Related papers: On central limit theorems for power variations of …
In this article, we investigate the asymptotic behavior of the solution to a one-dimensional stochastic heat equation with random nonlinear term generated by a stationary, ergodic random field. We extend the well-known central limit theorem…
We consider the one-dimensional stochastic heat equation driven by a multiplicative space-time white noise. We show that the spatial integral of the solution from $-R$ to $R$ converges in total variance distance to a standard normal…
In this article, we study the asymptotic behavior of the stochastic heat equation for large times.
We establish central limit theorems for general functionals on binomial point processes and their Poissonized version. As an application, a central limit theorem for Betti numbers of random geometric complexes in the thermodynamic regime is…
We investigate the rate of convergence in the central limit theorem for convex sets. We obtain bounds with a power-law dependence on the dimension. These bounds are asymptotically better than the logarithmic estimates which follow from the…
We prove a variant of the central limit theorem (CLT) for a sequence of i.i.d. random variables $\xi_j$, perturbed by a stochastic sequence of linear transformations $A_j$, representing the model uncertainty. The limit, corresponding to a…
We prove a central limit theorem for a sequence of random variables whose means are ambiguous and vary in an unstructured way. Their joint distribution is described by a set of measures. The limit is (not the normal distribution and is)…
We show that a large class of stochastic heat equations can be approximated by systems of interacting stochastic differential equations. As a consequence, we prove various comparison principles extending earlier results. Among other things,…
A central limit theorem is established for a sum of random variables belonging to a sequence of random fields. The fields are assumed to have zero mean conditional on the past history and to satisfy certain conditional $\alpha$-mixing…
This work concerns the limit behavior of the quartic variation (i.e., the power variation of order four) with respect to the time variable of the solution to the semilinear stochastic heat equation with space-time white noise. In a first…
We consider the solution to a stochastic heat equation. This solution is a random function of time and space. For a fixed point in space, the resulting random function of time, $F(t)$, has a nontrivial quartic variation. This process,…
Motivated by problems from statistical analysis for discretely sampled SPDEs, first we derive central limit theorems for higher order finite differences applied to stochastic process with arbitrary finitely regular paths. These results are…
We study the existence of global-in-time solutions for a nonlinear heat equation with nonlocal diffusion, power nonlinearity and suitably small data (either compared pointwisely to the singular solution or in the norm of a critical Morrey…
This article extends the work on stochastic constrained heat equation in \cite{brzezniak2020global}. We will show the existence of Martingale solutions to the stochastic-constrained heat equations. The proof is based on compactness,…
The energetics of the stochastic process has shown the balance of energy on the mesoscopic level. The heat and the energy defined there are, however, generally different from their macroscopic counterpart. We show that this discrepancy can…
These notes focus on the applications of the stochastic Taylor expansion of solutions of stochastic differential equations to the study of heat kernels in small times. As an illustration of these methods we provide a new heat kernel proof…
The long-time asymptotics of solutions of the Cauchy problem for the heat equation are constructed in the case when the initial function at infinity has power asymptotics.
We study the stochastic heat flow with constant initial data and analyze its spatial average on the scale of $\varepsilon\ll1$. We prove that the logarithm of the averaged process satisfies a pointwise central limit theorem: After being…
In this paper we investigate the asymptotic behavior and decay of the solution of the discrete in time $N$-dimensional heat equation. We give a convergence rate with which the solution tends to the discrete fundamental solution, and the…
We consider the stochastic heat equation which includes a fractional power of the Laplacian of order $\alpha \in (1, 2]$ and it is driven by a nonlinear space-time Gaussian white noise. We study two types of power variations for the…