Markov Selections for the Magnetohydrodynamics and the Hall-Magnetohydrodynamics Systems

The magnetohydrodynamics system has found rich applications in applied sciences and has history of intense investigation by researchers including mathematicians. On the other hand, the Hall-magnetohydrodynamics system consists of the former system added by a Hall term which is very singular, and the mathematical development on this system has seen its significant progress only in the last several years. In short, the magnetohydrodynamics system, similarly to the Navier–Stokes equations, is semilinear, while the Hall-magnetohydrodynamics system is quasilinear. In this manuscript, we consider the three-dimensional magnetohydrodynamics, as well as the Hall-magnetohydrodynamics systems, and prove that they have Markov selections. In the case of the magnetohydrodynamics system, we prove furthermore a weak–strong uniqueness result and that any Markov solution has the strong Feller property for regular initial conditions. Consequently, it is deduced that if the magnetohydrodynamics system is well posed starting from one initial data, then it is well posed starting from any initial data, verifying a sharp contrast to the deterministic case in which the well posedness for all time with small initial data is well known. In the case of the Hall-magnetohydrodynamics system, we are also able to prove the weak–strong uniqueness result; however, the proof of strong Feller property seems to break down due to the singularity of the Hall term, about which we elaborate in detail.

The author is grateful to Dr. Manil T. Mohan for fruitful discussions. The author expresses deep gratitude to the editor and the anonymous referees for valuable suggestions and comments which improved the manuscript significantly.

Authors and Affiliations

1. Department of Mathematics, University of Rochester, Rochester, NY, 14627, USA

   Kazuo Yamazaki

Corresponding author

Correspondence to Kazuo Yamazaki.

Communicated by Dr. Paul Newton.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix A: Definitions and Preliminaries Results

Here we leave general definitions and results on which the proofs throughout this manuscript relied. We let \({\mathcal {V}}, {\mathcal {H}}\) be separable Hilbert spaces such that \({\mathcal {V}} \subset {\mathcal {H}} \subset {\mathcal {V}}'\) with continuous injections. We fix

with \({\mathcal {B}}\) the Borel \(\sigma \)-field on \(\Omega \). We define \(\xi _{t}: \Omega \mapsto {\mathcal {V}}'\) by \(\xi _{t}(\omega ) = \omega (t)\). We denote for all \(t \ge 0\),

(cf. \({\mathcal {M}}_{t}, {\mathcal {M}}^{t}\) in Stroock and Varadhan 1997, p. 30, 139) and a mapping \(\Phi _{t}: \Omega \mapsto \Omega ^{t}\) for any \(t > 0\) as

Lemma 6.1

(Flandoli and Romito 2008, Lemma 2.1; see also Strauss 1966) The sets \(L^{\infty }_{loc} ([0,\infty ); {\mathcal {H}}) \cap \Omega \) and \(L_{loc}^{2}([0,\infty ); {\mathcal {V}}) \cap \Omega \) are both Borel sets in \(\Omega \). Moreover,

where \({\mathcal {H}}_{\sigma }\) is \({\mathcal {H}}\) endowed with weak topology.

Definition 6.1

1. (1)

   Since \(\Omega = C([0,\infty ); {\mathcal {V}}')\) is a Polish space and \({\mathcal {B}}_{t}\) is finitely generated, for any \(P \in Pr (\Omega )\) and \(t >0\), we may define, uniquely up to P-null sets, a regular conditional probability distribution (r.c.p.d.) \(\omega \mapsto P|_{{\mathcal {B}}_{t}}^{\omega }\) from \(\Omega \) to \(Pr(\Omega ^{t})\). In particular,

   $$\begin{aligned} \begin{aligned}&P |_{{\mathcal {B}}_{t}}^{\omega } (\{\xi _{t} = \omega (t)\}) = 1 \quad \forall \,\, \omega \in \Omega , \\&P(E_{1} \cap E_{2}) = \int _{E_{1}} P |_{{\mathcal {B}}_{t}}^{\omega } (E_{2}) P(d \omega ) \quad \forall \,\, E_{1} \in {\mathcal {B}}_{t}, E_{2} \in {\mathcal {B}}^{t}. \end{aligned} \end{aligned}$$

   (72)
2. (2)

   (Flandoli and Romito 2008, Definition 2.3; see also Stroock and Varadhan 1997, Lemma 6.1.1, Lemma 6.1.2) Let \(P \in Pr (\Omega ), t > 0\) and \({\mathcal {B}}_{t}\)-measurable map \(Q: \Omega \mapsto Pr(\Omega ^{t})\) satisfy \(Q_{\omega } ( \xi _{t} = \omega (t) ) = 1\) for all \(\omega \in \Omega \). Then we denote by \(P \otimes _{t}Q\) the unique probability measure on \(\Omega \) such that

   1. (a)

      \(P \otimes _{t}Q\) and P agree on \({\mathcal {B}}_{t}\),

   2. (b)

      \((Q_{\omega })_{\omega \in \Omega }\) is a r.c.p.d. of \(P \otimes _{t}Q\) on \({\mathcal {B}}_{t}\).

Next, we give a definition of a.e. Markov property. We point out again that it was actually called an a.s. Markov property in Flandoli and Romito (2008, Definition 2.4); however, it was brought to the author’s attention by a referee that an a.e. Markov property would be more appropriate considering that the exceptional set is a set of t’s rather than \(\omega \)’s.

Definition 6.2

(Flandoli and Romito 2008, p. 413, Definition 2.4)

1. (1)

   A family of probability measures (\(P_{x})_{x \in {\mathcal {H}}}\) is said to possess the Markov property if for all \(x \in {\mathcal {H}}\) and \(t \ge 0\), \(P_{x} |_{{\mathcal {B}}_{t}}^{\omega } = \Phi _{t} P_{\omega (t)}\) for \(P_{x}\)-almost every \(\omega \in \Omega \) where \(\Phi _{t}(\omega )(s) = \omega (s-t)\) for \(s \ge t\) as defined in (71).

2. (2)

   Let \((P_{x})_{x \in {\mathcal {H}}}\) be a family of probability measures such that a mapping \(x \mapsto P_{x}\) is measurable from \({\mathcal {H}}\) to \(Pr(\Omega )\) and it satisfies \(P_{x} (C([0,\infty ); {\mathcal {H}}_{\sigma }) \cap \Omega ) = 1\) for all \(x \in {\mathcal {H}}\), where \({\mathcal {H}}_{\sigma }\) denotes \({\mathcal {H}}\) endowed with the weak topology. Then \((P_{x})_{x \in {\mathcal {H}}}\) is said to possess the a.e. Markov property if for all \(x \in {\mathcal {H}}\), there exists a set \(T \subset (0,\infty )\) with null Lebesgue measure such that \(P_{x} |_{{\mathcal {B}}_{t}}^{\omega } = \Phi _{t} P_{\omega (t)}\) for \(P_{x}\)-almost every \(\omega \in \Omega \), for all \(t \notin T\).

We recall from Sect. 3 that \(Comp(Pr(\Omega ))\) is the family of all compact subsets of \(Pr(\Omega )\).

Definition 6.3

(Flandoli and Romito 2008, Definition 2.5; see also Stroock and Varadhan 1997, Theorem 12.2.4) Let \({\mathcal {C}}: {\mathcal {H}} \mapsto Comp(Pr(\Omega ))\) be a map that is Borel measurable and for all \(x \in {\mathcal {H}}\) and \(P \in {\mathcal {C}}(x)\), \(P ( C([0,\infty ); {\mathcal {H}}_{\sigma } ) \cap \Omega ) = 1\). The family \(({\mathcal {C}}(x))_{x \in {\mathcal {H}}}\) is said to be almost surely pre-Markov if for all \(x \in {\mathcal {H}}, P \in {\mathcal {C}}(x)\), there exists \(T \in (0, \infty )\) of null Lebesgue measure such that for all \(t \notin T\), the following properties hold:

1. (1)

   (disintegration) there exists \(N \in {\mathcal {B}}_{t}\) of null P-measure such that for all \(\omega \notin N\), it holds that \(\omega (t) \in {\mathcal {H}}\) and \(P |_{{\mathcal {B}}_{t}}^{\omega } \in \Phi _{t} {\mathcal {C}}(\omega (t))\),

2. (2)

   (reconstruction) for all \({\mathcal {B}}_{t}\)-measurable map \(\omega \mapsto Q_{\omega }\) from \(\Omega \) to \(Pr(\Omega ^{t})\) for which there exists \(N \in {\mathcal {B}}_{t}\) of null P-measure such that for all \(\omega \notin N, \omega (t) \in {\mathcal {H}}\) and \(Q_{\omega } \in \Phi _{t} {\mathcal {C}}(\omega (t))\), it holds that \(P \otimes _{t} Q \in {\mathcal {C}}(x)\).

Lemma 6.2

(Flandoli and Romito 2008, Theorem 2.8; see also Stroock and Varadhan 1997, Theorem 12.2.3) Let \(({\mathcal {C}}(x))_{x \in {\mathcal {H}}}\) be an a.e. pre-Markov family with non-empty convex values. Then, there exists a measurable mapping \(x \mapsto P_{x}\) on \({\mathcal {H}}\) with values in \(Pr(\Omega )\) such that \(P_{x} \in {\mathcal {C}}(x)\) for all \(x \in {\mathcal {H}}\) and \((P_{x})_{x \in {\mathcal {H}}}\) has the a.e. Markov property.

Lemma 6.3

(Flandoli and Romito 2008, Theorem 2.12) Let \(({\mathcal {C}}(x))_{x \in {\mathcal {H}}}\) be a pre-Markov family with non-empty convex values. Then there exists a measurable mapping \(x \mapsto P_{x}\) in \({\mathcal {H}}\) with values in \(Pr(\Omega )\) such that \(P_{x} \in {\mathcal {C}}(x)\) for all \(x \in {\mathcal {H}}\) and \((P_{x})_{x \in {\mathcal {H}}}\) has the Markov property.

We let \(B_{b}({\mathcal {H}})\) be the Banach space of all bounded Borel measurable mappings, \((P_{x})_{x \in {\mathcal {H}}}\) be an a.e. Markov process on \((\Omega , {\mathcal {B}})\) and \(({\mathcal {P}}_{t})_{t \ge 0}\) the associated transition semigroup on \(B_{b}({\mathcal {H}})\) defined by

The operator \({\mathcal {P}}_{t}: B_{b}({\mathcal {H}}) \mapsto B_{b}({\mathcal {H}})\) satisfies

1. (1)

   \({\mathcal {P}}_{0} = Id\),

2. (2)

   \(||{\mathcal {P}}_{t} ||_{{\mathcal {L}} (B_{b}({\mathcal {H}}), B_{b}({\mathcal {H}}))} = 1\),

but they are not semigroups as \({\mathcal {P}}_{t+s} = {\mathcal {P}}_{t} {\mathcal {P}}_{s}\) for all \(t \ge 0\) and almost every \(s \ge 0\). Let \({\mathcal {W}}\) be a Banach space where \({\mathcal {W}} \hookrightarrow {\mathcal {H}}\) continuously.

Definition 6.4

(Flandoli and Romito 2008, Definition 5.2) A semigroup \((\tilde{{\mathcal {P}}}_{t})_{t \ge 0}\) on \(B_{b}({\mathcal {H}})\) is \({\mathcal {W}}\)-strong Feller if for all \(t > 0, \psi \in B_{b}({\mathcal {H}})\), \(\tilde{{\mathcal {P}}}_{t} \psi \in C_{b}({\mathcal {W}})\), i.e. \(\tilde{{\mathcal {P}}}_{t} B_{b}({\mathcal {H}}) \subset C_{b}({\mathcal {W}})\).

Definition 6.5

(Flandoli and Romito 2008, Definition 5.3) A family \(({\tilde{P}}_{x})_{x \in {\mathcal {H}}}\) of probability measure on \((\Omega , {\mathcal {B}})\) is an a.s. \({\mathcal {W}}\)-Markov process if

1. (1)

   \({\tilde{P}}_{x} (C([0,\infty ); {\mathcal {W}})) = 1\) for all \(x \in {\mathcal {W}}\),

2. (2)

   for all \(t \ge 0\) and \(\phi \in B_{b}({\mathcal {H}})\), the mapping \(x \mapsto (\tilde{{\mathcal {P}}}_{t} \phi )(x) \triangleq {\mathbb {E}}^{{\tilde{P}}_{x}}[\phi (\xi _{t})]\) is Borel measurable,

3. (3)

   for all \(t \ge 0\), and almost every \(s \ge 0\), \(\tilde{{\mathcal {P}}}_{t+s} \phi = \tilde{{\mathcal {P}}}_{t} \tilde{{\mathcal {P}}}_{s} \phi \) for all \(\phi \in B_{b}({\mathcal {H}})\).

Lemma 6.4

(Flandoli and Romito 2008, Theorem 5.4) Let \((P_{x})_{x \in {\mathcal {H}}}\) be an a.e. Markov process on \((\Omega , {\mathcal {B}})\) and for all \(R > 0\), \((P_{x}^{(R)})_{x \in {\mathcal {W}}}\) an a.s. \({\mathcal {W}}\)-Markov process on \((\Omega , {\mathcal {B}})\) as defined in Definition 6.5. Suppose for all \(\rho > 0,\) there exists \(R_{\rho } > 0\) and a random time \(\tau _{\rho }\) on \((\Omega , {\mathcal {B}})\) such that for all \(x \in {\mathcal {W}}\) such that \(||x ||_{{\mathcal {W}}} \le \rho \),

1. (1)

   \(\lim _{\epsilon \rightarrow 0} P_{x+h}^{(R_{\rho })} (\tau _{\rho } \ge \epsilon ) = 1\) uniformly in \(h \in {\mathcal {W}}\) such that \(||h ||_{{\mathcal {W}}} \le 1\),

2. (2)

   for all \(t \ge 0, \phi \in B_{b}({\mathcal {H}})\), \({\mathbb {E}}^{P_{x}^{(R_{\rho })}}[\phi (\xi _{t})1_{\{ \tau _{\rho } \ge t \}}] = {\mathbb {E}}^{P_{x}}[ \phi (\xi _{t}) 1_{\{ \tau _{\rho } \ge t \}}]\).

Then if \(({\mathcal {P}}_{t}^{(R)})_{t \ge 0}\) is \({\mathcal {W}}\)-strong Feller for all \(R > 0\), then \(({\mathcal {P}}_{t})_{t \ge 0}\) is \({\mathcal {W}}\)-strong Feller.

Definition 6.6

A Borel probability measure \(\mu \) on H is fully supported on \({\mathcal {W}}\) if \(\mu (U) > 0\) for all \(U \subset {\mathcal {W}}\).

Lemma 6.5

(Flandoli and Romito 2008, Proposition B.1) Given \({\mathbb {P}} \in Pr(\Omega )\), two continuous adapted processes \(\theta , \zeta : [0,\infty ) \times \Omega \mapsto {\mathbb {R}}\) and \(t_{0} \ge 0\), the following are equivalent:

1. (1)

   \((\theta _{t}, {\mathcal {B}}_{t}, {\mathbb {P}})_{t \ge t_{0}}\) is a \({\mathbb {P}}\)-square-integrable martingale with quadratic variation \((\zeta _{t})_{t \ge t_{0}}\);

2. (2)

   there exists a \({\mathbb {P}}\)-null set \(N \in {\mathcal {B}}_{t_{0}}\) such that for all \(\omega \notin N\), the process \((\theta _{t}, {\mathcal {B}}_{t}, {\mathbb {P}} |_{{\mathcal {B}}_{t}}^{\omega })_{t \ge t_{0}}\) is a \({\mathbb {P}}|_{{\mathcal {B}}_{t}}^{\omega }\)-square-integrable martingale with quadratic variation \((\zeta _{t})_{t \ge t_{0}}\) and \({\mathbb {E}}^{{\mathbb {P}}}[{\mathbb {E}}^{{\mathbb {P}} |_{{\mathcal {B}}_{t}}^{\cdot }} [ \zeta _{t} ]] < \infty \).

Lemma 6.6

(Flandoli and Romito 2008, Corollary B.3) Let \(\theta : [0,\infty ) \times \Omega \mapsto {\mathbb {R}}\) be an adapted left lower semicontinuous process and assume that \((\theta _{t}, {\mathcal {B}}_{t}, {\mathbb {P}})_{t \ge 0}\) be an a.e. super-martingale. Let \(T_{\theta }\) be the set of exceptional times of \(\theta \) and \([a,b] \subset [0,\infty )\) with \(a, b \notin T_{\theta }\). Moreover, suppose that \(\theta _{t} = \alpha _{t} - \beta _{t}\) where \(\alpha _{t}, \beta _{t}\) are both positive, and \(\beta _{t}\) is non-decreasing. Then

for any \(\lambda > 0\).

Lemma 6.7

(Flandoli and Romito 2008, Proposition B.4) Let \(\alpha , \beta : [0,\infty ) \times \Omega \mapsto {\mathbb {R}}_{+}\) be two adapted processes such that \(\beta \) is non-decreasing and \(\theta = \alpha - \beta \) is left lower semicontinuous. Given \({\mathbb {P}} \in Pr(\Omega )\) and \(t_{0} \ge 0\), the following are equivalent.

1. (1)

   \((\theta _{t}, {\mathcal {B}}_{t}, {\mathbb {P}})_{t \ge t_{0}}\) is an a.e. super-martingale and for all \(t \ge t_{0}\), \({\mathbb {E}}^{{\mathbb {P}}}[\alpha _{t} + \beta _{t}] < \infty \),

2. (2)

   there is a \({\mathbb {P}}\)-null set \(N \in {\mathcal {B}}_{t_{0}}\) such that for all \(\omega \notin N\), the process \((\theta _{t}, {\mathcal {B}}_{t}, {\mathbb {P}}|_{{\mathcal {B}}_{t_{0}}}^{\omega })_{t \ge t_{0}}\) is an a.e. super-martingale and for all \(t \ge t_{0}\),

   $$\begin{aligned} {\mathbb {E}}^{{\mathbb {P}} |_{{\mathcal {B}}_{t_{0}}}^{\omega }} [\alpha _{t}+ \beta _{t}]< \infty \text { and } {\mathbb {E}}^{{\mathbb {P}}} [ {\mathbb {E}}^{{\mathbb {P}} |_{{\mathcal {B}}_{t_{0}}}^{\cdot }}[ \alpha _{t} + \beta _{t} ]] < \infty . \end{aligned}$$

Lemma 6.8

(Flandoli and Romito 2008, Lemma D.2) For all \(\alpha \in {\mathbb {R}}_{+} \backslash \{\frac{1}{2}\}, B_{j}, j \in \{1,2,3,4\}\), maps \(D(A^{\theta (\alpha )}) \times D(A^{\theta (\alpha )})\) continuously to \(D(A^{\alpha - \frac{1}{4}})\) where \(\theta (\alpha )\) is defined in (15). If \(\alpha = \frac{1}{2}\), then \(B_{j}\) maps \(D(A^{\frac{3}{4}}) \times D(A^{\frac{3}{4}})\) continuously to \(D(A^{\frac{1}{4} - \epsilon })\) for any \(\epsilon > 0\).

It is worth pointing out that (Flandoli and Romito 2008, Lemma D.2) is in fact a special case of the following (Yamazaki 2014, Lemma 2.5), (Iftimie 1999, Theorem 1.1) in case \(\alpha < \frac{1}{2}\).

Lemma 6.9

(Yamazaki 2014, Lemma 2.5 for \(x \in {\mathbb {R}}^{N}\), also Iftimie 1999, Theorem 1.1 for \(x \in {\mathbb {T}}^{3}\)) Suppose that the spatial variable x is an element in \({\mathbb {R}}^{N}, N \in {\mathbb {N}}\). Let \(\sigma _{1}, \sigma _{2} < \frac{N}{2}\) satisfy \(\sigma _{1} + \sigma _{2} > 0\). Then there exists a constant \(c(\sigma _{1}, \sigma _{2}) > 0\) such that

for all \(f \in {\dot{H}}^{\sigma _{1}}({\mathbb {R}}^{N})\), and \(g\in {\dot{H}}^{\sigma _{2}}({\mathbb {R}}^{N})\).

Indeed,

if \(\alpha < \frac{1}{2}\). Finally, we leave a key vector calculus identity that is not used for the NSE or the MHD system but is crucial for the HMHD system:

Indeed, denoting components by \(\Theta = (\Theta _{1}, \Theta _{2}, \Theta _{3})\) and \(\Psi = (\Psi _{1}, \Psi _{2}, \Psi _{3})\), a direct computation shows

Appendix B: Proofs of Selected Results

1.1 Proof of Proposition 3.2

The purpose of this subsection is to record the proof of Proposition 3.2 for completeness; they are all standard results, and thus, we only sketch it. From the literature on the standard well-posedness results of the MHD system, it is well known that the proof in case of the NSE in Flandoli and Romito (2008) may be extended to that of the MHD system. We focus on the HMHD system here, the case in which taking zero Hall parameter leads to the MHD system as well.

The first inequality follows from Yamazaki (2017a, Proposition 3.1) by considering an additive noise therein. The second inequality is verified within the proof of Yamazaki (2017a, Proposition 3.5) by considering an additive noise therein. For the third inequality, we also refer to Yamazaki [2017a, (48)]. It is clear from Yamazaki (2017a) that \(P_{n}^{G}\) also satisfies the properties \([MP1], \ldots , [MP4]\), associated with (13b) [see in particular Yamazaki 2017a, Proposition 3.1 for [MP1], Yamazaki 2017a, Section 3.6 for [MP2], and Yamazaki 2017a, equation (18) for [MP3], [MP4]]. Finally, because the initial distribution for the Galerkin approximation is the projection of \(\mu _{0}\) onto \(H_{n}\), it is clear that the marginal of \(P_{n}^{G}\) at \(t = 0\) is \(\mu _{0}\) so that \(P_{n}^{G}\) also satisfies [MP5]. This completes the proof of Proposition 3.2.

1.2 Proof of Proposition 3.3

The purpose of this subsection is to record the proof of Proposition 3.3 for completeness; it is similar to that of Flandoli and Romito (2008, Lemma A.2) and thus we only sketch it for the case of the HMHD system. We take \(\phi =(\phi _{1}, \phi _{2}) \in C^{\infty }\) and set \(\Phi (t) \triangleq t |Q^{\frac{1}{2}} \phi |_{H}^{2}\) and \(\Phi _{n}(t) \triangleq t |Q_{n}^{\frac{1}{2}} \phi |_{H}^{2}\) where \(Q_{n}\) is the projection of \(Q = \begin{pmatrix} Q_{1}\\ Q_{2} \end{pmatrix}\) onto \(H_{n}\). Since \((P_{n}^{G})_{n \in {\mathbb {N}}}\) satisfies [MP2] for all \(n \in {\mathbb {N}}\) and \((P_{k_{n}}^{G})_{n \in {\mathbb {N}}}\) is its subsequence, by [MP2]

is square-integrable and \((M_{t}^{\phi , k_{n}}, {\mathcal {B}}_{t}^{HMHD}, P_{k_{n}}^{G})\) is a continuous martingale with a quadratic variation of \(\langle \langle M_{t}^{\phi ,k_{n}} \rangle \rangle = t |Q_{n}^{\frac{1}{2}} \phi |_{H}^{2} = \Phi _{n}(t).\) By Lévy martingale characterization theorem (e.g. Karatzas and Shreve 1991, Chapter 3, Theorem 3.16), \((M_{t}^{\phi , k_{n}})_{t \ge 0}\) is a Brownian motion. If we let \(0 \le s < t\) and \(E \in {\mathcal {B}}_{s}^{HMHD}\), then using that \(M_{t}^{\phi , k_{n}}\rightarrow M_{t}^{\phi }\) as \(n\rightarrow \infty \) for all \(\omega \in \Omega _{HMHD}\), and \(P_{k_{n}}^{G} \rightarrow P_{\infty }^{G}\) weakly in \({\mathcal {U}}\), we may deduce

This implies that \(P_{\infty }^{G}\) satisfies [MP2]; thus, the proof of Proposition 3.3 is complete.

1.3 Proof of Proposition 3.4

The purpose of this subsection is to record the proof of Proposition 3.4 for completeness; it is similar to that of Flandoli and Romito (2008, Lemma A.3), and thus, we only sketch it for the case of the HMHD system. We see that for any \(m \ge 2\),

where we used that \(P_{n}\) satisfies [MP4]. Similar application of [MP3] leads to

Now for all \(T> 0, \lambda > 0\), we set

it is open in \(\Omega _{HMHD}\) by semicontinuity in the topology of \(\Omega _{HMHD}\). Then writing \(\alpha _{t} \triangleq |\xi _{t} |_{H}^{2} + 2 \int _{0}^{t} ||\xi _{s} ||_{V}^{2} \text {d}s\), \(\beta _{t} \triangleq |\xi _{0} |_{H}^{2} + t \sigma ^{2}, a \triangleq 0, b \triangleq T\) and \(\theta _{t} \triangleq \alpha _{t} - \beta _{t}\), we obtain

by (76), Lemma 6.6 and (75). This implies that P satisfies [MP1].

Next, by hypothesis (1), \(P_{n} \rightarrow P\) weakly in \({\mathcal {U}}\) as \(n \rightarrow \infty \). This implies by Skorokhod’s theorem that there exists a probability space \((\Sigma , {\mathcal {G}}, P)\) and a sequence of \({\mathcal {U}}\)-valued random variables \(\{X_{n}\}, X\) on \(\Sigma \) such that \(X_{n} \rightarrow X\) in \({\mathcal {U}}\) as \(n\rightarrow \infty \)P-almost surely and \(X_{n}, X\) have laws denoted by \(P_{n}, P\), respectively. It follows that

by that \(X_{n} \rightarrow X\) in \({\mathcal {U}}\)P-almost surely and (75). Similarly

by that \(X_{n} \rightarrow X\) in \({\mathcal {U}}\)P-almost surely as \(n\rightarrow \infty \), and (75). Thus, by (77), (78), (10), we see that \(E_{t}^{1}\) is P-integrable. To complete the proof that P satisfies [MP3], it suffices to show that \(E_{t}^{1}\) is an a.e. super-martingale, but this follows from Flandoli and Romito [2008, equation (A.4)]. Finally, the proof that P satisfies [MP4] is similar to that of [MP3].

1.4 Proof of Proposition 3.6

The purpose of this subsection is to leave the proof of Proposition 3.6 for completeness; it is similar to that of Flandoli and Romito (2008, Lemma 4.2) and thus we only sketch it for the case of the HMHD system. By Theorem 3.1, we know \({\mathcal {C}}_{HMHD}(x) \ne \emptyset \). Moreover, by Definition 3.3, as integration is linear, clearly \({\mathcal {C}}_{HMHD}(x)\) is convex. Finally, by Theorem 3.1 and [MP1], we know for all \({\mathbb {P}} \in {\mathcal {C}}_{HMHD}(x)\), \({\mathbb {P}}(L_{loc}^{\infty } ([0,\infty ); H) \cap L_{loc}^{2} ([0,\infty ); V) = 1\). This leads to

due to Lemma 6.1 with \({\mathcal {H}} = H\), \(\Omega = C([0,\infty ); {\mathcal {V}}') = C([0,\infty ); D(A^{\frac{\beta }{2}})') = \Omega _{HMHD}\) for \(\beta > \frac{5}{2}\) by (70) and (7). Therefore, we conclude that \({\mathbb {P}} (C([0,\infty ); H_{\sigma }) \cap \Omega _{HMHD} ) = 1\).

1.5 Proof of Proposition 3.8

The purpose of this subsection is to record the proof of Proposition 3.8 for completeness; it is similar to that of Flandoli and Romito (2008, Lemma 4.4) and thus we only sketch it only for the HMHD system. We fix \(x \in H\) and \({\mathbb {P}} \in {\mathcal {C}}_{HMHD}(x)\), denote by \(T_{{\mathbb {P}}}\) the set of exceptional times for \({\mathbb {P}}\) as defined in Definition 3.4. We may fix \(t \notin T_{{\mathbb {P}}}\) and let a mapping \(\omega \mapsto {\mathbb {P}} |_{{\mathcal {B}}_{t}^{HMHD}}^{\omega }\) be a r.c.p.d. of \({\mathbb {P}}\) on \({\mathcal {B}}_{t}^{HMHD}\) as defined in Definition 6.1. We set

By (9), \(S_{t} \in {\mathcal {B}}_{t}^{HMHD}\) and \(S^{t} \in {\mathcal {B}}_{HMHD}^{t}\). Moreover, \(1 = \int _{S_{t}} {\mathbb {P}} |_{{\mathcal {B}}_{t}^{HMHD}}^{\omega } [S^{t}] {\mathbb {P}} (d \omega )\) by [MP1] and (72). Therefore, there exists a \({\mathbb {P}}\)-null set \(N_{1} \in {\mathcal {B}}_{t}^{HMHD}\) such that \({\mathbb {P}}|_{{\mathcal {B}}_{t}^{HMHD}}^{\omega }[S^{t}] = 1\) for any \(\omega \notin N_{1}\). Next, as we fixed \({\mathbb {P}} \in {\mathcal {C}}_{HMHD}(x)\), we know [MP2] holds for \({\mathbb {P}}\); thus, if we let \((\phi _{n})_{n \in {\mathbb {N}}} \subset C^{\infty }\) be dense in \(C^{\infty }\) with the \(D(A^{\frac{\beta }{2}})\)-norm where \(\beta > \frac{5}{2}\), fix \(n \in {\mathbb {N}}\), define

then \((M_{t}^{\phi _{n}}, {\mathcal {B}}_{t}^{HMHD}, {\mathbb {P}})\) is a continuous, \({\mathbb {P}}\)-square-integrable martingale with a quadratic variation of \(\zeta _{t} = t |Q^{\frac{1}{2}} \phi _{n} |_{H}^{2}\). Thus by Lemma 6.5, there exists \(N_{2}^{n} \in {\mathcal {B}}_{t}^{HMHD}\) of null \({\mathbb {P}}\)-measure such that \((M_{s}^{\phi _{n}}, {\mathcal {B}}_{s}^{HMHD}, {\mathbb {P}} |_{{\mathcal {B}}_{t}^{HMHD}}^{\omega })_{s \ge t}\) is a continuous \({\mathbb {P}}|_{{\mathcal {B}}_{t}^{HMHD}}^{\omega }\)-square-integrable martingale with a quadratic variation \(\zeta _{t}\) for all \(\omega \notin N_{2}^{n}\). We set \(N_{2} \triangleq \cup _{n \in {\mathbb {N}}} N_{2}^{n}\) so that \({\mathbb {P}}(N_{2}) = 0\). Now if we set

then \(\alpha _{t}^{1}\) is lower semicontinuous, while \(\beta _{t}^{1}\) is clearly continuous and non-decreasing; hence, \(E_{t}^{1} = \alpha _{t}^{1} - \beta _{t}^{1}\) due to (11) and (80) is lower semicontinuous, \((E_{t}^{1}, {\mathcal {B}}_{t}^{HMHD}, {\mathbb {P}})\) is an a.e. super-martingale and \({\mathbb {E}}^{{\mathbb {P}}}[E_{t}^{1}] < \infty \) by [MP3]. Thus, by Lemma 6.7, there exists a \({\mathbb {P}}\)-null set \(N_{3} \in {\mathcal {B}}_{t}^{HMHD}\) such that \((E_{t}^{1}, {\mathcal {B}}_{t}^{HMHD}, {\mathbb {P}} |_{{\mathcal {B}}_{t_{0}}^{HMHD}}^{\omega })_{t \ge t_{0}}\) is an a.e. super-martingale for all \(\omega \notin N_{3}\).

If we also set

then identical application of Lemma 6.7 deduces the existence of a \({\mathbb {P}}\)-null set \(N_{4}^{n} \in {\mathcal {B}}_{t}^{HMHD}\) such that \((E_{t}^{n}, {\mathcal {B}}_{t}^{HMHD}, {\mathbb {P}} |_{{\mathcal {B}}_{t_{0}}^{HMHD}}^{\omega } )_{t \ge t_{0}}\) is an a.e. super-martingale for all \(\omega \notin N_{4}^{n}\). We take \(N_{4} \triangleq \cup _{n \in {\mathbb {N}}} N_{4}^{n}\).

Finally there exists a \({\mathbb {P}}\)-null set \(N_{5} \in {\mathcal {B}}_{t}^{HMHD}\) such that \({\mathbb {P}} |_{{\mathcal {B}}_{t}^{HMHD}}^{\omega } (\{\xi _{t} = \omega (t) \}) = 1\) for all \(\omega \notin N_{5}\). At last, the proof of Proposition 3.8 is complete once we set \(N \triangleq \cup _{j=1}^{5} N_{j}\) which is in \({\mathcal {B}}_{t}^{HMHD}\).

1.6 Proof of Proposition 3.9

The purpose of this subsection is to record the proof of Proposition 3.9 for completeness; it is similar to that of Flandoli and Romito (2008, Lemma 4.5), and thus, we only sketch it for the case of the HMHD system. We fix \(x \in H, {\mathbb {P}} \in {\mathcal {C}}_{HMHD}(x)\), let \(T_{{\mathbb {P}}}\) be the set of exceptional times of \({\mathbb {P}}\), and then fix \(t \notin T_{{\mathbb {P}}}\). We let a mapping \(\omega \mapsto Q_{\omega }\) be \({\mathcal {B}}_{t}^{HMHD}\)-measurable on \(\Omega _{HMHD}\), valued in \(Pr (\Omega _{HMHD}^{t})\) and denote by \(N_{Q} \in {\mathcal {B}}_{t}^{HMHD}\) the \({\mathbb {P}}\)-null set such that \(\omega (t) \in H\) and \(Q_{\omega } \in \Phi _{t} {\mathcal {C}}_{HMHD}(\omega (t))\) for all \(\omega \notin N_{Q}\). Firstly,

by (79), (72) and because \(Q_{\omega }[S^{t}] = 1\) by [MP1] as \(Q_{\omega } \in \Phi _{t} {\mathcal {C}}_{HMHD}(\omega (t))\). Thus, \({\mathbb {P}} \otimes _{t}Q\) satisfies [MP1] as \({\mathbb {P}} (S_{t}) = 1\) by (79) and that \({\mathbb {P}} \in {\mathcal {C}}_{HMHD}(x)\). Secondly, if we take \(\phi \in C^{\infty }\), then \((M_{s}^{\phi }, {\mathcal {B}}_{s}^{HMHD}, Q_{\omega })_{s \ge t}\) is a \(Q_{\omega }\)-square-integrable martingale for all \(\omega \notin N_{Q}\) due to [MP2]. Then Lemma 6.5 implies that \((M_{s}^{\phi }, {\mathcal {B}}_{s}^{HMHD}, {\mathbb {P}} \otimes _{t} Q)_{s \ge t}\) is a \({\mathbb {P}} \otimes _{t} Q\)-square-integrable martingale. As \({\mathbb {P}}\) and \({\mathbb {P}} \otimes _{t} Q\) are equal on \({\mathcal {B}}_{t}^{HMHD}\) due to Definition 6.1, and \((M_{s}^{\phi }, {\mathcal {B}}_{s}^{HMHD}, {\mathbb {P}})_{0\le s \le t}\) is a martingale, we see that \((M_{s}^{\phi }, {\mathcal {B}}_{s}^{HMHD}, {\mathbb {P}} \otimes _{t} Q)_{s \ge 0}\) is also a martingale; hence, \({\mathbb {P}} \otimes _{t} Q\) satisfies [MP2]. The proof that \({\mathbb {P}} \otimes _{t} Q\) satisfies [MP3] and [MP4] follows similarly using Lemma 6.7 and lower semicontinuity of \(E_{t}^{n}\) for \(n \ge 1\) as defined in (12) in the spirit of the proof of Proposition 3.8. Finally, \({\mathbb {P}}\) and \({\mathbb {P}} \otimes _{t} Q\) are equal on \({\mathcal {B}}_{t}^{HMHD}\), while \({\mathbb {P}} \in {\mathcal {C}}_{HMHD}(x)\) so that \(\delta _{x}\) is the marginal of \({\mathbb {P}}\) at \(t = 0\); it follows that \(\delta _{x}\) is the marginal of \({\mathbb {P}} \otimes _{t} Q\) at \(t= 0\).

Reprints and permissions