We consider nonlinear parabolic SPDEs of the form ∂tu = Lu+σ(u)ẇ, whereẇ denotes space-time white noise, σ : R → R is [globally] Lipschitz continuous, and L is the L 2 -generator of a Lévy process. We present precise criteria for existence as well as uniqueness of solutions. More significantly, we prove that these solutions grow in time with at most a precise exponential rate. We establish also that when σ is globally Lipschitz and asymptotically sublinear, the solution to the nonlinear heat equation is "weakly intermittent," provided that the symmetrization of L is recurrent and the initial data is sufficiently large.Among other things, our results lead to general formulas for the upper second-moment Liapounov exponent of the parabolic Anderson model for L in dimension (1 + 1). When L = κ∂xx for κ > 0, these formulas agree with the earlier results of statistical physics [25,29,30], and also probability theory [1,5] in the two exactly-solvable cases where u0 = δ0 and u0 ≡ 1.

We consider the stochastic heat equation of the following form: ∂ ∂ t u t ( x ) = ( L u t ) ( x ) + b ( u t ( x ) ) + σ ( u t ( x ) ) F ˙ t ( x ) for  t > 0 ,   x ∈ R d , \begin{equation*} \frac {\partial }{\partial t}u_t(x) = (\mathcal {L} u_t)(x) +b(u_t(x)) + \sigma (u_t(x))\dot {F}_t(x)\quad \text {for }t>0,\ x\in \mathbf {R}^d, \end{equation*} where L \mathcal {L} is the generator of a Lévy process and F ˙ \dot {F} is a spatially-colored, temporally white, Gaussian noise. We will be concerned mainly with the long-term behavior of the mild solution to this stochastic PDE. For the most part, we work under the assumptions that the initial data u 0 u_0 is a bounded and measurable function and σ \sigma is nonconstant and Lipschitz continuous. In this case, we find conditions under which the preceding stochastic PDE admits a unique solution which is also weakly intermittent. In addition, we study the same equation in the case where L u \mathcal {L}u is replaced by its massive/dispersive analogue L u − λ u \mathcal {L}u-\lambda u , where λ ∈ R \lambda \in \mathbf {R} . We also accurately describe the effect of the parameter λ \lambda on the intermittence of the solution in the case where σ ( u ) \sigma (u) is proportional to u u [the “parabolic Anderson model”]. We also look at the linearized version of our stochastic PDE, that is, the case where σ \sigma is identically equal to one [any other constant also works]. In this case, we study not only the existence and uniqueness of a solution, but also the regularity of the solution when it exists and is unique.

Consider non-linear time-fractional stochastic heat type equations of the following type,The operator ∂ β t is the Caputo fractional derivative while −(− ) α/2 is the generator of an isotropic stable process andis the Riesz fractional integral operator. The forcing noise denoted by · F (t, x) is a Gaussian noise. And the multiplicative non-linearity σ : R → R is assumed to be globally Lipschitz continuous. Mijena and Nane (Stochastic Process Appl 125(9):3301-3326, 2015) have introduced these time fractional SPDEs. These types of time fractional stochastic heat type equations can be used to model phenomenon with random effects with thermal memory. Under suitable conditions on the initial function, we study the asymptotic behaviour of the solution with respect to time and the parameter λ. In particular, our results are significant extensions of those in Ann Probab (to appear), Khoshnevisan (Electron J Probab 14(21): 548-568, 2009), Mijena andNane (2015) and Mijena and Nane (Potential Anal 44:295-312, 2016). Along the way, we prove a number of interesting properties about the deterministic counterpart of the equation.

Abstract. It is frequently the case that a white-noise-driven parabolic and/or hyperbolic stochastic partial differential equation (SPDE) can have randomfield solutions only in spatial dimension one. Here we show that in many cases, where the "spatial operator" is the L 2 -generator of a Lévy process X, a linear SPDE has a random-field solution if and only if the symmetrization of X possesses local times. This result gives a probabilistic reason for the lack of existence of random-field solutions in dimensions strictly larger than one.In addition, we prove that the solution to the SPDE is [Hölder] continuous in its spatial variable if and only if the said local time is [Hölder] continuous in its spatial variable. We also produce examples where the random-field solution exists, but is almost surely unbounded in every open subset of space-time. Our results are based on first establishing a quasi-isometry between the linear L 2 -space of the weak solutions of a family of linear SPDEs, on one hand, and the Dirichlet space generated by the symmetrization of X, on the other hand.We mainly study linear equations in order to present the local-time correspondence at a modest technical level. However, some of our work has consequences for nonlinear SPDEs as well. We demonstrate this assertion by studying a family of parabolic SPDEs that have additive nonlinearities. For those equations we prove that if the linearized problem has a randomfield solution, then so does the nonlinear SPDE. Moreover, the solution to the linearized equation is [Hölder] continuous if and only if the solution to the nonlinear equation is, and the solutions are bounded and unbounded together as well. Finally, we prove that in the cases where the solutions are unbounded, they almost surely blow up at exactly the same points.

Consider the following stochastic partial differential equation,where ξ is a positive parameter and σ is a globally Lipschitz continuous function. The stochastic forcing termḞ (t, x) is white in time but possibly colored in space. The operator L is a non-local operator. We study the behaviour of the solution with respect to the parameter ξ, extending the results in [8] and [11].