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In this paper, we establish the existence and uniqueness of solutions to stochastic heat equations with logarithmic nonlinearity driven by Brownian motion on a bounded domain $D$ in the setting of $L^2(D)$ space. The result is valid for all initial values in $L^2(D)$. The logarithmic Sobolev inequality plays an important role., 30 pages

We study the nonlinear Schrödinger equation with linear damping, i.e. a zero order dissipation, and additive noise. Working in $R^d$ with d = 2 or d = 3, we prove the uniqueness of the invariant measure when the damping coefficient is sufficiently large., 27 pages

We study the existence, uniqueness, and regularity of the solution to the stochastic reaction-diffusion equation (SRDE) with colored noise $\dot{F}$: $$ \partial_t u = a^{ij}u_{x^ix^j} + b^i u_{x^i} + cu - \bar{b} u^{1+\beta} + \xi u^{1+\gamma}\dot F,\quad (t,x)\in \mathbb{R}_+\times\mathbb{R}^d; \quad u(0,\cdot) = u_0, $$ where $a^{ij},b^i,c, \bar{b}$ and $\xi$ are $C^2$ or $L_\infty$ bounded random coefficients. Here $\beta>0$ denotes the degree of the strong dissipativity and $\gamma>0$ represents the degree of stochastic force. Under the reinforced Dalang's condition on $\dot{F}$, we show the well-posedness of the SRDE provided $\gamma < \frac{\kappa(\beta +1)}{d+2}$ where $\kappa>0$ is the constant related to $\dot F$. Our result assures that strong dissipativity prevents the solution from blowing up. Moreover, we provide the maximal H\"older regularity of the solution in time and space., Comment: 21 pages

In this paper, we investigate stochastic heat equation with sublinear diffusion coefficients. By assuming certain concavity of the diffusion coefficient, we establish non-trivial moment upper bounds and almost sure spatial asymptotic properties for the solutions. These results shed light on the smoothing intermittency effect under weak diffusion (i.e., sublinear growth) previously observed by Zeldovich et al. [Zel+87]. The sample-path spatial asymptotics obtained in this paper partially bridge a gap in earlier works of Conus et al. [CJK13; Con+13], which focused on two extreme scenarios: a linear diffusion coefficient and a bounded diffusion coefficient. Our approach is highly robust and applicable to a variety of stochastic partial differential equations, including the one-dimensional stochastic wave equation and the stochastic fractional diffusion equations., Comment: 50 pages, 6 figures

Existence, uniqueness, and regularity of a strong solution are obtained for stochastic PDEs with a colored noise $F$ and its super-linear diffusion coefficient: $$ du=(a^{ij}u_{x^ix^j}+b^iu_{x^i}+cu)dt+\xi|u|^{1+\lambda}dF, \quad (t,x)\in(0,\infty)\times\mathbb{R}^d, $$ where $\lambda \geq 0$ and the coefficients depend on $(\omega,t,x)$. The strategy of handling nonlinearity of the diffusion coefficient is to find a sharp estimation for a general Lipschitz case, and apply it to the super-linear case. Moreover, investigation for the estimate provides a range of $\lambda$, a sufficient condition for the unique solvability, where the range depends on the spatial covariance of $F$ and the spatial dimension $d$., Comment: 27 pages

We consider invariant measures for the stochastic Burgers equation on $\mathbb{R}$, forced by the derivative of a spacetime-homogeneous Gaussian noise that is white in time and smooth in space. An invariant measure is indecomposable, or extremal, if it cannot be represented as a convex combination of other invariant measures. We show that for each $a\in\mathbb{R}$, there is a unique indecomposable law of a spacetime-stationary solution with mean $a$, in a suitable function space. We also show that solutions starting from spatially-decaying perturbations of mean-$a$ periodic functions converge in law to the extremal space-time stationary solution with mean $a$ as time goes to infinity., 68 pages, to appear in Communications in Mathematical Physics

We introduce the uniqueness, existence, $L_p$-regularity, and maximal H\"older regularity of the solution to semilinear stochastic partial differential equation driven by a multiplicative space-time white noise: $$ u_t = au_{xx} + bu_{x} + cu + \bar b|u|^\lambda u_{x} + \sigma(u)\dot W,\quad (t,x)\in(0,\infty)\times\mathbb{R}; \quad u(0,\cdot) = u_0, $$ where $\lambda > 0$. The function $\sigma(u)$ is either bounded Lipschitz or super-linear in $u$. The noise $\dot W$ is a space-time white noise. The coefficients $a,b,c$ depend on $(\omega,t,x)$, and $\bar b$ depends on $(\omega,t)$. The coefficients $a,b,c,\bar{b}$ are uniformly bounded, and $a$ satisfies ellipticity condition. The random initial data $u_0 = u_0(\omega,x)$ is nonnegative. We have the maximal H\"older regularity by employing the H\"older embedding theorem. For example, if $\lambda \in(0,1]$ and $\sigma(u)$ has Lipschitz continuity, linear growth, and boundedness in $u$, for $T0$, $$u \in C^{1/4 - \varepsilon,1/2 - \varepsilon}_{t,x}([0,T]\times\mathbb{R})\quad(a.s.). $$ On the other hand, if $\lambda\in(0,1)$ and $\sigma(u) = |u|^{1+\lambda_0}$ with $\lambda_0\in[0,1/2)$, for $T0$, $$u \in C^{\frac{1/2-(\lambda -1/2) \vee \lambda_0}{2} - \varepsilon,1/2-(\lambda -1/2) \vee \lambda_0 - \varepsilon}_{t,x}([0,T]\times\mathbb{R})\quad (a.s.).$$ It should be noted that if $\sigma(u)$ is bounded Lipschitz in $u$, the H\"older regularity of the solution is independent of $\lambda$. However, if $\sigma(u)$ is super-linear in $u$, the H\"older regularities of the solution are affected by nonlinearities, $\lambda$ and $\lambda_0.$, Comment: 33 pages. arXiv admin note: text overlap with arXiv:2111.03302

For solutions of a certain class of SPDEs in divergence form we present some estimates of their $L_{p}$-norms and the $L_{p}$-norms of their first-order derivatives. The main novelty is that the low-order coefficients are supposed to belong to certain Morrey classes instead of $L_{p}$-spaces. Our results are new even if there are no stochastic terms in the equation., 16 pages

We study the compact support property for solutions of the following stochastic partial differential equations: $$\partial_t u = a^{ij}u_{x^ix^j}(t,x)+b^{i}u_{x^i}(t,x)+cu+h(t,x,u(t,x))\dot{F}(t,x),\quad (t,x)\in (0,\infty)\times{\bf{R}}^d,$$ where $\dot{F}$ is a spatially homogeneous Gaussian noise that is white in time and colored in space, and $h(t, x, u)$ satisfies $K^{-1}|u|^{\lambda}\leq h(t, x, u)\leq K(1+|u|)$ for $\lambda\in(0,1)$ and $K\geq 1$. We show that if the initial data $u_0\geq 0$ has a compact support, then, under the reinforced Dalang's condition on $\dot{F}$ (which guarantees the existence and the H\"older continuity of a weak solution), all nonnegative weak solutions $u(t, \cdot)$ have the compact support for all $t>0$ with probability 1. Our results extend the works by Mueller-Perkins [Probab. Theory Relat. Fields, 93(3):325--358, 1992] and Krylov [Probab. Theory Relat. Fields, 108(4):543--557, 1997], in which they show the compact support property only for the one-dimensional SPDEs driven by space-time white noise on $(0, \infty)\times \bf{R}$., Comment: 39pages. In v3 we have corrected some errors from v1. We mainly fixed Theorem 3.1 and Lemma 3.1 with a modification of the solution space (the equation (2.10) in v3). We added additional comments from v2 where the manuscript is the same as in v3

We study the dynamics of waves, oscillations, and other spatio-temporal patterns in stochastic evolution systems, including SPDE and stochastic integral equations. Representing a given pattern as a smooth, stable invariant manifold of the deterministic dynamics, we reduce the stochastic dynamics to a finite dimensional SDE on this manifold using the isochronal phase. The isochronal phase is defined by mapping a neighbourbhood of the manifold onto the manifold itself, analogous to the isochronal phase defined for finite-dimensional oscillators by A.T.~Winfree and J.~Guckenheimer. We then determine a probability measure that indicates the average position of the stochastic perturbation of the pattern/wave as it wanders over the manifold. It is proved that this probability measure is accurate on time-scales greater than $O(\sigma^{-2})$, but less than $O(\exp(C\sigma^{-2}))$, where $\sigma\ll1$ is the amplitude of the stochastic perturbation. Moreover, using this measure, we determine the expected velocity of the difference between the deterministic and stochastic motion on the manifold., Comment: 41 pages

In this paper we study the long time behavior of the solution to a certain class of space-time fractional stochastic equations with respect to the level $\lambda$ of a noise and show how the choice of the order $\beta \in (0, \,1)$ of the fractional time derivative affects the growth and decay behavior of their solution. We consider both the cases of white noise and colored noise. Our results extend the main results in "M. Foondun, \textit{Remarks on a fractional-time stochastic equation}, Proc. Amer. Math. Soc. 149 (2021), 2235-2247" to fractional Laplacian as well as higher dimensional cases.

We establish the $L_p$-regularity theory for a semilinear stochastic partial differential equation with multiplicative white noise: $$ du = (a^{ij}u_{x^ix^j} + b^{i}u_{x^i} + cu + \bar b^{i}|u|^\lambda u_{x^i})dt + \sigma^k(u)dw_t^k,\quad (t,x)\in(0,\infty)\times\bR^d; \quad u(0,\cdot) = u_0, $$ where $\lambda>0$, the set $\{ w_t^k,k=1,2,\dots \}$ is a set of one-dimensional independent Wiener processes, and the function $u_0 = u_0(\omega,x)$ is a nonnegative random initial data. The coefficients $a^{ij},b^i,c$ depend on $(\omega,t,x)$, and $\bar b^i$ depends on $(\omega,t,x^1,\dots,x^{i-1},x^{i+1},\dots,x^d)$. The coefficients $a^{ij},b^i,c,\bar{b}^i$ are uniformly bounded and twice continuous differentiable. The leading coefficient $a$ satisfies ellipticity condition. Depending on the diffusion coefficient $\sigma^k(u)$, we consider two different cases; (i) $\lambda\in(0,\infty)$ and $\sigma^k(u)$ has Lipschitz continuity and linear growth in $u$, (ii) $\lambda,\lambda_0\in(0,1/d)$ and $\sigma^k(u) = \mu^k |u|^{1+\lambda_0}$ ($\sigma^k(u)$ is super-linear). Each case has different regularity results. For example, in the case of $(i)$, for $\varepsilon>0$ $$u \in C^{1/2 - \varepsilon,1 - \varepsilon}_{t,x}([0,T]\times\bR^d)\quad \forall T0$ $$ u \in C^{\frac{1-(\lambda d) \vee (\lambda_0 d)}{2} - \varepsilon,1-(\lambda d) \vee (\lambda_0 d) - \varepsilon}_{t,x}([0,T]\times\bR^d)\quad \forall T, Comment: 28 pages

We study the scaling limit of a branching random walk in static random environment in dimension $d=1,2$ and show that it is given by a super-Brownian motion in a white noise potential. In dimension $1$ we characterize the limit as the unique weak solution to the stochastic PDE: \[\partial_t \mu = (\Delta {+} \xi) \mu {+} \sqrt{2\nu \mu} \tilde{\xi}\] for independent space white noise $\xi$ and space-time white noise $\tilde{\xi}$. In dimension $2$ the study requires paracontrolled theory and the limit process is described via a martingale problem. In both dimensions we prove persistence of this rough version of the super-Brownian motion., Comment: 30 Pages. This is a significantly shortened version of the original, a part of which was migrated to the article named "Killed rough super-Brownian motion"

In this paper we establish the optimal regularity estimates for the Cauchy problem of stochastic kinetic equations with random coefficients in anisotropic Besov spaces. As applications, we study the nonlinear filtering problem for a degenerate diffusion process, and obtain the existence and regularity of conditional probability densities under few assumptions. Moreover, we also show the well-posedness for a class of super-linear growth stochastic kinetic equations driven by velocity-time white noises, as well as a kinetic version of Parabolic Anderson Model with measure as initial values., Comment: 48pages

We prove optimal regularity estimates in Sobolev spaces in time and space for solutions to stochastic porous medium equations. The noise term considered here is multiplicative, white in time and coloured in space. The coefficients are assumed to be H\"older continuous and the cases of smooth coefficients of at most linear growth as well as $\sqrt{u}$ are covered by our assumptions. The regularity obtained is consistent with the optimal regularity derived for the deterministic porous medium equation in [Gess 2020] and [Gess, Sauer, Tadmor 2020] and the presence of the temporal white noise. The proof relies on a significant adaptation of velocity averaging techniques from their usual $L^1$ context to the natural $L^2$ setting of the stochastic case. We introduce a new mixed kinetic/mild representation of solutions to quasilinear SPDE and use $L^2$ based a priori bounds to treat the stochastic term., Comment: Accepted version, to appear in Annals of Probability

We study both strict and mild solutions to parabolic evolution equations of the form $dX+AXdt=F(t)dt+G(t)dW(t)$ in Banach spaces. First, we explore the deterministic case. The maximal regularity of solutions has been shown. Second, we investigate the stochastic case. We prove existence of strict solutions and show their space-time regularity. Finally, we apply our abstract results to a stochastic heat equation., Comment: 42 pages

We define a notion of a viscous shock solution of the stochastic Burgers equation that connects "top" and "bottom" spatially stationary solutions of the same equation. Such shocks generally travel in space, but we show that they admit time-invariant measures when viewed in their own reference frames. Under such a measure, the viscous shock is a deterministic function of the bottom and top solutions and the shock location. However, the measure of the bottom and top solutions must be tilted to account for the change of reference frame. We also show a convergence result to these stationary shock solutions from solutions initially connecting two constants, as time goes to infinity., Comment: 30 pages, 2 figures; minor corrections in this version; to appear in Archive for Rational Mechanics and Analysis

We obtain uniqueness and existence of a solution u to the following second-order stochastic partial differential equation: 1 $$\begin{aligned} du= \left( {\bar{a}}^{ij}(\omega ,t)u_{x^ix^j}+ f \right) dt + g^k dw^k_t, \quad t \in (0,T); \quad u(0,\cdot )=0, \end{aligned}$$ where $$T \in (0,\infty )$$ , $$w^k$$ $$(k=1,2,\ldots )$$ are independent Wiener processes, $$({\bar{a}}^{ij}(\omega ,t))$$ is a (predictable) nonnegative symmetric matrix valued stochastic process such that $$\begin{aligned} \kappa |\xi |^2 \le {\bar{a}}^{ij}(\omega ,t) \xi ^i \xi ^j \le K |\xi |^2 \quad \forall \;(\omega ,t,\xi ) \in \Omega \times (0,T) \times {\mathbf {R}}^d \end{aligned}$$ for some $$\kappa , K \in (0,\infty )$$ , $$\begin{aligned} f \in L_p\left( (0,T) \times {\mathbf {R}}^d, dt \times dx ; L_r(\Omega , {\mathscr {F}} ,dP) \right) , \end{aligned}$$ and $$\begin{aligned} g, g_x \in L_p\left( (0,T) \times {\mathbf {R}}^d, dt \times dx ; L_r(\Omega , {\mathscr {F}} ,dP; l_2) \right) \end{aligned}$$ with $$2 \le r \le p < \infty $$ and appropriate measurable conditions. Moreover, for the solution u, we obtain the following maximal regularity moment estimate 2 $$\begin{aligned}&\int _0^T \int _{{\mathbf {R}}^d}\left( \mathbb {E}\left[ |u(t,x)|^r\right] \right) ^{p/r} dx dt + \int _0^T \int _{{\mathbf {R}}^d}\left( \mathbb {E}\left[ |u_{xx}(t,x)|^r\right] \right) ^{p/r} dx dt \nonumber \\&\le N \bigg (\int _0^T \int _{{\mathbf {R}}^d}\left( \mathbb {E}\left[ |f(t,x)|^r\right] \right) ^{p/r} dx dt + \int _0^T \int _{{\mathbf {R}}^d}\left( \mathbb {E}\left[ |g(t,x)|_{l_2}^r\right] \right) ^{p/r} dx dt \nonumber \\&\quad + \int _0^T \int _{{\mathbf {R}}^d}\left( \mathbb {E}\left[ |g_x(t,x)|_{l_2}^r\right] \right) ^{p/r} dx dt \bigg ), \end{aligned}$$ where N is a positive constant depending only on d, p, r, $$\kappa $$ , K, and T. As an application, for the solution u to (1), the rth moment $$m^r(t,x):=\mathbb {E}|u(t,x)|^r$$ is in the parabolic Sobolev space $$W_{p/r}^{1,2}\left( (0,T) \times \mathbf {R}^d\right) $$ .

We consider approximations of the Stefan-type condition by imbalances of volume closely around the inner interface and study convergence of the solutions of the corresponding semilinear stochastic moving boundary problems. After a coordinate transformation, the problems can be reformulated as stochastic evolution equations on fractional power domains of linear operators. Here, the coefficients might fail to have linear growths and might be Lipschitz continuous only on bounded sets. We show continuity properties of the mild solution map in the coefficients and initial data, also incorporating the possibility of explosion of the solutions., 23 pages, typos corrected, abstract and introduction revised, remarks on possible convergence rates added

In this paper we study non-linear noise excitation for the following class of space-time fractional stochastic equations in bounded domains: $$\partial^\beta_tu_t(x)=-\nu(-\Delta)^{\alpha/2} u_t(x)+I^{1-\beta}_t[\lambda \sigma(u)\stackrel{\cdot}{F}(t,x)]$$ in $(d+1)$ dimensions, where $\nu>0, \beta\in (0,1)$, $\alpha\in (0,2]$. The operator $\partial^\beta_t$ is the Caputo fractional derivative, $-(-\Delta)^{\alpha/2} $ is the generator of an isotropic stable process and $I^{1-\beta}_t$ is the fractional integral operator. The forcing noise denoted by $\stackrel{\cdot}{F}(t,x)$ is a Gaussian noise. The multiplicative non-linearity $\sigma:\RR{R}\to\RR{R}$ is assumed to be globally Lipschitz continuous. These equations were recently introduced by Mijena and Nane(J. Mijena and E. Nane. Space time fractional stochastic partial differential equations. Stochastic Process Appl. 125 (2015), no. 9, 3301--3326). We first study the existence and uniqueness of the solution of these equations {and} under suitable conditions on the initial function, we {also} study the asymptotic behavior of the solution with respect to the parameter $\lambda$. In particular, our results are significant extensions of those in Foondun et al (M. Foondun, K. Tian and W. Liu. On some properties of a class of fractional stochastic equations. Preprint available at arxiv.org 1404.6791v1.), Foondun and Khoshnevisan (M. Foondun and D. Khoshnevisan. Intermittence and nonlinear parabolic stochastic partial differential equations, Electron. J. Probab. 14 (2009), no. 21, 548--568.), Nane and Mijena (J. Mijena and E. Nane. Space time fractional stochastic partial differential equations. Stochastic Process Appl. 125 (2015), no. 9, 3301--3326; J. B. Mijena, and E.Nane. Intermittence and time fractional partial differential equations. Submitted. 2014)., Comment: 23 pages. arXiv admin note: substantial text overlap with arXiv:1505.04615

In this paper, we obtain an explicit formula for the two-point correlation function for the solutions to the stochastic heat equation on $\mathbb{R}$. The bounds for $p$-th moments proved in [3] are simplified. We validate the Feynman-Kac formula for the $p$-point correlation function of the solutions to this equation with measure-valued initial data., Comment: 23 pages

We prove that the stochastic Burgers equation on $\mathbf{R}^{d}$, $d, 16 pages

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 $L^{p}(\Omega\times(0,T),W^{2,p}(G))$-norms of solutions. The H\"older continuity of solutions and their derivatives is also obtained by embedding., Comment: 20 pages

We consider the fractional stochastic heat type equation \begin{align*} \frac{\partial}{\partial t} u_t(x)=-(-\Delta)^{\alpha/2}u_t(x)+\xi\sigma(u_t(x))\dot{F}(t,x),\ \ \ x\in D, \ \ t>0, \end{align*} with nonnegative bounded initial condition, where $\alpha\in (0,2]$, $\xi>0$ is the noise level, $\sigma:\mathbb{R}\rightarrow\mathbb{R}$ 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 a change in the growth of the solution of these equations depending on the noise level $\xi$. On the other hand when the noise term behaves in time like the fractional Brownian motion with index $H\in (1/2,1)$, We also derive explicit bounds leading to a well-known intermittency property., Comment: Accepted at Stochastic Processes and Their Applications, 28 pages

The Dean-Kawasaki model consists of a nonlinear stochastic partial differential equation featuring a conservative, multiplicative, stochastic term with non-Lipschitz coefficient, and driven by space-time white noise; this equation describes the evolution of the density function for a system of finitely many particles governed by Langevin dynamics. Well-posedness for the Dean-Kawasaki model is open except for specific diffusive cases, corresponding to overdamped Langevin dynamics. There, it was recently shown by Lehmann, Konarovskyi, and von Renesse that no regular (non-atomic) solutions exist. We derive and analyse a suitably regularised Dean-Kawasaki model of wave equation type driven by coloured noise, corresponding to second order Langevin dynamics, in one space dimension. The regularisation can be interpreted as considering particles of finite size rather than describing them by atomic measures. We establish existence and uniqueness of a solution. Specifically, we prove a high-probability result for the existence and uniqueness of mild solutions to this regularised Dean-Kawasaki model., Comment: 48 pages, 1 figure

We discuss a class of stochastic second-order PDEs in one space-dimension with an inner boundary moving according to a possibly non-linear, Stefan-type condition. We show that proper separation of phases is attained, i.e., the solution remains negative on one side and positive on the other side of the moving interface, when started with the appropriate initial conditions. To extend results from deterministic settings to the stochastic case, we establish a Wong-Zakai type approximation. After a coordinate transformation the problems are reformulated and analysed in terms of stochastic evolution equations on domains of fractional powers of linear operators., Comment: 46 pages

The boundary behaviour of solutions of stochastic PDEs with Dirichlet boundary conditions can be surprisingly - and in a sense, arbitrarily - bad: as shown by Krylov, for any $��>0$ one can find a simple $1$-dimensional constant coefficient linear equation whose solution at the boundary is not $��$-H��lder continuous. We obtain a positive counterpart of this: under some mild regularity assumptions on the coefficients, solutions of semilinear SPDEs on $C^1$ domains are proved to be $��$-H��lder continuous up to the boundary with some $��>0$., 29 pages

Semilinear stochastic partial differential equations on bounded domains $\mathscr{D}$ are considered. The semilinear term may have arbitrary polynomial growth as long as it is continuous and monotone except perhaps near the origin. Typical examples are the stochastic Allen--Cahn and Ginzburg--Landau equations. The first main result of this article are $L^p$-estimates for such equations. The $L^p$-estimates are subsequently employed in obtaining higher regularity. This is motivated by ongoing work to obtain rate of convergence estimates for numerical approximations to such equations. It is shown, under appropriate assumptions, that the solution is continuous in time with values in the Sobolev space $H^2(\mathscr{D}')$ and $\ell^2$-integrable with values in $H^3(\mathscr{D}')$, for any compact $\mathscr{D}' \subset \mathscr{D}$. Using results from $L^p$-theory of SPDEs obtained by Kim~\cite{kim04} we get analogous results in weighted Sobolev spaces on the whole $\mathscr{D}$. Finally it is shown that the solution is H��lder continuous in time of order $\frac{1}{2} - \frac{2}{q}$ as a process with values in a weighted $L^q$-space, where $q$ arises from the integrability assumptions imposed on the initial condition and forcing terms.

In this paper, we introduce a new approach to rough and stochastic partial differential equations (RPDEs and SPDEs): we consider general Banach spaces as state spaces and — for the sake of simplicity — finite dimensional sources of noise, either rough or stochastic. By means of a time-dependent transformation of state space and rough path theory, we are able to construct unique solutions of the respective R- and SPDEs. As a consequence of our construction, we can apply the pool of results of rough path theory, in particular we can obtain strong and weak numerical schemes of high order converging to the solution process.

In this paper, we study the Cauchy problem for a quasilinear degenerate parabolic stochastic partial differential equation driven by a cylindrical Wiener process. In particular, we adapt the notion of kinetic formulation and kinetic solution and develop a well-posedness theory that includes also an $L^1$-contraction property. In comparison to the previous works of the authors concerning stochastic hyperbolic conservation laws [J. Funct. Anal. 259 (2010) 1014-1042] and semilinear degenerate parabolic SPDEs [Stochastic Process. Appl. 123 (2013) 4294-4336], the present result contains two new ingredients that provide simpler and more effective method of the proof: a generalized It\^{o} formula that permits a rigorous derivation of the kinetic formulation even in the case of weak solutions of certain nondegenerate approximations and a direct proof of strong convergence of these approximations to the desired kinetic solution of the degenerate problem., Comment: Published at http://dx.doi.org/10.1214/15-AOP1013 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org). arXiv admin note: text overlap with arXiv:1202.2031

We study quasilinear degenerate parabolic-hyperbolic stochastic partial differential equations with general multiplicative noise within the framework of kinetic solutions. Our results are twofold: First, we establish new regularity results based on averaging techniques. Second, we prove the existence and uniqueness of solutions in a full $L^1$ setting requiring no growth assumptions on the nonlinearities. In addition, we prove a comparison result and an $L^1$-contraction property for the solutions., Comment: 46 pages

We consider a stochastic extension of the nonlocal convective Cahn-Hilliard equation containing an additive Wiener process noise. We first introduce a suitable analytical setting and make some mathematical and physical assumptions. We then establish, in a variational context, the existence of a weak statistical solution for this problem. Finally we prove existence and uniqueness of a strong solution., Comment: 22 pages, no figures. Typos corrected, split Section 7 in two to better outline strategy of proof

We apply the well-known Banach-Necas-Babuska inf-sup theory in a stochastic setting to introduce a weak space-time formulation of the linear stochastic heat equation with additive noise. We give sufficient conditions on the the data and on the covariance operator associated to the driving Wiener process, in order to have existence and uniqueness of the solution. We show the relation of the obtained solution to the so-called mild solution and to the variational solution of the same problem. The spatial regularity of the solution is also discussed. Finally, an extension to the case of linear multiplicative noise is presented., Comment: 19 pages

International audience; We study the Cauchy problem for a semilinear stochastic partial differential equation driven by a finite-dimensional Wiener process. In particular, under the hypothesis that all the coefficients are sufficiently smooth and have bounded derivatives, we consider the equation in the context of power scale generated by a strongly elliptic differential operator. Application of semigroup arguments then yields the existence of a continuous strong solution.

We prove weak existence of Euler equation (or Navier-Stokes equation) perturbed by a multiplicative noise on bounded domains of $\mathbb R^2$ with Dirichlet boundary conditions and with periodic boundary conditions. Solutions are $H^1$ regular. The equations are of transport type.

We prove H��rmander's type hypoellipticity theorem for stochastic partial differential equations when the coefficients are only measurable with respect to the time variable. The need for such kind of results comes from filtering theory of partially observable diffusion processes, when even if the initial system is autonomous, the observation process enters the coefficients of the filtering equation and makes them time-dependent with no good control on the smoothness of the coefficients with respect to the time variable., 23 pages, localization on random events added

International audience; We study the Cauchy problem for a scalar semilinear degenerate parabolic partial differential equation with stochastic forcing. In particular, we are concerned with the well-posedness in any space dimension. We adapt the notion of kinetic solution which is well suited for degenerate parabolic problems and supplies a good technical framework to prove the comparison principle. The proof of existence is based on the vanishing viscosity method: the solution is obtained by a compactness argument as the limit of solutions of nondegenerate approximations.

We give a short proof of It��'s formula for stochastic Hilbert-space valued processes in the setting $V\subset H\subset V^{*}$ based on the possibility to lift the stochastic differentials, which are originally in $V^{*}$, into $H$. Using this result we also prove the maximum principle for second-order SPDEs in arbitrary domains., 20 pages

We prove pathwise uniqueness and strong existence of solutions for stochastic reaction-diffusion systems with locally Lipschitz continuous reaction term of polynomial growth and H��lder continuous multiplicative noise. Under additional assumptions on the coefficients, we also prove positivity of the solutions., 21 pages

We consider the nonlinear stochastic heat equation in one dimension. Under some conditions on the nonlinearity, we show that the "peaks" of the solution are rare, almost fractal like. We also provide an upper bound on the length of the "islands," the regions of large values. These results are obtained by analyzing the correlation length of the solution.

In this paper we give an $L_p$-theory for stochastic parabolic equations with random fractional Laplacian operator. The driving noises are general L��vy processes.

In this paper we study well-posedness of a second order SPDE with multiplicative noise on the torus $\T =[0,2\pi]$. The equation is considered in $L^p(\O\times(0,T);L^q(\T))$ for $p,q\in (1, \infty)$. It is well-known that if the noise is of gradient type, one needs a stochastic parabolicity condition on the coefficients for well-posedness with $p=q=2$. In this paper we investigate whether the well-posedness depends on $p$ and $q$. It turns out that this condition does depend on $p$, but not on $q$. Moreover, we show that if $1, Comment: Revision, accepted for publication in Electronic Journal of Probability

We consider a class of singular perturbations to the stochastic heat equation or semilinear variations thereof. The interesting feature of these perturbations is that, as the small parameter epsilon tends to zero, their solutions converge to the 'wrong' limit, i.e. they do not converge to the solution obtained by simply setting epsilon = 0. A similar effect is also observed for some (formally) small stochastic perturbations of a deterministic semilinear parabolic PDE. Our proofs are based on a detailed analysis of the spatially rough component of the equations, combined with a judicious use of Gaussian concentration inequalities., To appear in PTRF

In this article spatial and temporal regularity of the solution process of a stochastic partial differential equation (SPDE) of evolutionary type with nonlinear multiplicative trace class noise is analyzed., Comment: Published at http://www.sciencedirect.com/science/article/pii/S0022039611003706 in the Journal of Differential Equations

This article introduces a certain class of stochastic processes, which we suggest to call mild Ito processes, and a new - somehow mild - Ito type formula for such processes. Examples of mild Ito processes are mild solutions of SPDEs and their numerical approximation processes., Comment: 39 pages, 0 figures

In this paper, we study the regularities of solutions of nonlinear stochastic partial differential equations in the framework of Hilbert scales. Then we apply our general result to several typical nonlinear SPDEs such as stochastic Burgers and Ginzburg-Landau's equations on the real line, stochastic 2D Navier-Stokes equations in the whole space and a stochastic tamed 3D Navier-Stokes equation in the whole space, and obtain the existence of their respectively smooth solutions., 26Pages

In this paper, we first study the existence-uniqueness and large deviation estimate of solutions for stochastic Volterra integral equations with singular kernels in 2-smooth Banach spaces. Then, we apply them to a large class of semilinear stochastic partial differential equations (SPDE) driven by Brownian motions as well as by fractional Brownian motions, and obtain the existence of unique maximal strong solutions (in the sense of SDE and PDE) under local Lipschitz conditions. Lastly, high order SPDEs in a bounded domain of Euclidean space, second order SPDEs on complete Riemannian manifolds, as well as stochastic Navier-Stokes equations are investigated., Comment: 65Pages

The maximum principle for SPDEs is established in multidimensional $C^{1}$ domains. An application is given to proving the H\"older continuity up to the boundary of solutions of one-dimensional SPDEs., Comment: 26 pages

In this article we study the solution of the Kuramoto-Sivashinsky equation (for surface erosion or surface growth) on a bounded interval subject to a random forcing term. We show that a unique solution to the equation exists for all time and depends continuously on the initial data., To appear in: Nonlinear Analysis