Motivated by probabilistic representations of the Navier–Stokes equations, we introduce a new class of stochastic differential equations whose coefficients depend on the entire flow of time marginals. Under one-sided Lipschitz conditions and in the presence of singular drifts, we establish the existence and uniqueness of both strong and weak solutions. These distribution-flow dependent stochastic differential equations are closely related to quasilinear backward Kolmogorov equations and Fokker–Planck equations. We also study distribution-flow dependent SDEs associated with generalized two-dimensional Navier–Stokes equations and driven by fractional Brownian noise. When the Hurst parameter H∈(0,1 2) and the initial vorticity ν0∈M(R2), we prove global well-posedness and smoothness of solutions. In the case H=1 2, we establish global strong well-posedness and smoothness for the corresponding SDEs associated with the backward Navier–Stokes equations, and clarify their relation to the Navier–Stokes equations.
In this paper, we establish Wong-Zakai approximation and support theorem for quasilinear parabolic stochastic partial differential equations. Since this class of equations is not locally monotone, we implement a regularization strategy using a sequence of heat kernel to modify the matrix-valued diffusion term, thereby recovering the monotonicity. As a result, the problem is reduced to prove two results of further layer approximations. To this end, we further employ a kinetic formulation of the approximating solutions in the L1 setting and fully use the doubling variables techniques.
In this paper, we study the asymptotic behaviors of supercritical branching Markov processes {X-t, t > 0} with spatial motions being L & eacute;vy processes having regularly renormalized by some appropriate function h(t), {X-t, t > 0} converges weakly to some random measure. In this paper, we first establish the large deviations of {X-t, t > 0}. More precisely, we prove that, (i) when renormalized by a function A(t) which grows faster than h(t), {X-t, t > 0}, conditioned on {R-t > A(t)}, converges weakly; (ii) when renormalized by a function A(t) which grows slower than h(t), {X-t, t > 0}, conditioned on {Rt < A(t)}, converges weakly, where Rt is the maximum of X-t. We also characterize the almost sure envelopes of Rt.
Let X beta be a real symmetric or complex Hermitian matrix whose entries are independent anisotropic Gaussian random fields. We provide the sufficient and necessary conditions such that multiple collisions of eigenvalue processes of A beta+T beta X beta T & lowast;beta occur with positive probability. In addition, for a real or complex rectangular matrix W beta with independent Gaussian random field entries, we obtain the sufficient and necessary conditions under which the probability of multiple collisions of non-trivial singular value processes of B beta+T beta W beta T beta is positive. In both cases, the size of the set of collision times is characterized via Hausdorff dimension. Moreover, we recover the corresponding results where the Gaussian random fields are isotropic.
This is the first part in a series of papers devoted to studying the tail profiles of the bulk/boundary quotients of Gaussian multiplicative chaos measures appearing in boundary Liouville conformal field theory, for which we establish preliminary joint moment bounds. These moment bounds will serve as a crucial ingredient in establishing the right tail profile of the bulk Gaussian multiplicative chaos measure in subsequent papers. The main idea is to implement the so-called localization trick at the boundary, and we also record a useful generalization of Kahane's convexity inequality, which is of independent interest.
We study an additive-noise approximation to Keller-Segel-Dean-Kawasaki dynamics, which is proposed as an approximate model to the fluctuating hydrodynamics of chemotactically interacting particles around their mean-field limit. As such, the interaction potential is given by the Green's function associated to Poisson's equation, which is singular around the origin. Two parameters play a key r & ocirc;le in the approximation: the noise intensity e which captures the amplitude of fluctuations (tending to zero as the effective system size tends to infinity) and the correlation length S which represents the effective scale under consideration. Let S(e) -* 0 as e -* 0. Under the relative scaling assumption lim epsilon -> 0 e log(S(e)-1) = 0 we obtain analogues of law of large numbers and large deviation principles in irregular spaces of distributions using methods of singular stochastic partial differential equations. The same techniques also yield a central limit theorem under the relative scaling lim epsilon -> 0 e1/2 log(S(e)-1) = 0. Assuming the more restrictive relative scaling lim epsilon -> 0 e1/2S-gamma-2 = 0 for some ry is an element of (-1/2, 0), we also obtain analogues of law of large numbers and large deviation principles in regular function spaces using a mixture of pathwise and probabilistic tools. We further describe consequences of these results relevant to applications of our approximation in studying continuum fluctuations of particle systems.
We study the spectral gap behavior of an operator obtained by summing a random permutation M and a deterministic bistochastic matrix Q. We are interested in the asymptotic in terms of dimension. In the case where (M, Q) are asymptotically free with amalgamation over the diagonal, we can compute limit operators (u, q) which give the limit distribution. Therefore we introduce free with amalgamation operators that are suitable for computing the spectral gap limit of our operator in high dimensions. We then approximate the spectral radius of the corresponding limit operator and finally give an upper bound for the spectral radius of the finite-dimensional operator. In particular, we show that if the deterministic matrix underlying graph is an expander, then the underlying Markov chain associated to the sum with a random permutation is again an expander.
We consider a one-dimensional random walk Sn with i.i.d. increments with zero mean and finite variance. We derive asymptotic expansions for the tail distribution P(tau(x) > n) of the first passage times tau x := inf{n > 1: x + Sn <= 0}, x > 0. We also derive asymptotic expansion for local probabilities P(S-n = x, tau(0) > n). Studying the asymptotic expansions we obtain a sequence of discrete polyharmonic functions and obtain analogues of renewal theorem for them.
Let W-n, n E N, be the Wishart matrix associated to an n & times; d(n) random matrix F-n with independent rows (F-n)i, i = 1,.. . , n, whose components are in the q(i)-th, q(i) E N, Wiener chaos of an isonormal Gaussian process X-i. In the case when the random matrix F(n )is "tall", i.e., d(n) = o(n), as n ->infinity, we provide sufficient conditions on the kernels and the orders of the Wiener chaoses which ensure that for any epsilon > 0 (small enough) there exists n(epsilon) such that P(||W-n - Id(n)||op <= epsilon) > L-n,L-epsilon for any n > n epsilon, and L-n,L-epsilon -> 1, as n ->infinity. Here Id(n) denotes the d(n) & times; d(n) identity matrix and the symbol || * ||(op) denotes the operator norm of a square matrix. We explicitly compute the lower bound L-n,L-epsilon in terms of n, d(n) and epsilon. As a by-product, we get that the random matrix F-n/\/n is an approximate isometry with high probability, as n ->infinity. We apply the main result to a covariance estimation problem, which is of potential interest in data analysis. The theory is illustrated by means of matrices F-n whose entries are suitable stochastic integrals of Ornstein-Uhlenbeck processes.
We introduce a random barrier to a supercritical branching random walk in an i.i.d. random environment {L-n} indexed by time n, i.e., in each generation, only the individuals born below the barrier can survive and reproduce. At generation n (n is an element of N), the barrier is given by chi(n )+ epsilon n, where {chi n} is a random walk determined by the random environment. Lv & Hong (2024) showed that for almost every L :={L-n}, the quenched survival probability (denoted by (lL)(epsilon)) of the particle system will be 0 (resp., positive) when epsilon <= 0 (resp., epsilon > 0). In the present paper, we prove that root epsilon log lL(epsilon) will converge in Probability/ almost surely/ in L-p to an explicit negative constant (depending on the environment) as epsilon down arrow 0 under some integrability conditions respectively. This result extends the scope of the result of Gantert et al. (2011) to the random environment case.
We provide a complete theory of reflected backward stochastic differential equations driven by G-Brownian motion (reflected G-BSDEs), suggested in Li et al. (2018) [15]. We put forward a new definition of reflected G-BSDEs by introducing a non-Skorohodtype minimality condition. Unlike Li et al. (2018) [15], the alternative formulation of this paper is nice in the sense that it is a natural extension of the definitions of standard G-BSDEs as well as reflected BSDEs. We then prove the existence, uniqueness and some properties of solution under Lipschitz-type assumptions on generator. We also provide some applications including probabilistic interpretation of obstacle problems for fully nonlinear PDEs and super-hedging strategy for American options in nonlinear markets with volatility uncretainty. The key argument is the dynamic programming approach, analyzed in the accompanying paper (O et al. (2023) [22]), and we develop it to the G-expectation framework. Our approach is completely different from the penalization method in Li et al. (2018) [15].
We study the expected transition frequency between the two metastable states of a stochastic wave equation with double-well potential. By transition state theory, the frequency factorizes into two components: one depends only on the invariant measure, given by the phi 4d quantum field theory, and the other takes the dynamics into account. We compute the first component with the variational approach to stochastic quantization when d = 2, 3. For the two-dimensional equation with random data but no stochastic forcing, we also compute the transmission coefficient leading to an Eyring-Kramers law.
We study the l infinity -> l infinity operator norm of products of independent random matrices with independent and identically distributed entries. For n-by-n matrices whose entries are centered, have unit variance, and have a finite moment of order 4 alpha for some alpha > 1, we find that the operator norm of the product of p matrices behaves asymptotically like n (P+1 /2) root 2/pi. The case of products of possibly non-square matrices with possibly non-centered entries is also covered.
We consider a Markovian growth process on a partially ordered set Λ, equivalent to last passage percolation (LPP) with independent (not necessarily identical) exponentially distributed weights on the elements of Λ. Such a process includes inhomogeneous exponential LPP on the Euclidean lattice ℕ_0^d. We give non-asymptotic bounds on the mean and variance, as well as higher, central, and exponential moments of the passage time τ_A to grow any set A ⊆ Λ in terms of characteristics of A. We also give a limit shape theorem when Λ is equipped with a monoid structure. Methods involve making use of the backward equation associated to the Markovian evolution and comparison inequalities with respect to the time-reversed generator.
In this article, we prove the Quantitative Central Limit Theorem (QCLT) for the spatial average of the solution of the nonlinear stochastic heat equation with constant initial condition, driven by space-time Gaussian white noise in dimension 1. The novelty is that the equation contains a drift term. We assume that the drift and diffusion coefficients are twice differentiable with bounded first and second order derivatives. For the proof, we use Malliavin calculus, and the second-order Poincaré inequality due to Vidotto (2020). To estimate the moment of the second Malliavin derivative of the solution, we develop a novel estimate for the product of two heat kernels, which is of independent interest. Finally, we provide the functional result corresponding to this CLT.
In this paper, we proved moderate deviation principles for a fully coupled two-time-scale stochastic systems, where the slow process is given by stochastic differential equations with small noise, while the fast process is a rapidly changing purely jump process on finite state space. The system is fully coupled in that the drift and diffusion coefficients of the slow process, as well as the jump distribution of the fast process, depend on states of both processes. Moreover, the diffusion component in the slow process can be degenerate, and the drift is allowed to have polynomial growth. Our approach is based on the combination of the weak convergence method from [A. Budhiraja, P. Dupuis, and A. Ganguly, Electron. J. Probab. 23 (2018), pp. 1-33; Ann. Probab. 44 (2016), pp. 1723-1775] with Poisson equation for the fast-varying purely jump process.
In this paper, we investigate the number of real zeros of random Weyl polynomials of degree \(n \to \infty\) with general coefficient distributions. Motivated by the results of arXiv:1409.4128 and arXiv:1402.4628 as well as arXiv:1711.03316 and arXiv:1912.11901, we determine how the expected number of real zeros and their variance, over various natural intervals, depend on the moments of the common coefficient distribution. Our main finding is that while the first-order asymptotic of the expectation is universal, the next-order correction depends on the third and fourth moments of the distribution, and may grow linearly with \(\log n\), depending on the interval under consideration. In contrast, for the variance we show that the leading-order term is universal, which differs from the behavior observed for random trigonometric polynomials in arXiv:1711.03316 and arXiv:1912.11901. Our approach relies on an Edgeworth expansion for random walks arising from Weyl polynomials, a result of independent interest.
We develop a Malliavin calculus for nonlinear Hawkes processes in the sense of Carlen and Pardoux. This approach, based on perturbations of the jump times of the process, enables the construction of a local Dirichlet form. As an application, we establish criteria for the absolute continuity of solutions to stochastic differential equations driven by Hawkes processes. We also derive sensitivity formulas for the valuation of financial derivatives with respect to model parameters.
We prove that the probability the cluster of the origin in a subcritical Poisson random connection model (RCM) has size at least $n$ decays exponentially as $n$ increases, under minimal assumptions. We extend a recent method of Vanneuville (arXiv:2304.12110) from Bernoulli percolation on vertex-transitive graphs to the RCM. The key idea is that the subcritical RCM can be constructed by site percolation on a very high-intensity RCM. The latter RCM becomes ``almost vertex-transitive'' in a certain sense at very high intensities, which is a new method that we expect to be useful for other problems. We obtain the result for connection functions with unbounded support, a setting in which it was not previously known.
