We consider stochastic partial differential equations under minimal assumptions: the coefficients are merely bounded and measurable and satisfy the stochastic parabolicity condition. In particular, the diffusion term is allowed to be scaling-critical. We derive local supremum estimates with a stochastic adaptation of De Giorgi's iteration and establish a weak Harnack inequality for the solutions. The latter is then used to obtain pointwise almost sure continuity.
The existence and uniqueness of the mild solutions for a class of degenerate functional SPDEs are obtained, where the drift is assumed to be Hölder-Dini continuous. Moreover, the non-explosion of the solution is proved under some reasonable conditions. In addition, the Harnack is derived by the coupling by change of measure. Finally, the shift Harnack inequality is obtained for the equations without delay, which is new even in the non-degenerate case.
Shift Harnack and integration by part formula are establish for semilinear spde with delay and a class of stochastic semilinear evolution equation which cover the hyperdissipative Naiver-Stokes/Burges equation. For the case of stochastic equation with delay, an extension to path space is given.
By a new approximate method, dimensional free Harnack inequalities are established for a class of semilinear stochastic differential equations in Hilbert space with multiplicative noise. These inequalities are applied to study the strong Feller property for the semigroup and some properties of invariant measure.
The existence and uniqueness of the mild solution for a class of functional SPDEs with multiplicative noise and a locally Dini continuous drift are proved. In addition, under a reasonable condition the solution is non-explosive. Moreover, Harnack inequalities are derived for the associated semigroup under certain global conditions, which is new even in the case without delay.
Due to technical reasons, existing results concerning Harnack type inequalities for SPDEs with multiplicative noise apply only to the case where the coefficient in the noise term is an Hilbert-Schmidt perturbation of a fixed bounded operator. In this paper we investigate a class of semi-linear SPDEs with strongly multiplicative noise whose coefficient is even allowed to be unbounded which is thus no way to be Hilbert-Schmidt. Gradient estimates, log-Harnack inequality and applications are derived. Applications to stochastic reaction-diffusion equations driven by space-time white noise are pres…
We mainly investigate the log-Harnack inequality for the reflected stochastic partial differential equation driven by multiplicative noises based on the gradient estimate of the associated Markov semigroup. To do it, the penalization method and the comparison principle for stochastic partial differential equations are adopted. As its applications, we also establish the entropy-cost inequality and study some important estimates of the transition density relative to its invariant measure.
By using Malliavin calculus, explicit derivative formulae are established for a class of semi-linear functional stochastic partial differential equations with additive or multiplicative noise. As applications, gradient estimates and Harnack inequalities are derived for the semigroup of the associated segment process. Keywords: Bismut formula, Malliavin calculus, gradient estimate, Harnack inequality, functional SPDE
In this paper, the distribution dependent stochastic differential equation in a separable Hilbert space with a Dini continuous drift is investigated. The existence and uniqueness of weak and strong solutions are obtained. Moreover, some regularity results as well as gradient estimates and log-Harnack inequality are derived for the associated semigroup. In addition, dimensional free Harnack inequality with power and shift Harnack inequality are also proved when the noise is additive. All of the results extend the ones in the distribution independent situation.
The Hölder continuity of the solution to a nonlinear stochastic partial differential equation arising from one dimensional super process is obtained. It is proved that the Hölder exponent in time variable is as close as to 1/4, improving the result of 1/10 in a recent paper by Li et al [3]. The method is to use the Malliavin calculus. The Hölder continuity in spatial variable x of exponent 1/2 is also obtained by using this new approach. This Hölder continuity result is sharp since the corresponding linear heat equation has the same Hölder continuity.
The asymptotic log-Harnack inequality is established for several different models of stochastic differential systems with infinite memory: non-degenerate SDEs, Neutral SDEs, semi-linear SPDEs, and stochastic Hamiltonian systems. As applications, the following properties are derived for the associated segment Markov semigroups: asymptotic heat kernel estimate; uniqueness of the invariant probability measure; asymptotic gradient estimate and hence, asymptotically strong Feller property; and asymptotic irreducibilty.
In this paper, we focus on the existence of the density for the law of the solutions to parabolic stochastic partial differential equations with two reflecting walls. The main tool is Malliavin calculus.
By using the local dimension-free Harnack inequality established on incomplete Riemannian manifolds, integrability conditions on the coefficients are presented for SDEs to imply the non-explosion of solutions as well as the existence, uniqueness and regularity estimates of invariant probability measures. These conditions include a class of drifts unbounded on compact domains such that the usual Lyapunov conditions can not be verified. The main results are extended to second order differential operators on Hilbert spaces and semi-linear SPDEs.
In this article, we study a class of semilinear stochastic partial differential equations driven by an additive space time white noise. We establish Harnack inequalities for the semigroup associated with the solution by using coupling method, which implies the strong Feller property. Our results generalize the results of Zhang [Potential Analysis 33 (2010), no. 2, 137-151.] and can be applied to some types of SPDE such as reaction-diffusion equation and transport-diffusion equation perturbed by space time white noise.
We give a proof of Lipschitz continuity of p-harmonious functions, that are tug-of-war game analogies of ordinary p-harmonic functions. This result is used to obtain a new proof of Harnack's inequality for p-harmonic functions in the case $p>2$ that avoids classical techniques like Moser iteration, but instead relies on suitable choices of strategies for the stochastic tug-of-war game.
We present several results on the smoothness in $L_{p}$ sense of filtering densities under the Lipschitz continuity assumption on the coefficients of a partially observable diffusion processes. We obtain them by rewriting in divergence form filtering equation which are usually considered in terms of formally adjoint to operators in nondivergence form.
In this article, we consider the quasi-linear stochastic wave and heat equations on the real line and with an additive Gaussian noise which is white in time and behaves in space like a fractional Brownian motion with Hurst index $H\in (0,1)$. The drift term is assumed to be globally Lipschitz. We prove that the solution of each of the above equations is continuous in terms of the index $H$, with respect to the convergence in law in the space of continuous functions.
We investigate the optimal Hölder continuity and hitting probabilities for systems of stochastic heat equations and stochastic wave equations driven by an additive fractional Brownian sheet with temporal index $1/2$ and spatial index $H\le1/2$. Using stochastic calculus for fractional Brownian motion, we prove that these systems are well-posed and the solutions are Hölder continuous. Furthermore, the optimal Hölder exponents are obtained, which is the first result, as far as we knew, on the optimal Hölder continuity of SHEs and SWEs driven by fractional Brownian sheet that is rough in space. B…
We consider the stochastic wave and heat equations with affine multiplicative Gaussian noise which is white in time and behaves in space like the fractional Brownian motion with index $H \in (\frac14,\frac12)$. The existence and uniqueness of the solution to these equations has been proved recently by the authors. In the present note we show that these solutions have modifications which are Hölder continuous in space of order smaller than $H$, and Hölder continuous in time of order smaller than $γ$, where $γ=H$ for the wave equation and $γ=H/2$ for the heat equation.
In this article, we consider the one-dimensional stochastic wave and heat equations driven by a linear multiplicative Gaussian noise which is white in time and behaves in space like a fractional Brownian motion with Hurst index $H\in (\frac 14,1)$. We prove that the solution of each of the above equations is continuous in terms of the index $H$, with respect to the convergence in law in the space of continuous functions. The proof is based on a tightness criterion on the plane and Malliavin calculus techniques in order to identify the limit law.
For the class of stochastic partial differential equations studied in [Conus-Dalang,2008], we prove the existence of density of the probability law of the solution at a given point $(t,x)$, and that the density belongs to some Besov space. The proof relies on the method developed in [Debussche-Romito, 2014]. The result can be applied to the solution of the stochastic wave equation with multiplicative noise, Lipschitz coefficients and any spatial dimension $d\ge 1$, and also to the heat equation. This provides an extension of the results proved in [Sanz-Solé and Süß, 2013].
A new coupling argument is introduced to establish Driver's integration by parts formula and shift Harnack inequality. Unlike known coupling methods where two marginal processes with different starting points are constructed to move together as soon as possible, for the new-type coupling the two marginal processes start from the same point but their difference is aimed to reach a fixed quantity at a given time. Besides the integration by parts formula, the new coupling method is also efficient to imply the shift Harnack inequality. Differently from known Harnack inequalities where the values o…
We prove the existence and the Besov regularity of the density of the solution to a general parabolic SPDE which includes the stochastic Burgers equation on an unbounded domain. We use an elementary approach based on the fractional integration by parts.
We prove existence and uniqueness of solutions to a class of stochastic semilinear evolution equations with a monotone nonlinear drift term and multiplicative noise, considerably extending corresponding results obtained in previous work of ours. In particular, we assume the initial datum to be only measurable and we allow the diffusion coefficient to be locally Lipschitz-continuous. Moreover, we show, in a quantitative fashion, how the finiteness of the $p$-th moment of solutions depends on the integrability of the initial datum, in the whole range $p \in ]0,\infty[$. Lipschitz continuity of t…
We consider a $d$-dimensional branching particle system in a random environment. Suppose that the initial measures converge weakly to a measure with bounded density. Under the Mytnik-Sturm branching mechanism, we prove that the corresponding empirical measure $X_t^n$ converges weakly in the Skorohod space $D([0,T];M_F(\mathbb{R}^d))$ and the limit has a density $u_t(x)$, where $M_F(\mathbb{R}^d)$ is the space of finite measures on $\mathbb{R}^d$. We also derive a stochastic partial differential equation $u_t(x)$ satisfies. By using the techniques of Malliavin calculus, we prove that $u_t(x)$ i…
We establish exact, dimension-dependent, spatio-temporal, uniform and local moduli of continuity for (1) the fourth order L-Kuramoto-Sivashinsky (L-KS) SPDEs and for (2) the time-fractional stochastic partial integro-differential equations (SPIDEs), driven by space-time white noise in one-to-three dimensional space. Both classes were introduced---with Brownian-time-type kernel formulations---by Allouba in a series of articles starting in 2006, where he presented class (2) in its rigorous stochastic integral equations form. He proved existence, uniqueness, and sharp spatio-temporal Hölder regul…
We consider a class of semilinear stochastic evolution equations driven by an additive cylindrical stable noise.We investigate structural properties of the solutions like Markov, irreducibility, stochastic continuity, Feller and strong Feller properties, and study integrability of trajectories. The obtained results can be applied to semilinear stochastic heat equations with Dirichlet boundary conditions and bounded and Lipschitz nonlinearities.
We study the sample path regularity of the solutions of a class of spde's which are second order in time and that includes the stochastic wave equation. Non-integer powers of the spatial Laplacian are allowed. The driving noise is white in time and spatially homogeneous. Continuing with the work initiated in Dalang and Mueller (2003), we prove that the solutions belong to a fractional $L^2$-Sobolev space. We also prove Hölder continuity in time and therefore, we obtain joint Hölder continuity in the time and space variables. Our conclusions rely on a precise analysis of the properties of the s…
We prove global well-posedness for a class of dissipative semilinear stochastic evolution equations with singular drift and multiplicative Wiener noise. In particular, the nonlinear term in the drift is the superposition operator associated to a maximal monotone graph everywhere defined on the real line, on which no continuity nor growth assumptions are imposed. The hypotheses on the diffusion coefficient are also very general, in the sense that the noise does not need to take values in spaces of continuous, or bounded, functions in space and time. Our approach combines variational techniques …
Strong Feller property and irreducibility are study for a class of non-linear monotone stochastic partial differential equations with multiplicative noise. Hölder continuity of the associated Markov semigroups are discussed in some special cases. The main results are applied to several examples such as stochastic porous media equations, stochastic fast diffusion equations.
The Dirichlet problem for a class of stochastic partial differential equations is studied in Sobolev spaces. The existence and uniqueness result is proved under certain compatibility conditions that ensure the finiteness of $L^{p}(Ω\times(0,T),W^{2,p}(G))$-norms of solutions. The Hölder continuity of solutions and their derivatives is also obtained by embedding.
We prove existence and uniqueness of strong solutions, as well as continuous dependence on the initial datum, for a class of fully nonlinear second-order stochastic PDEs with drift in divergence form. Due to rather general assumptions on the growth of the nonlinearity in the drift, which, in particular, is allowed to grow faster than polynomially, existing techniques are not applicable. A well-posedness result is obtained through a combination of a priori estimates on regularized equations, interpreted both as stochastic equations as well as deterministic equations with random coefficients, an…
We consider the Harnack inequality for harmonic functions with respect to three types of infinite dimensional operators. For the infinite dimensional Laplacian, we show no Harnack inequality is possible. We also show that the Harnack inequality fails for a large class of Ornstein-Uhlenbeck processes, although functions that are harmonic with respect to these processes do satisfy an a priori modulus of continuity. Many of these processes also have a coupling property. The third type of operator considered is the infinite dimensional analog of operators in Hörmander's form. In this case a Harnac…
We study master equations of the form $$(\partial_t+L)^su=f\quad\hbox{in}~\mathbb{R}\timesΩ$$ where $L$ is a divergence form elliptic operator and $Ω\subseteq\mathbb{R}^n$. These are nonlocal equations of order $2s$ in space and $s$ in time that take into account the values of $u$ everywhere in $Ω$ and for past times. We show parabolic interior and boundary Harnack inequalities and local parabolic Hölder continuity of solutions. To this end, we prove a characterization of fractional powers of parabolic operators $\partial_t+L$ with a degenerate parabolic extension problem.
In this paper we show that the Cahn-Hilliard stochastic SPDE has a function valued solution in dimension 4 and 5 when the perturbation is driven by a space-correlated Gaussian noise. This is done proving general results on SPDEs with globally Lipschitz coefficients associated with operators on smooth domains of $\mathbb{R}^d$ which are parabolic in the sense of Petrovskii}, and do not necessarily define a semi-group of operators. We study the regularity of the trajectories of the solutions and the absolute continuity of the law at some given time and position.
We solve a time-dependent linear SPDE with additive Levy noise in the mild and weak sense. Existence of a generalized invariant measure for the associated transition semigroup is established and the generator is characterized on the corresponding L^2-space. The square field operator is calculated, allowing to derive a Poincare and a Harnack inequality.
We study strictly parabolic stochastic partial differential equations on $\R^d$, $d\ge 1$, driven by a Gaussian noise white in time and coloured in space. Assuming that the coefficients of the differential operator are random, we give sufficient conditions on the correlation of the noise ensuring Hölder continuity for the trajectories of the solution of the equation. For self-adjoint operators with deterministic coefficients, the mild and weak formulation of the equation are related, deriving path properties of the solution to a parabolic Cauchy problem in evolution form.
We exhibit a class of properties of an spde that guarantees existence, uniqueness and bounds on moments of the solution. These moment bounds are expressed in terms of quantities related to the associated deterministic homogeneous p.d.e. With these, we can, for instance, obtain solutions to the stochastic heat equation on the real line for initial data that falls in a certain class of Schwartz distributions, but our main focus is the stochastic wave equation on the real line with irregular initial data. We give bounds on higher moments, and for the hyperbolic Anderson model, explicit formulas f…
We consider divergence-form parabolic equation with measurable uniformly elliptic matrix and the vector field in a large class containing, in particular, the vector fields in $L^p$, $p>d$, as well as some vector fields that are not even in $L_{\rm loc}^{2+\varepsilon}$, $\varepsilon>0$. We establish Hölder continuity of the bounded soutions, sharp two-sided Gaussian bound on the heat kernel, Harnack inequality.
In this paper we study the pathwise uniqueness of solution to the following stochastic partial differential equation (SPDE) with Hölder continuous coefficient: \begin{eqnarray*} \frac{\partial X_t(x)}{\partial t}=\frac{1}{2} ΔX_t(x) +G(X_t(x))+H(X_{t-}(x)) \dot{L}_t(x),~~~ t>0, ~x\in\mathbb{R}, \end{eqnarray*} where $\dot{L}$ denotes an $α$-stable white noise on $\mathbb{R}_+\times \mathbb{R}$ without negative jumps, $G$ satisfies the Lipschitz condition and $H$ is nondecreasing and $β$-Hölder continuous for $1<α<2$ and $0<β<1$. For $G\equiv0$ and $H(x)=x^β$, in Mytnik (2002) a weak solution t…
We prove a new uniqueness result for solutions to Fokker-Planck-Kolmogorov (FPK) equations for probability measures on infinite-dimensional spaces. We consider infinite-dimensional drifts that admit certain finite-dimensional approximations. In contrast to most of the previous work on FPK-equations in infinite dimensions, we include cases with non-constant coefficients in the second order part and also include degenerate cases where these coefficients can even be zero. Also a new existence result is proved. Some applications to Fokker-Planck-Kolmogorov equations associated with SPDEs are prese…
We show uniqueness in law for the critical SPDE \begin{eqnarray} \label{qq1} dX_t = AX_t dt + (-A)^{1/2}F(X(t))dt + dW_t,\;\; X_0 =x \in H, \end{eqnarray} where $A$ $ : \text{dom}(A) \subset H \to H$ is a negative definite self-adjoint operator on a separable Hilbert space $H$ having $A^{-1}$ of trace class and $W$ is a cylindrical Wiener process on $H$. Here $F: H \to H $ can be locally Hölder continuous with at most linear growth (some functions $F$ which grow more than linearly can also be considered). This leads to new uniqueness results for generalized stochastic Burgers equations and for…
We consider singular quasilinear stochastic partial differential equations (SPDEs) studied in \cite{FHSX}, which are defined in paracontrolled sense. The main aim of the present article is to establish the global-in-time solvability for a particular class of SPDEs with origin in particle systems and, under a certain additional condition on the noise, prove the convergence of the solutions to stationary solutions as $t\to\infty$. We apply the method of energy inequality and Poincaré inequality. It is essential that the Poincaré constant can be taken uniformly in an approximating sequence of the…
We study Malliavin differentiability of solutions to sub-critical singular parabolic stochastic partial differential equations (SPDEs) and we prove the existence of densities for a class of singular SPDEs. Both of these results are implemented in the setting of regularity structures. For this we construct renormalized models in situations where some of the driving noises are replaced by deterministic Cameron-Martin functions, and we show Lipschitz continuity of these models with respect to the Cameron-Martin norm. In particular, in many interesting situations we obtain a convergence and stabil…
The Heston stochastic volatility process, which is widely used as an asset price model in mathematical finance, is a paradigm for a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half-plane. The generator of this process with killing, called the elliptic Heston operator, is a second-order, degenerate-elliptic partial differential operator whose coefficients have linear growth in the spatial variables and where the degeneracy in the operator symbol is proportional to the distance to the bou…
