Stationary Solutions of Stochastic Partial Differential Equations in the Space of Tempered Distributions

Abstract

Abstract. In Rajeev (Indian J. Pure Appl. Math. 44 (2013), no. 2, 231-258.), 'Translation invariant diffusion in the space of tempered distributions', it was shown that there is an one-to-one correspondence between solutions of a class of finite dimensional stochastic differential equations (SDEs) and solutions of a class of stochastic parial differential equations (SPDEs) in the space of tempered distributions, which are driven by the same Brownian motion. Coefficients of the SDEs were related to coefficients of the SPDEs through convolution with the initial value of the SPDEs.In this paper, we consider the situation where solutions of the SDEs are stationary and ask whether solutions of the corresponding SPDEs are also stationary. We provide an affirmative answer, when the initial random variable takes value in a certain set C, which ensures that coefficients of the SDEs are related to coefficients of the SPDEs in the above manner.

Translation Invariant Diffusions and Stochastic Partial Differential Equations in ${\cal S}^{\prime}

“…In the context of our results this raises the question of wether the existence of an invariant measure and questions of ergodicity can be answered by randomising both x and y. We refer to [4], Chapter (5), for some results in this direction.…”

“…Our method relies on three ingredients viz. one, a quasi-linear extension of linear differential operators by identifying the coefficients σ ij (x) as a restriction of the functional σ ij , φ , φ ∈ S −p = S ′ p , p > d, σ ij ∈ S p to the distribution φ = δ x ; two, an Itô formula for translations of tempered distributions by semimartingales (see [36], [4], [46]); and finally, the monotonicity inequality (see [5], [20]). Indeed, this last inequality, whose abstract version has been known for some time (see [28], [25], [18]), has proved to be an indispensable tool for proving uniqueness results for SPDE's in the framework of a scale of Hilbert spaces of the type discussed above(see [20], [39]).…”

Invariant manifolds for stochastic partial differential equations in continuously embedded Hilbert spaces

“…In this appendix we provide the required results about Hermite Sobolev spaces. References on this topic are, for example, [20], [22] and [2].…”

“…If the A j are smooth, then necessary and sufficient conditions for local invariance are that (1) The vector fields A j (y) ∈ T y M , the tangent space of M at y, and (2) L(y) − 1 see, for example [11,28], and also [13] for the more general situation of jumpdiffusions and submanifolds with boundary. The conditions (1) and (2) above are referred to in [11] as 'Nagumo type consistency' conditions. However the term 1 2 r j=1 DA j (y)A j (y) in condition (2) can also be viewed as a 'Stratonovic' correction term (see Remark 4.22 and Remark 6.8).…”

“…The conditions (1) and (2) above are referred to in [11] as 'Nagumo type consistency' conditions. However the term 1 2 r j=1 DA j (y)A j (y) in condition (2) can also be viewed as a 'Stratonovic' correction term (see Remark 4.22 and Remark 6.8). While conditions (1) and ( 2) suffice for the quasi-semilinear and semilinear cases mentioned above, they no longer suffice for the Itô type SPDEs.…”