We define linear stochastic heat equations (SHE) on p.c.f.s.s. sets equipped with regular harmonic structures. We show that if the spectral dimension of the set is less than two, then function-valued "random-field" solutions to these SPDEs exist and are jointly Hölder continuous in space and time. We calculate the respective Hölder exponents, which extend the well-known results on the Hölder exponents of the solution to SHE on the unit interval. This shows that the "curse of dimensionality" of the SHE on $\mathbb{R}^n$ depends not on the geometric dimension of the ambient space but on the analytic properties of the operator through the spectral dimension. To prove these results we establish generic continuity theorems for stochastic processes indexed by these p.c.f.s.s. sets that are analogous to Kolmogorov's continuity theorem. We also investigate the long-time behaviour of the solutions to the fractal SHEs.
A stochastic heat equation on an unbounded nested fractal driven by a general stochastic measure is investigated. Existence, uniqueness and continuity of the mild solution are proved provided that the spectral dimension of the fractal is less than 4/3.
We study stochastic heat equations in the sense of Walsh defined by fractal Laplacians on Cantor-like sets. For this purpose, we first investigate the corresponding heat kernels. Then, we prove existence and uniqueness of mild solutions to stochastic heat equations provided some Lipschitz and linear growth conditions. We establish Hölder continuity in space and time and compute the Hölder exponents. Moreover, we address the question of weak intermittency.
This paper provides explicit pointwise formulas for the heat kernel on compact metric measure spaces that belong to a $(\mathbb{N}\times\mathbb{N})$-parameter family of fractals which are regarded as projective limits of metric measure graphs and do not satisfy the volume doubling property. The formulas are applied to obtain uniform continuity estimates of the heat kernel and to derive an expression of the fundamental solution of the free Schrödinger equation. The results also open up the possibility to approach infinite dimensional spaces based on this model.
We study the spectral heat content for a class of open sets with fractal boundaries determined by similitudes in $\mathbb{R}^{d}$, $d\geq 1$, with respect to subordinate killed Brownian motions via $α/2$-stable subordinators and establish the asymptotic behavior of the spectral heat content as $t\to 0$ for the full range of $α\in (0,2)$. Our main theorems show that these asymptotic behaviors depend on whether the sequence of logarithms of the coefficients of the similitudes is arithmetic when $α\in [d-\b,2)$, where $\b$ is the interior Minkowski dimension of the boundary of the open set. The m…
The main result of the present paper is a statement on existence, uniqueness and regularity for mild solutions to a parabolic transport diffusion type equation that involves a non-smooth coefficient. We investigate related Cauchy problems on bounded smooth domains with Dirichlet boundary conditions by means of semigroup theory and fixed point arguments. Main ingredients are the definition of a product of a function and a (not too irregular) distribution as well as a corresponding norm estimate. As an application, transport stochastic partial differential equations driven by fractional Brownian…
We consider on-diagonal heat kernel estimates and the laws of the iterated logarithms for a switch-walk-switch random walk on a lamplighter graph under the condition that the random walk on the underlying graph enjoys sub-Gaussian heat kernel estimates.
We consider the linear stochastic heat and wave equations with generalized Gaussian noise that is white in time and spatially correlated. Under the assumption that the homogeneous spatial correlation $f$ satisfies some mild conditions, we show that the solutions to the linear stochastic heat and wave equations exhibit tall peaks in macroscopic scales, which means they are macroscopically multi-fractal. We compute the macroscopic Hausdorff dimension of the peaks for Gaussian random fields with vanishing correlation and then apply this result to the solution of the linear stochastic heat and wav…
This paper proves the strong parabolic Harnack inequality for local weak solutions to the heat equation associated with time-dependent (nonsymmetric) bilinear forms. The underlying metric measure Dirichlet space is assumed to satisfy the volume doubling condition, the strong Poincaré inequality, and a cutoff Sobolev inequality. The metric is not required to be geodesic. Further results include a weighted Poincaré inequality, as well as upper and lower bounds for non-symmetric heat kernels.
This paper presents a detailed analysis of the heat kernel on an $(\mathbb{N}\times\mathbb{N})$-parameter family of compact metric measure spaces, which do not satisfy the volume doubling property. In particular, uniform bounds of the heat kernel and its Lipschitz continuity, as well as the continuity of the corresponding heat semigroup are studied; a specific example is presented revealing a logarithmic correction. The estimates are further applied to derive several functional inequalities of interest in describing the convergence to equilibrium of the diffusion process.
We survey some of our recent results on existence, uniqueness and regularity of function solutions to parabolic and transport type partial differential equations driven by non-differentiable noises. When applied pathwise to random situations, they provide corresponding statements for stochastic partial differential equations driven by fractional noises of sufficiently high regularity order. The approach is based on semigroup theory.
We study the stochastic heat equation driven by an additive infinite dimensional fractional Brownian noise on the unit sphere $\mathbb{S}^{2}$. The existence and uniqueness of its solution in certain Sobolev space is investigated and sample path regularity properties are established. In particular, the exact uniform modulus of continuity of the solution in time/spatial variable is derived.
We prove the existence of density for the solution to the multiplicative semilinear stochastic heat equation on an unbounded spatial domain, with drift term satisfying a half-Lipschitz type condition. The methodology is based on a careful analysis of differentiability for a map defined on weighted functional spaces.
We study the smoothness of the density of a semilinear heat equation with multiplicative spacetime white noise. Using Malliavin calculus, we reduce the problem to a question of negative moments of solutions of a linear heat equation with multiplicative white noise. Then we settle this question by proving that solutions to the linear equation have negative moments of all orders.
In this paper, we study the stochastic heat equation with a general multiplicative Gaussian noise that is white in time and colored in space. Both regularity and strict positivity of the densities of the solution have been established. The difficulty, and hence the contribution, of the paper lie in three aspects, which include rough initial conditions, degenerate diffusion coefficient, and weakest possible assumptions on the correlation function of the noise. In particular, our results cover the parabolic Anderson model starting from a Dirac delta initial measure.
We study the smoothness of the density of the solution to the nonlinear heat equation u_t=Lu(t,x)+σ(u(t,x))W on a torus with a periodic boundary condition, where L is the generator of a Levy process on the torus, and W is white noise. We use Malliavin calculus techniques to show that the law of the solution has a density with respect to the Lebesgue measure for all t >0 and x in R.
We consider a system of $d$ linear stochastic heat equations driven by an additive infinite-dimensional fractional Brownian noise on the unit circle $S^1$. We obtain sharp results on the Hölder continuity in time of the paths of the solution $u=\{u(t, x)\}_{t \in \mathbb{R}_+, x \in S^1}$. We then establish upper and lower bounds on hitting probabilities of $u$, in terms of respectively Hausdorff measure and Newtonian capacity.
We prove a scale-invariant boundary Harnack principle for inner uni- form domains in metric measure Dirichlet spaces. We assume that the Dirichlet form is symmetric, strongly local, regular, and that the volume doubling property and two-sided sub-Gaussian heat kernel bounds are satisfied. We make no assumptions on the pseudo-metric induced by the Dirichlet form, hence the underlying space can be a fractal space.
We establish existence, uniqueness and higher order weighted $L_p$-Sobolev regularity for the stochastic heat equation with zero Dirichlet boundary condition on angular domains and on polygonal domains in $\mathbb{R}^2$. We use a system of mixed weights consisting of appropriate powers of the distance to the vertexes and of the distance to the boundary to measure the regularity with respect to the space variable. In this way we can capture the influence of both main sources for singularities: the incompatibility between noise and boundary condition on the one hand and the singularities of the …
We provide conditions which yield a strong law of large numbers for expressions of the form $1/N\sum_{n=1}^{N}F\big(X(q_1(n)),..., X(q_\ell(n))\big)$ where $X(n),n\geq 0$'s is a sufficiently fast mixing vector process with some moment conditions and stationarity properties, $F$ is a continuous function with polinomial growth and certain regularity properties and $q_i,i>m$ are positive functions taking on integer values on integers with some growth conditions. Applying these results we study certain multifractal formalism type questions concerning Hausdorff dimensions of some sets of numbers wi…
We investigate the Hölder continuity of solutions to stochastic partial differential equations of the form $\frac{\partial u}{\partial t}=\mathcal{L}u+σ(u)\dot{F}$, subject to a suitable initial condition. The noise term $\dot{F}$ is white in time, colored in space, and $\mathcal{L}$ is the $\mathcal{L}^{2}$-generator of a Lévy process. Under a growth assumption on the characteristic exponent of the Lévy process, we derive sufficient conditions for the solution to be locally Hölder continuous. Moreover, we show that these conditions are equivalent to those derived in related papers by Khoshnev…
In this paper, we study the following stochastic heat equation \[ \partial_tu=\mathcal{L} u(t,x)+\dot{B},\quad u(0,x)=0,\quad 0\le t\le T,\quad x\in\mathbb{R}d, \] where $\mathcal{L}$ is the generator of a Lévy process $X$ taking value in $\mathbb{R}^d$, $B$ is a fractional-colored Gaussian noise with Hurst index $H\in\left(\frac12,\,1\right)$ for the time variable and spatial covariance function $f$ which is the Fourier transform of a tempered measure $μ.$ After establishing the existence of solution for the stochastic heat equation, we study the regularity of the solution $\{u(t,x),\, t\ge 0…
A solution of the heat equation with a distribution-valued potential is constructed by regularization. When the potential is the generalized derivative of a Hölder continuous function, regularity of the resulting solution is in line with the standard parabolic theory.
We prove the existence of the reflected diffusion on a complex of an arbitrary size for a large class of planar simple nested fractals. Such a process is obtained as a folding projection of the free Brownian motion from the unbounded fractal. We give sharp necessary geometric conditions on the fractal under which this projection can be well defined. They are illustrated by various specific examples. We first construct a proper version of the transition probability densities for reflected process and we prove that it is a continuous, bounded and symmetric function which satisfies the Chapman-Ko…
In this paper, we establish a necessary and sufficient condition for the existence and regularity of the density of the solution to a semilinear stochastic (fractional) heat equation with measure-valued initial conditions. Under a mild cone condition for the diffusion coefficient, we establish the smooth joint density at multiple points. The tool we use is Malliavin calculus. The main ingredient is to prove that the solutions to a related stochastic partial differential equation have negative moments of all orders. Because we cannot prove $u(t,x)\in \mathbb{D}^\infty$ for measure-valued initia…
Even though the heat equation with random potential is a well-studied object, the particular case of time-independent Gaussian white noise in one space dimension has yet to receive the attention it deserves. The paper investigates the stochastic heat equation with space-only Gaussian white noise on a bounded interval. The main result is that the space-time regularity of the solution is the same for additive noise and for multiplicative noise in the Wick-Itô-Skorokhod interpretation.
We initiate in this paper the study of analytic properties of the Liouville heat kernel. In particular, we establish regularity estimates on the heat kernel and derive non trivial lower and upper bounds.
Sharp Besov regularities in time and space variables are investigated for $\left(u(t,x),\; t\in [0,T],\; x\in \mathbb{R}\right)$, the mild solution to the stochastic heat equation driven by space-time white noise. Existence, Hölder continuity, and Besov regularity of local times are established for $u(t,x)$ viewed either as a process in the space variable or time variable. Hausdorff dimensions of their corresponding level sets are also obtained.
We prove the existence and establish the Lifschitz singularity of the integrated density of states for certain random Hamiltonians $H^ω=H_0+V^ω$ on fractal spaces of infinite diameter. The kinetic term $H_0$ is given by $ϕ(-\mathcal L),$ where $\mathcal L$ is the Laplacian on the fractal and $ϕ$ is a completely monotone function satisfying some mild regularity conditions. The random potential $V^ω$ is of alloy-type.
In this article, we consider the problem of estimating the heat kernel on measure-metric spaces equipped with a resistance form. Such spaces admit a corresponding resistance metric that reflects the conductivity properties of the set. In this situation, it has been proved that when there is uniform polynomial volume growth with respect to the resistance metric the behaviour of the on-diagonal part of the heat kernel is completely determined by this rate of volume growth. However, recent results have shown that for certain random fractal sets, there are global and local (point-wise) fluctuation…
Let $D$ be a domain in $R^d$ and $u$ be the solution to the stochastic heat equation $$ du=Δu dt+ g\,dW_t, \quad t>0, x\in D, $$ with zero initial and boundary data. Here $W_t$ is a one-dimensional Wiener process on a probability space $Ω$. It has been proved (see below for references) that for any $p\geq 2$ the inequality $$ \|\nabla u\|_{L_p(Ω\times [0,T]\times D)} \leq c \|g\|_{L_p(Ω\times [0,T]\times D)} $$ holds if $\partial D\in C^1$. In this note we prove that if $p>4$ then this inequality fails in any polygon in $R^2$ having an angle greater than or equal to $\frac{pπ}{2(p-2)}$. We als…
We study space-time regularity of the solution of the nonlinear stochastic heat equation in one spatial dimension driven by space-time white noise, with a rough initial condition. This initial condition is a locally finite measure $μ$ with, possibly, exponentially growing tails. We show how this regularity depends, in a neighborhood of $t=0$, on the regularity of the initial condition. On compact sets in which $t>0$, the classical Hölder-continuity exponents $\frac{1}{4}-$ in time and $\frac{1}{2}-$ in space remain valid. However, on compact sets that include $t=0$, the Hölder continuity of th…
We use the formalism of Hairer's regularity structures theory \cite{hai-14} to study a heat equation with non-linear perturbation driven by a space-time fractional noise. Different regimes are observed, depending on the global pathwise roughness of the noise. To this end, and following the procedure exhibited in \cite{hai-14}, the equation is first "lifted" into some abstract "regularity structure" and therein solved through a fixed-point argument. Then we construct a consistent "model" above the fractional noise, by relying on a smooth Fourier-type approximation of the process.
We present a concrete family of fractals, which we call the (two-dimensional) thin scale irregular Sierpiński gaskets and each of which is equipped with a canonical strongly local regular symmetric Dirichlet form. We prove that any fractal $K$ in this family satisfies the full off-diagonal heat kernel estimates with some space-time scale function $Ψ_{K}$ and the singularity of the associated energy measures with respect to the canonical volume measure (uniform distribution) on $K$, and also that the decay rate of $r^{-2}Ψ_{K}(r)$ to $0$ as $r\downarrow 0$ can be made arbitrarily slow by suitab…
Fine regularity of stochastic processes is usually measured in a local way by local Hölder exponents and in a global way by fractal dimensions. Following a previous work of Adler, we connect these two concepts for multiparameter Gaussian random fields. More precisely, we prove that almost surely the Hausdorff dimensions of the range and the graph in any ball $B(t_0,ρ)$ are bounded from above using the local Hölder exponent at $t_0$. We define the deterministic local sub-exponent of Gaussian processes, which allows to obtain an almost sure lower bound for these dimensions. Moreover, the Hausdor…
The family of $V$-variable fractals provides a means of interpolating between two families of random fractals previously considered in the literature; scale irregular fractals ($V=1$) and random recursive fractals ($V=\infty$). We consider a class of $V$-variable affine nested fractals based on the Sierpinski gasket with a general class of measures. We calculate the spectral exponent for a general measure and find the spectral dimension for these fractals. We show that the spectral properties and on-diagonal heat kernel estimates for $V$-variable fractals are closer to those of scale irregular…
We study the time regularity of local weak solutions of the heat equation in the context of local regular symmetric Dirichlet spaces. Under two basic and rather minimal assumptions, namely, the existence of certain cut-off functions and a very weak $L^2$ Gaussian type upper-bound for the heat semigroup, we prove that the time derivatives of a local weak solution of the heat equation are themselves local weak solutions. This applies, for instance, to local weak solutions of parabolic equations with uniformly elliptic symmetric divergence form second order operators with measurable coefficients.…
Bergman-space regularity for the heat equation with white-noise boundary forcing
University of Craiova（克赖奥瓦大学）；Gheorghe Mihoc-Caius Iacob Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy（罗马尼亚科学院格奥尔基·米胡克-卡尤斯·雅各布数学统计与应用数学研究所）；Institut de Mathématiques de Bordeaux, UMR 5251, Université de Bordeaux/Bordeaux INP/CNRS（波尔多数学研究所，UMR 5251，波尔多大学/波尔多国立理工学院/CNRS）；Institut Universitaire de France (IUF)（法国高等研究院）
With a view toward fractal spaces, by using a Korevaar-Schoen space approach, we introduce the class of bounded variation (BV) functions in a general framework of strongly local Dirichlet spaces with a heat kernel satisfying sub-Gaussian estimates. Under a weak Bakry-Émery curvature type condition, which is new in this setting, this BV class is identified with a heat semigroup based Besov class. As a consequence of this identification, properties of BV functions and associated BV measures are studied in detail. In particular, we prove co-area formulas, global $L^1$ Sobolev embeddings and isope…
This article studies potential theory and spectral analysis on compact metric spaces, which we refer to as fractal quantum graphs. These spaces can be represented as a (possibly infinite) union of 1-dimensional intervals and a totally disconnected (possibly uncountable) compact set, which roughly speaking represents the set of junction points. Classical quantum graphs and fractal spaces such as the Hanoi attractor are included among them. We begin with proving the existence of a resistance form on the Hanoi attractor, and go on to establish heat kernel estimates and upper and lower bounds on t…
