/or killing locations. We investigate the uSBPs corresponding to four choices of $\psi$ in detail which reflect different levels of information available to an observer. We also provide a rigorous analysis of the connections and the comparisons among the outcomes of these four cases. Our results are novel in the field of uSBP. The regime-switching approach proposed in this work provides a unified framework for tackling different uSBP scenarios, which not only reconciles but also extends the existing literature on uSBP.

math.PR↗

arXiv2025

Two behavioural pseudometrics for continuous-time Markov processes

Bisimulation is a concept that captures behavioural equivalence of states in a variety of types of transition systems. It has been widely studied in discrete-time settings where a key notion is the bisimulation metric which quantifies "how similar two states are". In [ 11], we generalized the concept of bisimulation metric in order to metrize the behaviour of continuous-time Markov processes. Similarly to the discrete-time case, we constructed a pseudometric following two iterative approaches - through a functional and through a real-valued logic, and showed that the outcomes coincide: the pseudometric obtained from the logic is a specific fixpoint of the functional which yields our first pseudometric. However, different from the discrete-time setting, in which the process has a step-by-step dynamics, the behavioural pseudometric we constructed applies to Markov processes that evolve continuously through time, such as diffusions and jump diffusions. While our treatment of the pseudometric in [11] relied on the time-indexed Markov kernels, in [ 8 , 9, 10 ], we showed the importance of trajectories in the consideration of behavioural equivalences for true continuous-time Markov processes. In this paper, we take the work from [11 ] further and propose a second behavioural pseudometric for diffusions based on trajectories. We conduct a similar study of this pseudometric from both the perspective of a functional and the viewpoint of a real-valued logic. We also compare this pseudometric with the first pseudometric obtained in [11].

cs.LO↗

arXiv2025

The Schr\"odinger Bridge Problem for Jump Diffusions with Regime Switching

The Schr\"odinger bridge problem (SBP) aims at finding the measure $\hat{\mathbf{P}}$ on a certain path space which possesses the desired state-space distributions $\rho_0$ at time $0$ and $\rho_T$ at time $T$ while minimizing the KL divergence from a reference path measure $\mathbf{R}$. This work focuses on the SBP in the case when $\mathbf{R}$ is the path measure of a jump diffusion with regime switching, which is a Markov process that combines the dynamics of a jump diffusion with interspersed discrete events representing changing environmental states. To the best of our knowledge, the SBP in such a setting has not been previously studied. In this paper, we conduct a comprehensive analysis of the dynamics of the SBP solution $\hat{\mathbf{P}}$ in the regime-switching jump-diffusion setting. In particular, we show that $\hat{\mathbf{P}}$ is again a path measure of a regime-switching jump diffusion; under proper assumptions, we establish various properties of $\hat{\mathbf{P}}$ from both a stochastic calculus perspective and an analytic viewpoint. In addition, as an demonstration of the general theory developed in this work, we examine a concrete unbalanced SBP (uSBP) from the angle of a regime-switching SBP, where we also obtain novel results in the realm of uSBP.

math.PR↗

arXiv2025

A behavioural pseudometric for continuous-time Markov processes

In this work, we generalize the concept of bisimulation metric in order to metrize the behaviour of continuous-time processes. Similarly to what is done for discrete-time systems, we follow two approaches and show that they coincide: as a fixpoint of a functional and through a real-valued logic. The whole discrete-time approach relies entirely on the step-based dynamics: the process jumps from state to state. We define a behavioural pseudometric for processes that evolve continuously through time, such as Brownian motion or involve jumps or both.

cs.LO↗

arXiv2024

Schr\"odinger Bridge Problem for Jump Diffusions

The Schr\"odinger bridge problem (SBP) seeks to find the measure $\hat{\mathbf{P}}$ on a certain path space which interpolates between state-space distributions $\rho_0$ at time $0$ and $\rho_T$ at time $T$ while minimizing the KL divergence (relative entropy) to a reference path measure $\mathbf{R}$. In this work, we tackle the SBP in the case when $\mathbf{R}$ is the path measure of a jump diffusion. Under mild assumptions, with both the operator theory approach and the stochastic calculus techniques, we establish an $h$-transform theory for jump diffusions and devise an approximation method to achieve the jump-diffusion SBP solution $\hat{\mathbf{P}}$ as the strong-convergence limit of a sequence of harmonic $h$-transforms. To the best of our knowledge, these results are novel in the study of SBP. Moreover, the $h$-transform framework and the approximation method developed in this work are robust and applicable to a relatively general class of jump diffusions. In addition, we examine the SBP of particular types of jump diffusions under additional regularity conditions and extend the existing results on the SBP from the diffusion case to the jump-diffusion setting.

math.PR↗

arXiv2024

Stochastic Kimura Equations

In this work we study the one-dimensional stochastic Kimura equation $\partial_{t}u\left(z,t\right)=z\partial_{z}^{2}u\left(z,t\right)+u\left(z,t\right)\dot{W}\left(z,t\right)$ for $z,t>0$ equipped with a Dirichlet boundary condition at $0$, with $\dot{W}$ being a Gaussian space-time noise. This equation can be seen as a degenerate analog of the parabolic Anderson model. We combine the Wiener chaos theory from Malliavin calculus, the Duhamel perturbation technique from PDEs, and the kernel analysis of (deterministic) degenerate diffusion equations to develop a solution theory for the stochastic Kimura equation. We establish results on existence, uniqueness, moments, and continuity for the solution $u\left(z,t\right)$. In particular, we investigate how the stochastic potential and the degeneracy in the diffusion operator jointly affect the properties of $u\left(z,t\right)$ near the boundary. We also derive explicit estimates on the comparison under the $L^{2}-$ norm between $u\left(z,t\right)$ and its deterministic counterpart for $\left(z,t\right)$ within a proper range.

math.PR↗

arXiv2023

Behavioural pseudometrics for continuous-time diffusions

Bisimulation is a concept that captures behavioural equivalence of states in a variety of types of transition systems. It has been widely studied in a discrete-time setting where the notion of a step is fundamental. In our setting we are considering "flow"-processes emphasizing that they evolve in continuous time. In such continuous-time settings, the concepts are not straightforward adaptations of their discrete-time analogues and we restrict our study to diffusions that do not lose mass over time and with additional regularity constraints. In previous work we proposed different definitions of behavioural equivalences for continuous-time stochastic processes where the evolution is a flow through time. That work only addressed equivalences. In this work, we aim at quantifying how differently processes behave. We present two pseudometrics for diffusion-like processes. These pseudometrics are fixpoints of two different functionals on the space of 1-bounded pseudometrics on the state space. We also characterize these pseudometrics in terms of real-valued modal logics; this is a quantitative analogue of the notion of logical characterization of bisimulation. These real-valued modal logics indicate that the two pseudometrics are different and thus yield different notions of behavioural equivalence.

cs.LO↗

arXiv2020

Fundamental Solution to 1D Degenerate Diffusion Equation with Locally Bounded Coefficients

In this work we study the degenerate diffusion equation $\partial_{t}=x^{\alpha}a\left(x\right)\partial_{x}^{2}+b\left(x\right)\partial_{x}$ for $\left(x,t\right)\in\left(0,\infty\right)^{2}$, equipped with a Cauchy initial data and the Dirichlet boundary condition at $0$. We assume that the order of degeneracy at 0 of the diffusion operator is $\alpha\in\left(0,2\right)$, and both $a\left(x\right)$ and $b\left(x\right)$ are only locally bounded. We adopt a combination of probabilistic approach and analytic method: by analyzing the behaviors of the underlying diffusion process, we give an explicit construction to the fundamental solution $p\left(x,y,t\right)$ and prove several properties for $p\left(x,y,t\right)$; by conducting a localization procedure, we obtain an approximation for $p\left(x,y,t\right)$ for $x,y$ in a neighborhood of 0 and $t$ sufficiently small, where the error estimates only rely on the local bounds of $a\left(x\right)$ and $b\left(x\right)$ (and their derivatives). There is a rich literature on such a degenerate diffusion in the case of $\alpha=1$. Our work extends part of the existing results to cases with more general order of degeneracy, both in the analysis context (e.g., heat kernel estimates on fundamental solutions) and in the probability view (e.g., wellposedness of stochastic differential equations).

math.AP↗

arXiv2020

A diffusion-driven Characteristic Mapping method for particle management

We present a novel particle management method using the Characteristic Mapping framework. In the context of explicit evolution of parametrized curves and surfaces, the surface distribution of marker points created from sampling the parametric space is controlled by the area element of the parametrization function. As the surface evolves, the area element becomes uneven and the sampling, suboptimal. In this method we maintain the quality of the sampling by pre-composition of the parametrization with a deformation map of the parametric space. This deformation is generated by the velocity field associated to the diffusion process on the space of probability distributions and induces a uniform redistribution of the marker points. We also exploit the semigroup property of the heat equation to generate a submap decomposition of the deformation map which provides an efficient way of maintaining evenly distributed marker points on curves and surfaces undergoing extensive deformations.

math.NA↗

arXiv2019

The Fundamental Solution to One-Dimensional Degenerate Diffusion Equation, I

In this work we adopt a combination of probabilistic approach and analytic methods to study the fundamental solutions to variations of the Wright-Fisher equation in one dimension. To be specific, we consider a diffusion equation on $\left(0,\infty\right)$ whose diffusion coefficient vanishes at the boundary 0, equipped with the Cauchy initial data and the Dirichlet boundary condition. One type of diffusion operator that has been extensively studied is the one whose diffusion coefficient vanishes linearly at 0. Our main goal is to extend the study to cases when the diffusion coefficient has a general order of degeneracy. We primarily focus on the fundamental solution to such a degenerate diffusion equation. In particular, we study the regularity properties of the fundamental solution near 0, and investigate how the order of degeneracy of the diffusion operator and the Dirichlet boundary condition jointly affect these properties. We also provide estimates for the fundamental solution and its derivatives near 0.

math.AP↗

arXiv2019

Bisimulation for Feller-Dynkin Processes

Bisimulation is a concept that captures behavioural equivalence. It has been studied extensively on nonprobabilistic systems and on discrete-time Markov processes and on so-called continuous-time Markov chains. In the latter time is continuous but the evolution still proceeds in jumps. We propose two definitions of bisimulation on continuous-time stochastic processes where the evolution is a \emph{flow} through time. We show that they are equivalent and we show that when restricted to discrete-time, our concept of bisimulation encompasses the standard discrete-time concept. The concept we introduce is not a straightforward generalization of discrete-time concepts.

cs.LO↗

arXiv2018

Asymptotic Properties of Random Voronoi Cells with Arbitrary Underlying Density

We consider the Voronoi diagram generated by $n$ i.i.d. $\mathbb{R}^{d}$-valued random variables with an arbitrary underlying probability density function $f$ on $\mathbb{R}^{d}$, and analyse the asymptotic behaviours of certain geometric properties, such as the measure, of the Voronoi cells as $n$ tends to infinity. We adapt the methods used by Devroye et al (2017) to conduct a study of the asymptotic properties of two types of Voronoi cells: 1, Voronoi cells that have a fixed nucleus; 2, Voronoi cells that contain a fixed point. For the first type of Voronoi cells, we show that their geometric properties resemble those in the case when the Voronoi diagram is generated by a homogeneous Poisson point process. For the second type of Voronoi cells, we determine the limiting distribution, which is universal in all choices of $f$, of the rescaled measure of the cells. For both types, we establish the asymptotic independence of measures of disjoint cells.

math.PR↗

arXiv2018

Steep Points of Gaussian Free Fields in Any Dimension

This work aims to extend the existing results on the Hausdorff dimension of the classical thick point sets of a Gaussian free field (GFF) to a more general class of exceptional sets. We adopt the circle or sphere averaging regularization to treat a singular GFF in any dimension, and introduce the notion of "$f-$steep point" of the GFF for certain test function $f$. Roughly speaking, the $f-$steep points of a generic element of the GFF are locations where, when weighted by the function $f$, the "steepness", or in other words, the "rate of change" of the regularized field element becomes unusually large. Different choices of $f$ lead to the study of various exceptional behaviors of the GFF. We investigate the Hausdorff dimension of the set consisting of $f-$steep points, from which we can recover the existing results on thick point sets for both log-correlated and polynomial-correlated GFFs, and also obtain new results on exceptional sets that, to our best knowledge, have not been previously studied. Our method is inspired by the one used to study the thick point sets of the classical 2D log-correlated GFF.

math.PR↗

arXiv2015

Thick Points of High-Dimensional Gaussian Free Fields

This work aims to extend the existing results on thick points of logarithmic-correlated Gaussian Free Fields to Gaussian random fields that are more singular. To be specific, we adopt a sphere averaging regularization to study polynomial-correlated Gaussian Free Fields in higher-than-two dimensions. Under this setting, we introduce the definition of thick points which, heuristically speaking, are points where the value of the Gaussian Free Field is unusually large. We then establish a result on the Hausdorff dimension of the sets containing thick points.

math.PR↗

arXiv2012

Gaussian Free Fields and KPZ Relation in R^4

This work aims to extend part of the two dimensional results of Duplantier and Sheffield on Liouville quantum gravity to four dimensions, and indicate possible extensions to other even-dimensional spaces R^(2n) as well as Riemannian manifolds. Let "\Theta" be the Gaussian free field on R^4 with the underlying Hilbert space being the Sobolev space H^2 witb the inner product determined by the operator (I-\Delta)^2. Assume "\theta" is a generic element from \Theta. We consider a sequence of random Borel measures on R^4, each of which is absolutely continuous with respect to the Lebesgue measure dx and the density function is given by the exponential of a centered Gaussian family parametrized by x in R^4. We show that with probability 1, this sequence of measures weakly converges to a limit random measure which can be "formally" written as "exp(2\gamma\theta(x)dx". In this setting, we also prove a KPZ relation, which is the quadratic relation between the scaling exponent of a bounded Borel set on R^4 under the Lebesgue measure and its counterpart under the random measure obtained above. Our approach is similar to the one used by Duplantier and Sheffield in 2D but with adaptations to R^4.

math.PR↗