In this paper, we investigate the hyperbolic Anderson equation generated by a time-independent Gaussian noise with two objectives: The solvability and intermittency. First, we prove that Dalang's condition is necessary and sufficient for existence of the solution. Second, we establish the precise long time and high moment asymptotics for the solution under the usual homogeneity assumption of the covariance of the Gaussian noise. Our approach is fundamentally different from the ones existing in literature. The main contributions in our approach include the representation of Stratonovich moment under Laplace transform via the moments of the Brownian motions in Gaussian potentials and some large deviation skills developed in dealing effectively with the Stratonovich chaos expansion.

This paper is concerned with a wave equation in dimension $d\in \{1,2, 3\}$, with a multiplicative space-time Gaussian noise which is fractional in time and homogeneous in space. We provide necessary and sufficient conditions on the space-time covariance of the Gaussian noise, allowing the existence and uniqueness of a mild Skorohod solution.

In this note, we establish the bounds cε ^2/3≤ P{∫ _0^1∫ _0^1δ _0(B_s-B̃_r)dsdr≤ε}≤ C ε ^2/3 for the mutual intersection local time of two independent 1-dimensional Brownian motions B and B̃ .

We study a wave equation in dimension $d\in \{1,2\}$ with a multiplicative space-time Gaussian noise. The existence and uniqueness of the Stratonovich solution is obtained under some conditions imposed on the Gaussian noise. The strategy is to develop some Strichartz-type estimates for the wave kernel in weighted Besov spaces, by which we can prove the well-posedness of an associated Young-type equation. Those Strichartz bounds are of independent interest.

UDC 519.21 We provide the Feynman–Kac representation for the parabolic Anderson equations driven by a general Gaussian noise. As a feature of the idea, we can mention the argument of subadditivity in establishing the required exponential integrability.

In this article, we study the stochastic wave equation in all dimensions $d\leq 3$, driven by a Gaussian noise $\dot{W}$ which does not depend on time. We assume that either the noise is white, or the covariance function of the noise satisfies a scaling property similar to the Riesz kernel. The solution is interpreted in the Skorohod sense using Malliavin calculus. We obtain the exact asymptotic behaviour of the $p$-th moment of the solution either when the time is large or when $p$ is large. For the critical case, that is the case when $d=3$ and the noise is white, we obtain the exact transition time for the second moment to be finite.

UDC 519.21 Given the i.i.d. -valued stochastic processes with the stationary increments, a minimal condition is provided for the occupation measure to be absolutely continuous with respect to the Lebesgue measure on An isometry identity related to the resulting density (known as intersection local time) is also established.

The theory of regularity structures enables the definition of the following parabolic Anderson model in a very rough environment: $\partial_{t} u_{t}(x) = \frac12 \Delta u_{t}(x) + u_{t}(x) \, \dot W_{t}(x)$, for $t\in\mathbb{R}_{+}$ and $x\in \mathbb{R}^{d}$, where $\dot W_{t}(x)$ is a Gaussian noise whose space time covariance function is singular. In this rough context, we shall give some information about the moments of $u_{t}(x)$ when the stochastic heat equation is interpreted in the Skorohod as well as the Stratonovich sense. Of special interest is the critical case, for which one observes a blowup of moments for large times.

We construct a K-rough path [along the terminology of Deya (Probab Theory Relat Fields 166:1–65, 2016)] above either a space-time or a spatial fractional Brownian motion, in any space dimension d. This allows us to provide an interpretation and a unique solution for the corresponding parabolic Anderson model, understood in the renormalized sense. We also consider the case of a spatial fractional noise.

This paper concerns the parabolic Anderson equation ∂u ∂t = 1 2 ∆u+ u ∂d+1WH ∂t∂x1 · · · ∂xd generated by a (d + 1)-dimensional fractional noise with the Hurst parameter H = (H0, H1, · · · , Hd) with special interest in the setting that some of H0, · · · , Hd are less than half. In the recent work [9], the case of the spatial roughness has been investigated. To put the last piece of the puzzle in place, this work investigates the case when H0 < 1/2 with the concern on solvability, Feynman-Kac’s moment formula and intermittency of the system. Key-words: parabolic Anderson equation, Dalang’s condition, fractional, rough and critical Gaussian noises, Feynman-Kac’s representation, Brownian motion, moment asymptotics AMS subject classification (2010): 60F10, 60H15, 60H40, 60J65, 81U10. ∗Research partially supported by the Simons Foundation #585506. 1

This paper considers the parabolic Anderson equation partial derivative u/partial derivative t = 1/2 Delta u + u partial derivative W-d+1(H)/partial derivative t partial derivative x(1) ... partial derivative x(d) generated by a (d + 1)-dimensional fractional noise with the Hurst parameter H = (H-0, H-1 (, ...,) H-d). The existence/uniqueness, Feynman-Kac's moment formula and the precise intermittency exponents are formulated in the case when some of H-1, ..., H-d are less than one half, and in the case when the Dalang's condition d - Sigma H-n (k = 1)j < 1 is repeated by d - Sigma H-n (k=1)j = 1. Some partial result is also achieved for the case when H-0 < 1/2 which brings insight on what to expect as the Gaussian noise is rough in time.

We compute the limit of the free energy 1/Nt(N) log E exp {1/N Sigma(1 <= j <= k <= N )integral(tN )(0)gamma(B(j(s) - B)k(s))ds} (N -> infinity) of the mean field generated by the independent Brownian particles {B-j(s)} interacting through the non-negative definite function gamma(.). Our main theorem is relevant to the high moment asymptotics for the parabolic Anderson models with Gaussian noise that is white in time, white or colored in space. Our approach makes a novel connection to the celebrated Donsker-Varadhan's large deviation principle for the i.i.d. random variables in infinite dimensional spaces. As an application of our main theorem, we provide a probabilistic treatment to the Hartree's theory on the asymptotics for the ground state energy of bosonic quantum system.

In this paper, we provide the exact forms of large and moderate deviations for the empirical mean of population and the centered total population of a sub-critical branching process with immigration. The rate functions in our large and moderate deviations are explicitly identified. Our theorems also apply to the models of the integer-valued autoregression. In computing the generating function requested by Gärtner-Ellis theorem, our treatment substantially relies on an algorithm specifically designed for the autoregressive structure of our models.

The aim of this paper is to establish the almost sure asymptotic behavior as the space variable becomes large, for the solution to the one spatial dimensional stochastic heat equation driven by a Gaussian noise which is white in time and which has the covariance structure of a fractional Brownian motion with Hurst parameter greater than 1/4 and less than 1/2 in the space variable.

. In this paper, we consider the parabolic Anderson equation that is driven by a Gaussian noise fractional in time and white or fractional in space, and is solved in a mild sense defined by Skorokhod integral. Our objective is the precise moment Lyapunov exponent and high moment asymptotics. As far as the long term asymptotics are concerned, some feature given in our theorems is different from what have been observed in the Stratonovich-regime and in the setting of the white time noise. While the difference disappears when it comes to the high moment asymptotics. To achieve our goal, we introduce a variational inequality and use some newly developed tools such as time-space LDP of Feynman–Kac type, linearization by tangent approximation, together with some techniques developed along the line of probability in Banach spaces. Résumé. lorsque l’on considère les asymptotiques des grands moments. Nos résultats sont obtenus en introduisant une nouvelle inégalité variationnelle, et à l’aide d’outils nouveaux tels qu’un principe de grandes déviations de type Feynman–Kac, la linéarisation par des approximations tangentes, et des techniques inspirées des probabilités dans les espaces de Banach. MSC:

Partially motivated by the recent papers of Conus, Joseph and Khoshnevisan [Ann. Probab. 41 (2013) 2225-2260] and Conus et al. [Probab. Theory Related Fields 156 (2013) 483-533], this work is concerned with the precise spatial asymptotic behavior for the parabolic Anderson equation{partial derivative u/partial derivative t (t, x) = 1/2 Delta u(t, x) + V (t, x)u(t, x),u(0, x) = u(0)(x),where the homogeneous generalized Gaussian noise V (t, x) is, among other forms, white or fractional white in time and space. Associated with the ColeHopf solution to the KPZ equation, in particular, the precise asymptotic formlim R ->infinity (logR)(-2/3) log max(vertical bar x vertical bar <= R) u(t, x) = 3/4 3 root 2t/3 a.s.is obtained for the parabolic Anderson model partial derivative(t)u = 1/2 partial derivative(2)(xx)u + (W) over dotu with the (1 + 1)-white noise (W) over dot(t, x). In addition, some links between time and space asymptotics for the parabolic Anderson equation are also pursued.

Motivated by the study of the directed polymer model with mobile Poissonian traps or catalysts and the stochastic parabolic Anderson model with time-dependent potential, we investigate the asymptotic behavior of 𝔼⊗𝔼_0exp{± θ∫ ^t_0V̅(s,B_s)ds} (t→∞ ) where θ >0 is a constant, V is the renormalized Poisson potential of the form V(s,x)=∫ _ℝ^d1/|y-x|^p( ω _s(dy)-dy) , and ω _s is the measure-valued process consisting of independent Brownian particles whose initial positions form a Poisson random measure on ℝ^d with Lebesgue measure as its intensity. Different scaling limits are obtained according to the parameter p and dimension d . For the logarithm of the negative exponential moment, the range of d/22) , the exponential moments become infinite for all t>0 .

The moment Lyapunov exponent is computed for the solution of the parabolic Anderson equation with an (1 + 1)-dimensional time-space white noise. Our main result positively confirms an open problem posted in (Ann. Probab. (2015) to appear) and originated from the observations made in the physical literature (J. Statist. Phys. 78 (1995) 1377-1401) and (Nuclear Physics B 290 (1987) 582-602). By a link through the Feynman-Kac's formula, our theorem leads to the evaluation of the ground state energy for the n-body problem with Dirac pair interaction.