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Register a new userFrechet differentiability of mild solutions to SPDEs with respect to the initial datum
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We establish n-th order Frechet differentiability with respect to the initial datum of mild solutions to a class of jump-diffusions in Hilbert spaces. In particular, the coefficients are Lipschitz continuous, but their derivatives of order higher than one can grow polynomially, and the (multiplicative) noise sources are a cylindrical Wiener process and a quasi-left-continuous integer-valued random measure. As preliminary steps, we prove well-posedness in the mild sense for this class of equations, as well as first-order G^ateaux differentiability of their solutions with respect to the initial datum, extending previous results in several ways. The differentiability results obtained here are a fundamental step to construct classical solutions to non-local Kolmogorov equations with sufficiently regular coefficients by probabilistic means.
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We consider semilinear stochastic evolution equations on Hilbert spaces with multiplicative Wiener noise and linear drift term of the type $A + varepsilon G$, with $A$ and $G$ maximal monotone operators and $varepsilon$ a small parameter, and study the differentiability of mild solutions with respect to $varepsilon$. The operator $G$ can be a singular perturbation of $A$, in the sense that its domain can be strictly contained in the domain of $A$.
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 Holder 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 spaces can be classical as well as given by various fractal structures. The whole approach is low dimensional and works for spectral dimensions less than 4.
We start by introducing a new definition of solutions to heat-based SPDEs driven by space-time white noise: SDDEs (stochastic differential-difference equations) limits solutions. In contrast to the standard direct definition of SPDEs solutions; this new notion, which builds on and refines our SDDEs approach to SPDEs from earlier work, is entirely based on the approximating SDDEs. It is applicable to, and gives a multiscale view of, a variety of SPDEs. We extend this approach in related work to other heat-based SPDEs (Burgers, Allen-Cahn, and others) and to the difficult case of SPDEs with multi-dimensional spacial variable. We focus here on one-spacial-dimensional reaction-diffusion SPDEs; and we prove the existence of a SDDEs limit solution to these equations under less-than-Lipschitz conditions on the drift and the diffusion coefficients, thus extending our earlier SDDEs work to the nonzero drift case. The regularity of this solution is obtained as a by-product of the existence estimates. The uniqueness in law of our SPDEs follows, for a large class of such drifts/diffusions, as a simple extension of our recent Allen-Cahn uniqueness result. We also examine briefly, through order parameters $epsilon_1$ and $epsilon_2$ multiplied by the Laplacian and the noise, the effect of letting $epsilon_1,epsilon_2to 0$ at different speeds. More precisely, it is shown that the ratio $epsilon_2/epsilon_1^{1/4}$ determines the behavior as $epsilon_1,epsilon_2to 0$.
We use Yosida approximation to find an It^o formula for mild solutions $left{X^x(t), tgeq 0right}$ of SPDEs with Gaussian and non-Gaussian coloured noise, the non Gaussian noise being defined through compensated Poisson random measure associated to a Levy process. The functions to which we apply such It^o formula are in $C^{1,2}([0,T]times H)$, as in the case considered for SDEs in [9]. Using this It^o formula we prove exponential stability and exponential ultimate boundedness properties in mean square sense for mild solutions. We also compare such It^o formula to an It^o formula for mild solutions introduced by Ichikawa in [8], and an It^o formula written in terms of the semigroup of the drift operator [11] which we extend before to the non Gaussian case.
We study the nonlinear stochastic heat equation driven by space-time white noise in the case that the initial datum $u_0$ is a (possibly signed) measure. In this case, one cannot obtain a mild random-field solution in the usual sense. We prove instead that it is possible to establish the existence and uniqueness of a weak solution with values in a suitable function space. Our approach is based on a construction of a generalized definition of a stochastic convolution via Young-type inequalities.
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