scholarly journals Finite dimensional reducing and smooth approximating for a class of stochastic partial differential equations

2020 ◽  
Vol 25 (4) ◽  
pp. 1565-1581
Author(s):  
Min Yang ◽  
◽  
Guanggan Chen
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Partial Differential ◽  
Finite Dimensional

scholarly journals Variational principles for stochastic soliton dynamics

2016 ◽  
Vol 472 (2187) ◽  
pp. 20150827 ◽  
Author(s):  
Darryl D. Holm ◽  
Tomasz M. Tyranowski
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Variational Principles ◽  
Soliton Solutions ◽  
Stochastic Modelling ◽  
Partial Differential ◽  
Stochastic Perturbations ◽  
Finite Dimensional ◽  
Stochastically Perturbed

We develop a variational method of deriving stochastic partial differential equations whose solutions follow the flow of a stochastic vector field. As an example in one spatial dimension, we numerically simulate singular solutions (peakons) of the stochastically perturbed Camassa–Holm (CH) equation derived using this method. These numerical simulations show that peakon soliton solutions of the stochastically perturbed CH equation persist and provide an interesting laboratory for investigating the sensitivity and accuracy of adding stochasticity to finite dimensional solutions of stochastic partial differential equations. In particular, some choices of stochastic perturbations of the peakon dynamics by Wiener noise (canonical Hamiltonian stochastic deformations, CH-SD) allow peakons to interpenetrate and exchange order on the real line in overtaking collisions, although this behaviour does not occur for other choices of stochastic perturbations which preserve the Euler–Poincaré structure of the CH equation (parametric stochastic deformations, P-SD), and it also does not occur for peakon solutions of the unperturbed deterministic CH equation. The discussion raises issues about the science of stochastic deformations of finite-dimensional approximations of evolutionary partial differential equation and the sensitivity of the resulting solutions to the choices made in stochastic modelling.

THE MONOTONICITY INEQUALITY FOR LINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

2009 ◽  
Vol 12 (04) ◽  
pp. 575-591 ◽  
Author(s):  
L. GAWARECKI ◽  
V. MANDREKAR ◽  
B. RAJEEV
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Evolution Equations ◽  
Differential Operators ◽  
Probabilistic Representation ◽  
Partial Differential ◽  
Finite Dimensional ◽  
Monotonicity Inequality ◽  
Elliptic Differential Operators

We prove the monotonicity inequality for differential operators A and L that occur as coefficients in linear stochastic partial differential equations associated with finite-dimensional Itô processes. We characterize the solutions of such equations. A probabilistic representation is obtained for solutions to a class of evolution equations associated with time dependent, possibly degenerate, second-order elliptic differential operators.

scholarly journals ANOTHER APPROACH TO SOME ROUGH AND STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

2011 ◽  
Vol 11 (02n03) ◽  
pp. 535-550 ◽  
Author(s):  
JOSEF TEICHMANN
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Solution Process ◽  
Partial Differential ◽  
State Spaces ◽  
New Approach ◽  
Finite Dimensional ◽  
Rough Path ◽  
Rough Path Theory

In this paper, 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 simplicity — 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 respective R- and SPDEs. As a consequence of our construction, we can apply the pool of results of rough path theory, in particular we can obtain strong and weak numerical schemes of high order converging to the solution process.

scholarly journals Existence of weak solutions to SPDEs with fractional Laplacian and non-Lipschitz coefficients

2021 ◽  
Author(s):  
Shohei Nakajima
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Weak Solutions ◽  
Stochastic Partial Differential Equations ◽  
Fractional Laplacian ◽  
Existence Of Solutions ◽  
Partial Differential ◽  
Continuity Theorem ◽  
Existence Of Weak Solutions ◽  
One Dimensional

AbstractWe prove existence of solutions and its properties for a one-dimensional stochastic partial differential equations with fractional Laplacian and non-Lipschitz coefficients. The method of proof is eatablished by Kolmogorov’s continuity theorem and tightness arguments.

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