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Register a new userSchauder-type estimates for higher-order parabolic SPDEs
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In this paper we consider the Cauchy problem for $2m$-order stochastic partial differential equations of parabolic type in a class of stochastic Hoelder spaces. The Hoelder estimates of solutions and their spatial derivatives up to order $2m$ are obtained, based on which the existence and uniqueness of solution is proved. An interesting finding of this paper is that the regularity of solutions relies on a coercivity condition that differs when $m$ is odd or even: the condition for odd $m$ coincides with the standard parabolicity condition in the literature for higher-order stochastic partial differential equations, while for even $m$ it depends on the integrability index $p$. The sharpness of the new-found coercivity condition is demonstrated by an example.
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Under various conditions, we establish Schauder estimates for both divergence and non-divergence form second-order elliptic and parabolic equations involving Holder semi-norms not with respect to all, but only with respect to some of the independent variables. A novelty of our results is that the coefficients are allowed to be merely measurable with respect to the other independent variables.
In this note, we use the non-homogeneous Poisson stochastic process to show how knowing Schauder and Sobolev estimates for the one-dimensional heat equation allows one to derive their multidimensional analogs. The method is probability. We generalize the result of Krylov-Priola [7].
The study of intermittency for the parabolic Anderson problem usually focuses on the moments of the solution which can describe the high peaks in the probability space. In this paper we set up the equation on a finite spatial interval, and study the other part of intermittency, i.e., the part of the probability space on which the solution is close to zero. This set has probability very close to one, and we show that on this set, the supremum of the solution over space is close to 0. As a consequence, we find that almost surely the spatial supremum of the solution tends to zero exponentially fast as time increases. We also show that if the noise term is very large, then the probability of the set on which the supremum of the solution is very small has a very high probability.
Consider an infinite system [partial_tu_t(x)=(mathscr{L}u_t)(x)+ sigmabigl(u_t(x)bigr)partial_tB_t(x)] of interacting It^{o} diffusions, started at a nonnegative deterministic bounded initial profile. We study local and global features of the solution under standard regularity assumptions on the nonlinearity $sigma$. We will show that, locally in time, the solution behaves as a collection of independent diffusions. We prove also that the $k$th moment Lyapunov exponent is frequently of sharp order $k^2$, in contrast to the continuous-space stochastic heat equation whose $k$th moment Lyapunov exponent can be of sharp order $k^3$. When the underlying walk is transient and the noise level is sufficiently low, we prove also that the solution is a.s. uniformly dissipative provided that the initial profile is in $ell^1(mathbf {Z}^d)$.
We obtain Dini and Schauder type estimates for concave fully nonlinear nonlocal parabolic equations of order $sigmain (0,2)$ with rough and non-symmetric kernels, and drift terms. We also study such linear equations with only measurable coefficients in the time variable, and obtain Dini type estimates in the spacial variable. This is a continuation of the work [10, 11] by the first and last authors.
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