Local mild solutions for rough stochastic partial differential equations

Mild Solutions of Neutral Stochastic Partial Functional Differential Equations

Exact Solution for Nonlinear Local Fractional Partial Differential Equations

In this work, we extend the existing local fractional Sumudu decomposition method to solve the nonlinear local fractional partial differential equations. Then, we apply this new algorithm to resolve the nonlinear local fractional gas dynamics equation and nonlinear local fractional Klein-Gordon equation, so we get the desired non-differentiable exact solutions. The steps to solve the examples a...

The Yamada-Watanabe theorem for mild solutions to stochastic partial differential equations

We prove the Yamada-Watanabe Theorem for semilinear stochastic partial differential equations with path-dependent coefficients. The so-called “method of the moving frame” allows us to reduce the proof to the Yamada-Watanabe Theorem for stochastic differential equations in infinite dimensions.

Another Approach to Some Rough and Stochastic Partial Differential Equations

In this note 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 simiplicity – 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 r...

Strong solutions to stochastic differential equations with rough coefficients

We study strong existence and pathwise uniqueness for stochastic differential equations in R with rough coefficients, and without assuming uniform ellipticity for the diffusion matrix. Our approach relies on direct quantitative estimates on solutions to the SDE, assuming Sobolev bounds on the drift and diffusion coefficients, and L bounds for the solution of the corresponding Fokker-Planck PDE,...