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

Abstract

In this article we show that the ordinary stochastic differential equations of K.Itô maybe considered as part of a larger class of second order stochastic PDE's that are quasi linear and have the property of translation invariance. We show using the 'monotonicity inequality' and the Lipshitz continuity of the coefficients σ ij and b i , existence and uniqueness of strong solutions for these stochastic PDE's. Using pathwise uniqueness, we prove the strong Markov property.

Stochastic PDEs in 𝒮' for SDEs driven by Lévy noise

“…As an application of [15, Theorem 1], we get the required existence and pathwise uniqueness in this case. Extension to arbitrary initial condition φ can be done as in [26].…”

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

“…The problem, related to the computability of 'forward interest rate models', is also known as the 'consistency problem' for such models [12]. In this paper we study the mathematical problem of finding invariant submanifolds for a class of SPDEs that include apart from the quasi-semilinear and semilinear SPDEs (see, for example [11,28,41]) a more recent class of SPDEs studied in [33,34]. We will refer to this latter class as Itô type SPDEs.…”

“…While diffusions on manifolds in R d is a well studied topic (see for a partial list [19,8,9,17,18,38]), the submanifolds considered above in [5,11,28] are finite dimensional submanifolds of a single Hilbert space H. On the other hand the Itô type SPDEs have the following feature: The solution (Y t ) lies in a Hilbert space G ⊂ H and the equation holds in H since the mappings L, A j : G → H. In the framework of [33,34], the spaces G and H are realized as G = S p+1 (R d ) and H = S p (R d ) where S p (R d ), p ∈ R are the Hermite Sobolev spaces and L, A j are quasi-linear second and first order differential operators respectively. In the present paper, our analysis is carried out in a framework that we call a (G, H)-submanifold, where G is continuously embedded in H; see Section 3 (in particular Definition 3.20) for details.…”

“…An important feature of the solutions (Y t ) of the SPDE considered in [33,34] is that they are translation invariant i.e. Y t = τ Xt Φ, where (X t ) is a R d -valued diffusion and Φ ∈ G. In other words, Y t = ψ(X t ) where ψ(x) := τ x Φ : R d → G := S p+1 (R d ) is the orbit map associated with the translation operators τ x : S p (R d ) → S p (R d ).…”