Under partial dissipativity and super-linear growth conditions on the coefficients, we investigate the numerical approximation of random periodic solutions for McKean-Vlasov stochastic differential equations. By combining reflection coupling with the continuous-time backward Euler–Maruyama scheme on an infinite-time horizon, we establish the existence and uniqueness in distribution of numerical random periodic solutions. We further derive discretization error bounds on the infinite-time horizon, which yield the convergence rate of the backward Euler–Maruyama approximation to the random periodic solution in the L^1 -Wasserstein distance. The results show that the convergence of the numerical random periodic solutions is improved by taking a smaller step size and a larger number of particles.
In the simple random walk, the steps are independent. In contrast, in the elephant random walk every step depends on the whole past. One extension, as suggested by Bercu et al. [4], is to allow for stops/delays, that is, to put mass at zero. The present paper is devoted to, what we call, the Bernoulli elephant random walk, which comes up naturally in that context.
We show that the small-time asymptotics of the sub-Riemannian heat kernel, its derivatives, and its logarithmic derivatives can be localized, allowing them to be studied even on incomplete manifolds, under essentially optimal conditions on the distance to infinity. Away from abnormal minimizers, we show that the asymptotics are closely connected to the structure of the minimizing geodesics between the two relevant points (which is non-trivial on the cut locus). This gives uniform heat kernel bounds on compacts, and also allows a complete expansion of the heat kernel, and its derivatives, in a wide variety of cases. The method extends naturally to logarithmic derivatives of the heat kernel, where we again get uniform bounds on compacts and a more precise expansion for any particular pair of points, in most cases. In particular, we determine the measure giving the law of large numbers for the corresponding diffusion bridge, and the leading terms of the logarithmic derivatives are given by the cumulants of geometrically natural random variables with respect to this measure. One consequence is that the non-abnormal cut locus is characterized by the behavior of the log-Hessian of the heat kernel.
In this work, we establish a strong averaging principle for a class of McKean–Vlasov stochastic partial differential equations (SPDEs) with two-timescale structures, driven by Lévy noise. By employing Khasminskii’s time discretization method in combination with a variational approach, we prove that the slow component converges strongly to the solution of the corresponding averaged equation. Furthermore, we derive explicit rates of convergence. The main results can be applied to a set of nonlinear McKean–Vlasov SPDEs, including the stochastic porous medium equation, the stochastic p-Laplace equation, as well as McKean–Vlasov stochastic differential equations.
This paper is devoted to establishing the strong averaging principle for a class of slow–fast McKean–Vlasov multivalued stochastic evolution equations in the variational framework. We first establish existence and uniqueness of solutions for this type of stochastic evolution equations. Then we study the averaging principle and prove that the slow component converges in strong sense to the solution of the averaged equation. The main results of this paper are applicable to slow–fast McKean–Vlasov stochastic differential equations with normal reflections in convex domains and slow–fast McKean–Vlasov stochastic variational inequalities. In particular, convergence order is obtained for McKean–Vlasov stochastic evolution equations reflected in convex domains.
Motivated by a problem from incompressible fluid mechanics of Brenier [7], and its recent entropic relaxation by [3], we study a problem of entropic minimisation on the path space when the reference measure is a generic Feller semimartingale. Under mild regularity conditions on the minimizers, we show that our problem connects naturally with a version, possibly non-local, of the Hamilton–Jacobi–Bellman equation involving a pressure term.
A Markov chain X(i )on a finite state space S has transition matrix P and initial state i. We may run the chains (X-i : i is an element of S) in parallel, while insisting that any two such chains coalesce whenever they are simultaneously at the same state. There are Strajectories which evolve separately, but not necessarily independently, prior to coalescence. What can be said about the number k(& micro;) of coalescence classes of the process, and what is the set K (P) of such numbers k(& micro;), as the coupling & micro; of the chains ranges over couplings that are consistent with P? We continue earlier work of Grimmett and Holmes (In: In and out of equilibrium 3, Birkhauser/Springer, Cham, 2021) on these two fundamental questions, which have special importance for the "coupling from the past" algorithm. We concentrate partly on a family of couplings termed block measures, which may be viewed as couplings of lumpable chains with coalescing lumps. Constructions of such couplings are presented and also of non-block measure with similar properties.
This paper investigates the asymptotic distribution of order statistics in generalized allocation schemes. We focus on scenarios where n, the number of particles, and N, the number of cells, satisfy n = θ N^a for some a > 1 and a positive constant θ . Building on foundational work by Kolchin and others, where the joint distribution of the number of particles in each cell, (η _1 = k_1, … , η _N = k_N) , follows a conditional joint distribution (ξ _1 = k_1, … , ξ _N = k_N |ξ _1 + ⋯ + ξ _N = n) , we assume that ξ _i are independent geometric random variables and k_1 + ⋯ + k_N = n . We extend the analysis to determine the limiting distributions and the moment generating functions of the order statistics η _(1)≤η _(2)≤⋯≤η _(N) under these allocation models. We present the limiting distributions for η _(m) and the moment generating functions of η _(m) / N^a-2 for 1≤ m ≤ N under certain conditions. These findings have significant implications for applications in random graphs and sequences of runs, providing deeper insights into their asymptotic behavior.
Random iterated function systems of one-dimensional maps on a compact interval X⊂ℝ often fail to be globally contractive in any fixed metric, even when sample Lyapunov exponents are negative. A classical workaround is to twist the geometry by a positive weight w and require a weighted-derivative condition of the form ∑ _k=1^m p_k | f_k'(x) w(x)/w(f_k(x))| ≤ r<1, ∀ x∈ X, which ensures that the associated Markov operator is a strict contraction in a weighted Wasserstein–Kantorovich metric d_W^(w) . In this paper, we replace ad hoc constructions of w by a spectral theory for the derivative transfer operator (ℒφ )(x) = ∑ _k=1^m p_k |f_k'(x)| φ (f_k(x)), φ∈ C(X). Our main result proves that the optimal contraction constant, minimised over all continuous weights w>0 , coincides with the spectral radius ρ (ℒ) . Equivalently, ρ (ℒ)<1 if and only if there exists a strictly positive continuous supersolution ϕ =1/w of ℒϕ≤ rϕ for some r<1 , and in this case the corresponding d_W^(w) yields exponential convergence of every trajectory to a unique invariant probability measure. Conversely, if ρ (ℒ)≥ 1 then no choice of w can produce global contractivity. We illustrate the framework on random logistic and Ricker families, providing parameter-dependent bounds on ρ (ℒ) and showing how loss of weighted contractivity organises the transition between ergodic and null-recurrent regimes. The spectral construction also prepares the ground for controlled variants, where one seeks uniform spectral bounds for the corresponding derivative transfer operators over admissible control laws.
In this paper, we study McKean–Vlasov stochastic differential equations driven by fractional stable processes dX_t=b(t,X_t,ℒ_X_t)dt+σ (t, ℒ_X_t)dZ_t^H,α, where ℒ_X_t denotes the law of X_t , {Z_t^H,α, t∈ [0,T]} is a fractional stable process with parameters α∈ (1,2) and H∈ (1/α ,1) . We establish the existence and uniqueness theorem for solutions of these type of equations, and then give the theory of chaos propagation. Moreover, we show that the solutions can be approximated by the solutions of the associated averaged McKean–Vlasov stochastic differential equations in the sense of moment convergence, and provide an example of numerical simulation. These results not only generalize the corresponding results about stable processes to fractional stable processes, but also extend the fractional Brownian motion case to the non-Gaussian case.
We introduce a framework for stochastic differential equations (SDEs) with interaction on compact, connected, d-dimensional manifolds. For SDEs whose drift and diffusion coefficients may depend on both the state variable and the empirical distribution, we establish existence and uniqueness of strong solutions under general regularity assumptions. We study the associated measure-valued process on the Wasserstein space over the manifold, deriving an explicit Itô–Wiener decomposition. We prove Malliavin differentiability of the solution and, using directional derivatives in the Wasserstein space, establish smooth dependence of the solution on the measure component for a class of coefficients.
In this paper, we investigate the two-color nonlinear unbalanced urn model where the drawing rule is governed by a (concave) function with values in ℝ^+ and the replacement mechanism is described by an unbalanced matrix. By connecting this model to stochastic approximation theory, we derive the law of the iterated logarithm for this model and provide illustrative examples.
In this paper, by Malliavin calculus for Wiener–Poisson functionals, Bismut formulas for the parameter of mean-field stochastic differential equations (SDEs) with jumps are established. As applications, formulas for the Greeks are derived for asset price processes described by mean-field SDEs with or without jumps. Both European and Asian options are considered. Numerical results illustrate that the obtained formulas have better effects than the finite difference method in computing the Greeks.
We study the landscape complexity of the Hamiltonian X_N(x) +μ/2‖ x‖ ^2, where μ >0 and X_N is an isotropic Gaussian random field on ℝ^N . We derive asymptotic formulas for the expected number of critical points of the Hamiltonian, as the dimension N tends to infinity. These results complement the corresponding findings for locally isotropic Gaussian random fields, as well as the work of Auffinger and Zeng (Auffinger and Zeng 402:951–993 2023, 2022) on non-isotropic Gaussian random fields with isotropic increments.
Characterization problems, in general, hold a special place in probability literature. The present paper, inspired by the seminal works of Vinogradov (Theory Probab Appl 18:811–813, 1974) and Block and Savits (Ann Probab 465–474, 1980), focuses on using the Laplace transform to obtain characterization results for certain classes of life distributions. We also highlight an interesting interpretation of such characterization results in terms of a specific type of shock model and also their potential applications in establishing results involving convolutions.
In this paper, we study the long-time behavior of time-inhomogeneous diffusion processes in one dimension. Utilizing the probabilistic coupling method, we introduce a sufficient condition, inspired by Chen and Wang (Trans Am Math Soc, 349(3):1239–1267, 1997), to analyze the convergence of the transition kernels starting from different initial locations under a specific Wasserstein metric. Subsequently, we apply the main results to derive quantitative convergence rates for time-inhomogeneous diffusion processes by establishing rigorous conditions on the drift and diffusion coefficients. Our framework encompasses classical examples characterized by logarithmic, algebraic, and stretched exponential convergence rates. In particular, we offer a new perspective for evaluating the non-exponential convergence of time-homogeneous diffusions.
This paper is devoted to studying the averaging principle for a system of stochastic partial differential equations (SPDEs) that has a slow component driven by fractional Brownian motion (fBm) with a Hurst parameter H>1/2 , and a fast component driven by fast-varying diffusion. The optimal orders for the slow component that converges to the solution of the corresponding averaged equation have been obtained by using the Poisson equation method under some appropriate conditions. More precisely, the optimal orders are 1/2 and (1/2) - ε ^* (for all sufficiently small ε ^* ).
We consider the stochastic partial differential equation (SPDE) ∂ _t u = 12∂ ^2_x u + b(u) + σ (u) Ẇ, where u=u(t,x) is defined for (t,x)∈ (0,∞ )×ℝ and Ẇ denotes space-time white noise. We prove that this SPDE is well posed solely under the assumptions that the initial condition u(0) is bounded and measurable, and b and σ are locally Lipschitz continuous functions having at most linear growth with regularly behaved local Lipschitz constants. Our method is based on a truncation argument together with moment bounds and tail estimates of the truncated solution. The novelty of our method is in the pointwise nature of the truncation argument.
We consider a weighted sum of a series of independent Poisson random variables and show that it results in a new compound Poisson distribution which includes the Poisson distribution and Poisson distribution of order k and Poisson distribution of order infinity. An explicit representation for its distribution is obtained in terms of Bell polynomials. We then extend it to a compound Poisson process (CPP) and time-fractional compound Poisson process (TFCPP). It is shown that the one-dimensional distributions of the TFCPP exhibit over-dispersion property, are not infinitely divisible and possess the long-range dependence property. Also, their moments and factorial moments are derived. The martingale characterization results for the CPP and the TFCPP are established. Finally, the fractional diffserential equation associated with the TFCPP is also obtained. Some possible applications to insurance are pointed out.
We consider the local times of (1 + β ) -stable d-dimensional super-Brownian motion with 0< β < 1 . It is known from Sugitani (J Math Soc Jpn 41(3):437–462, 1989) that for β = 1 , the local time is differentiable for d=1 . For 0< β < 1 , Mytnik and Perkins (Ann Probab 31(3): 1413–1440, 2003) proved that the local time, denoted by L(t, x), is jointly continuous for d = 1 , while it is locally unbounded in x for d ≥ 2 where it exists. This paper strengthens the results of Mytnik and Perkins for d=1 by showing that the local time L(t, x) is continuously differentiable in the spatial parameter x. Moreover, we give a representation of the spatial derivative, denoted by ∂/∂ xL(t, x) , and further prove that the derivative is locally γ -Hölder continuous in x for any index γ∈ (0, β/1+β) .
