In this paper, we study intermittency properties for various stochastic PDEs with varieties of space time Gaussian noises via matching upper and lower moment bounds of the solution. Due to the absence of the powerful Feynman Kac formula, the lower moment bounds have been missing for many interesting equations except for the stochastic heat equation. This work introduces and explores the Feynman diagram formula for the moments of the solution and the small ball nondegeneracy for the Green's function to obtain the lower bounds for all moments which match the upper moment bounds. Our upper and lower moments are valid for various interesting equations, including stochastic heat equations, stochastic wave equations, stochastic heat equations with fractional Laplcians, and stochastic diffusions which are both fractional in time and in space.
Let $ξ$ denote space-time white noise, and consider the following stochastic partial differential equations: (i) $\dot{u}=\frac{1}{2} u" + uξ$, started identically at one; and (ii) $\dot{Z}=\frac12 Z" + ξ$, started identically at zero. It is well known that the solution to (i) is intermittent, whereas the solution to (ii) is not. And the two equations are known to be in different universality classes. We prove that the tall peaks of both systems are multifractals in a natural large-scale sense. Some of this work is extended to also establish the multifractal behavior of the peaks of stochastic…
This paper studies the stochastic heat equation driven by time fractional Gaussian noise with Hurst parameter $H\in(0,1/2)$. We establish the Feynman-Kac representation of the solution and use this representation to obtain matching lower and upper bounds for the $L^p(Ω)$ moments of the solution.
The goal of the present note is to study intermittency properties for the solution to the fractional heat equation $$\frac{\partial u}{\partial t}(t,x) = -(-Δ)^{β/2} u(t,x) + u(t,x)\dot{W}(t,x), \quad t>0,x \in \bR^d$$ with initial condition bounded above and below, where $β\in (0,2]$ and the noise $W$ behaves in time like a fractional Brownian motion of index $H>1/2$, and has a spatial covariance given by the Riesz kernel of index $α\in (0,d)$. As a by-product, we obtain that the necessary and sufficient condition for the existence of the solution is $α<β$.
We obtain existence results for the solution u of nonlocal semilinear parabolic PDEs on $\mathbb{R}^d$ with polynomial nonlinearities in $(u, \nabla u)$, using a tree-based probabilistic representation. This probabilistic representation applies to the solution of the equation itself, as well as to its partial derivatives by associating one of d marks to the initial tree branch. Partial derivatives are dealt with by integration by parts and subordination of Brownian motion. Numerical illustrations are provided in examples for the fractional Laplacian in dimension up to 10, and for the fractiona…
We study the nonlinear fractional stochastic heat equation in the spatial domain $\mathbb{R}$ driven by space-time white noise. The initial condition is taken to be a measure on $\mathbb{R}$, such as the Dirac delta function, but this measure may also have non-compact support. Existence and uniqueness, as well as upper and lower bounds on all $p$-th moments $(p\ge 2)$, are obtained. These bounds are uniform in the spatial variable, which answers an open problem mentioned in Conus and Khoshnevisan [9]. We improve the weak intermittency statement by Foondun and Khoshnevisan [14], and we show tha…
We study the nonlinear stochastic time-fractional diffusion equations in the spatial domain $\mathbb{R}$, driven by multiplicative space-time white noise. The fractional index $β$ varies continuously from $0$ to $2$. The case $β=1$ (resp. $β=2$) corresponds to the stochastic heat (resp. wave) equation. The cases $β\in \:]0,1[\:$ and $β\in \:]1,2[\:$ are called {\it slow diffusion equations} and {\it fast diffusion equations}, respectively. Existence and uniqueness of random field solutions with measure-valued initial data, such as the Dirac delta measure, are established. Upper bounds on all $…
We consider time fractional stochastic heat type equation $$\partial^β_tu_t(x)=-ν(-Δ)^{α/2} u_t(x)+I^{1-β}_t[σ(u)\stackrel{\cdot}{W}(t,x)]$$ in $(d+1)$ dimensions, where $ν>0$, $β\in (0,1)$, $α\in (0,2]$, $d<\min\{2,β^{-1}\}\a$, $\partial^β_t$ is the Caputo fractional derivative, $-(-Δ)^{α/2} $ is the generator of an isotropic stable process, $\stackrel{\cdot}{W}(t,x)$ is space-time white noise, and $σ:\R \to\RR{R}$ is Lipschitz continuous. Mijena and Nane proved in \cite{JebesaAndNane1} that : (i) absolute moments of the solutions of this equation grows exponentially; and (ii) the distances t…
In this article, we consider the stochastic wave and heat equations driven by a Gaussian noise which is spatially homogeneous and behaves in time like a fractional Brownian motion with Hurst index $H>1/2$. The solutions of these equations are interpreted in the Skorohod sense. Using Malliavin calculus techniques, we obtain an upper bound for the moments of order $p\geq2$ of the solution. In the case of the wave equation, we derive a Feynman-Kac-type formula for the second moment of the solution, based on the points of a planar Poisson process. This is an extension of the formula given by Dalan…
In this paper we introduce a stochastic integral with respect to the solution X of the fractional heat equation on [0,1], interpreted as a divergence operator. This allows to use the techniques of the Malliavin calculus in order to establish an Itô-type formula for the process X.
In this paper, we study the parabolic Anderson model of Skorohod type driven by a fractional Gaussian noise in time with Hurst parameter $H \in (0, 1/2)$. By using the Feynman-Kac representation for the $L^p(Ω)$ moments of the solution, we find the upper and lower bounds for the moments.
Lately, many phenomena in both applied and abstract mathematics and related disciplines have been expressed in terms of high order and fractional PDEs. Recently, Allouba introduced the Brownian-time Brownian sheet (BTBS) and connected it to a new system of fourth order interacting PDEs. The interaction in this multiparameter BTBS-PDEs connection is novel, leads to an intimately-connected linear system variant of the celebrated Kuramoto-Sivashinsky PDE, and is not shared with its one-time-parameter counterpart. It also means that these PDEs systems are to be solved for a family of functions, a …
We introduce an approach to study certain singular PDEs which is based on techniques from paradifferential calculus and on ideas from the theory of controlled rough paths. We illustrate its applicability on some model problems like differential equations driven by fractional Brownian motion, a fractional Burgers type SPDE driven by space-time white noise, and a non-linear version of the parabolic Anderson model with a white noise potential.
We consider a class of parabolic stochastic PDEs on bounded domains $D\subseteq\mathbb{R}^d$ that includes the stochastic heat equation, but with a fractional power $γ$ of the Laplacian. Viewing the solution as a process with values in a scale of fractional Sobolev spaces $H_r$, with $r < γ- d/2$, we study its power variations in $H_r$ along regular partitions of the time-axis. As the mesh size tends to zero, we find a phase transition at $r=-d/2$: the solutions have a nontrivial quadratic variation when $r<-d/2$ and a nontrivial $p$th order variation for $p= 2γ/(γ-d/2-r)>2$ when $r>-d/2$. Mor…
In this paper we study dynamic backward problems, with the computation of conditional expectations as a main objective, in a framework where the (forward) state process satisfies a Volterra type SDE, with fractional Brownian motion as a typical example. Such processes are neither Markov processes nor semimartingales, and most notably, they feature a certain time inconsistency which makes any direct application of Markovian ideas, such as flow properties, impossible without passing to a path-dependent framework. Our main result is a functional Itô formula, extending the seminal work of Dupire \…
The work in this paper is four-fold. Firstly, we introduce an alternative approach to solve fractional ordinary differential equations as an expected value of a random time process. Using the latter, we present an interesting numerical approach based on Monte Carlo integration to simulate solutions of fractional ordinary and partial differential equations. Thirdly, we show that this approach allows us to find the fundamental solutions for fractional partial differential equations (PDEs), in which the fractional derivative in time is in the Caputo sense and the fractional in space one is in the…
In this paper we study the regularity of non-linear parabolic PDEs and stochastic PDEs on metric measure spaces admitting heat kernels. In particular we consider mild function solutions to abstract Cauchy problems and show that the unique solution is Hölder continuous in time with values in a suitable fractional Sobolev space. As this analysis is done via a-priori estimates, we can apply this result to stochastic PDEs on metric measure spaces and solve the equation in a pathwise sense for almost all paths. The main example of noise term is of fractional Brownian type and the metric measure spa…
In this article, we consider the stochastic wave equation on the real line driven by a linear multiplicative Gaussian noise, which is white in time and whose spatial correlation corresponds to that of a fractional Brownian motion with Hurst index $H\in (\frac14,\frac12)$. Initial data are assumed to be constant. First, we prove that this equation has a unique solution (in the Skorohod sense) and obtain an exponential upper bound for the $p$-th moment of the solution, for any $p\geq 2$. Condition $H>\frac14$ turns out to be necessary for the existence of solution. Secondly, we show that this so…
We show that introducing an exponential cut-off on a suitable Sobolev norm facilitates the proof of quasi-invariance of Gaussian measures with respect to Hamiltonian PDE flows and allows us to establish the exact Jacobi formula for the density. We exploit this idea in two different contexts, namely the periodic fractional Benjamin-Bona-Mahony (BBM) equation with dispersion~$γ>1$ and the periodic one dimensional quintic defocussing nonlinear Schrödinger equation (NLS). For the BBM equation we study the transport of the cut-off Gaussian measures on fractional Sobolev spaces, while for the NLS eq…
The characterization of intermittency in turbulence has its roots in the K62 theory, and if no proper definition is to be found in the literature, statistical properties of intermittency were studied and models were developed in attempt to reproduce it. The first contribution of this work is to propose a requirement list to be satisfied by models designed within the Lagrangian framework. Multifractal stochastic processes are a natural choice to retrieve multifractal properties of the dissipation. Among them, following the proposition of \cite{Mandelbrot1968}, we investigate the Gaussian Multip…
In this paper, we prove a sample-path comparison principle for the nonlinear stochastic fractional heat equation on $\mathbb{R}$ with measure-valued initial data. We give quantitative estimates about how close to zero the solution can be. These results extend Mueller's comparison principle on the stochastic heat equation to allow more general initial data such as the (Dirac) delta measure and measures with heavier tails than linear exponential growth at $\pm\infty$. These results generalize a recent work by Moreno Flores [25], who proves the strict positivity of the solution to the stochastic …
We introduce $n$-parameter $\Rd$-valued Brownian-time Brownian sheet (BTBS): a Brownian sheet where each "time" parameter is replaced with the modulus of an independent Brownian motion. We then connect BTBS to a new system of $n$ linear, fourth order, and interacting PDEs and to a corresponding fourth order interacting nonlinear PDE. The coupling phenomenon is a result of the interaction between the Brownian sheet, through its variance, and the Brownian motions in the BTBS; and it leads to an intricate, intriguing, and random field generalization of our earlier Brownian-time-processes (BTPs) c…
The paper is devoted to the numerical solutions of fractional PDEs based on its probabilistic interpretation, that is, we construct approximate solutions via certain Monte Carlo simulations. The main results represent the upper bound of errors between the exact solution and the Monte Carlo approximation, the estimate of the fluctuation via the appropriate central limit theorem(CLT) and the construction of confidence intervals. Moreover, we provide rates of convergence in the CLT via Berry-Esseen type bounds. Concrete numerical computations and illustrations are included.
In this paper we study the Poisson and heat equations on bounded and unbounded domains with smooth boundary with random Dirichlet boundary conditions. The main novelty of this work is a convenient framework for the analysis of such equations excited by the white in time and/or space noise on the boundary. Our approach allows us to show the existence and uniqueness of weak solutions in the space of distributions. Then we prove that the solutions can be identified as smooth functions inside the domain, and finally the rate of their blow up at the boundary is estimated. A large class of noises in…
We introduce the notion of classical fractional query algorithms, which generalize decision trees in the average-case setting, and can potentially perform better than them. We show that the limiting run-time complexity of a natural class of these algorithms obeys the non-linear partial differential equation $\min_{k}\partial^{2}u/\partial x_{k}^{2}=-2$, and that the individual bit revealment satisfies the Schramm-Steif bound for Fourier weight, connecting noise sensitivity with PDEs. We discuss relations with other decision tree results.
We study the connection between PDEs and Lévy processes running with clocks given by time-changed Poisson processes with stochastic drifts. The random times we deal with are therefore given by time-changed Poissonian jumps related to some Frobenious-Perron operators $K$ associated to random translations. Moreover, we also consider their hitting times as a random clock. Thus, we study processes driven by equations involving time-fractional operators (modelling memory) and fractional powers of the difference operator $I-K$ (modelling jumps). For this large class of processes we also provide, in …
The fractional Laplacian has been strongly studied during past decades. In this paper we present a different approach for the associated Dirichlet problem, using recent deep learning techniques. In fact, intensively PDEs with a stochastic representation have been understood via neural networks, overcoming the so-called curse of dimensionality. Among these equations one can find parabolic ones in $\mathbb{R}^d$ and elliptic in a bounded domain $D \subset \mathbb{R}^d$. In this paper we consider the Dirichlet problem for the fractional Laplacian with exponent $α\in (1,2)$. We show that its solut…
In this paper we study general nonlinear stochastic differential equations, where the usual Brownian motion is replaced by a Lévy process. We also suppose that the coefficient multiplying the increments of this process is merely Lipschitz continuous and not necessarily linear in the time-marginals of the solution as is the case in the classical McKean-Vlasov model. We first study existence, uniqueness and particle approximations for these stochastic differential equations. When the driving process is a pure jump Lévy process with a smooth but unbounded Lévy measure, we develop a stochastic cal…
In this paper, we consider the stochastic singular integral operators and obtain the BMO estimates. As an application, we consider the fractional Laplacian equation with additive noises \bess du_t(x)=Δ^{\fracα{2}}u_t(x)dt+\sum_{k=1}^\infty\int_{\mathbb{R}^m}g^k(t,x)z\tilde N_k(dz,dt),\ \ \ u_0=0,\ 0\leq t\leq T, \eess where $Δ^{\fracα{2}}=-(-Δ)^{\fracα{2}}$, and $\int_{\mathbb{R}^m}z\tilde N_k(t,dz)=:Y_t^k$ are independent $m$-dimensional pure jump Lévy processes with Lévy measure of $ν^k$. Following the idea of \cite{Kim}, we obtain the $q$-th order BMO quasi-norm of the $\fracα{q_0}$-order d…
We study the simple hypothesis testing problem for the drift coefficient for stochastic fractional heat equation driven by additive noise. We introduce the notion of asymptotically the most powerful test, and find explicit forms of such tests in two asymptotic regimes: large time asymptotics, and increasing number of Fourier modes. The proposed statistics are derived based on Maximum Likelihood Ratio. Additionally, we obtain a series of important technical results of independent interest: we find the cumulant generating function of the log-likelihood ratio; obtain sharp large deviation type re…
In this work, we prove a version of Hörmander's theorem for a stochastic evolution equation driven by a trace-class fractional Brownian motion with Hurst exponent $\frac{1}{2} < H < 1$ and an analytic semigroup on a given separable Hilbert space. In contrast to the classical finite-dimensional case, the Jacobian operator in typical solutions of parabolic stochastic PDEs is not invertible which causes a severe difficulty in expressing the Malliavin matrix in terms of an adapted process. Under a Hörmander's bracket condition and some algebraic constraints on the vector fields combined with the r…
We consider the fractional stochastic heat type equation \begin{align*} \frac{\partial}{\partial t} u_t(x)=-(-Δ)^{α/2}u_t(x)+ξσ(u_t(x))\dot{F}(t,x),\ \ \ x\in D, \ \ t>0, \end{align*} with nonnegative bounded initial condition, where $α\in (0,2]$, $ξ>0$ is the noise level, $σ:\mathbb{R}\rightarrow\mathbb{R}$ is a globally Lipschitz function satisfying some growth conditions and the noise term behaves in space like the Riez kernel and is possibly correlated in time and $D$ is the unit open ball centered at the origin in $\mathbb{R}^d$. When the noise term is not correlated in time, we establish…
