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Fully Nonlinear Stochastic Partial Differential Equations [PDF]

SIAM Journal on Mathematical Analysis, 1996
The authors are concerned with the following stochastic partial differential equation: \[ du(t, .)= L(t, ., u, Du, D^2u) dt+ \langle b(t, .)Du+ h(t, .)u, dW(t) \rangle, \qquad u(0)= u_0, \tag{1} \] where \(L\), \(b\) and \(h\) are suitable functions and \(W\) is an \(\mathbb{R}^N\)-valued Brownian motion.
G. Da Prato, Tubaro, Luciano
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Stochastic Partial Differential Equations and Applications - VII

1989
Weak, Strong, and 4 Semigroup Solutions of Classical SDE: An Example. Feynman Path Integrals for Time-Dependent Potentials. The Irreducibility of Transition Semigroups and Approximate Controllability. Gradient Bounds for Solutions of Elliptic and Parabolic Equations.
G. Da Prato, Tubaro, Luciano
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Stochastic partial differential equations and diffusion processes [PDF]

Russian Mathematical Surveys, 1982
CONTENTS § 1. Introduction § 2. Solubility of the direct and inverse Cauchy problems § 3. The direct equation of inverse diffusion. The method of variation of constants § 4. The method of characteristics. First integrals and the Liouville equations for diffusion processes § 5.
Nicolai V Krylov, Boris Rozovskii
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Stochastic Partial Differential Equations

2003
The purpose of this chapter is to give an introduction to stochastic partial differential equations from a computational point of view. The presented tools provide a consistent quantitative way of relating uncertainty in input to uncertainty in output for PDE-based models.
H. P. Langtangen, H. P. Langtangen, H. Osnes, H. Osnes   +3 more
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Stochastic Partial Differential Equations

2015
A natural generalisation of the finite-dimensional diffusions are stochastic partial differential equations. In this chapter we focus on the Allen-Cahn equation introduced in Section 5.7 in one spatial dimension. Section 5.7 gives the main theorem and a rough outline of its proof.
Frank den Hollander, Anton Bovier
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Stochastic Partial Differential Equations

2007
Preliminaries Introduction Some Examples Brownian Motions and Martingales Stochastic Integrals Stochastic Differential Equations of Ito Type Levy Processes and Stochastic Integrals Stochastic Differential Equations of Levy Type Comments Scalar Equations of First Order Introduction Generalized Ito's Formula Linear Stochastic Equations Quasilinear ...
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Stochastic Partial Differential Equations

2016
So far, we have discussed discrete interface models. Taking their (mesoscopic) continuum limit, as a time evolution of interfaces or some other related physical order parameters, one would expect to obtain stochastic partial differential equations (SPDEs), which are partial differential equations having stochastic terms such as a space-time Gaussian ...
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Stochastic Partial Differential Equations

2012
A brief discussion on the relevance of stochastic partial differential equations (SPDEs) in Sect. 9.1 is followed by a review of the type of SPDEs studied in the mathematical literature (Sect. 9.2). Section 9.3 shows that SPDEs can be solved by the methods in Chaps. 7 and 8 via time and space discretization.
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Stochastic partial differential equations with delays [PDF]

Stochastics, 1982
We consider a rather general type of linear stochastic partial differential equations (PDE's) with delays where stochastic integral terms with respect to Wiener proceses and random measures are present. We prove existence and uniquencess of solutions under a coercivity hypothesis that is the generalization to the delay case of the hypothesis in [10].
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Stochastic Bilinear Partial Differential Equations

1975
We prove existence and uniqueness theorems for a class of partial differential equations with a bilinear stochastic forcing term. We give both white noise and Wiener process [Ito integral] versions and indicate the interrelationships. Another feature is the use of semigroup theory, in contrast to the Lions-Magenes variational theory.
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