Nondegenerate Diffusion Research Articles

Optimal Relaxed Control for a Decoupled G-FBSDE

In this paper we study a system of decoupled forward-backward stochastic differential equations driven by a G-Brownian motion (G-FBSDEs) with non-degenerate diffusion. Our objective is to establish the existence of a relaxed optimal control for a non-smooth stochastic optimal control problem. The latter is given in terms of a decoupled G-FBSDE. The cost functional is the solution of the backward stochastic differential equation at the initial time. The key idea to establish existence of a relaxed optimal control is to replace the original control problem by a suitably regularised problem with mollified coefficients, prove the existence of a relaxed control, and then pass to the limit.

Non-adaptive estimation for degenerate diffusion processes

We consider a degenerate system of stochastic differential equations. The first component of the system has a parameter θ 1 \theta _1 in a non-degenerate diffusion coefficient and a parameter θ 2 \theta _2 in the drift term. The second component has a drift term with a parameter θ 3 \theta _3 and no diffusion term. Parametric estimation of the degenerate diffusion system is discussed under a sampling scheme. We investigate the asymptotic behavior of the joint quasi-maximum likelihood estimator for ( θ 1 , θ 2 , θ 3 ) (\theta _1,\theta _2,\theta _3) . The estimation scheme is non-adaptive. The estimator incorporates information of the increments of both components, and under this construction, we show that the asymptotic variance of the estimator for θ 1 \theta _1 is smaller than the one for standard estimator based on the first component only, and that the convergence of the estimator for θ 3 \theta _3 is much faster than for the other parameters. By simulation studies, we compare the performance of the joint quasi-maximum likelihood estimator with the adaptive and one-step estimators investigated in Gloter and Yoshida [Electron. J. Statist 15 (2021), no. 1, 1424–1472].

The large-time and vanishing-noise limits for entropy production in nondegenerate diffusions

The large-time and vanishing-noise limits for entropy production in nondegenerate diffusions

Stochastic generalized Kolmogorov systems with small diffusion: I. Explicit approximations for invariant probability density function

This paper presents Part I of a two-part series on studying the long-term coexistence states of stochastic generalized Kolmogorov systems with small diffusion. Part I establishes a mathematical framework for approximating the invariant probability measures (IPMs) and density functions (IPDFs) of these systems, while Part II will focus on analyzing their non-autonomous periodic counterparts. Compared with the existing approximation methods available only for systems with non-degenerate linear diffusion, this paper introduces two new and easily implementable approximation methods, the log-normal approximation (LNA) and updated normal approximation (uNA), which can be used for systems with not only non-degenerate but also degenerate diffusion. Moreover, we utilize the Kolmogorov-Fokker-Planck (KFP) operator and matrix algebra to develop algorithms for calculating the associated covariance matrix and verifying its positive definiteness. Our new approximation methods exhibit good accuracy in approximating the IPM and IPDF at both local and global levels, and significantly relaxes the minimal criteria for positive definiteness of the solution of the continuous-type Lyapunov equation. We demonstrate the utility of our methods in several application examples from biology and ecology.

Robustness of Stochastic Optimal Control to Approximate Diffusion Models Under Several Cost Evaluation Criteria

In control theory, typically a nominal model is assumed based on which an optimal control is designed and then applied to an actual (true) system. This gives rise to the problem of performance loss because of the mismatch between the true and assumed models. A robustness problem in this context is to show that the error because of the mismatch between a true and an assumed model decreases to zero as the assumed model approaches the true model. We study this problem when the state dynamics of the system are governed by controlled diffusion processes. In particular, we discuss continuity and robustness properties of finite and infinite horizon α-discounted/ergodic optimal control problems for a general class of nondegenerate controlled diffusion processes as well as for optimal control up to an exit time. Under a general set of assumptions and a convergence criterion on the models, we first establish that the optimal value of the approximate model converges to the optimal value of the true model. We then establish that the error because of the mismatch that occurs by application of a control policy, designed for an incorrectly estimated model, to a true model decreases to zero as the incorrect model approaches the true model. We see that, compared with related results in the discrete-time setup, the continuous-time theory lets us utilize the strong regularity properties of solutions to optimality (Hamilton–Jacobi–Bellman) equations, via the theory of uniformly elliptic partial differential equations, to arrive at strong continuity and robustness properties. Funding: The research of S. Yüksel was partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).

Threshold dynamics and probability density functions of a stochastic predator–prey model with general distributed delay

Predator–prey interactions are among the most complicated interactions between biological species, in which there may be both delay digestion and stochastic fluctuation. In the literature, due to the computational complexity and the lack of available methods, few papers have directly considered the predator–prey model for the combination of general infinite time delays with stochasticity. In this sense, this work is devoted to studying the long-term qualitative behavior of a stochastic predator–prey system with general infinite distributed delay. First, we establish sufficient and necessary conditions for exponential extinction and persistence of species. Notably, not only do our results cover the predator–prey model without stochastic noises and time delays, but also reveal the impact of environmental noises on population dynamics. Moreover, there are some challenges to analyze the stationary distribution of the numbers of prey and predator under species persistence. Based on the recent work (Zhou et al., 2020) where the density function of a three-dimensional avian influenza model with non-degenerate diffusion is concerned, we further generalize the relevant theories to a five-dimensional population system with degenerate diffusion. Via solving the corresponding Fokker–Planck equations, several approximate expressions of the density function of the stochastic predator–prey system with special distribution delay kernels are obtained. Finally, we provide some numerical simulations to supplement the analytical results and study the impact of stochastic noises.

Convergence of Weak Euler Approximation for Nondegenerate Stochastic Differential Equations Driven by Point and Martingale Measures

This paper studies the weak Euler approximation for solutions to stochastic differential equations (SDEs) driven by point and martingale measures, with Hölder-continuous coefficients. The equation under consideration includes a nondegenerate main part whose jump intensity measure is absolutely continuous with respect to the Lévy measure of a spherically symmetric stable process. It encompasses a broad range of stochastic processes including the nondegenerate diffusions and SDEs driven by Lévy processes. To investigate the dependence of the convergence rate on the regularity of the coefficients and driving processes, the regularity of a solution to the associated backward Kolmogorov equation is considered. In particular, for the first time the Hölder norm of the subordinated part of the corresponding generator is rigorously estimated.

Flux in Tilted Potential Systems: Negative Resistance and Persistence

Many real-world systems are well-modeled by Brownian particles subject to gradient dynamics plus noise arising, e.g., from the thermal fluctuations of a heat bath. Of central importance to many applications in physics and biology (e.g., molecular motors) is the net steady-state particle current or “flux” enabled by the noise and an additional driving force. However, this flux cannot usually be calculated analytically. Motivated by this, we investigate the steady-state flux generated by a nondegenerate diffusion process on a general compact manifold; such fluxes are essentially equivalent to the stochastic intersection numbers of Manabe (Osaka Math J 19(2):429–457, 1982). In the case that noise is small and the drift is “gradient-like” in an appropriate sense, we derive a graph-theoretic formula for the small-noise asymptotics of the flux using Freidlin–Wentzell theory. When additionally the drift is a local gradient sufficiently close to a generic global gradient, there is a natural flux equivalent to the entropy production rate—in this case our graph-theoretic formula becomes Morse-theoretic, and the result admits a description in terms of persistent homology. As an application, we provide a mathematically rigorous explanation of the paradoxical “negative resistance” phenomenon in Brownian transport discovered by Cecchi and Magnasco (Phys Rev Lett 76(11):1968, 1996).

Asymptotic Bismut formulae for stochastic functional differential equations with infinite delay

Using Malliavin calculus, this paper establishes asymptotic Bismut formulae for stochastic functional differential equations with infinite delay. Both nondegenerate and degenerate diffusion coefficients are treated. In addition, combined with the corresponding exponential ergodicity, stabilization bounds for ∇ P t f \nabla P_{t}f as t → ∞ t\rightarrow \infty are derived.

On Diffusion Processes with Drift in Ld+ 1

This paper is a natural continuation of Krylov (2020 and 2021) where strong Markov processes are constructed in time inhomogeneous setting with Borel measurable uniformly bounded and uniformly nondegenerate diffusion and drift in $L_{d+1}(\mathbb {R}^{d+1})$ and some properties of their Green’s functions and probability of passing through narrow tubes are investigated. On the basis of this here we study some further properties of these processes such as Harnack inequality, Hölder continuity of potentials, Fanghua Lin estimates and so on.

Some properties of solutions of Itô equations with drift in [formula omitted

This paper is a natural continuation of Krylov (2020), where strong Markov processes are constructed in time inhomogeneous setting with Borel measurable uniformly bounded and uniformly nondegenerate diffusion and drift in Ld+1(Rd+1). Here we study some properties of these processes such as higher summability of Green’s functions, a maximum principle for related transport equations with irregular coefficients, boundedness of resolvent operators in Lebesgue spaces, establish Itô’s formula, and so on.

On potentials of Itô’s Processes with Drift in Ld+ 1

This paper is a natural continuation of Krylov (2020), where strong Markov processes are constructed in the time inhomogeneous setting with Borel measurable uniformly bounded and uniformly nondegenerate diffusion and drift in \(L_{d+1}(\mathbb {R}^{d+1})\). Here we study some properties of these processes such as the probability to pass through narrow tubes, higher summability of Green’s functions, and so on. The results seem to be new even if the diffusion is constant.

On fractional and nonlocal parabolic mean field games in the whole space

We study Mean Field Games (MFGs) driven by a large class of nonlocal, fractional and anomalous diffusions in the whole space. These non-Gaussian diffusions are pure jump Lévy processes with some σ-stable like behaviour. Included are σ-stable processes and fractional Laplace diffusion operators (−Δ)σ2, tempered nonsymmetric processes in Finance, spectrally one-sided processes, and sums of subelliptic operators of different orders. Our main results are existence and uniqueness of classical solutions of MFG systems with nondegenerate diffusion operators of order σ∈(1,2). We consider parabolic equations in the whole space with both local and nonlocal couplings. Our proofs use pure PDE-methods and build on ideas of Lions et al. The new ingredients are fractional heat kernel estimates, regularity results for fractional Bellman, Fokker-Planck and coupled Mean Field Game equations, and a priori bounds and compactness of (very) weak solutions of fractional Fokker-Planck equations in the whole space. Our techniques require no moment assumptions and use a weaker topology than Wasserstein.

Saturated Fronts in Crowds Dynamics

Abstract We consider a degenerate scalar parabolic equation, in one spatial dimension, of flux-saturated type. The equation also contains a convective term. We study the existence and regularity of traveling-wave solutions; in particular we show that they can be discontinuous. Uniqueness is recovered by requiring an entropy condition, and entropic solutions turn out to be the vanishing-diffusion limits of traveling-wave solutions to the equation with an additional non-degenerate diffusion. Applications to crowds dynamics, which motivated the present research, are also provided.

Adaptive estimation for degenerate diffusion processes

We discuss parametric estimation of a degenerate diffusion system from time-discrete observations. The first component of the degenerate diffusion system has a parameter 𝜃1 in a non-degenerate diffusion coefficient and a parameter 𝜃2 in the drift term. The second component has a drift term parameterized by 𝜃3 and no diffusion term. Asymptotic normality is proved in two different situations for an adaptive estimator for 𝜃3 with some initial estimators for (𝜃1,𝜃2), and an adaptive one-step estimator for (𝜃1,𝜃2,𝜃3) with some initial estimators for them. Our estimators incorporate information of the increments of both components. Thanks to this construction, the asymptotic variance of the estimators for 𝜃1 is smaller than the standard one based only on the first component. The convergence of the estimators for 𝜃3 is much faster than the other parameters. The resulting asymptotic variance is smaller than that of an estimator only using the increments of the second component.

Does spatial homogeneity ultimately prevail in nutrient taxis systems? A paradigm for structure support by rapid diffusion decay in an autonomous parabolic flow

This work is concerned with the doubly degenerate cross-diffusion system (*) { u t = ( u v u x ) x − ( u 2 v v x ) x + u v , v t = v x x − u v , \begin{equation}\tag {*} \left \{ \begin {array}{l} u_t = (uvu_x)_x - (u^2 vv_x)_x + uv, \\[1mm] v_t = v_{xx}-uv, \end{array} \right . \end{equation} that has been proposed as a model for experimentally observable quite complex pattern formation phenomena in bacterial populations. It is shown that for any initial data satisfying adequate regularity and positivity assumptions, a no-flux initial-boundary value problem for the above in a bounded real interval possesses a global weak solution which is continuous in its first and essentially smooth in its second component. This solution is seen to asymptotically stabilize in the sense that (**) u ( ⋅ , t ) → u ∞ and v ( ⋅ , t ) → 0 as t → ∞ \begin{equation}\tag {**} u(\cdot ,t) \to u_\infty \quad \text {and} \quad v(\cdot ,t)\to 0 \qquad \text {as } t\to \infty \end{equation} with some nonnegative u ∞ ∈ C 0 ( Ω ¯ ) u_\infty \in C^0(\overline {\Omega }) which can be obtained as the evaluation of a weak solution z ∈ C 0 ( Ω ¯ × [ 0 , 1 ] ) z\in C^0(\overline {\Omega }\times [0,1]) to a porous medium-type parabolic problem at the finite time 1 1 . It is, moreover, revealed that for each suitably regular nonnegative function u ⋆ u_\star on Ω \Omega , the pair ( u ⋆ , 0 ) (u_\star ,0) , formally constituting an equilibrium of \eqref{01}, is stable in an appropriate sense. This finally implies a sufficient criterion for the limit u ∞ u_\infty in \eqref{02} to be spatially heterogeneous. The latter properties are in sharp contrast to known asymptotic features of corresponding nutrient taxis systems involving linear nondegenerate diffusion, as for which the literature appears to exclusively provide results on solutions which approach spatially constant states in the large time limit.

Density function analysis for a stochastic SEIS epidemic model with non-degenerate diffusion

In this paper, we construct a stochastic SEIS epidemic model that incorporates constant recruitment, non-degenerate diffusion and infectious force in the latent period and infected period. By solving the corresponding Fokker-Planck equation, we obtain the exact expression of the density function around the endemic equilibrium of the deterministic system provided that the basic reproduction number is greater than one. Our work greatly improves the result of Chen [A new idea on density function and covariance matrix analysis of a stochastic SEIS epidemic model with degenerate diffusion, Appl. Math. Lett., 2020, 106200].

Long-time asymptotics of non-degenerate non-linear diffusion equations

We study the long-time asymptotics of prototypical non-linear diffusion equations. Specifically, we consider the case of a non-degenerate diffusivity function that is a (non-negative) polynomial of the dependent variable of the problem. We motivate these types of equations using Einstein’s random walk paradigm, leading to a partial differential equation in non-divergence form. On the other hand, using conservation principles leads to a partial differential equation in divergence form. A transformation is derived to handle both cases. Then, a maximum principle (on both an unbounded and a bounded domain) is proved in order to obtain bounds above and below for the time-evolution of the solutions to the non-linear diffusion problem. Specifically, these bounds are based on the fundamental solution of the linear problem (the so-called Aronson’s Green function). Having thus sandwiched the long-time asymptotics of solutions to the non-linear problems between two fundamental solutions of the linear problem, we prove that, unlike the case of degenerate diffusion, a non-degenerate diffusion equation’s solution converges onto the linear diffusion solution at long times. Select numerical examples support the mathematical theorems and illustrate the convergence process. Our results have implications on how to interpret asymptotic scalings of potentially anomalous diffusion processes (such as in the flow of particulate materials) that have been discussed in the applied physics literature.

Lipschitz regularity for viscous Hamilton-Jacobi equations with Lp terms

We provide Lipschitz regularity for solutions to viscous time-dependent Hamilton-Jacobi equations with right-hand side belonging to Lebesgue spaces. Our approach is based on a duality method, and relies on the analysis of the regularity of the gradient of solutions to a dual (Fokker-Planck) equation. Here, the regularizing effect is due to the non-degenerate diffusion and coercivity of the Hamiltonian in the gradient variable.

Strong time periodic solutions to Keller-Segel systems: An approach by the quasilinear Arendt-Bu theorem

It is shown that the classical as well as quasilinear Keller-Segel systems with non-degenerate diffusion possess for given T-periodic and sufficiently small forcing functions a unique, strong T-time periodic solution. The proof given relies on the existence of strong T-periodic solutions for the linearized system, its characterization in terms of maximal Lp-regularity of the underlying operator and a quasilinear version of the Arendt-Bu Theorem. The latter is of independent interest and yields the existence of strong T-periodic solutions to general quasilinear evolution equations under suitable conditions on the operators and the forcing terms.