Stochastics and Partial Differential Equations-Analysis and Computations最新文献

In this work, we are interested in establishing weak and strong well-posedness for McKean–Vlasov SDEs with additive stable noise and a convolution type non-linear drift with singular interaction kernel in the framework of Lebesgue–Besov spaces. We prove that the well-posedness of the system holds for the thresholds (in terms of regularity indexes) deriving from the scaling of the noise and that the corresponding SDE can be understood in the classical sense. Especially, we characterize quantitatively how the non-linearity allows to go beyond the stronger thresholds, coming from Bony’s paraproduct rule, usually obtained for linear SDEs with singular interaction kernels. We also specifically characterize in function of the stability index of the driving noise and the parameters of the drift when the dichotomy between weak and strong uniqueness occurs.

Pathwise uniqueness is established for a class of one-dimensional stochastic Volterra equations driven by Brownian motion with singular kernels and Hölder continuous diffusion coefficients. Consequently, the existence of unique strong solutions is obtained for this class of stochastic Volterra equations.

We consider stochastic PDEs on the d-dimensional torus with fractional Laplacian of parameter (rho in (0,2]), quadratic nonlinearity and driven by space-time white noise. These equations are known to be locally subcritical, and thus amenable to the theory of regularity structures, if and only if (rho > d/3). Using a series of recent results by the second named author, A. Chandra, I. Chevyrev, M. Hairer and L. Zambotti, we obtain precise asymptotics on the renormalisation counterterms as the mollification parameter (varepsilon ) becomes small and (rho ) approaches its critical value. In particular, we show that the counterterms behave like a negative power of (varepsilon ) if (varepsilon ) is superexponentially small in ((rho -d/3)), and are otherwise of order (log (varepsilon ^{-1})). This work also serves as an illustration of the general theory of BPHZ renormalisation in a relatively simple situation.

In this paper, we investigate the long-term dynamics of fractional stochastic delay reaction-diffusion equations on unbounded domains with a polynomial drift term of arbitrary order driven by nonlinear noise. We first define a mean random dynamical system in a Hilbert space for the solutions of the equation and prove the existence and uniqueness of weak pullback mean random attractors. We then establish the existence and regularity of invariant measures of the system under further conditions on the nonlinear delay and diffusion terms. We also prove the tightness of the set of all invariant measures of the equation when the time delay varies in a bounded interval. We finally show that every limit of a sequence of invariant measures of the delay equation must be an invariant measure of the limiting system as delay approaches zero. The uniform tail-estimates and the Ascoli–Arzelà theorem are used to derive the tightness of distribution laws of solutions in order to overcome the non-compactness of Sobolev embeddings on unbounded domains.

A stochastic SIR epidemic model taking into account the heterogeneity of the spatial environment is constructed. The deterministic model is given by a partial differential equation and the stochastic one by a space-time jump Markov process. The consistency of the two models is given by a law of large numbers. In this paper, we study the deviation of the spatial stochastic model from the deterministic model by a functional central limit theorem. The limit is a distribution-valued Ornstein–Uhlenbeck Gaussian process, which is the mild solution of a stochastic partial differential equation.

We present two approaches to establish the exponential decay of correlation functions of Euclidean quantum field theories (EQFTs) via stochastic quantization (SQ). In particular we consider the elliptic stochastic quantization of the Høegh–Krohn (or (exp (alpha phi )_2)) EQFT in two dimensions. The first method is based on a path-wise coupling argument and PDE apriori estimates, while the second on estimates of the Malliavin derivative of the solution to the SQ equation.

We study the fully degenerate second-order evolution equation

given with the zero initial data. Here (a^{ij}(t)), (b^i(t)), c(t) are merely locally integrable functions, and ((a^{ij}(t))_{d times d}) is a nonnegative symmetric matrix with the smallest eigenvalue (delta (t)ge 0). We show that there is a positive constant N such that

where (p,q in (1,infty )), (alpha (t)=int _0^t delta (s)ds), and w is Muckenhoupt’s weight.

This paper establishes (L_p)-solvability for stochastic time fractional Burgers’ equations driven by multiplicative space-time white noise:

where (alpha in (0,1)), (beta < 3alpha /4+1/2), and (d< 4--2(2beta -1)_+/alpha ). The operators (partial _t^alpha ) and (partial _t^beta ) are the Caputo fractional derivatives of order (alpha ) and (beta ), respectively. The process (W_t) is an (L_2(mathbb {R}^d))-valued cylindrical Wiener process, and the coefficients (a^{ij}, b^i, c, {bar{b}}^{i}) and (sigma (u)) are random. In addition to the uniqueness and existence of a solution, the Hölder regularity of the solution is also established. For example, for any constant (T<infty ), small (varepsilon >0), and almost sure (omega in varOmega ),

and

The Hölder regularity of the solution in time changes behavior at (beta = 1/2). Furthermore, if (beta ge 1/2), then the Hölder regularity of the solution in time is (alpha /2) times that in space.

Let (U={U(t,x), (t,x)in mathring{{mathbb {R}}}_+times {mathbb {R}}^d}) and (partial _{x}U={partial _{x}U(t,x), (t,x)in mathring{{mathbb {R}}}_+times {mathbb {R}}}) be the solution and gradient solution of the fourth order linearized Kuramoto–Sivashinsky (L-KS) SPDE, driven by the space-time white noise in one-to-three dimensional spaces, respectively. We use the underlying explicit kernels to prove the exact global temporal moduli and temporal LILs for the L-KS SPDEs and gradient, and utilize them to prove that the sets of temporal fast points where exceptional oscillation of (U(cdot ,x)) and (partial _{x}U(cdot ,x)) occur infinitely often are random fractals, and evaluate their Hausdorff dimensions and their hitting probabilities. It has been confirmed that these points of (U(cdot ,x)) and (partial _{x}U(cdot ,x)), in time, are everywhere dense with power of the continuum almost surely, and their hitting probabilities are determined by the packing dimension (dim _{_{p}}(B)) of the target set B.

We consider a nonlinear SPDE approximation of the Dean–Kawasaki equation for independent particles. Our approximation satisfies the physical constraints of the particle system, i.e. its solution is a probability measure for all times (preservation of positivity and mass conservation). Using a duality argument, we prove that the weak error between particle system and nonlinear SPDE is of the order (N^{-1-1/(d/2+1)}log N). Along the way we show well-posedness, a comparison principle, and an entropy estimate for a class of nonlinear regularized Dean–Kawasaki equations with Itô noise.