Abstract
A class of (possibly) degenerate stochastic integro-differential equations of parabolic type is considered, which includes the Zakai equation in nonlinear filtering for jump diffusions. Existence and uniqueness of the solutions are established in Bessel potential spaces.
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1 Introduction
We consider the equation
for \((t,x)\in [0,T]\times \mathbb {R}^d:=H_T\) with initial condition
on a filtered probability space \((\Omega ,\mathcal {F},P,(\mathcal {F}_t)_{t\ge 0})\), carrying a sequence \(w=(w^i_t)_{i=1}^{\infty }\) of independent \(\mathcal {F}_t\)-Wiener processes and an \(\mathcal {F}_t\)-Poisson martingale measure \(\tilde{\pi }(dz,dt)=\pi (dz,dt)-\mu (dz)\otimes dt\), where \(\pi (dz,dt)\) is an \(\mathcal {F}_t\)-Poisson random measure with a \(\sigma \)-finite characteristic measure \(\mu (dz)\) on a measurable space \((Z,\mathcal {Z})\) with a countably generated \(\sigma \)-algebra \(\mathcal {Z}\). We note that here, and later on, the summation convention is used with respect to repeated (integer-valued) indices and multi-numbers.
In the above equation \(\mathcal {A}_t\) is an integro-differential operator of the form \(\mathcal {A}_t=\mathcal {L}_t+\mathcal {N}_t^{\xi }+\mathcal {N}^{\eta }_t+\mathcal {R}_t\) with a “zero-order” linear operator \(\mathcal {R}_t\) specified later, a second order differential operator
and integral operators \(\mathcal {N}^{\xi }_t\) and \(\mathcal {N}^{\eta }_t\) defined by
for a suitable class of real-valued functions \(\varphi \) on \(\mathbb {R}^d\) for each \(t\in [0,T]\), where \(\nu (dz)\) is a fixed \(\sigma \)-finite measure on \((Z,\mathcal {Z})\). The coefficients \(a^{ij}\), \(b^i\) and c are real functions on \(\Omega \times H_T\) for \(i,j=1,2,\ldots ,d\), and \(\lambda ^\xi \), \(\lambda ^\eta \) and \(\gamma \) are real functions on \(\Omega \times H_T \times Z\), and \(\eta =(\eta ^i_{t,z}(x))\) and \(\xi =(\xi ^i_{t,z}(x))\) are \(\mathbb {R}^d\)-valued functions of \((\omega ,t,x,z)\in \Omega \times H_T\times Z\). Under “zero-order operators” we mean bounded linear operators \(\mathcal {R}\) mapping the Sobolev spaces \(W^k_p\) into themselves for some \(k\ge 0\). For each integer \(r\ge 1\) the operator \(\mathcal {M}_t^r\) is a first order differential operator of the form
The coefficients \(\sigma ^{ir}\) and \(\beta ^{r}\) are real functions on \(\Omega \times H_T\) for \(i,j=1,2,\ldots ,d\) and integers \(r\ge 1\). The free terms f and \(g^r\) are real functions defined on \(\Omega \times H_T\) for every \(r\ge 1\), and h is a real function defined on \(\Omega \times H_T\times Z\). The stochastic differentials in Eq. (1.1) are understood in Itô’s sense, see the definition of a solution in the next section.
We are interested in the solvability of the above problem in \(L_p\)-spaces. We note that Eq. (1.1) may degenerate, i.e., the pair of linear operators \((\mathcal {L},\mathcal {M})\) satisfies only the stochastic parabolicity condition, Assumption 2.1 below, and the operators \(\mathcal {N}^{\xi }\) and \(\mathcal {N}^{\eta }\) may also degenerate. Our main result, Theorem 2.1 states that under the stochastic parabolicity condition on the operators \((\mathcal {L},\mathcal {M})\), \(\mathcal {N}^{\xi }\), \(\mathcal {N}^{\eta }\), and appropriate regularity conditions on their coefficients and on the initial and free data, the Cauchy problem (1.1)–(1.2) has a unique generalised solution \(u=(u_t)_{t\in [0,T]}\) for any given T. Moreover, this theorem describes the temporal and spatial regularity of u in terms of Bessel potential spaces \(H^n_p\), and presents also a supremum estimate in time. The uniqueness of the solution is proved by an application of a theorem on Itô’s formula from [17], which generalises a theorem of Krylov in [24] to the case of jump processes. The existence of a generalised solution is proved in several steps. First we obtain a priori estimates in Sobolev spaces \(W^n_p\) for integers \(n\in [0,m]\), where m is a parameter measuring the spatial smoothness of the coefficients and the data in (1.1)–(1.2). These estimates allow us to construct a generalised solution by approximating (1.1)–(1.2) with non-degenerate equations with smooth coefficients and compactly supported smooth data in \(x\in \mathbb {R}^d\). Thus we see that a solution operator, mapping the initial and free data into the solution of (1.1)–(1.2), exists and it is a bounded linear operator in appropriate \(L_p\)-spaces. Hence by interpolation we get our a priori estimates in Bessel potential spaces \(H^n_p\) for any given \(p\ge 2\) and real number \(n\in [0,m]\). We obtain essential supremum estimates in time for the solution from integral estimates, by using the simple fact that the essential supremum of Lebesgue functions over an interval [0, T] is the limit of their \(L_r([0,T])\)-norm as \(r\rightarrow \infty \). Hence we get the temporal regularity of the solution formulated in our main theorem by using Theorem 2.2 on Itô’s formula in [17], an extension of Lemma 5.3 in [9] and a well-known interpolation inequality, Theorem 4.1(v) below.
In the literature there are many results on stochastic integral equations with unbounded operators, driven by jump processes and martingale measures. A general existence and uniqueness theorem for stochastic evolution equations with nonlinear operators satisfying stochastic coercivity and monotonicity conditions is proved in [15], which generalises some results in [25] and [32] to stochastic evolution equations driven by semimartingales and random measures. Further generalisations are obtained and the asymptotic behaviour of the solutions are investigated in [28]. For a monograph on stochastic evolution equations driven by Lévy noise we refer to [33].
The main theorem in [15] implies the existence of a unique generalised solution to (1.1)–(1.2) in \(L_2\)-spaces when instead of the stochastic parabolicity condition (2.1) in Assumption 2.1 below, the strong stochastic parabolicity condition,
with a constant \(\lambda >0\) is assumed on \((\mathcal {L},\mathcal {M})\). Under the weaker condition of stochastic parabolicity the solvability of (1.1)–(1.2) in \(L_2\)-spaces is investigated and existence and uniqueness theorems are presented in [8] and [27]. The first result on solvability in \(L_p\)-spaces for the stochastic PDE problem (1.1)–(1.2) with \(\xi =\eta =0\) and \(h=0\) was obtained in [26], and was improved in [16]. However, there is a gap in the proof of the crucial a priori estimate in [26]. This gap is filled in and more general results on solvability in \(L_p\)-spaces for systems of stochastic PDEs driven by Wiener processes are presented in [12]. As far as we know Theorem 2.1 below is the first result on solvability in \(L_p\)-spaces of stochastic integro-differential equations (SIDEs) without any non-degeneracy conditions. It generalises the main result of [9] on deterministic integro-differential equations to SIDEs. Our motivation to study Eq. (1.1) comes from nonlinear filtering of jump-diffusion processes, and we want to apply Theorem 2.1 to filtering problems in a continuation of the present paper.
We note that under non-degeneracy conditions SIDEs have been investigated with various generalities in the literature, and very nice results on their solvability in \(L_p\)-spaces have recently been obtained. In particular, \(L_p\)-theories for such equations have been developed in [20, 21, 29, 30] and [31], which extend some results of the \(L_p\)-theory of Krylov [22] to certain classes of equations with nonlocal operators. See also [7] for an \(L_p\)-theory for stochastic PDEs driven by Lévy processes, [36] for an existence and uniqueness theorem for stochastic quasi-geostrophic equations driven by Poisson martingale measures, and [6, 10] and [35] for \(L_p\) theory of deterministic equations with nonlocal operators. Nonlinear filtering problems and the related equations describing the conditional distributions have been extensively studied in the literature. For results in the case of jump-diffusion models see, for example, [2, 4, 11] and [14].
In conclusion, we introduce some notions and notations used throughout this paper. All random elements are given on the filtered probability space \((\Omega ,\mathcal {F},P,(\mathcal {F}_t)_{t\ge 0})\). We assume that \(\mathcal {F}\) is P-complete, the filtration \((\mathcal {F}_t)_{t\ge 0}\) is right-continuous, and \(\mathcal {F}_0\) contains all P-zero sets of \(\mathcal {F}\). The \(\sigma \)-algebra of the predictable subsets of \(\Omega \times [0,\infty )\) is denoted by \(\mathcal {P}\). For notations, notions and results concerning Lévy processes, Poisson random measures and stochastic integrals we refer to [1, 3] and [19].
For vectors \(v=(v^i)\) and \(w=(w^i)\) in \(\mathbb {R}^d\) we use the notation \(vw=\sum _{i=1}^mv^iw^i\) and \(|v|^2=\sum _i|v^i|^2\). For real-valued Lebesgue measurable functions f and g defined on \(\mathbb {R}^d\) the notation (f, g) means the integral of the product fg over \(\mathbb {R}^d\) with respect to the Lebesgue measure on \(\mathbb {R}^d\). A finite list \(\alpha =\alpha _1\alpha _2,\ldots ,\alpha _n\) of numbers \(\alpha _i\in \{1,2,\ldots ,d\}\) is called a multi-number of length \(|\alpha |:=n\), and the notation
is used for integers \(n\ge 1\), where
is the generalised derivative of a real-valued function v with respect to \(x_i\), \(i=1,2,\ldots ,d\). We use also the multi-number \(\epsilon \) of length 0 such that \(D_{\epsilon }\) means the identity operator. For an integer \(n\ge 0\) and functions v on \(\mathbb {R}^d\), whose generalised derivatives up to order n are functions, we use the notation \(D^nv\) for the collection \(\{D_{\alpha }v:|\alpha |=n\},\) and define
For differentiable functions \(v=(v^1,\ldots ,v^d):\mathbb {R}^d\rightarrow \mathbb {R}^d\) the notation Dv means the Jacobian matrix whose jth entry in the i-th row is \(D_jv^i\). When we talk about “derivatives up to order n” of a function for some nonnegative integer n, then we always include the zero-order derivative, i.e. the function itself. The space of smooth functions \(\varphi =\varphi (x)\) with compact support on the d-dimensional Euclidean space \(\mathbb {R}^d\) is denoted by \(C_0^{\infty }\).
For \(p, q\ge 1\) we denote by \(\mathcal {L}_{q}=\mathcal {L}_{q}(Z, \mathbb {R})\) the Banach spaces of \(\mathbb {R}\)-valued \(\mathcal {Z}\)-measurable functions \(h=h(z)\) of \(z\in Z\) such that
The notation \(\mathcal {L}_{p,q}\) means the space \(\mathcal {L}_{p}\cap \mathcal {L}_{q}\) with the norm
The space of sequences \(\nu =(\nu ^{1},\nu ^{2},\ldots )\) of real numbers \(\nu ^{k}\) with finite norm
is denoted by \(l_2\).
The Borel \(\sigma \)-algebra of a separable Banach space V is denoted by \(\mathcal {B}(V)\), and for \(p\ge 0\) the notations \(L_p([0,T],V)\) and \(L_p(\mathbb {R}^d,V)\) are used for the space of V-valued Borel-measurable functions f on [0, T] and g on \(\mathbb {R}^d\), respectively, such that \(|f|_V^p\) and \(|g|_V^p\) have finite Lebesgue integral over [0, T] and \(\mathbb {R}^d\), respectively. For \(p\ge 1\) and \(f\in L_p(\mathbb {R}^d,V)\) we use the notation \(|f|_{L_p}\), defined by
In the sequel, V will be \(\mathbb {R}\), \(l_2\) or \(\mathcal {L}_{p,q}\). For integer \(n\ge 0\) the space of functions from \(L_p(\mathbb {R}^d,V)\), whose generalised derivatives up to order n are also in \(L_p(\mathbb {R}^d,V)\), is denoted by \(W^n_p=W^n_p(\mathbb {R}^d,V)\) with the norm
By definition \(W^0_p(\mathbb {R}^d,V)=L_p(\mathbb {R}^d,V)\).
For \(m\in \mathbb {R}\) and \(p\in (1,\infty )\) we use the notation \(H^m_p=H^m_p(\mathbb {R}^d;V)\) for the Bessel potential space with exponent p and order m, defined as the space of V-valued generalised functions \(\varphi \) on \(\mathbb {R}^d\) such that
where \(\Delta =\sum _{i=1}^dD_i^2\). Moreover, we use \(\mathbb {H}^m_p\) to denote the space of \(\mathcal {P}\)-measurable functions from \(\Omega \times [0,T]\) to \(H^m_p\) such that
We will often omit the target space V in the notations \(W^n_p(V)\), \(H^m_p(V)\), and \(\mathbb {H}^m_p(V)\) for convenience if \(V=\mathbb {R}\), and we use \(\mathbb {L}_p\) to denote \(\mathbb {H}^0_p\). When \(V=\mathcal {L}_{p,q}\) we use \(W^n_{p,q}\), \(H^m_{p,q}\) and \(\mathbb {H}^m_{p,q}\) to denote \(W^n_p(\mathcal {L}_{p,q})\), \(H^m_p(\mathcal {L}_{p,q})\) and \(\mathbb {H}^m_p(\mathcal {L}_{p,q})\) respectively, and we use \(\mathbb {L}_{p,q}\) to denote \(\mathbb {H}^0_{p,q}\).
Remark 1.1
If V is a UMD space, see for example [18] for the definition of UMD spaces, then by Theorem 5.6.11 in [18] for \(p>1\) and integers \(n\ge 1\) we have \(W^n_p(V)=H^n_p(V)\) with equivalent norms. Clearly, \(\mathcal {L}_{p,q}\) is a UMD space for \(p, q\in (1,\infty )\), which implies \(W^n_{p,q}=H^n_{p,q}\) for non-negative integers n and \(p,q\in (1,\infty )\).
2 Formulation of the results
To formulate our assumptions we fix a constant K, a non-negative real number m, an exponent \(p\in [2,\infty )\), and non-negative \(\mathcal {Z}\)-measurable real-valued functions \(\bar{\eta }\) and \(\bar{\xi }\) on Z such that they are bounded by K and
We denote by \(\lceil m \rceil \) the smallest integer which is greater than or equal to m, and \(\lfloor m \rfloor \) the largest integer which is less than or equal to m.
Assumption 2.1
The derivatives in \(x\in \mathbb {R}^d\) of \(a^{ij}\), \(b^i\) and c up to order \(\max \{\lceil m \rceil ,2\}\), \(\max \{\lceil m \rceil , 1\}\) and \(\lceil m \rceil \), respectively, are \(\mathcal {P}\otimes \mathcal {B}(\mathbb {R}^d)\)-measurable functions on \(\Omega \times H_T\), bounded by K for all \(i,j=1,2,\ldots d\). The functions \(\sigma ^i=(\sigma ^{ir})_{r=1}^\infty \) and \(\beta =(\beta ^r)_{r=1}^\infty \) and their derivatives up to order \(\lceil m \rceil +1\) are \(l_2\)-valued \(\mathcal {P}\otimes \mathcal {B}(\mathbb {R}^d)\)-measurable functions, bounded by K. Moreover \(a^{ij}=a^{ji}\) for all \(i,j=1,\ldots d\), and for \(P\otimes dt \otimes dx\)-almost all \((\omega ,t,x)\in \Omega \times H_T\)
where
Assumption 2.2
The mapping \(\xi =(\xi ^i)\) is an \(\mathbb {R}^d\)-valued \(\mathcal {P}\otimes \mathcal {B}(\mathbb {R}^d)\otimes \mathcal {Z}\)-measurable function on \(\Omega \times [0,T]\times \mathbb {R}^d\times Z\). Its derivatives in \(x\in \mathbb {R}^d\) up to order \(\max \{\lceil m \rceil ,3\}\) exist and are continuous in \(x\in \mathbb {R}^d\) such that
for all \((\omega ,t,x,z)\in \Omega \times H_T\times Z\). Moreover,
for all \((\omega ,t,x,z,\theta )\in \Omega \times H_T\times Z \times [0,1]\), where \(\mathbf {I}\) is the \(d\times d\) identity matrix, and \(D\xi \) denotes the Jacobian matrix of \(\xi \) in \(x\in \mathbb {R}^d\).
Assumption 2.3
The function \(\eta =(\eta ^i)\) maps \(\Omega \times [0,T]\times \mathbb {R}^d\times Z\) into \(\mathbb {R}^d\) such that Assumption 2.2 holds with \(\eta \) and \(\bar{\eta }\) in place of \(\xi \) and \(\bar{\xi }\), respectively.
Remark 2.1
Assumptions 2.2 and 2.3 imply that for each \((\omega ,t,z,\theta )\in \Omega \times [0,T]\times Z\times [0,1]\) the mappings
are \(C^{k}\)-diffeomorphisms on \(\mathbb {R}^d\) with \(k=\bar{m}\). Note that a \(C^k\)-diffeomorphism \(\rho \) for an integer \(k\ge 1\) means a one-to-one mapping from \(\mathbb {R}^d\) onto \(\mathbb {R}^d\) such that \(\rho \) and its inverse \(\rho ^{-1}\) are continuously differentiable up to order k.
Assumption 2.4
The functions \(\lambda ^\xi \), \(\lambda ^\eta \) and \(\gamma \) are real-valued \(\mathcal {P}\otimes \mathcal {B}(\mathbb {R}^d)\otimes \mathcal {Z}\)-measurable function on \(\Omega \times [0,T]\times \mathbb {R}^d\times Z\). The derivatives in \(x\in \mathbb {R}^d\) of \(\lambda ^\xi \), \(\lambda ^\eta \) up to order \(\max \{\lceil m \rceil ,1\}\) exist and are continuous in \(x\in \mathbb {R}^d\) such that
The derivatives in \(x\in \mathbb {R}^d\) of \(\bar{\gamma }:=\gamma -\lambda ^\eta \) up to order \(\lceil m \rceil \) exist and are continuous in \(x\in \mathbb {R}^d\) such that
Assumption 2.5
The operator \(\mathcal {R}_t\) is a linear mapping from \(L_p(\mathbb {R}^d)\) into \(L_p(\mathbb {R}^d)\) for every \(t\in [0,T]\) and \(\omega \in \Omega \), such that for every \(\varphi \in C_0^{\infty }\) the function \(\mathcal {R}\varphi \) is \(\mathcal {P}\)-measurable and
Assumption 2.6
The free data \(f=(f_t)_{t\in [0,T]}\), \(g=(g^r_t)_{t\in [0,T]}\) and \(h=(h_t)_{t\in [0,T]}\) are \(\mathcal {P}\)-measurable processes with values in \(H^m_p\), \(H^{m+1}_p(l^2)\) and \(H^{m+1}_{p,2}=H_p^{m+1}(\mathcal {L}_{p,2})\), respectively, such that almost surely \(\mathcal {K}^p_{p,m}(T)<\infty \), where
The initial value \(\psi \) is an \(\mathcal {F}_0\)-measurable random variable with values in \(H^m_p\).
Remark 2.2
By Taylor’s formula we have
where to ease notation we omit the arguments t and z, and write \(v_k\) instead of \(D_kv\) for functions v. Due to Assumption 2.3 these equations extend to \(v\in W^1_p\) for \(p\ge 2\) as well. Hence after changing the order of integrals, by integration by parts we obtain
for \(\varphi \in C_0^{\infty }\), with
where for the sake of short notation the arguments t, z of \(\eta \) and of \(\eta _k=D_k\eta \) have been omitted, and \(\tau _{\theta \eta }\) is defined in (2.2). Operators \(\mathcal {J}^{k}_{\xi }\) and \(\mathcal {J}^{0}_{\xi }\) are defined as \(\mathcal {J}^{k}_{\eta }\) and \(\mathcal {J}^{0}_{\eta }\) in (2.3) and (2.4) but with \(\xi \) everywhere in place of \(\eta \).
Definition 2.1
An \(L_p\)-valued cadlag \(\mathcal {F}_t\)-adapted process \(u=(u_t)_{t\in [0,T]}\) is a generalised solution to Eqs. (1.1)–(1.2) with initial value \(u_0=\psi \), if \(u_t\in W^1_p\) for \(P\otimes dt\)-almost every \((\omega , t)\in \Omega \times [0,T]\), such that almost surely \(u\in L_p([0,T],W^1_p)\) and
for all for each \(\varphi \in C^\infty _0\) and \(t\in [0,T]\), where \(\tau _{\eta _{s,z}}= x+\eta _{s,z}(x)\), \(x\in \mathbb {R}^d\),
with \(\bar{b}^i_s=b^i_s-D_ja^{ij}_s\) for all \((s,\omega )\in [0,T]\times \Omega \), and the stochastic integrals are Itô integrals.
The next theorem is the main result of this paper.
Theorem 2.1
If Assumptions 2.1 through 2.6 hold with \(m\ge 0\), then there is at most one generalised solution to (1.1). If Assumptions 2.1 through 2.6 hold with \(m\ge 1\), then there is a unique generalised solution \(u=(u_t)_{t\in [0,T]}\), which is a weakly cadlag \(H^m_p\)-valued adapted process, and it is a strongly cadlag \(H_p^s\)-valued process for any \(s\in [0,m)\). Moreover,
with a constant \(N=N(d,m,p,q,T,K,K_{\bar{\xi }},K_{\bar{\eta }})\).
3 Preliminaries
For vectors \(v=(v^1,\ldots ,v^d)\in \mathbb {R}^d\) we define the operators \(T^v\), \(I^v\) and \(J^v\) by
acting on functions \(\varphi \) and \(\phi \) defined on \(\mathbb {R}^d\) such that the generalised derivatives \(D_i\phi \) exist. If \(v=v(x)\) is a function of \(x\in \mathbb {R}^d\) then the notation \(T^v\), \(I^v\) and \(J^v\) mean the operators defined by (3.1) with v(x) in place v. For example, \(J^{\xi }\) and \(J^{\eta }\) mean for each \(\omega \in \Omega \), \(t\in [0,T]\) and \(z\in Z\) the operators defined on differentiable functions \(\varphi \) on \(\mathbb {R}^d\) by
for each fixed variable \((\omega ,t,z)\) suppressed in this notation. We will often use the Taylor formulas
and
with \(\varphi _i:=D_i\varphi \) and \(\phi _{ij}:=D_{i}D_{ j}\phi \), which hold for every \(x\in \mathbb {R}^d\) when \(\varphi \) and \(\phi \) are continuous functions on \(\mathbb {R}^d\) with continuous derivatives up to first order and up to second order, respectively. These equations hold for dx-almost every \(x\in \mathbb {R}^d\) when \(\varphi \in W^1_p\) and \(\phi \in W^2_p\).
We fix a non-negative smooth function \(k=k(x)\) with compact support on \(\mathbb {R}^d\) such that \(k(x)=0\) for \(|x|\ge 1\), \(k(-x)=k(x)\) for \(x\in \mathbb {R}^d\), and \(\int _{\mathbb {R}^d}k(x)\,dx=1\). For \(\varepsilon >0\) and locally integrable functions v of \(x\in \mathbb {R}^d\) we use the notation \(v^{(\varepsilon )}\) for the mollification of v, defined by
Note that if \(v=v(x)\) is a locally Bochner integrable function on \(\mathbb {R}^d\), with respect to the Lebesgue measure, which takes values in a separable Banach space, then the mollification of v is defined as in (3.4) in the sense of Bochner integral.
The following lemmas are taken from [9] and for their proof we refer to [9].
Lemma 3.1
Let Assumption 2.3 hold. Then for every \((\omega ,t,z)\in \Omega \times [0,T]\times Z\) the operators \(T^{\eta }\), \(I^{\eta }\) and \(J^{\eta }\) are bounded linear operators from \(W^k_p\) to \(W^k_p\), from \(W^{k+1}_p\) to \(W^k_p\) and from \(W^{k+2}_p\) to \(W^k_p\) respectively, for \(k=0,1,\ldots ,\bar{m}\), such that \(T^{\eta }\varphi \), \(I^{\eta }f\) and \(J^{\eta }g\) are \(\mathcal {P}\otimes \mathcal {Z}\)-measurable \(W^k_p\)-valued functions of \((\omega , t,z)\), and
for all \(\varphi \in W^k_p\), \(f\in W^{k+1}_p\) and \(g\in W^{k+2}_p\), where N is a constant only depending on K, m, d, p.
Lemma 3.2
Let \(\rho \) be a \(C^k(\mathbb {R}^d)\)-diffeomorphism for some \(k\ge 1\), such that
for some constants \(M>0\) and \(N>0\). Then there are positive constants \(M'=M'(N,d)\) and \(N'=N'(N,M,d,k)\) such that (3.5) holds with \(g:=\rho ^{-1}\), the inverse of \(\rho \), in place of \(\rho \), with \(M'\) and \(N'\) in place of M and N, respectively.
The following lemma is a slight generalisation of Lemma 3.4 in [9].
Lemma 3.3
Let \(\rho \) be a \(C^k\)-diffeomorphism for \(k\ge 2\), such that (3.5) holds for some positive constants M and N. Then there is a positive constant \(\varepsilon _0=\varepsilon _0(M,N,d)\) such that \(\rho _{\varepsilon ,\vartheta }:=\vartheta \rho +(1-\vartheta )\rho ^{(\varepsilon )}\) is a \(C^{k}\)-diffeomorphism for every \(\varepsilon \in (0,\varepsilon _0)\) and \(\vartheta \in [0,1]\), and (3.5) remains valid for \(\rho _{\varepsilon ,\vartheta }\) in place of \(\rho \), with \(M{''}=M/2\) in place of M. Moreover, \(\rho ^{(\varepsilon )}\) is a \(C^{\infty }\)-diffeomorphism for \(\varepsilon \in (0,\varepsilon _0)\).
Proof
We show first that \(|\det D\rho _{\varepsilon ,\vartheta }|\) is separated away from zero for sufficiently small \(\varepsilon >0\). To this end observe that for bounded Lipschitz functions \(v=(v^1,v^2,\dots ,v^d)\) on \(\mathbb {R}^d\) and \(v_{\varepsilon ,\vartheta }:=\vartheta v+(1-\vartheta )v^{(\varepsilon )}\) we have
where L is the Lipschitz constant of v and K is a bound for |v|. Using this observation and taking into account that \(D_i\rho ^l\) is bounded by N and it is Lipschitz continuous with a Lipschitz constant not larger than N, we get
Thus setting \(\varepsilon '=M/(2d!\,N^{d})\), for \(\varepsilon \in (0,\varepsilon ')\) and \(\vartheta \in [0,1]\) we have
Clearly, \(\rho _{\varepsilon ,\vartheta }\) is a \(C^{k}\) function. Hence by the implicit function theorem \(\rho _{\varepsilon ,\vartheta }\) is a local \(C^{k}\)-diffeomorphism for \(\varepsilon \in (0,\varepsilon ')\) and \(\vartheta \in [0,1]\). We prove now that \(\rho _{\varepsilon ,\vartheta }\) is a global \(C^{k}\)-diffeomorphism for sufficiently small \(\varepsilon \). Since by the previous lemma \(|D\rho ^{-1}|\le N'\), we have
for all \(x,y\in \mathbb {R}^d\) and \(\varepsilon >0\) and \(\vartheta \in [0,1]\). Observe that
Thus \(|x-y| \le N'|\rho _{\varepsilon ,\vartheta }(x)-\rho _{\varepsilon ,\vartheta }(y)| +\varepsilon N'N|x-y|\). Therefore setting \(\varepsilon ''=1/(2NN')\), for all \(\varepsilon \in (0,\varepsilon '')\) and \(\vartheta \in [0,1]\) we have
which implies \(\lim _{|x|\rightarrow \infty }|\rho _{\varepsilon ,\vartheta }(x)|=\infty \), i.e., under \(\rho _{\varepsilon ,\vartheta }\) the pre-image of any compact set is a compact set for each \(\varepsilon \in (0,\varepsilon '')\) and \(\vartheta \in [0,1]\). A continuous function with this property is called a proper function, and by Theorem 1 in [13] a local \(C^1\)- diffeomorphism from \(\mathbb {R}^d\) into \(\mathbb {R}^d\) is a global diffeomorphism if and only if it is a proper function. Thus we have proved that \(\rho _{\varepsilon ,\vartheta }\) is a global \(C^{k}\)-diffeomorphism for \(\varepsilon \in (0,\varepsilon _0)\) and \(\vartheta \in [0,1]\), where \(\varepsilon _0=\min (\varepsilon ',\varepsilon '')\). Clearly, \(\rho _{\varepsilon ,0}=\rho ^{(\varepsilon )}\) is a \(C^{\infty }\) function and hence it is a \(C^{\infty }\)-diffeomorphism for every \(\varepsilon \in (0,\varepsilon _0)\).
Now we can complete the proof of the lemma by noting that since \(D_{j}\rho ^{(\varepsilon )}=(D_{j}\rho )^{(\varepsilon )}\), the condition \(|D^i\rho |\le N\) implies \(|D^i\rho _{\varepsilon ,\vartheta }|\le N\) for any \(\varepsilon >0\) and \(\vartheta \in [0,1]\). \(\square \)
For fixed \(\varepsilon >0\) and \(\vartheta \in [0,1]\) let \(\rho =\rho _{\varepsilon ,\vartheta }\) denote any of the functions
for each \((\omega ,t,z)\in \Omega \times [0,T]\times Z\), and assume that Assumptions 2.2 and 2.3 hold. Then by the inverse function theorem \(\rho \) is a local \(C^1\)-diffeomorphism for each t, \(\theta \) and z. Since
we have \(\lim _{|x|\rightarrow \infty }|\rho _{\varepsilon ,\vartheta }(x)|=\infty .\) Hence \(\rho _{\varepsilon ,\vartheta }\) is a global \(C^1\)-diffeomorphism on \(\mathbb {R}^d\), for \(\varepsilon >0\), \(\vartheta \in [0,1]\), for each \(t\in [0,T]\), \(z\in Z\) and \(\theta \in [0,1]\), by Theorem 1 in [13]. Note that by the formula on the derivative of inverse functions, a \(C^1\)-diffeomorphism and its inverse have continuous derivatives up to the same order. Thus Lemmas 3.2 and 3.3 imply the following statement, which is a slight generalisation of Corollary 3.6 in [9].
Corollary 3.4
Let Assumptions 2.2 and 2.3 hold. Then there is a positive constant \(\varepsilon _0=\varepsilon _0(K,d)\) such that \(\rho =\rho _{\varepsilon ,\vartheta }\) is a \(C^k\)-diffeomorphism on \(\mathbb {R}^d\) for \(k=\bar{m}\), for any \(\varepsilon \in (0,\varepsilon _0)\), \(\vartheta \in [0,1]\) and \((\omega ,t,z)\in \Omega \times [0,T]\times Z\). Moreover, for some constants \(M=M(K,d)\) and \(N=N(K,d,\bar{m})\)
for any \(\varepsilon \in (0,\varepsilon _0)\), \(\vartheta \in [0,1]\), \((\omega ,t,z)\in \Omega \times [0,T]\times Z\) and for \(k=1,2,\ldots ,\bar{m}\). Furthermore, if \(\vartheta =0\) then \(\rho \) is a \(C^{\infty }\)-diffeomorphism for each \(\varepsilon \in (0,\varepsilon _0)\), \((\omega ,t,z)\in \Omega \times [0,T]\times Z\), and for each integer \(m\ge 1\) there are constants \(M=M(K,d)\) and \(N=N(K,d,m)\) such that the estimates in (3.6) hold for all \(\varepsilon \in (0,\varepsilon _0)\), \((\omega ,t,z)\in \Omega \times [0,T]\times Z\) and \(k=1,2,\ldots ,m\).
Lemma 3.5
Let V be a separable Banach space, and let \(f=f(x)\) be a V-valued function of \(x\in \mathbb {R}^d\) such that \(f\in L_p(V)=L_p(\mathbb {R}^d,V)\) for some \(p\ge 1\). Then we have
Proof
This lemma is well-known. Its proof can be found, e.g., in [17], see Lemma 4.4 therein. For the convenience of the readers, we present the proof below. By the properties of Bochner integrals, Jensen’s inequality and Fubini’s theorem
Since V is separable, it has a countable dense subset \(V_0\). Denote by \(\mathcal {H}\subset L_p(V)\) the space of functions h of the form
for some integer \(k\ge 1\), \(v_i\in V_0\) and continuous real functions \(\varphi _i\) on \(\mathbb {R}^d\) with compact support. Then for such an h we have
where \(L_p=L_p(\mathbb {R}^d,\mathbb {R})\). For \(f\in L_p(V)\) and \(h\in \mathcal {H}\) we have
Letting here \(\varepsilon \rightarrow 0\) for each \(f\in L_p(V)\) we obtain
Since \(\mathcal {H}\) is dense in \(L_p(V)\), we can choose \(h\in \mathcal {H}\) to make \(|f-h|_{L_p(V)}\) arbitrarily small, which proves \(\lim _{\varepsilon \rightarrow 0}|f-f^{(\varepsilon )}|_{L_p(V)}=0\). \(\square \)
Recall that \(L_p(\mathcal {L}_{q_1}\cap \mathcal {L}_{q_2})\) denotes the \(L_p\)-space of \(\mathcal {L}_{q_1}\cap \mathcal {L}_{q_2}\)-valued functions on \(\mathbb {R}^d\) with respect to the Lebesgue measure on \(\mathbb {R}^d\). Since \((Z,\mathcal {Z},\mu )\) is a \(\sigma \)-finite separable measure space, \(\mathcal {L}_{q_1}\cap \mathcal {L}_{q_2}\) is a separable Banach space for any \(q_1,q_2\in [1,\infty )\). Hence, by Lemma 3.6 in [17] for each \(v\in L_p(\mathcal {L}_{q_1}\cap \mathcal {L}_{q_2})\), \(p>1\) there is a \(\mathcal {B}(\mathbb {R}^d)\otimes \mathcal {Z}\)-measurable function \(\bar{v}=\bar{v}(x,z)\) such that for every \(x\in \mathbb {R}^d\) we have \(v(x,z)=\bar{v}(x,z)\) for \(\mu \)-almost every \(z\in Z\). Therefore if \(v\in L_p(\mathcal {L}_{q_1}\cap \mathcal {L}_{q_2})\) for some \(p>1\), then we may assume that v is a \(\mathcal {B}(\mathbb {R}^d)\otimes \mathcal {Z}\)-measurable real-valued function. Moreover, we will often use the following characterisation of \(W^n_p(\mathcal {L}_{q_1}\cap \mathcal {L}_{q_2})\).
Lemma 3.6
Let \(v\in L_p(\mathcal {L}_{p}\cap \mathcal {L}_{q})\) for some \(p,q\in (1,\infty )\), and let \(\alpha \) be a multi-index. Then the following statements hold.
-
(i)
If \(v_{\alpha }\), the \(\mathcal {L}_{p}\cap \mathcal {L}_{q}\)-valued generalised \(D_{\alpha }\)-derivative, belongs to \(L_p(\mathcal {L}_p\cap \mathcal {L}_q)\), then for \(\mu \)-almost every \(z\in Z\) the function \(v_{\alpha }(\cdot ,z)\) belongs to \(L_p(\mathbb {R}^d,\mathbb {R})\) and it is the generalised \(D_{\alpha }\)-derivative of \(v(\cdot ,z)\).
-
(ii)
If \(v_{\alpha }(\cdot ,z)\), the generalised \(D_{\alpha }\)-derivative of the function \(v(\cdot ,z)\), belongs to \(L_p(\mathbb {R}^d,\mathbb {R})\) for \(\mu \)-almost every \(z\in Z\) such that
$$\begin{aligned} \int _{\mathbb {R}^d} \Big (\int _{Z}|v_{\alpha }(x,z)|^{r}\,\mu (dz)\Big )^{p/r}\,dx<\infty \quad \text {for }r=p,q, \end{aligned}$$(3.7)then \(v_{\alpha }\) belongs to \(L_p(\mathcal {L}_p\cap \mathcal {L}_q)\), and it is the \(\mathcal {L}_p\cap \mathcal {L}_q\)-valued generalised \(D_{\alpha }\)-derivative of v.
Proof
(i) Let \(\bar{v}_{\alpha }\) denote the \(\mathcal {L}_{p}\cap \mathcal {L}_{q}\)-valued generalised \(D_{\alpha }\)-derivative of v. Then
for every \(\varphi \in C_0^{\infty }\), where the integrals are understood as Bochner integrals of \(\mathcal {L}_{p}\cap \mathcal {L}_{q}\)-valued functions. Hence
for all \(\varphi \in C_0^{\infty }\) and bounded \(\mathcal {Z}\)-measurable functions \(\psi \) supported on sets of finite \(\mu \)-measure. We can use Fubini’s theorem to get
Thus for each \(\varphi \in C_0^{\infty }\) we have
for \(\mu \)-almost every \(z\in Z\). Consequently, for \(\mu \)-almost every \(z\in Z\) Eq. (3.8) holds for all \(\varphi \in \Phi \) for a separable dense set \(\Phi \subset C_0^{\infty }\) in \(L_{p/(p-1)}(\mathbb {R}^d,\mathbb {R})\). Notice that for \(\mu \)-almost every \(z\in Z\) the functions \(\bar{v}_{\alpha }(\cdot ,z)\) and \(v(\cdot ,z)\) belong to \(L_{p}(\mathbb {R}^d,\mathbb {R})\). Hence there is a set \(S\subset Z\) of full \(\mu \)-measure such that for \(z\in S\) Eq. (3.8) holds for all \(\varphi \in C_0^{\infty }\), which proves that for \(z\in S\) the function \(\bar{v}_{\alpha }(\cdot ,z)\) is the generalised \(D_{\alpha }\)-derivative of \(v(\cdot , z)\). To prove (ii) notice that if for \(\mu \)-almost every \(z\in Z\) the function \(v_{\alpha }(\cdot ,z)\) belongs to \(L_p(\mathbb {R}^d,\mathbb {R})\) and it is the \(D_{\alpha }\) generalised derivative of the function \(v(\cdot ,z)\), then for \(\mu \)-almost every \(z\in Z\) we have
for every \(\varphi \in C_0^{\infty }\). Using condition (3.7) and that \(v\in L_p(\mathcal {L}_p\cap \mathcal {L}_q)\), it is easy to check that, as functions of z, both sides of the above equation are functions in \(\mathcal {L}_p\cap \mathcal {L}_q\), and hence that these integrals define the same functions as the corresponding \(\mathcal {L}_p\cap \mathcal {L}_q\)-valued Bochner integrals. This proves that \(v_{\alpha }\) is the \(\mathcal {L}_{p}\cap \mathcal {L}_q\)-valued generalised \(D_{\alpha }\)-derivative of v. \(\square \)
Lemma 3.7
Let Assumptions 2.3 hold with \(m=0\). Then the following statements hold.
-
(i)
Let \(\zeta \) be a \(\mathcal {F}\otimes \mathcal {B}([0,T]\times \mathbb {R}^d)\otimes \mathcal {Z}\)-measurable function on \(\Omega \times H_T\times Z\) such that it is continuously differentiable in \(x\in \mathbb {R}^d\) and
$$\begin{aligned} |\zeta |+|D\zeta |\le K\bar{\eta }\quad \text {for all } (\omega ,t,x,z)\in \Omega \times H_T\times Z. \end{aligned}$$(3.9)Then there is a constant \(N=N(K,d)\) such that for \(\varphi \in W^1_1\)
$$\begin{aligned} A:=\int _{\mathbb {R}^d}\zeta (t,x,z)I^{\eta }\varphi (x)\,dx \le N\bar{\eta }^2(z)\,|\varphi |_{L_1} \quad \text {for all }(\omega ,t,z)\in \Omega \times [0,T]\times Z. \end{aligned}$$(3.10) -
(ii)
There is a constant \(N=N(K,d)\) such that for all \(\phi \in W^2_1\)
$$\begin{aligned} B:=\int _{\mathbb {R}^d}J^{\eta }\phi (x)\,dx \le N\bar{\eta }^2(z)|\phi |_{L_1}. \end{aligned}$$(3.11) -
(iii)
There is a constant \(N=N(K,d)\) such that for all \(\phi \in W^1_1\)
$$\begin{aligned} C:=\int _{\mathbb {R}^d}I^{\eta }\phi (x)\,dx \le N\bar{\eta }(z)|\phi |_{L_1}. \end{aligned}$$(3.12)
Proof
The proof of (3.11) and (3.12) is given in [8] and [9]. For the convenience of the reader we prove each of the above estimates here. We may assume that \(\varphi , \phi \in C_0^{\infty }\). For each \((\omega ,t,z,\theta )\in \Omega \times [0,T]\times Z\times [0,1]\) let \(\tau ^{-1}_{\theta \eta }\) denote the inverse of the function \(x\rightarrow x+\theta \eta _{t,z}(x)\). Using (3.2) and (3.3) by change of variables we have
and
Due to (3.9) and Assumption 2.3 we have a constant \(N=N(K,d)\) such that
for all \((\omega ,x,t,z,\theta )\in \Omega \times \mathbb {R}^d\times [0,T]\times Z\times [0,1]\). Thus from (3.13) and (3.14) by integration by parts we get (3.10), (3.11) and (3.12). \(\square \)
Lemma 3.8
Let Assumption 2.3 hold with an integer \(m\ge 0\). Then for all \(\omega \in \Omega \), \(t\in [0,T]\) and \(r>1\) we have
for \(\varphi ,\phi \in L_{r/(r-1)}\) and \(h\in W^2_r(\mathcal {L}_{r,2})\) and \(g\in W^2_r(\mathcal {L}_{r,1})\) with a constant \(N=N(r,d,K,K_{\bar{\eta }})\).
Proof
First assume \(\varphi ,\phi \in C^\infty _0\), and notice that
with
and
By Fubini’s theorem and Hölder’s inequality
By Taylor’s formula, Fubini’s theorem, change of variables, integration by parts and using Assumption 2.3 we get
with \(N=N(d,K)\), where \(\tau _{\theta \eta }\) is defined by (2.2), and \(\tau ^{-1}_{\theta \eta }\) is its inverse. Hence by Hölder’s inequality, change of variables, Fubini’s theorem and using \(|\bar{\eta }|\le K\) we obtain
and in the same way
with a constant \(N=N(K,d,r,K_{\bar{\eta }})\). By integration by parts, using Assumption 2.3, Cauchy–Schwarz and Hölder inequalities we get
and in the same way
with a constant \(N=N(K,d,r,K_{\bar{\eta }})\). Combining this with (3.17) through (3.20), we get (3.15) and (3.16) for \(\varphi ,\phi \in C^\infty _0\). Assume now that \(\varphi ,\phi \in L_{r/(r-1)}\) and \(\mu (Z)<\infty \). It is easy to see that by Hölder inequality we have
for \(\varphi ,\phi \in L_{r/(r-1)}\) and \(h,g\in L_r(\mathcal {L}_r)\) with a constant \(C=C(K,K_{\bar{\eta }},d,r,\mu (Z))\), which implies that estimates (3.15) and (3.16) for \(\varphi ,\phi \in C^\infty _0\) can be extended by continuity to \(\varphi ,\phi \in L_{r/(r-1)}\) for finite measures \(\mu \). In the general case of a \(\sigma \)-finite measure \(\mu \), there exist \(Z_n\in \mathcal {Z}\), \(n=1,2,\ldots \), such that \(\mu (Z_n)<\infty \) and \(\cup _{n=1}^\infty Z_n=Z\). We define measures \(\mu _n\) for integers \(n\ge 1\) such that \(d\mu _n=\mathbf{1}_{Z_n}d\mu \), where \(\mathbf{1}_{Z_n}\) is the indicator for \(Z_n\). By the previous argument, for \(\varphi ,\phi \in L_{r/(r-1)}\) we have
and
with a constant \(N=N(r,d,K,K_{\bar{\eta }})\). Then an application of Fatou’s lemma finishes the proof of this lemma. \(\square \)
Next we present two important Itô’s formulas from [17] for the pth power of \(L_p\)-norm of \(L_p\)-valued stochastic processes.
Lemma 3.9
Let \((u^i_t)_{t\in 0,T}\) be a progressively measurable \(L_p\)-valued process such that there exist \(f^i\in \mathbb {L}_p\), \(g^i=(g^{ir})_{r=1}^\infty \in \mathbb {L}_p\), \(h^i\in \mathbb {L}_{p,2}\), and an \(L_p\)-valued \(\mathcal {F}_0\)-measurable random variable \(\psi ^i\) for each \(i=1,2,\ldots ,M\) for some integer M, such that for every \(\varphi \in C_0^{\infty }\)
for \(P\otimes dt\)-almost every \((\omega ,t)\in \Omega \times [0,T]\) and all \(i=1,2,\ldots ,M\). Then there are \(L_p\)-valued adapted cadlag processes \(\bar{u}=(\bar{u}^1,\bar{u}^2,\ldots ,\bar{u}^M)\) such that Eq. (3.21), with \(\bar{u}^i\) in place of \(u^i\), holds for every \(i=1,2,\ldots ,M\) and each \(\varphi \in C_0^{\infty }\) almost surely for all \(t\in [0,T]\). Moreover, \(u^i=\bar{u}^i\) for \(P\otimes dt\)-almost every \((\omega ,t)\in \Omega \times [0,T]\), and
holds almost surely for all \(t\in [0,T]\), where the definition \(0/0:=0\) is used where 0/0 occurs, and \({\bar{u}_{s-}}\) denotes the left-hand limit in \(L_{p}({\mathbb {R}^{d}}) \) of \({\bar{u}}\) at \(s \in (0,T]\).
Lemma 3.10
Let \(u=(u_{t})_{t\in 0,T}\) be a progressively measurable \(W^1_p\)-valued process such that the following conditions hold: (i)
(ii) there exist \(f\in \mathbb {L}_p\) for \(\alpha \in \{0,1,\ldots ,d\}\), \(g\in \mathbb {L}_p\), \(h\in \mathbb {L}_{p,2}\), and an \(L_p\)-valued \(\mathcal {F}_0\)-measurable random variable \(\psi \), such that for every \(\varphi \in C_0^{\infty }\) we have
for \(P\otimes dt\)-almost every \((\omega ,t)\in \Omega \times [0,T]\), where \(D_{\alpha }^{*}=-D_{\alpha }\) for \(\alpha =1,2,\ldots ,d\), and \(D^{*}_{\alpha }\) is the identity operator for \(\alpha =0\). Then there is an \(L_p\)-valued adapted cadlag process \(\bar{u}=(\bar{u}_t)_{t\in [0,T]}\) such that for each \(\varphi \in C_0^{\infty }\) Eq. (3.22) holds with \(\bar{u}\) in place of u almost surely for all \(t\in [0,T]\). Moreover, \(u=\bar{u}\) for \(P\otimes dt\)-almost every \((\omega ,t)\in \Omega \times [0,T]\), and almost surely
for all \(t\in [0,T]\), where \(\bar{u}_{s-}\) denotes the left-hand limit in \(L_p(\mathbb {R}^d)\) of \(\bar{u}\) at \(s\in (0,T]\).
The following slight generalisation of Lemma from [16] will play a role in obtaining supremum estimates.
Lemma 3.11
Let \(T\in [0,\infty ]\) and let \(f=(f_t)_{t\ge 0}\) and \(g=(g_{t})_{t\ge 0}\) be nonnegative \(\mathcal {F}_{t}\)-adapted processes such that f is a cadlag and g is a continuous process. Assume
for any constant \(c>0\) and bounded stopping time \(\tau \le T\). Then, for any bounded stopping time \(\tau \le T\), for \(q\in (0,1)\)
Proof
This lemma is proved in [16] when both processes f and g are continuous. A word by word repetition of the proof in [16] extends it to the case when f is cadlag. For the convenience of the reader we present the proof below. By replacing \(f_{t}\) and \(g_{t}\) with \(f_{t\wedge T}\) and \(g_{t\wedge T}\), respectively, we see that we may assume that \(T=\infty \). Then we replace \(g_{t}\) with \(\max _{s\le t}g_{s}\) and see that without losing generality we may assume that \(g_{t}\) is nondecreasing. In that case fix a constant \(c>0\) and let \(\theta _{f}=\inf \{t\ge 0: f_{t}\ge c\}\), \(\theta _{g}=\inf \{t\ge 0: g_{t}\ge c\}\). Then
In the light of (3.23) we replace the expectation with
Hence
and it remains to substitute \(c^{1/q}\) in place of c and integrate with respect to c over \((0,\infty )\). The lemma is proved. \(\square \)
Finally we present a slight modification of Lemma 5.3 from [9] which we will use in proving regularity in time of solutions to (1.1)–(1.2).
Lemma 3.12
Let V be a reflexive Banach space, embedded continuously and densely into a Banach space U. Let f be a U-valued weakly cadlag function on [0, T] such that the weak limit in U at T from the left is f(T). Assume there is a dense subset S of [0, T] such that \(f(s)\in V\) for \(s\in S\) and \(\sup _{s\in S}|f(s)|_V<\infty \). Then f is a V-valued function, which is cadlag in the weak topology of V, and hence \(\sup _{s\in [0,T]}|f(s)|_V=\sup _{s\in S}|f(s)|_V\).
Proof
Since S is dense in [0, T], for a given \(t\in [0,T)\) there is a sequence \(\{t_n\}_{n=1}^{\infty }\) with elements in S such that \(t_n\downarrow t\). Due to \(\sup _{n\in \mathbb {{N}}}|f(t_n)|_V<\infty \) and the reflexivity of V there is a subsequence \(\{t_{n_k}\}\) such that \(f(t_{n_k})\) converges weakly in V to some element \(v\in V\). Since f is weakly cadlag in U, for every continuous linear functional \(\varphi \) over U we have \(\lim _{k\rightarrow \infty }\varphi (f(t_{n_k}))=\varphi ({f}(t))\). Since the restriction of \(\varphi \) in V is a continuous functional over V we have \(\lim _{k\rightarrow \infty }\varphi (f(t_{n_k}))=\varphi (v)\). Hence \(f(t)=v\), which proves that f is a V-valued function over [0, T). Moreover, by taking into account that
we obtain \(K:=\sup _{t\in [0,T)}|f(s)|_V<\infty \). Let \(\phi \) be a continuous linear functional over V. Due to the reflexivity of V, the dual \(U^*\) of the space U is densely embedded into \(V^*\), the dual of V. Thus for \(\phi \in V^{*}\) and \(\varepsilon >0\) there is \(\phi _\varepsilon \in U^*\) such that \(|\phi -\phi _\varepsilon |_{V^*}\le \varepsilon \). Hence for arbitrary sequence \(t_n\downarrow t\), \(t_n\in [0,T]\) we have
Letting here \(n\rightarrow \infty \) and then \(\varepsilon \rightarrow 0\), we get
which proves that f is right-continuous in the weak topology in V. We can prove in the same way that at each \(t\in [0,T]\) the function f has weak limit in V from the left at each \(t\in (0,T]\), which finishes the proof of the lemma. \(\square \)
4 Some results on interpolation spaces
A pair of complex Banach spaces \(A_0\) and \(A_1\), which are continuously embedded into a Hausdorff topological vector space \(\mathcal {H}\), is called an interpolation couple, and \(A_{\theta }=[A_0,A_1]_{\theta }\) denotes the complex interpolation space between \(A_0\) and \(A_1\) with parameter \(\theta \in (0,1)\). For an interpolation couple \(A_0\) and \(A_1\) the notations \(A_0\cap A_1\) and \(A_0+A_1\) is used for the subspaces
equipped with the norms \(|v|_{A_0\cap A_1}=\max (|v|_{A_0},|v|_{A_1})\) and
respectively. Then the following theorem lists some well-known facts about complex interpolation, see e.g., [5] or 1.9.3, 1.18.4 and 2.4.2 in [34] and 5.6.9 in [18].
Theorem 4.1
-
(i)
If \(A_0,A_1\) and \(B_0,B_1\) are two interpolation couples and \(S:A_0+A_1\rightarrow B_0+B_1\) is a linear operator such that its restriction onto \(A_i\) is a continuous operator into \(B_i\) with operator norm \(C_i\) for \(i=0,1\), then its restriction onto \(A_{\theta }=[A_0,A_1]_{\theta }\) is a continuous operator into \(B_{\theta }=[B_0,B_1]_{\theta }\) with operator norm \(C_0^{1-\theta }C_1^{\theta }\) for every \(\theta \in (0,1)\).
-
(ii)
For a \(\sigma \)-finite measure space \(\mathfrak M\) and an interpolation couple of separable Banach spaces \(A_0\), \(A_1\) we have
$$\begin{aligned}{}[L_{p_0}(\mathfrak M,A_0),L_{p_1}(\mathfrak M,A_1)]_{\theta }=L_{p}(\mathfrak M,[A_0,A_1]_{\theta }), \end{aligned}$$for every \(p_0,p_1\in [1,\infty )\), \(\theta \in (0,1)\), where \(1/p=(1-\theta )/p_0+\theta /p_1\).
-
(iii)
Let \(H^m_p\) denote the Bessel potential spaces of complex-valued functions. Then for \(m_0,m_1\in \mathbb {R}\) and \(1<p_0,p_1<\infty \)
$$\begin{aligned}{}[H^{m_0}_{p_0},H^{m_1}_{p_1}]_{\theta }=H^m_p, \end{aligned}$$where \(m=(1-\theta )m_0+\theta m_1\), and \(1/p=(1-\theta )/p_0+\theta /p_1\). Moreover, for integers m one has \(H^m_p=W^m_p\) with equivalent norms.
-
(iv)
For a UMD Banach space V, denote by \(H^m_p(V)\) the Bessel potential spaces of V-valued functions. Then for \(1<p<\infty \) and \(m_0, m_1\in \mathbb {R}\)
$$\begin{aligned}{}[H^{m_0}_p(V),H^{m_1}_p(V)]_\theta =H^m_p(V) \end{aligned}$$for every \(\theta \in (0,1)\), where \(m=(1-\theta )m_0+\theta m_1\).
-
(v)
For \(\theta \in [0,1]\) there is a constant \(c_{\theta }\) such that
$$\begin{aligned} |v|_{A_{\theta }}\le c_{\theta }|v|_{A_0}^{1-\theta }|v|^{\theta }_{A_1} \end{aligned}$$for all \(v\in A_0\cap A_1\).
5 \(L_p\) estimates
Let Assumptions 2.1 through 2.6 hold with an integer \(m\ge 0\), and assume in this section that \(\mathcal {R}\equiv 0\). Let \(u=(u_t)_{t\in [0,T]}\) be a \(W^{m+2}_p\)-valued solution to (1.1)–(1.2). Then for each \(\varphi \in C^\infty _0\) and any multi-number \(\alpha \) with \(|\alpha |\le m\), from the definition of the generalised solution and by integrating by parts we have
for all \(t\in [0,T]\) and almost all \(\omega \in \Omega \). To shorten some expressions we introduce the operators \(\mathbb {J}^\xi \), \(\mathbb {J}^\eta \) and \(\mathbb {I}^\eta \), defined by
on functions \(\varphi \). For functions \(v\in W^m_p\) let \(\mathbf{v}=\{v_{\alpha }:|\alpha |\le m\}\) denote the vector whose coordinates are the derivatives \(v_{\alpha }:=D_{\alpha } v\) for \(|\alpha |\le m\). We use also the notation \(I^{\eta }{} \mathbf{v}\), \(J^{\eta }{} \mathbf{v}\) and \(T^{\eta }{} \mathbf{v}\) for the vectors with coordinates \(I^{\eta }{v}_{\alpha }\), \(J^{\eta }{v}_{\alpha }\) and \(T^{\eta }{v}_{\alpha }\), respectively for \(|\alpha |\le m\). Then applying Lemma 3.9 to \(\mathbf{u}_t\) we have
almost surely for all \(t\in [0,T]\), where we use the notation
for \(v\in W^{m+2}_p\), for each \(f\in W^m_p\), \(g\in W^{m+1}_p(l_2)\), \(\omega \in \Omega \) and \(t\in [0,T]\). In order to estimate \(E|\mathbf{u}_t|^p_{L_p}\) we define also the “p-form”
for \(v\in W^{m+2}_p\), \(h\in W_p^{m+2}(\mathcal {L}_{p,2})\), for each \(\omega \in \Omega \) and \(t\in [0,T]\).
Proposition 5.1
Let Assumption 2.1 hold. Then for any \(p\in [2,\infty )\) there is a constant \(N=N(d,p,m,K)\) such that
for all \(v\in W^{m+2}_p\), \(f\in W^m_p\), \(g\in W^{m+1}_p(l_2)\), \(\omega \in \Omega \) and \(t\in [0,T]\).
Proof
This estimate is proved in [12] in a more general setting. \(\square \)
Proposition 5.2
Let Assumption 2.3 hold with an integer \(m\ge 0\). Then for \(\omega \in \Omega \) and \(t\in [0,T]\)
for \(v\in W^{m+2}_p\) and \(h\in W^{m+2}_p(\mathcal {L}_{p,2})\) with constants \(N=N(d,m,p,K, K_{\bar{\eta }})\).
Proof
First we claim that for any \(|\alpha |=n\le m\)
with some functions \(b^{\alpha \beta }\), \(c^{\alpha \beta }\) and \(\bar{c}^{\alpha \beta }\) of \((\omega ,t,z,x)\), such that for \(|\beta |\le n\)
with a constant \(N=N(d,m,K)\). The reader can prove this easily by induction on n, noticing that
where, as before, the subscripts indicates derivatives in the corresponding coordinates. Using equations (5.7) and (5.8) we have
where \(\mathbf{b}\) and \(\mathbf{c}\) denote the matrices with entries \(b^{\alpha \beta }\) and \(c^{\alpha \beta }\), respectively for \(|\alpha |\le m\) and \(|\beta |\le m\). Clearly,
and
with
where to ease notation we write I and T in place of \(I^{\eta }\) and \(T^{\eta }\), respectively. Hence
By Lemma 3.7
and due to (5.9)
with constants \(N=N(d,m,p,K,K_{\bar{\eta }})\). By Taylor’s formula, (5.9), and Assumption 2.3
It is easy to see that
and
Thus, from the above estimates we have
It remains to estimate
by the right-hand side of (5.6). If \(p=2\), then we have
To estimate \(H_2(v,h)\) notice that by Taylor’s formula, the change of variable \(y=\tau _{\theta \eta }(x)=x+\theta \eta (x)\) and by integration by parts we have
where \(\beta =0,1,2,\ldots ,d\) and \(\zeta ^{\alpha \beta }\) are some functions of \((\omega ,t,z,x,\theta )\) such that
with a constant \(N=N(d,m,K)\). Hence by the Cauchy–Schwarz and Young inequalities we get
with \(N'=N'(d,m,K)\), which gives the estimate for \(H_2(v,h)\) and finishes the proof of (5.6) for \(p=2\). If \(p>2\) then by taking \(r=p\), \(\varphi =|\mathbf{v}|^{p-2}v_\alpha \) and \(h=h_\alpha \) for each \(\alpha \) in estimate (3.15) and using Young’s inequality, we get
Similarly, by taking \(r=p/2\), \(\phi =|\mathbf{v}|^{p-2}\) and \(g=|\mathbf{h}|^2\) in estimate (3.16) and using Young’s inequality, we have
which completes the proof of the proposition. \(\square \)
Corollary 5.1
Let Assumption 2.3 hold with an integer \(m\ge 0\). Then for any \(p\ge 2\) and \(v\in W^{m+2}_p\)
for all \(\omega \in \Omega \) and \(t\in [0,T]\) with a constant \(N=N(d,p,m,K,K_{\bar{\eta }})\).
Proof
Notice that
where
By Proposition 5.2, we have
with a constant \(N=N(d,p,m,K,K_{\bar{\eta }})\). Moreover by the convexity of the function \(f(x)=|x|^p\), we have \(\mathfrak {Q}_p(t,v)\ge 0\), which finishes the poof of this corollary. \(\square \)
For integers \(m\ge 0\) set
for \(v\in W^{m+1}_p\), \(g\in W^{m+1}_p(l_2)\), \(h\in W^{m+1}_p(\mathcal {L}_{p,2})\), \(\omega \in \Omega \) and \(t\in [0,T]\), where repeated indices \(\alpha \) mean summation over all multi-numbers of length m. (Recall that for functions \(v\in W^m_p\) we use the notation \(\mathbf{v}\) with coordinate \(v_{\alpha }=D_{\alpha }v\) for \(|\alpha |\le m\)).
Proposition 5.3
Let \(m\ge 0\) be an integer and \(p\in [2,\infty )\). Then the following estimates hold for all \((\omega ,t)\in \Omega \times [0,T]\).
-
(i)
If Assumption 2.1 is satisfied with \(m\ge 0\) then
$$\begin{aligned} {\mathfrak P}^2_{m,p}(t,v,g) \le N\left( |v|^{2p}_{W^m_p}+|v|^{2p-2}_{W^m_p}|g|^{2}_{W^{m}_p}\right) \end{aligned}$$(5.14)for all \(v\in W^{m+1}_p\) and \(g\in W^{m+1}_p(l_2)\), with a constant \(N=N(d,m,p,K)\).
-
(ii)
If Assumption 2.3 is satisfied then
$$\begin{aligned} {\mathfrak Q}_{m,p}(t,v,h) \le N\left( |v|^{p}_{W^{m+1}_p} +|h|^{p}_{W^m_{p,2}}\right) \end{aligned}$$(5.15)for all \(v\in W^{m+1}_p\) and \(h\in W^{m}_p(\mathcal {L}_{p,2})\) with \(N=N(d,m,p,K,K_{\bar{\eta }})\).
Proof
Noticing that \(p|\mathbf{v}|^{p-2}v_{\alpha }\sigma ^{ir}D_iv_{\alpha } =\sigma ^{ir}D_i|\mathbf{v}|^p\), by integration by parts and by Minkowski’s and Hölder’s inequalities we can see that
can be estimated by the right-hand side of (5.14). By Minkowski and Hölder’s inequalities it is easy to show that
can also be estimated by the right-hand side of (5.14). To prove (ii) let \(\mathbf{y}\) denote the vector with coordinates \(y_{\alpha }=(\mathbb {I}^{\eta }v+h)_{\alpha }\) for \(|\alpha |\le m\). Then the integrand in (5.13) can be written as
By Taylor’s formula
with constants \(N=N(d,p,m)\) and \(N'=N'(d,p,m)\), where \(\mathbf{e}\) denotes the vector with coordinates \(e_{\alpha }:=(\mathbb {I}^{\eta }v)_{\alpha }\) for \(|\alpha |\le m\). By Fubini’s theorem and Hölder’s inequality
Using Hölder’s inequality, taking into account (5.7) and using Lemma 3.1 we obtain
By Lemma 3.1 and (5.7) we have
Combining these inequalities and using Young’s inequality we get (5.15). \(\square \)
6 Proof of the main result
6.1 Uniqueness of the generalised solution
Let Assumptions 2.1 through 2.4 hold with \(m=0\). For a fixed \(p\in [2,\infty )\) let \(u^{(i)}=(u^{(i)}_t)_{t\in [0,T]}\) be generalised solutions to Eq. (1.1) with initial condition \(u^{(i)}_0=\psi \in L_p\) for \(i=1,2\). Then for \(v=u^{(2)}-u^{(1)}\) by Lemma 3.10 on Itô formula we have that almost surely
for all \(t\in [0,T]\), where \(\zeta _1\) and \(\zeta _2\) are local martingales defined by
\(Q_s(\cdot )\), \(\bar{Q}^{\xi }_s(\cdot )\), \(\bar{Q}^{\eta }_s(\cdot )\) and \(P^{\eta }_s(z,\cdot )\) are functionals on \(W^1_p\), for each \((\omega ,s)\) and z, defined by
Recall that \(\bar{b}^i=b-D_ja^{ij}\), \(\mathcal {J}_{\eta }^{i}\) and \(\mathcal {J}_{\eta }^{0}\) are defined by (2.3)–(2.4), and (v, w) denotes the Lebesgue integral over \(\mathbb {R}^d\) of the product vw for real functions v and w on \(\mathbb {R}^d\).
Note that due to the convexity of the function \(|r|^p\), \(r\in \mathbb {R}\), we have
for real-valued functions \(v=v(x)\), \(x\in \mathbb {R}^d\). Together with the above functionals we need also estimate the functionals \(\mathfrak Q_s(\cdot )\) and \(\hat{Q}^{\eta }_s(\cdot )\) defined for each \((\omega ,s)\in \Omega \times [0,T]\) by
for \(v\in W^1_p\).
Proposition 6.1
Let Assumptions 2.1, 2.2 and 2.3 hold with \(m=0\). Then for \(p\ge 2\) there are constants \(N=N(d,p,K)\), \(N_1=N_2(d,p,K,K_{\bar{\xi }})\) and \(N_2=N_2(d,p,K,K_{\bar{\eta }})\) such that
for all \(v\in W^1_p\) and \((\omega ,s)\in \Omega \times [0,T]\).
Proof
Notice that the estimate (6.4) is the special case of Proposition 5.3 (ii), and for \(v\in W^2_p\) the second and third estimates in (6.3) follow from the estimate (5.11) in Corollary 5.1. Notice also that for \(v\in W^2_p\) the first estimate in (6.3) is a special case of (5.5) in Proposition 5.1, and the last estimate in (6.3) is a special case of the estimate in Proposition 5.2. It is an easy exercise to show that the functionals on the left-hand side of the inequalities in (6.3) are continuous in \(v\in W^1_p\), that completes the proof of the proposition. \(\square \)
Define now the stochastic process
and the stopping time
for every integer \(n\ge 1\), where \((\rho _n)_{n=1}^{\infty }\) is an increasing sequence of stopping times, converging to infinity such that \((\zeta _i(t\wedge \rho _n))_{t\in [0,T]}\) is a martingale for each \(n\ge 1\) and \(i=1,2\). Then clearly, \(E\zeta _i(t\wedge \tau _n)=0\) for \(t\in [0,T]\) and \(i=1,2\). Due to (6.2) and the estimate in (6.4) we have
which implies
Thus, substituting \(t\wedge \tau _n\) in place of t in (6.1) and then taking expectation and using Proposition 6.1 and Assumption 2.5 we obtain
for \(t\in [0,T]\). Hence by Gronwall’s lemma \(Ey(t\wedge \tau _n)=0\) for each \(t\in [0,T]\) and integer \(n\ge 1\), which implies that almost surely \(y_t=0\) for all \(t\in [0,T]\), and completes the proof of the uniqueness.
6.2 A priori estimates
Proposition 6.2
Let Assumptions 2.1 through 2.4 and assume Assumption 2.6 hold with an integer \(m\ge 0\) and \(\mathcal {R}\equiv 0\). Let \(u=(u_t)_{t\in [0,T]}\) be a \(W^{m+2}_p\)-valued generalised solution to (1.1)–(1.2) such that it is cadlag as a \(W^m_p\)-valued process and
Then
with a constant \(N=N(m,d,p,T, K,K_{\bar{\xi }},K_{\bar{\eta }})\).
Proof
We may assume that the right-hand side of the inequality (6.5) is finite. For a fixed integer \(n\in [0,m]\), multi-numbers \(|\alpha |\le n\) and \(\varphi \in C_0^{\infty }\), we have
For an integer \(n\le m\) let \(\mathbf{u}\) denote the vector with coordinates \(u_{\alpha }:=D_{\alpha }u\) for \(|\alpha |\le n\). Recall, see (5.1), that by Lemma 3.9 on Itô’s formula we have
where the \(Q_{p}\), \(Q^{\xi }_{p}\) and \(\hat{Q}_{n,p}\) are defined in (5.2), (5.3) and (5.4), with n in place of m, and \(\zeta _i=(\zeta _i(t))_{t\in [0,T]}\) is a cadlag local martingale starting from zero for each \(i=1,2,3\), such that
and
where
for \(v\in W^{m+2}_p\) and \(h\in W^{m+2}_{p,2}\). By Proposition 5.1 and Corollary 5.1 we obtain
Hence using the estimate (5.15) in Proposition 5.3 we have
for all \(t\in [0,T]\), for a localising sequence \((\tau _k)_{k=1}^{\infty }\) of stopping times for \(\zeta _i\), \(i=1,2,3\). Hence by Gronwall’s lemma
for \(t\in [0,T]\) and \(k\ge 1\) with a constant \(N=N(d,m,p,T,K,K_{\bar{\xi }}, K_{\bar{\eta }})\), which implies
by Fatou’s lemma. To show that we can interchange the supremum and expectation it suffices to prove that for every \(\varepsilon >0\)
and
with a constant \(N=N(\varepsilon ,d,m,p,T,K,K_{\bar{\xi }},K_{\bar{\eta }})\). The proof of (6.9) is well-known and it goes as follows. Recall the notation \(\mathfrak P^2_{p}\) in (5.12) and notice that by the Davis inequality, using the estimate in (5.14) we have
which gives (6.9) by virtue of (6.8). To prove (6.10) we first assume that \(\mu \) is a finite measure. Notice that
where A(s, v, h) is defined by
for \(v\in W^m_p\), \(\mathbf{v}=(v_{\alpha })_{|\alpha |\le n}\) and \(h\in W^{m+2}_p(\mathcal {L}_{p,2})\). By similar calculations to those in the proof of Proposition 5.2 we can easily see that
with
where \(\mathbf{b}=(b^{\alpha \beta })\) and \(\mathbf{c}=(c^{\alpha \beta })\) are from (5.7). Hence
with
Note that for \(\rho _i(t):=\rho _{i1}(t)-\rho _{i2}(t)\) one can always have the supremum estimate
This, however, is not always useful, and when almost surely
then we can view \(\rho _i(t)\) as the stochastic Itô integral
and apply the Davis inequality
By Minkowski’s and Hölder’s inequalities
Thus we can view \(\zeta _2(t)\) as a stochastic Itô integral, and applying the Davis inequality we get
for \(i=2\) in the same way as estimate in (6.9) is proved. By Lemma 3.7 (iii) we have
which, as before, allows us to get the estimate (6.13) for \(i=1\). By estimate (3.15), Young’s inequality and (6.8), we have
with constants N and \(N'\) depending only on K, d, m, p, T, \(K_{\bar{\xi }}\) and \(K_{\bar{\eta }}\). Thus we can use estimate (6.12) to get
for \(i=4\) with a constant \(N=N(K,d,m,p,T,K_{\bar{\xi }},K_{\bar{\eta }})\). Similarly, using the estimate for \(A_3\) in (5.10) and the estimate (3.16) we obtain (6.14) for \(i=3\). Due to (5.9)
for \(v\in W^m_p\). Consequently, viewing \(\zeta _5(t)\) as Itô integral we get the estimate (6.13) for \(i=5\), and applying estimate (6.12) to \(\zeta _6(t)\) we obtain the estimate (6.14) for \(i=6\). Combining the estimates (6.13) for \(i=1,2,5\) and the estimate (6.13) for \(i=3,4,6\) we obtain (6.11).
In the general case of \(\sigma \)-finite measure \(\mu \) we have a nested sequence \((Z_k)_{k=1}^{\infty }\) of sets \(Z_k\in \mathcal {Z}\) such that \(\mu (Z_k)<\infty \) for every k and \(\cup _{k=1}^{\infty }Z_k=Z\). For each integer \(k\ge 1\) define the measures
for \(F\in \mathcal {Z}\otimes \mathcal {B}((0,T])\) and \(G\in \mathcal {Z}\), and set \(\tilde{\pi }_k(dz,dt)=\pi _k(dz,dt)-\mu _k\otimes dt\). Let \(\zeta ^{(k)}_2\) and \(\zeta ^{(k)}_3\) be defined as \(\zeta _2\) and \(\zeta _3\), respectively, but with \(\tilde{\pi }_k\), \(\pi _k\) and \(\mu _k\) in place of \(\tilde{\pi }\), \(\pi \) and \(\mu \), respectively, in (6.6) and (6.7). By virtue of what we have proved above, for each k we have
for \(\varepsilon >0\) with a constant \(N=N(\varepsilon ,m,p,T,K,K_{\bar{\xi }}, K_{\bar{\eta }})\). Note that for a subsequence \(k'\rightarrow \infty \)
for \(i=2,3\). Hence letting \(k=k'\rightarrow \infty \) in (6.15) by Fatou’s lemma we obtain (6.10), which completes the proof of the proposition. \(\square \)
6.3 Existence of a generalised solution
Before the construction of a generalised solution to (1.1)–(1.2), we introduce some notations. For integers \(r>1\), real numbers \(n\ge 0\) and \(p\ge 2\) let \(\mathbb {U}^{n}_{r,p}\) denote the space of \(H^n_p\)-valued \(\mathcal {F}\otimes \mathcal {B}([0,T])\)-measurable functions v on \(\Omega \times [0,T]\) such that
The subspace of well-measurable functions \(v:\Omega \times [0,T]\rightarrow H^n_p\) in \(\mathbb {U}^{n}_{r,p}\) is denoted by \(\mathbb {V}^{n}_{r,p}\), and we will use \(\mathbb {V}^n_p\) to denote \(\mathbb {V}^n_{p,p}\). Set \(\Psi ^n_p:=L_p(\Omega , \mathcal {F}_0; H^n_p)\), and recall from the Introduction the definition of the spaces \(\mathbb {H}^n_p=\mathbb {H}^n_p(\mathbb {R})\) and \(\mathbb {H}^n_p(V)\) for separable Banach spaces V.
In the whole section we assume that Assumptions 2.1 through 2.6 with \(m\ge 1\) are in force. By a standard stopping time argument we may assume that
First we assume that m is an integer, \(\mathcal {R}\equiv 0\), and make also the following additional assumption
Assumption 6.1
The initial condition \(\psi \) and the free data f, g and h vanish if \(|x|\ge R\) for some \(R>0\).
Under the above conditions we approximate the Cauchy problem (1.1)–(1.2) by mollifying (in \(x\in \mathbb {R}^d\)) all data and coefficients involved in it. For \(\varepsilon \in (0,\varepsilon _0)\) we consider the equation
with initial condition
where \(\varepsilon _0\) is given in Corollary 3.4,
with operators
and
for real-valued differentiable functions \(\varphi \). Recall that \(\mathbf {I}\) denotes the identity matrix, and \(v^{(\varepsilon )}\) denotes the mollification \(v^{(\varepsilon )}=S^\varepsilon v\) of v in \(x\in \mathbb {R}^d\) defined in (3.4).
Note that by virtue of standard properties of mollifications and by Corollary 3.4 and Lemma 3.5, we have that Assumptions 2.1 through 2.6 are satisfied for (6.17)–(6.18) with every integer \(m\ge 0\) with non-negative functions \(\bar{\xi }=\bar{\xi }_m(z)\), \(\bar{\eta }=\bar{\eta }_m(z)\) of \(z\in Z\) and constants \(K=K_m\),
Moreover, there is a constant \(\delta >0\) such that \(P\otimes dt \otimes dx\)-almost all \((\omega ,t,x)\in \Omega \times H_T\)
Due to (6.16) and Assumption 6.1
for each n. Hence by [15] the Cauchy problem (6.17)–(6.18) has a unique generalised solution \(u^\varepsilon \), which is a \(W^n_2\)-valued cadlag process and for each integer \(n\ge 0\), and there is a constant N such that
Thus by Sobolev’s embedding \(u^\varepsilon \) is a cadlag \(W^n_p\)-valued process for every n such that
Moreover, by Proposition 6.2 and Lemma 3.5 for \(m\ge 1\) and \(n=0,1,\ldots ,m\) we have
for every integer \(r>1\) with a constant \(N=N(d,p,m,T,K,K_{\bar{\xi }},K_{\bar{\eta }})\), where \(i=1\) when \(p=2\) and \(i=2\) when \(p>2\). Recall that \(\mathbb {V}^{n}_{r,p}\) denotes the subspace of well-measurable functions \(v:\Omega \times [0,T]\rightarrow H^n_p\) in \(\mathbb {U}^{n}_{r,p}\). Since \(\mathbb {V}^n_{r,p}\) is reflexive, there exists a sequence \(\{\varepsilon _k\}_{k=1}^\infty \) and a process \(u\in \mathbb {V}^{n}_{r,p}\) such that \(\lim _{k\rightarrow \infty }\varepsilon _k=0\) and \(u^{\varepsilon _k}\) converges weakly to some u in \(\mathbb {V}^{n}_{r,p}\). To show that a modification of u is a solution to (1.1)–(1.2) we pass to the limit in the equation
where \(\varphi \in C_0^{\infty }\), and \(\langle \mathcal {A}_s^\varepsilon u_s^{\varepsilon },\varphi \rangle \) is defined as \(\langle \mathcal {A}_s u_s,\varphi \rangle \) in (2.5) but with \(u^\varepsilon \), \(a^\varepsilon \), \(b^{(\varepsilon )}\), \(c^{(\varepsilon )}\), \(\xi ^{(\varepsilon )}\), \(\eta ^{(\varepsilon )}\), \((\lambda ^\xi )^{(\varepsilon )}\) and \((\lambda ^\eta )^{(\varepsilon )}\) in place of u, a, b, c, \(\xi \), \(\eta \), \(\lambda ^\xi \) and \(\lambda ^\eta \) respectively. To this end we take a bounded well-measurable real-valued process \(\zeta =(\zeta _t)_{t\in [0,T]}\), multiply both sides of Eq. (6.20) with \(\zeta _t\) and then integrate the expressions we get against \(P\otimes dt\) over \(\Omega \times [0,T]\). Thus we obtain
where F and \(F^i_{\varepsilon }\), \(i=1,2,3\), are linear functionals of \(v\in \mathbb {V}^1_p\), defined by
and
For each \(i=1,2,3\) we also define the functional \(F^i\) in the same way as \(F^i_{\varepsilon }\) is defined above, but with \(\mathcal {A}\), \(\mathcal {M}\), \(\eta \) and \(\gamma \) in place of \(\mathcal {A}^{\varepsilon }\), \(\mathcal {M}^{\varepsilon }\), \(\eta ^{(\varepsilon )}\) and \(\gamma ^{(\varepsilon )}\) respectively. Obviously, by Hölder’s inequality and the boundedness of \(\zeta \), for all \(v\in \mathbb {V}^1_p\) we have
with \(q=p/(p-1)\) and a constant C independent of v and \(\varepsilon \), which means \(F\in {\mathbb {V}^1_p}^*\), the space of all bounded linear functionals on \(\mathbb {V}^1_p\). Next we show that \(F^i_{\varepsilon }\) and \(F^i\) are also in \({\mathbb {V}^1_p}^*\) for each \(\varepsilon >0\), and \(F^i_{\varepsilon }\rightarrow F^i\) strongly in \({\mathbb {V}^1_p}^*\) as \(\varepsilon \rightarrow 0\) for \(i=1,2,3\).
Lemma 6.1
For sufficiently small \(\varepsilon >0\) the functionals \(F^i\) and \(F^i_{\varepsilon }\) are in \({\mathbb {V}^1_p}^*\) for \(i=1,2,3\).
Proof
To show \(F^1_{\varepsilon }\in {\mathbb {V}^1_p}^*\) we notice that \( F^1_\varepsilon =\sum _{k=1}^7R^k_\varepsilon (v) \) with
where \(\mathcal {J}^i_{\xi ^{(\varepsilon )}}\), \(\mathcal {J}^i_{\eta ^{(\varepsilon )}}\), \(\mathcal {J}^0_{\xi ^{(\varepsilon )}}\), \(\mathcal {J}^0_{\eta ^{(\varepsilon )}}\) are defined by (2.3) and (2.4) with \(\xi ^{(\varepsilon )}\) and \(\eta ^{(\varepsilon )}\) in place of \(\eta \) respectively, for \(i=1,2,...,d\). Since the functions \(\zeta \), \(a^{\varepsilon }\), \({\bar{b}}^{(\varepsilon )}\) and \(c^{(\varepsilon )}\) are in magnitude bounded by a constant, by Hölder’s inequality we have
for \(i=1\) with \(q=p/(p-1)\) and a constant N independent of v and \(\varepsilon \). which shows that \(R^1_\varepsilon \in {\mathbb {V}^1_p}^*\) for all \(\varepsilon \). Using Taylor’s formula
and taking into account that \(|\zeta |\) is bounded by a constant, we have
Hence by Hölder’s inequality and then the change of variable \(y=x+\vartheta \theta \eta _{s,z}^{(\varepsilon )}(x)\), by Corollary 3.4 we get (6.23) for \(i=2\) and \(\varepsilon \in (0,\varepsilon _0)\), with \(\varepsilon _0\) given in Corollary 3.4, which proves that \(R^2_\varepsilon \in {\mathbb {V}^1_p}^*\) for \(\varepsilon \in (0,\varepsilon _0)\). We can prove in the same way that \(R^i_\varepsilon \in {\mathbb {V}^1_p}^*\) for \(i=3,4,5\) and \(\varepsilon \in (0,\varepsilon _0)\). Similarly, due to the boundedness of \(\zeta \), Assumptions 2.2 and 2.4, using Lemma 3.1 and Hölder’s inequality we have (6.23) for \(i=6, 7\) for \(\varepsilon \in (0,\varepsilon _0)\). Hence \(F^1_{\varepsilon }\in {\mathbb {V}^1_p}^*\) for \(\varepsilon \in (0,\varepsilon _0)\). Due to the boundedness of \(\zeta \), \(\sigma ^{(\varepsilon )r}\) and \(\beta ^{(\varepsilon )r}\), by Davis’ and Hölder’s inequalities, we get
for \(v\in \mathbb {V}^1_p\) with constants C and \(C'\) independent of v and \(\varepsilon \), which shows \(F^2_\varepsilon \in {\mathbb {V}^1_p}^*\). By the boundedness of \(\zeta \), using Davis’ and Hölder’s inequalities, we obtain
for \(v\in \mathbb {V}^1_p\) with a constant C independent of \(\varepsilon \) and v, where
and
Due to Assumptions 2.3 and 2.4 by Lemma 3.1 we have
for \(i=1,2\) with constants C and \(C'\) independent of v and \(\varepsilon \). Consequently, \(F^3_\varepsilon \in {\mathbb {V}^1_p}^*\). In the same way we obtain \(F^i\in {\mathbb {V}^1_p}^*\) for \(i=1,2,3\). \(\square \)
Lemma 6.2
For each \(i=1,2,3\)
Proof
We define the functionals \(R^i\) for \(i=1,2,\ldots ,7\) in the same way as \(R^i_\varepsilon \) are defined in (6.22), but with a, b, c, \(\xi \), \(\eta \), \(\lambda ^{\xi }\) and \(\lambda ^{\eta }\) in place of \(a^\varepsilon ,b^{(\varepsilon )},c^{(\varepsilon )}\), \(\xi ^{(\varepsilon )}\) \(\eta ^{(\varepsilon )}\), \((\lambda ^{\xi })^{(\varepsilon )}\) and \((\lambda ^{\eta })^{(\varepsilon )}\), respectively. To prove \(F^1_\varepsilon \rightarrow F^1\) strongly in \({\mathbb {V}^1_p}^*\), we notice that
Since \(\zeta \) is bounded, for a constant N independent of v and \(\varepsilon \), we have
for all \(\varepsilon \ge 0\) with
By Hölder’s inequality and well-known properties of mollifications
with a constant \(N=N(K,d)\), where \(q=p/(p-1)\). Hence we get
for \(i=1\). Clearly,
with
Note that \(|\eta ^{(\varepsilon )}-\eta |\le \varepsilon \bar{\eta }\) for all \(t\in [0,T]\), \(x\in \mathbb {R}^d\), \(z\in Z\) and \(\varepsilon >0\). Moreover, by Taylor’s formula, Minkowski’s inequality and Corollary 3.4
for \(s\in [0,T]\), \(z\in Z\), \(\omega \in \Omega \) and \(\varepsilon \in (0,\varepsilon _0)\), with a constant \(N=N(K,d,p)\). Hence by Hölder’s inequality we have
which by virtue of (6.26) proves (6.25) for \(i=2\). We can prove similarly (6.25) for \(i=3,4,5,6,7\), which proves \(F^1_\varepsilon \rightarrow F^1\) strongly. By the boundedness of \(\zeta \) and using Davis’ and Hölder’s inequalities we have
for \(v\in \mathbb {V}^1_p\) and integers \(k\ge 1\) with a constant C independent of v and \(\varepsilon \), where
and
with \(q=p/(p-1)\). By standard properties of mollification
for all \(\varepsilon >0\) and \((x,t,\omega )\in H_T\times \Omega \) with a constant \(N=N(K,d)\). Thus,
with \(q=p/(p-1)\) and a constant \(N=N(K,d)\). Consequently, letting here \(\varepsilon \rightarrow 0\) we obtain (6.24) for \(i=2\). By the boundedness of \(\zeta \), using Davis’ inequality we get
for \(v\in \mathbb {V}^1_p\) with a constant C independent of \(\varepsilon \) and v, where
and
Notice that by Taylor’s formula
where \(v_i:=D_iv\) and
for \(\theta \in (0,1)\), \(\varepsilon >0\) and \((s,z,\omega )\in [0,T]\times Z \times \Omega \). By Corollary 3.4 there are positive constants \(\varepsilon _0\) and \(M=M(K,d,m)\) such that the function \(\chi ^{\varepsilon }_{\theta }(s,z,\cdot )\) is a \(C^{\bar{m}}\)-diffeomorphism on \(\mathbb {R}^d\) for every \(\varepsilon \in (0,\varepsilon _0)\), \(\theta \in (0,1)\), \(\omega \in \Omega \), \(t\in [0,T]\) \(z\in Z\), and
for all \((x,s,z,\omega )\in H_T\times Z \times \Omega \). Due to Assumption 2.3 we have
Thus using Hölder’s inequality we get
with a constant C independent of \(\varepsilon \) and v. Furthermore we notice that by Assumption 2.4
with a constant \(N=N(K,d)\) for all \(\varepsilon >0\) and \((s,z,\omega ,z)\in H_T\times \Omega \times Z\). Hence, in a similar way as the estimate of \(B^1_\varepsilon (v)\) is obtained, we can show
with a constant C independent of \(\varepsilon \) and v. Hence, combining (6.27) and (6.28) we get (6.24) for \(i=3\). \(\square \)
Since \(F^i_{\varepsilon }\rightarrow F^i\) strongly in \({\mathbb {V}^1_p}^*\) as \(\varepsilon \rightarrow 0\) and \(u^{\varepsilon _k}\rightarrow u\) weakly in \(\mathbb {V}^1_p\) as \(\varepsilon _k\rightarrow 0\), we have
By well-known properties of mollifications and using Lemma 3.5 it is easy to show
and
Hence, taking \(k\rightarrow \infty \) in Eq. (6.21) we get
for every bounded well-measurable process \(\zeta \) and every \(\varphi \in C^\infty _0\), which implies that for every \(\varphi \in C^\infty _0\) Eq. (1.1) holds \(P\otimes dt\) almost everywhere. Hence, by Lemma 3.10u has an \(L_p\)-valued cadlag modification, denoted also by u, which is a generalised solution to (1.1)–(1.2). Moreover, from (6.19) we obtain
for \(n=0,1,\ldots ,m\) for every integer \(r>1\) with a constant \(N=N(d,p,m,T,K,K_{\bar{\xi }},K_{\bar{\eta }})\), where \(i=1\) when \(p=2\) and \(i=2\) for \(p>2\). Letting here \(r\rightarrow \infty \) we obtain
for \(n=0,1,2,\ldots m\) with a constant \(N=N(d,p,m,T,K,K_{\bar{\xi }},K_{\bar{\eta }})\). We already know that u is and \(L_p\)-valued cadlag process. Hence, applying Lemma 3.12 with \(V=H^m_p\), \(U=H^0_p\) and we obtain that u is weakly cadlag as an \(H^m_p\)-valued process, and we can change the essential supremum into supremum in (6.29), i.e.,
Thus we can also see that u is strongly cadlag as an \(H^{m-1}_p\)-valued process. To dispense with Assumption 6.1 we take a non-negative function \(\chi \in C^\infty _0(\mathbb {R}^d)\) such that \(\chi (x)=1\) for \(|x|\le 1\) and \(\chi (x)=0\) for \(|x|\ge 2\), and for integers \(n\ge 1\) define
for all \(t\in [0,T]\), \(x\in \mathbb {R}^d\), \(z\in Z\), where \(\chi _n(x)=\chi (x/n)\). Then for each n there is a unique generalised solution \(u^n=(u^n_t)_{t\in [0,T]}\) to Eqs. (1.1)–(1.2) with \(\psi ^n\), \(f^n\), \(g^n\) and \(h^n\) in place of \(\psi \), f, g and h, respectively. Moreover by (6.30)
with a constant \(N=N(d, p,m,T, K,K_{\bar{\xi }}, K_{\bar{\eta }})\). Letting here \(l,n\rightarrow \infty \) we get
Consequently, there is an \(H^m_p\)-valued adapted process \(u=(u_t)_{t\in [0,T]}\) such that for a subsequence \(n'\rightarrow \infty \) we have \(\sup _{t\le T}|u^{n'}_t-u_t|_{H^m_p}\rightarrow 0\) almost surely. Hence u is an \(H^{m-1}_p\)-valued cadlag process, and it is easy to show that it is a generalised solution to (1.1)–(1.2), such that (6.30) holds and u is weakly cadlag as an \(H^m_p\)-valued process.
The next theorem extends the above result to Eqs. (1.1)–(1.2) with \(\mathcal {R}\) satisfying Assumption 2.5.
Theorem 6.1
Let Assumptions 2.1 through 2.6 hold with an integer \(m\ge 1\) and a real number \(p\ge 2\). Assume also (6.16). Then Eq. (1.1)–(1.2) has a unique generalised solution u, such that u is an \(H^{m}_p\)-valued weakly cadlag process, satisfying estimate (6.30), and it is cadlag as an \(H^{m-1}_p\)-valued process.
Proof
We use the standard method of continuity, see, e.g., [23]. For \(\lambda \in [0,1]\), we consider the equation
for \((t,x)\in H_T\) with initial condition
where \( \mathcal {A}^0_t=\mathcal {L}_t+\mathcal {N}_t^\xi +\mathcal {N}_t^\eta \) for every \(t\in [0,T]\). We look for a solution u from the space \(\mathcal {H}^m_p\) of \(\mathcal {F}_t\)-adapted \(H^m_p\)-valued weakly cadlag processes which are strongly cadlag as \(H^{m-1}_p\)-valued processes such that \(|u|^p_{\mathcal {H}^m_p}=E\sup _{t\le T}|u_t|^p_{H^m_p}<\infty \). Notice that \(\mathcal {H}^m_p\) is a Banach space. If \(u\in \mathcal {H}^m_p\) is a generalised solution to (6.31), then by Assumption 2.5 and estimate (6.30) we have
with a constant \(N=N(m,d,p,T,K,K_{\bar{\xi }},K_{\bar{\eta }})\). Hence by Gronwall’s lemma we have estimate (6.30) for u. Let \(\Lambda \) denote the set of \(\lambda \in [0,1]\) such that for any \(\psi \in \Psi ^m_p=L_p(\Omega ,\mathcal {F}_0;H^m_p)\), \(f\in \mathbb {H}^m_p\), \(g\in \mathbb {H}^{m+1}_p(l^2)\) and \(h\in \mathbb {H}^{m+i}_p(\mathcal {L}_{p,2})\), with \(i=1\) when \(p=2\) and \(i=2\) when \(p>2\), Eqs. (6.31)–(6.32) has a unique generalised solution in \(\mathcal {H}^m_p\). Clearly \(0\in \Lambda \), and we need to prove \(1\in \Lambda \). To this end, it suffices to show that there is an \(\delta >0\) such that for any \(\lambda _0\in \Lambda \),
Fix \(\lambda _0\in \Lambda \), \(\psi \in \Psi ^m_p\), \(f\in \mathbb {H}^m_p\), \(g\in \mathbb {H}^{m+1}_p(l^2)\) and \(h\in \mathbb {H}^{m+i}_p(\mathcal {L}_{p,2})\). For \(v\in \mathcal {H}^m_p\) and \(\lambda \in [0,1]\) we consider the equation
for \((t,x)\in H_T\), with initial condition \( u_0=\psi . \) Since \(\lambda _0\in \Lambda \), this problem has a unique generalised solution \(u\in \mathcal {H}^m_p\). Define the operator \(Q_\lambda \) by \( u=Q_\lambda v. \) Then \(Q_\lambda \) maps \(\mathcal {H}^m_p\) into itself, and \(\lambda \in \Lambda \) if and only if there is a fixed point of \(Q_\lambda \). If \(v^i\in \mathcal {H}^m_p\) and \(u^i=Q_\lambda v^i\) for \(i=1,2\), then for \(u:=u^2-u^1\) we have
with \( u_0=0. \) Hence, using estimate (6.30) for u, due to Assumption 2.5 on \(\mathcal {R}\) we get
with constants N and \(N'\) depending only on \(m,d,p,T,K,K_{\bar{\xi }}\) and \(K_{\bar{\eta }}\). Taking \(\delta =(2N')^{-1}\) we obtain that \(Q_\lambda \) is a contraction mapping on \(\mathcal {H}^m_p\) if \(\lambda \in [\lambda _0-\delta ,\lambda _0+\delta ]\cap [0,1]\). Consequently, (6.31) has a unique solution u in \(\mathcal {H}^m_p\), and it satisfies (6.30). \(\square \)
If \(m\ge 1\) is not an integer, then we set \(\theta =m-\lfloor m\rfloor \) and by Theorem 4.1 we have
for \(i=1,2\), and
for integers \(r>1\). If Assumptions 2.1 through 2.6 with \(m\ge 1\) hold then we have shown above that the solution operator \(\mathbb {S}\), which maps the data \((\psi ,f,g,h)\) into the generalised solution u of (1.1)–(1.2), is continuous from
to \(\mathbb {U}^{\lfloor m\rfloor }_{p,r}\), and from
to \(\mathbb {U}^{\lceil m\rceil }_{p,r}\), for \(i=1\) when \(p=2\) and for \(i=2\) when \(p>2\), with operator norms bounded by a constant \(N=N(d,p,m,T,K,K_{\bar{\xi }},K_{\bar{\eta }})\). Hence by Theorem 4.1 (i) we have
with a constant \(N=(p,d,m,T,K,K_{\bar{\eta }})\). In the same way we get
Now, like before, letting here \(r\rightarrow \infty \) we obtain (6.29) for real numbers \(s\in [0,m]\), and using Lemma 3.12 we get that u is an \(H^m_p\)-valued weakly cadlag process such that (6.30) holds for any \(s\in [0,m]\). Taking into account that u is a strongly cadlag \(L_p\)-valued process and using the interpolation inequality Theorem 4.1(v) with \(A_0:=L_p\) and \(A_1:=H^m_p\), we get that u is strongly cadlag as an \(H^s_p\)-valued process for every real number \(s<m\).
Finally we can prove estimate (2.6) for \(q\in (0, p)\) by applying Lemma 3.11 in the same way as it is used in [16] to prove the corresponding supremum estimate therein.
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Acknowledgements
The main result of this paper was presented at the conference on “Harmonic Analysis for Stochastic PDEs” in Delft, 10-13 July, 2018 and at the “9th International Conference on Stochastic Analysis and Its Applications” in Bielefeld, 3-7 September, 2018. The authors are grateful to the organisers of these meetings, and to Konstantinos Dareiotis, Stefan Geiss and Mark Veraar for useful discussions and precious remarks. Finally we would like to thank the referees for their thorough work, comments and suggestions, and for pointing out several typos in the manuscript.
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Gyöngy, I., Wu, S. On \(L_p\)-solvability of stochastic integro-differential equations. Stoch PDE: Anal Comp 9, 295–342 (2021). https://doi.org/10.1007/s40072-019-00160-8
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DOI: https://doi.org/10.1007/s40072-019-00160-8