1 Introduction↩︎

Let \(\varphi:{\mathbb{R}}^d\rightarrow{\mathbb{R}}\), \(f:[0,T]\times{\mathbb{R}}^d\times{\mathbb{R}}\) be measurable functions and \(u\) be a solution of the semilinear Cauchy problem \[\label{eq1461} \begin{cases} \partial_su+L u=-f(s,x,u),\quad (s,x)\in Q_T:=(0,T)\times{\mathbb{R}}^d,\\ u(T,x)=\varphi(x),\quad x\in{\mathbb{R}}^d, \end{cases}\tag{1}\] where \(L\) is a Lévy-type integro-differential operator of the form \[\label{eq1464} L={\mathcal{L}}+{\mathcal{I}}.\tag{2}\] Here, for \(u\in C^{\infty}_c({\mathbb{R}}^d)\), \[{\mathcal{L}}u(x)=\sum^d_{i=1}b_i(x)\partial_{x_i}u(x) +\frac{1}{2}\sum^d_{i,j=1}a_{ij}(x)\partial^2_{x_ix_j}u(x), \qquad a=\sigma\sigma^T,\] for some bounded Lipschitz continuous coefficients \(\sigma:{\mathbb{R}}^d\rightarrow{\mathbb{R}}^d\times{\mathbb{R}}^d\), \(b:{\mathbb{R}}^d\rightarrow{\mathbb{R}}^d\), and \[{\mathcal{I}}u(x)=\int_{{\mathbb{R}}^d}(u(x+y)-u(x)-y\cdot\nabla u(x) {\mathbf{1}}_{|y|\le1})\,\nu(dy)\] for some Lévy measure \(\nu\) on \({\mathbb{R}}^d\). Under these assumptions on \(\sigma,b,\nu\) the operator \(L\) generates a conservative Feller semigroup such that \(C^{\infty}_c({\mathbb{R}}^d)\subset D(L)\).

Suppose that \(\{L^n={\mathcal{L}}^n+{\mathcal{I}}^n\}\) is a sequence of operators of the same form as \(L\) such that each \(L^n\) generates a Feller semigroup and for each \(n\) there exists a unique solution of (1 ) with \(L\) replaced by \(L^n\). The aim of the present paper is to show that if \(L^n\rightarrow L\) in the sense described by (7 ) below, then \[\label{eq1463} u_n\rightarrow u,\tag{3}\] and the form of convergence (pointwise, almost everywhere, weak in \(L^2\), etc.) depends on the assumptions on the data \(\varphi\) and \(f\). We are mainly interested in the case where the nonlocal parts \({\mathcal{I}}^n\) are present. The model simple example is \[\label{eq1465} L=\Delta,\qquad L^n={\mathcal{I}}^n=\Delta^{\alpha_n/2}\quadwith \alpha_n\rightarrow 2^{-}\tag{4}\] (the fractional Laplace operator \(\Delta^{\alpha_n/2}:=-(-\Delta)^{\alpha_n/2}\) corresponds to the Lévy measure \(\nu_n(dy)=c_{d,\alpha_n}|y|^{-d-\alpha_n}\,dy\) for some constant \(c_{d,\alpha_n}>0\)). This model example was recently addressed in [1]. In that paper it is proved that under some assumptions on \(\varphi,f\) the convergence (3 ) holds weakly star in \(L^{\infty}(0,T;L^2({\mathbb{R}}^d))\) and weakly in \(L^2(0,T;L^2({\mathbb{R}}^d))\).

In the present paper we consider much more general situation and give stronger convergence results even in the model example (4 ) considered in [1]. To describe them it is convenient to regard \(L,L^n\) as pseudodifferential operators. Since \(C^{\infty}_c({\mathbb{R}}^d)\subset D(L)\), by Courrège theorem, \(L\) restricted to \(C^{\infty}_c({\mathbb{R}}^d)\) has the form \[Lu(x)=-p(x,D)u(x):=-\int_{{\mathbb{R}}^d}e^{i(x,\xi)}p(x,\xi)\hat{u}(\xi)\,d\xi, \quad u\in C^{\infty}_c({\mathbb{R}}^d),\] with symbol \(p(x,\xi):{\mathbb{R}}^d\times{\mathbb{R}}^d\rightarrow{\mathbb{C}}\) defined by \[\label{eq1466} p(x,\xi)=-i(b(x),\xi)+\frac{1}{2}(a(x)\xi,\xi)+\psi(\xi),\quad x,\xi\in{\mathbb{R}}^d,\tag{5}\] where \(\psi:{\mathbb{R}}^d\rightarrow{\mathbb{C}}\) is the characteristic exponent of the pure-jump Lévy process with Lévy measure \(\nu\), i.e. \(\psi\) is given by the Lévy–Khinchine formula \[\psi(\xi)=\int_{y\neq0}(1-e^{i(\xi,y)}+i(\xi,y){\mathbf{1}}_{\{|y|\le1\}})\,\nu(dy).\] In general, by a solution of (1 ) we mean a probabilistic solution, i.e. a function \(u:[0,T)\times{\mathbb{R}}^d\) which satisfies the following “nonlinear Feynman–Kac formula": \[\label{eq1467} u(s,x)=E\Big(\varphi(X^{s,x}_T) +\int^T_sf(t,X^{s,x}_t,u(r,X^{s,x}_t))\,dt\Big),\quad (s,x)\in[0,T)\times{\mathbb{R}}^d,\tag{6}\] where \(X^{s,x}\) is the jump-diffusion process with generator \(L\) starting at time \(s\) from \(x\).

Let \(p_n\) be the symbol of \(L^n\), which is defined by (5 ) with \(a,b,\psi\) replaced by some \(a^n,b^n\) and \(\psi_n\). We assume that \(a^n,b^n\) are uniformly bounded and Lipschitz continuous, and \[\label{eq1462} a^n\rightarrow a,\,b^n\rightarrow b locally uniformly on {\mathbb{R}}^d,\quad \psi_n\rightarrow\psi pointwise in {\mathbb{R}}^d.\tag{7}\] As for \(f\), we assume that \((x,y)\rightarrow f(t,x,y)\) is continuous, \(y\rightarrow f(t,x,y)\) is decreasing (in fact this condition can be somewhat relaxed), \(\bar f(t,x,y):=f(t,x,y)-f(t,x,0)\) is bounded and \(f(t,x,0)\) is bounded or satisfies some integrability conditions. Examples are \[\label{eq1468} f(t,x,y)=h_1(t,x)-h_2(t,x)\psi(y),\quad \psi(y)=y|y|^{p-1}\quador\quad \psi(y)=e^y-1,\tag{8}\] where \(h_1,h_2\) are continuous in \(x\), \(h_2\) is bounded and nonnegative and \(h_1\) is bounded or satisfies some integrability condition. We show that then, if \(\varphi\) is bounded continuous and \(f(\cdot,\cdot,0)\) is bounded, then the probabilistic solutions \(u_n\) of (1 ) with \(L\) replaced by \(L^n\) (they exist and are unique) converge pointwise to the probabilistic solution \(u\).

The assumptions on \(\varphi,f(\cdot,\cdot,0)\) can be relaxed if \(a^n,b^n\) are constant, i.e. if \(L^n\) are Lévy operators, and, roughly speaking, \(\psi_n\) satisfy the Hartman–Wintner condition uniformly in \(n\). One can then show that some form of convergence (3 ) holds for \(\varphi, f(\cdot,\cdot,0)\) satisfying integrability conditions. Furthermore, for such subclass of operators \(u_n,u\) become weak solutions (for \(L^2\)-data) or renormalized solutions (for \(L^1\)-data) and we get results on stability of weak or renormalized solutions.

In the present paper, we also consider operators with symbols of the form (5 ) with \(\psi(\xi)\) replaced by \(\psi(\gamma^T\xi)\) for some bounded Lipschitz \(\gamma:{\mathbb{R}}^d\rightarrow{\mathbb{R}}^{d\times k}\). This means that we allow operators (2 ) with the integral part defined as \({\mathcal{I}}\), but with the measure \(\nu\) depending in a suitable way on \(x\). As an example may serve the operator \[Lu(x)=|\gamma(x)|^{\alpha}\Delta^{\alpha/2}u(x),\quad u\in C^{\infty}_c({\mathbb{R}}^d).\] For such operators we prove some convergence results but only for bounded continuous \(\varphi\) and bounded \(f(\cdot,\cdot,0)\).

To prove our results we use probabilistic methods. The solution \(u_n\) associated with \(L^n\) is given by (6 ) with \(X^{s,x}\) replaced by the process \(X^{s,x,n}\) with generator \(L^n\). Therefore, roughly speaking, the proof of (3 ) reduces to the proof that the right-hand side of formula (6 ) representing \(u_n\) (it involves \(u_n\) and \(X^{s,x,n}\)) converges in a suitable way to the right-hand side of (6 ). To this end, we show the convergence of solutions of backward stochastic differential equations (BSDEs) associated with the Cauchy problem involving \(L^n\) to the solution of BSDE associated with (1 ). In fact, we show some abstract, more general than needed stability result for solution of BSDEs, which may be of independent interest. It is worth noting that our stability result pertains to the BSDEs with different filtrations (generated by the processes \(X^{s,x,n}\) in our applications) and its proof uses in an essential way the notion of weak convergence of filtrations introduced in [2].

2 Stability of solutions of BSDEs↩︎

In proofs of our main theorems we will need a stability result for solutions of Markov-type backward stochastic differential equations (BSDEs in abbreviation) with forward driving processes associated with operators \(L^n\). However, it is convenient to state it for more general abstract BSDEs.

Let \((\Omega,{\mathcal{F}},{\mathbb{F}}=({\mathcal{F}}_t)_{t\ge0},P)\) be a filtered probability space satisfying the usual conditions. We denote by \({\mathcal{D}}\) the space of all \(({\mathcal{F}}_t)\)-progressively measurable càdlàg processes. \({\mathcal{M}}\) (resp. \({\mathcal{M}}_{loc}\)) is the space of all càdlàg \((({\mathcal{F}}_t),P)\)-martingales (resp. local martingales) \(M\) such that \(M_0=0\).

Recall that a càdlàg \(({\mathcal{F}}_t)\)-adapted process \(Y\) is of Doob class (D) if the collection \(\{Y_{\tau},\tau\in{\mathcal{T}}\}\), where \({\mathcal{T}}\) is the set a finite valued \({\mathbb{F}}\)-stopping times, is uniformly integrable.

In what follows \(T>0\) is an arbitrary but fixed number, \(\xi\) is an \({\mathcal{F}}_T\)-measurable random variable, \(X\) is a process of class \({\mathcal{D}}\) and \(f:[0,T]\times{\mathbb{R}}^d\times{\mathbb{R}}\rightarrow{\mathbb{R}}\) is a measurable function. As for \(\xi\) and \(f\) we will need the following hypotheses.

1. \({\mathbb{R}}^d\times{\mathbb{R}}\ni(x,y)\mapsto f(t,x,y)\) is continuous for every \(t\in[0,T]\).

2. There is \(\mu\ge0\) such that \((f(t,x,y)-f(t,x,y'))(y-y')\le\mu|y-y'|^2\) for all \(t\in[0,T]\), \(x\in{\mathbb{R}}^d\) and \(y,y'\in{\mathbb{R}}\).

3. \(E\int^T_0F_r(t,X_t)\,dt<\infty\) for every \(r>0\), where \(F_r(t,x)=\sup_{|y|\le r}|f(t,x,y)-f(t,x,0)|\).

4. \(E|\xi|+E\int^T_0|f(t,X_t,0)|\,dt<\infty\).

Suppose now that for each \(n\ge1\) we are given a filtration \({\mathbb{F}}^n=({\mathcal{F}}^n_t)_{t\ge0}\) such that \({\mathcal{F}}^n_t\subset{\mathcal{F}}\) for \(t\ge0\), an \({\mathcal{F}}^n_T\)-measurable random variable \(\xi^n\) and an \({\mathbb{F}}^n\)-measurable processes \(X^n\). Suppose also that for each \(n\ge1\) there exists a unique solution \((Y^n,M^n)\) of \(BSDE(\xi^n,X^n,f,{\mathbb{F}}^n)\), i.e. \[\label{eq24618} Y^n_t=\xi^n+\int^T_tf(s,X^n_s,Y^n_s)\,ds-\int^T_tdM^n_s,\quad t\in[0,T].\tag{10}\] In our stability result a key role will be played by the concept of weak convergence of filtration introduced in [2]. To state it we denote by \(D([0,T];{\mathbb{R}}^d)\) the space of \({\mathbb{R}}^d\)-valued functions on \([0,T]\) which are right continuous and have left-limits. We equip it with the Skorokhod \(J_1\)-topology. We use “\(\mathop{\rightarrow}_P\)" to denote the convergence in probability \(P\).

In the stability theorem our basic assumption on the processes associated with operators \(L^n\) is the following:

1. \(X^n\mathop{\rightarrow}_PX\) in \([D([0,T];{\mathbb{R}})]^d\) and \({\mathbb{F}}^n\mathop{\rightarrow}_{w}{\mathbb{F}}\) on \([0,T]\).

We will also need the following assumptions relating \(\{X^n\}\) to \(f\) and \(\varphi\) (below \(dt\) denotes the Lebesgue measure on \((0,T)\)):

1. For every \(r>0\) the sequence \(\{F_r(\cdot,X^n_{\cdot})\}_{n\ge1}\) is uniformly \(dt\otimes P\)-integrable on \((0,T)\times\Omega\).

2. \(\{f(\cdot,X^n,0)\}_{n\ge1}\) is uniformly \(dt\otimes P\)-integrable on \((0,T)\times\Omega\).

3. \(\xi^n\longrightarrow\xi\) in \(L^1(P)\), where \(\xi^n=\varphi(X^n)\), \(\xi=\varphi(X)\).

It is well known that the study of backward equations with \(f\) satisfying (H2) can be reduced by a simple transformation of variables to the study of equations satisfying (H2) with \(\mu=0\). For the reader’s convenience, we describe the transformation below.

3 Probabilistic solutions↩︎

In what follows we will assume that \(\sigma:{\mathbb{R}}^d\rightarrow{\mathbb{R}}^{d\times d}\), \(b:{\mathbb{R}}^d\rightarrow{\mathbb{R}}^d\), \(\gamma:{\mathbb{R}}^d\rightarrow{\mathbb{R}}^{d\times k}\) are bounded Lipschitz continuous functions and \(\nu\) is a Lévy measure on \({\mathbb{R}}^d\), i.e. a measure such that \(\nu(\{0\})=0\) and \(\int_{{\mathbb{R}}^d}(1\wedge|y|^2)\,\nu(dy)<\infty\). We let \(a=\sigma\cdot\sigma^T\), where \(\sigma^T\) denotes the transpose of \(\sigma\).

3.1 SDEs and Feller semigroups↩︎

Let \(s\in[0,T)\), \(x=(x_1,\dots,x_d)\in{\mathbb{R}}^d\), \(W=(W^1,\dots,W^d)\) be a standard Wiener process and \(N=(N^1,\dots,N^k)\) be a pure-jump \(k\)-dimensional Lévy proces independent of \(W\) starting from 0. Consider the stochastic differential equation (SDE) \[\label{eq3461} X^{s,x}_t=x+\int^t_sb(X^{s,x}_r)\,dr+\int^t_s\sigma(X^{s,x}_r)\,dW_ +\int^t_s\gamma(X^{s,x}_{r-})\,dN_r,\quad t\in[s,T].\tag{33}\] It can be rewritten as \[X^{s,x,i}_t=x_i+ \sum^{d+1+k}_{j=1}\int^t_s\Phi_{ij}(X^{s,x}_{r-})\,dZ^{j}_r, \quad t\in[s,T],\quad i=1,\dots,d,\] or succinctly as \[\label{eq34618} X^{s,x}_t=x+\int^t_s\Phi(X^{s,x}_{r-})\,dZ_r, \quad t\in[s,T],\tag{34}\] where \[Z_t=(t, W_t,N_t),\,\,t\ge0,\qquad \Phi_{ij}(x)=\begin{cases} b_i(x) & j=1,\\ \sigma_{ij}(x), & j=2,\dots,d+1,\\ \gamma_{i,j-d-1}(x), & j=d+2,\dots,d+1+k. \end{cases}\] It is well known (see, e.g., [7]) that for each \((s,x)\in[0,T)\times {\mathbb{R}}^d\) there exists a unique strong solution \(X^{s,x}\) of (33 ). Furthermore, by [9] (see also [10]), \({\mathbb{X}}=(X^{0,x})_{x\in{\mathbb{R}}^d}\) is a Feller process, i.e. it is a Markov process whose transition semigroup defined by \[P_tu(x)=Eu(X^{0,x}_t),\quad u\in{\mathcal{B}}_b({\mathbb{R}}^d),\] is a Feller semigroup. Let \((L, D(L))\) denote its generator. By [9] (or [10]), \(C^{\infty}_c({\mathbb{R}}^d)\subset D(L)\), where \(C^{\infty}_c({\mathbb{R}}^d)\) is the space of smooth functions on \({\mathbb{R}}^d\) with compact support. Therefore, by the Courrège theorem, \(L\) restricted to \(C_c^{\infty}({\mathbb{R}}^d)\) is a pseudodifferential operator of the form \[\label{eq3463} Lu(x)=-p(x,D)u(x):=-\int_{{\mathbb{R}}^d}e^{i(x,\xi)}p(x,\xi)\hat{u}(\xi)\,d\xi, \quad u\in C^{\infty}_c({\mathbb{R}}^d),\tag{35}\] with some symbol \(p(x,\xi):{\mathbb{R}}^d\times{\mathbb{R}}^d\rightarrow{\mathbb{C}}\) (in (35 ), \(\hat{u}\) is the Fourier transform of \(u\)). In fact, by [9], \(p\) has the form \[p(x,\xi)=\psi_Z(\Phi^T(x)\xi), \quad x,\xi\in{\mathbb{R}}^d,\] where \(\psi_Z\) is the characteristic exponent of the Lévy process \(Z\). For later use, we give below more explicit form of \(p\). Since \[\psi_Z(\bar\xi)=-i\xi^{(1)}+\frac{1}{2}(\xi^{(2)},\xi^{(2)})+\psi(\xi^{(3)}), \quad \bar\xi=(\xi^{(1)},\xi^{(2)},\xi^{(3)})\in{\mathbb{R}}\times{\mathbb{R}}^d\times{\mathbb{R}}^k,\] where \(\psi\) is the characteristic exponent of \(N\), and \[\Phi^T(x)\xi=((b(x),\xi),\sigma^T(x)\xi,\gamma^T(x)\xi)^T\in{\mathbb{R}}^{1+d+k},\quad \xi\in{\mathbb{R}}^d,\] we see that \[\label{eq34620} p(x,\xi)=-i(b(x),\xi)+\frac{1}{2}(a(x)\xi,\xi)+\psi(\gamma^T(x)\xi),\quad x,\xi\in{\mathbb{R}}^d.\tag{36}\]

From the the Riesz representation theorem it follows that there exists a family of kernels \(\{p_t(x,dy), (t,x)\in(0,\infty)\times{\mathbb{R}}^d\}\) such that \[P_tu(x)=\int_{{\mathbb{R}}^d}u(y)p_t(x,dy),\quad u\in{\mathcal{B}}_b({\mathbb{R}}^d).\] Furthermore, for each \((t,x)\in(0,\infty)\times{\mathbb{R}}^d\), \(p_t(x,\cdot)\) is a uniquely defined positive Radon measure on \({\mathbb{R}}^d\) (for more details see [11]). In case the kernels are absolutely continuous with respect to the Lebesgue measure on \({\mathbb{R}}^d\), i.e. are represented as \(p_t(x,dy)=p_t(x,y)\,dy\), we call \(p_t(x,y)\) the transition densities of \({\mathbb{X}}\).

3.2 Probabilistic solutions and BSDEs, Lévy-type operators↩︎

In this subsection, we first consider the case where \(L\) is a Lévy type operator of the form (35 ).

Note that (38 ) can be considered as a nonlinear version of the Feynman–Kac formula.

In many interesting situations Definition 3 is too strong, because for some natural assumptions on \(\varphi,f\) we do not know whether (37 ) is satisfied for every \((s,x)\). On the other hand, for some subclasses of operators of the form (2 ) we have additional information on the associated process \(X^{s,x}\) which allows us to give a meaningful weaker version of the definition of a solution. One example we given below. The second will be given in the next subsection.

Suppose that the transition kernels of \({\mathbb{X}}\) are absolutely continuous with respect to the Lebesque measure and let \(p_t(x,y)\) denote the transition densities. Then (38 ) can be written as \[\label{eq34624} u(s,x)=\int_{{\mathbb{R}}^d}\varphi(y)p_{T-s}(x,y)\,dy +\int^T_s\!\int_{{\mathbb{R}}^d}f(t,y,u(t,y))p_{t-s}(x,y)\,dt\,dy.\tag{39}\] Of course, to define the right-hand side of (39 ) we need not know that \(u\) is defined everywhere on \([0,T)\times{\mathbb{R}}^d\). This leads to the following slightly weakened version of Definition 3:

Theorem 5 below provides very useful relation between probabilistic solutions of (1 ) and solutions of BSDEs of the form \[\label{eq3462} Y^{s,x}_t=\varphi(X^{s,x}_T)+\int^T_tf(r,X^{s,x}_r,Y^{s,x}_r)\,dr -\int^T_tdM^{s,x}_r,\quad t\in[s,T].\tag{40}\]

In what follows, for \(0\le s<T\) we denote by \(({\mathcal{F}}^s_t)_{t\in[s,T]}\) the standard augmentation of the filtration \(({\mathcal{F}}^{0,s}_t)_{t\in[s,T]}\), where \({\mathcal{F}}^{0,s}_t=\sigma((W_r-W_s,N_r-N_s), r\in[s,t])\). By slight abuse of notation, we say that \(f\) satisfies (H3) (resp. (H4)) with \(X\) replaced by \(X^{s,x}\) if \(E\int^T_s|f(t,X^{s,x}_t)\,dt<\infty\) for \(r>0\) (resp. \(E\int^T_s|f(t,X^{s,x}_t,0)|\,dt<\infty\)).

In the next theorem we assume that the transiltion kernels of \({\mathbb{X}}\) are absolutely continuous with respect to the Lebesgue measure.

3.3 Probabilistic solutions and BSDEs, Lévy operators↩︎

In this section, we confine ourselves to Lévy operators, i.e we assume that \(a, b\) are constant and \(\gamma=I_d\) (i.e. \(k=d\)). We shall see that in that case probabilistic solutions are renormalized solutions or weak solutions depending on the integrability conditions imposed on \(\varphi,f\). We also provide stability results for these kind of solutions.

In the case under consideration \(X^{s,x}\) is a Lévy process starting at time \(s\) from \(x\) with the Lévy triplet \((b,a,\nu)\). Therefore the characteristic function of \(X^{s,x}_t\) has the form \[Ee^{i(\xi,X^{s,x}_t)}= e^{i(\xi,x)}e^{-(t-s)\psi(\xi)},\quad \xi\in{\mathbb{R}}^d,\] where the characteristic exponent \(\psi:{\mathbb{R}}^d\rightarrow{\mathbb{C}}\) is given by the celebrated Lévy-Khintchine formula \[\psi(\xi)=-i(b,\xi)+\frac{1}{2}(\xi,a\xi) +\int_{{\mathbb{R}}^d}(1-e^{i(\xi,y)}+i(\xi,y){\mathbf{1}}_{\{|y|\le1\}})\,\nu(dy).\] The generator of the semigroup \((P_t)\) generated by \({\mathbb{X}}\) restricted to \(C^{\infty}_c({\mathbb{R}}^d)\) can be written in the form \[\label{eq34612} Lu(x)=- \psi(D)u(x):=-\int_{{\mathbb{R}}^d}e^{i(x,\xi)}\psi(\xi)\hat{u}(\xi)\,d\xi,\quad u\in C^{\infty}_c({\mathbb{R}}^d),\tag{44}\] which is the special case of (35 ). For a characteristic exponent \(\psi\) we consider the bilinear form \[\Psi(u,v)=\int_{{\mathbb{R}}^d}\hat{u}(\xi)\overline{\hat{v}(\xi)}\psi(\xi)\,d\xi, \quad u,v\in H^{\psi,1}({\mathbb{R}}^d),\] where \[H^{\psi,1}({\mathbb{R}}^d)=\Big\{u\in L^2({\mathbb{R}}^d):\int_{{\mathbb{R}}^d}(1+|\psi(\xi)|)|\hat{u}(\xi)|^2\,d\xi<\infty\Big\}.\]

Note that the operator \(L=-(-\Delta)^{\alpha/2}=:\Delta^{\alpha/2}\) considered in [1] corresponds to \(\psi(\xi)=|\xi|^{\alpha}\) and in that case the form \((\Psi,H^{\psi,1}({\mathbb{R}}^d))\) admits equivalent description as \[\Psi(u,v)=\frac{c_{d,\alpha}}{2} \int_{{\mathbb{R}}^d\times{\mathbb{R}}^d\setminus\Delta} \frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{d+\alpha}}\,dx\,dy\] for some constant \(c_{d,\alpha}>0\) (see, e.g., [12] for more details).

Let \({\mathcal{E}}\) be the time-dependent Dirichlet form \({\mathcal{E}}\) on \(L^2({\mathbb{R}}\times{\mathbb{R}}^d)\) associated with \(\Psi\) in the sense described in [13] or [14] (see also [15] or [16]). We define capacity Cap associated with \({\mathcal{E}}\) and quasi-continuous functions as in [13] or [14]. Let \(Q_T=(0,T)\times{\mathbb{R}}^d\). We say that some property holds quasi-everywhere (q.e. in abbreviation) in \(Q_T\) if it holds in \(Q_T\) except on a set whose capacity Cap equals zero. For other notions of capacity which lead to the same notion of exceptional subsets of \(Q_T\), i.e. subsets of capacity zero, we refer to [16].

If the transition kernels of \({\mathbb{X}}\) have strictly positive densities, then by Remark 7, generalized probabilistic solutions are integral solutions.

In many cases one can show that probabilistic or generalized probabilistic solutions coincide with other notions of solutions considered in the theory of PDEs, for instance there are viscosity solutions, weak solutions or renormalized solutions. As for viscosity solutions, we refer the reader to [18]. Weak and renormalized solutions will be addressed in Section 5.

4 Stability of probabilistic solutions↩︎

For \(n\ge1\) let \(\sigma^n,\gamma^n:{\mathbb{R}}^d\rightarrow{\mathbb{R}}^{d\times k}\), \(b^n:{\mathbb{R}}^d\rightarrow{\mathbb{R}}^d\), \(\gamma^n:{\mathbb{R}}^d\rightarrow{\mathbb{R}}^{d\times k}\), be bounded Lipschitz continuous functions and \(\nu_n\) be a Lévy measure on \({\mathbb{R}}^d\). We set \(a^n=\sigma^n\cdot(\sigma^n)^T\). We consider operators of the form \[\begin{align} \label{eq4462} L^nu_n(x)&=\sum^d_{i=1}b^n_i(x)\partial_{x_i}u_n(x) +\frac{1}{2}\sum^d_{i,j=1}a^n_{ij}(x)\partial^2_{x_ix_j}u_n(x) \nonumber\\ &\quad+\int_{{\mathbb{R}}^d}(u_n(x+y)-u_n(x)-y\cdot\nabla u_n(x) {\mathbf{1}}_{|y|\le1})\,\nu_n(dy). \end{align}\tag{45}\] We denote by \(u_n\) the probabilistic solution of the problem \[\label{eq4463} \begin{cases} \partial_su_n+L^nu_n=-f(s,x,u_n),\quad (s,x)\in Q_T:=(0,T)\times{\mathbb{R}}^d,\\ u_n(T,x)=\varphi(x),\quad x\in{\mathbb{R}}^d. \end{cases}\tag{46}\] It exists and is unique by Theorem 5.

Since \(N\) is a pure-jump symmetric Lévy process in \({\mathbb{R}}^k\) with the characteristics \((0,0,\nu)\), the Lévy–Khinchine formula says that the characteristic function of \(N_t-N_s\) is of the form \[Ee^{i(\xi,N_t-N_s)}=e^{-(t-s)\psi(\xi)},\quad \xi\in{\mathbb{R}}^k,\] for some \(\psi:{\mathbb{R}}^k\rightarrow{\mathbb{R}}\) called the characteristic exponent of \(N\) (see Section 5 for more details). Suppose that \(N^n\) is another pure-jump Lévy processes. In what follows we denote its characteristic exponent by \(\psi_n\).

4.1 Stability of solutions of SDEs↩︎

We begin with some stability results for solutions of SDEs of the form (34 ). We will need the following assumptions: \[\label{eq4466} \begin{cases} a^n,b^n,\gamma^n are uniformly bounded, \\ \sigma^n\rightarrow\sigma, \,b^n\rightarrow b,\,\gamma^n\rightarrow\gamma locally uniformly on {\mathbb{R}}^d. \end{cases}\tag{47}\] and \[\label{eq4464} \lim_{n\rightarrow\infty}|\psi_n(\xi)-\psi(\xi)|=0,\quad \xi\in{\mathbb{R}}^k,\tag{48}\]

Let \(\tilde{W},\tilde{W}^n\) and \(\tilde{N},\tilde{N}^n\) be the processes of Proposition 9. For \((s,x)\in[0,T)\times{\mathbb{R}}^d\) we denote by \(\tilde{X}^{s,x}\) be the unique solution of (33 ) with \(W,N\) replaced by \(\tilde{W},\tilde{N}\), and by \(\tilde{X}^{s,x,n}\) the unique strong solution of the SDE \[\tilde{X}^{s,x,n}_t=x+\int^t_s\sigma^n(\tilde{X}^{s,x,n}_r)\,d\tilde{W}^n_r+\int^t_sb^n(X^{s,x,n}_r)\,dr +\int^t_s\gamma^n(X^{s,x}_{r-})\,d\tilde{N}^{n}_r,\quad t\in[s,T].\]

We can now state a useful sufficient condition for hypotheses (C) to hold with \(X^n\) repaced by \(\tilde{X}^{s,x,n}\) and \(X\) replaced by \(\tilde{X}^{s,x}\).

4.2 Stability of solutions of PDEs↩︎

We first give a general remark concernig applications of results of Section 2 to the convergence of solutions of PDEs.

Fix \((s,x)\in[0,T)\times{\mathbb{R}}^d\). Define \(\tilde{X}^{s,x},\tilde{X}^{s,x,n}\) as in Proposition 10 and denote by \(\tilde{Y}^{s,x}\) (resp. \(\tilde{Y}^{s,x,n}\)) the first component of the solution of BSDE\((\varphi(\tilde{X}^{s,x}_T),\tilde{X}^{s,x},f,\tilde{\mathbb{F}})\) (resp. BSDE\((\varphi(\tilde{X}^{s,x,n}_T),\tilde{X}^{s,x,n},f,\tilde{\mathbb{F}}^n)\)). By uniqueness, we have \(X^{s,x}=_d\tilde{X}^{s,x}\) and \(X^{s,x,n}=_d\tilde{X}^{s,x,n}\). It follows that the probabilistic solution \(u\) of (1 ) satisfies the equation \[\label{eq4468} u(s,x)=E\Big(\varphi(\tilde{X}^{s,x}_T) +\int^T_sf(t,\tilde{X}^{s,x}_t,u(t,\tilde{X}^{s,x}_t))\,dt\Big)\tag{49}\] and similarly, the probabilistic solution \(u_n\) of (46 ) satisfies \[\label{eq4469} u_n(s,x)=E\Big(\varphi(\tilde{X}^{s,x,n}_T) +\int^T_sf(t,\tilde{X}^{s,x,n}_t,u_n(t,\tilde{X}^{s,x,n}_t))\,dt\Big).\tag{50}\] Since \(E\tilde{Y}^{s,x}_s\) equals the right-hand side of (49 ) and \(E\tilde{Y}^{s,x,n}_s\) equals the right-hand side of (50 ), we see that pointwise convergence of \(u_n\) to \(u\) will be proved once we prove that \[\label{eq44610} \tilde{Y}^{s,x,n}_s\mathop{\rightarrow}_d\tilde{Y}^{s,x}_s\quadin {\mathbb{R}}.\tag{51}\] Similarly, the convergence of \(u_n\) to \(u\) in other topologies is related to (51 ). Such idea of proving stability of solutions, but for linear equations and Neumann problem, has appeared in [22].

To show (51 ) we will use Theorem 4 with the data \[\label{eq44611} X=\tilde{X}^{s,x},\quad X^n=\tilde{X}^{s,x,n},\quad \xi=\varphi(\tilde{X}^{s,x}_T),\quad \xi^n=\varphi(\tilde{X}^{s,x,n}_T).\tag{52}\]

Below we present some sufficient conditions which together with (?? ) imply (H5)–(H7). As for \(f\), we will assume that \[\label{eq44612} |f(t,x,y)-f(t,x,0)|\le h(t,x)g(|y|)\tag{53}\] for some nonnegative measurable \(h\) and nondecreasing \(g:{\mathbb{R}}_+\rightarrow{\mathbb{R}}\). Examples of \(f\) satisfying (H1), (H2) and (53 ) are functions defined by (8 ). In the first result we require no additional assumptions on the underlying processes. The price is that we require that \(\varphi\) is bounded and continuous and our assumptins on \(f\), when applied to (8 ), say that \(h_1,h_2\) are bounded. Next we show a result which covers integrable \(\varphi\) and \(f\) of the form (8 ) with integrable \(\varphi,h_1\) and essentially bounded \(h_2\).

We shall see in Section 6 that the following theorem, in which we impose additional assumptions on the underlying processes, is applicable in many interesting situations.

5 Weak and renormalized solutions↩︎

In this section, as in Subsection 3.3, we confine ourselves to Lévy operators. Recall that the classic Hartman–Wintner condition \[\label{eq5461} \lim_{|\xi|\rightarrow\infty} \frac{Re\,\psi(\xi)}{\ln(1+|\xi|)}=\infty\tag{64}\] is sufficient for the existence of transition densities of \({\mathbb{X}}\). Furthermore, if (64 ) is satisfied, then \[p_t(x,y)=(2\pi)^{-d/2}\int_{{\mathbb{R}}^d}e^{-i(x-y,\xi)}e^{-t\psi(\xi)}\,d\xi.\] In particular, for every \(t>0\), \[\label{eq5462} \sup_{x,y\in{\mathbb{R}}^d}p_t(x,y)\le(2\pi)^{-d/2} \int_{{\mathbb{R}}^d}e^{-t\mathrm{Re}\,\psi(\xi)}\,d\xi,\tag{65}\] so the first part of (54 ) is satisfied. The first part of (55 ) is also satisfied since \(\hat{p}_t(x,y):=p_t(x,y)\) is the transition density of the Lévy process with the Lévy triplet \((-b,a,\nu)\). From the above it follows in particular, that if (64 ) is satisfied and \(\varphi\in L^q({\mathbb{R}}^d)\), \(f(\cdot,\cdot,0)\in L^q(Q_T)\) for some \(q\in[1,\infty)\), then \[\label{eq5463} E|\varphi(X^{s,x}_T)|<\infty,\,\,\,(s,x)\in Q_T,\quad E\int^T_s|f(t,X^{s,x}_t,0)|\,dt<\infty\, q.e. (s,x)\in Q_T.\tag{66}\] Indeed, the first part of (66 ) is a consequence of (54 ) and Hölder’s inequality. As for the second part of (66 ), we note that by (59 ) and (60 ), \(R^{0,T}|f(\cdot,\cdot,0)|<\infty\) a.e. in \(Q_T\), where \(R^{0,T}f(\cdot,\cdot,0)(s,x):=E\int^T_s|f(t,X^{s,x}_t,0)|\,dt\). In fact, \(R^{0,T}|f(\cdot,\cdot,0)|<\infty\) q.e. in \(Q_T\), because by [17], \(R^{0,T}|f(\cdot,\cdot,0)|\) is quasi-continuous.

5.1 Weak solutions↩︎

Let \(V=H^{\psi,1}({\mathbb{R}}^d)\), \({\mathcal{W}}=\{u\in L^2(0,T;V):\partial_tu\in L^2(0,T;V')\}\), where \(V'\) is the dual space of \(V\). We denote by \(\langle\cdot,\cdot\rangle\) the duality pairing between \(V'\) and \(V\) and by \(D(0,T;L^2({\mathbb{R}}^d))\) the set of Borel functions on \((0,T]\times{\mathbb{R}}^d\) such that the mapping \((0,T]\ni t\mapsto u(t):=u(t,\cdot)\in L^2({\mathbb{R}}^d)\) is càdlàg, i.e. right continuous with left limits.

5.2 Renormalized solutions↩︎

In what follows \(T_k\), \(k>0\), is the truncation operator defined by \(T_k(y)=((-k)\vee y)\wedge k\) and \({\mathcal{M}}_{0,b}(Q_T)\) is the space of signed Borel measures on \(Q_T\) of finite total variation which do not charge Borel measures of capacity Cap zero. We denote by \(\|\mu\|_{TV}\) the total variation norm of a bounded (signed) Borel measure \(\mu\) on \(Q_T\).

As in [17] we call a Borel measurable function \(u\) defined q.e. on \(Q_T\) quasi-càdlàg if for q.e. \((s,x)\in Q_T\) the proces \([s,T]\ni t\mapsto u(t,X^{s,x}_t)\) is càdlàg.

Following [15] we adopt the following definition of renormalized solution of (1 ). In the case of local operators, this is essentially [23].

Note that by [15], if (H2) is satisfied, then there exists at most one renormalized solution of (1 ).

It is worth noting that as compared with Proposition 13, in Proposition 14 we do not assume that (53 ) is satisfied with \(g(|y|)=|y|\). It follows in particular that Proposition 14 covers equations with nonlinearities given in (8 ).

5.3 Stability results↩︎

We start with a simple observation. W know that (64 ) implies (65 ). Therefore if \(\psi,\psi_n\) satisfy the Hartman–Wintner condition and condition (48 ), and moreover there is \(N\ge1\) such that for every \(t\in(0,T]\), \[\label{eq5466} \sup_{n\ge N}\int_{{\mathbb{R}}^d}e^{-t\mathrm{Re}\,\psi_n(\xi)}\,d\xi<\infty,\tag{70}\] then (54 ) is satisfied (the estimate for \(p_t\) follows by Fatou’s lemma).

In case \(L=\Delta^{\alpha/2}\) weak convergence \(u_n\rightarrow u\) in \(L^2(Q_T)\) was proved in [1] by completely different methods and with additional assumption [1].

6 Applications↩︎

We will give examples of applications of our general convergence result stated in Theorems 11 and 12. Clearly, in the case of Lévy operators, if the transition densities exist, then (55 ) is satisfied. We shall see that in many interesting cases one can easily check (48 ) and (54 ). Therefore, in these cases, one can apply the convergence result to equations with \(\varphi\), \(f\) satisfying various assumptions of Theorem 12, in particular to weak and renormalized solutions consedered in Propositions 13 and 14. In the case of more general pseudodifferential operators of the form (35 ) the situation is quite different. It is known that in some interesting situations there exist transition densities (see, e.g., [24]), but the problem is whether, for instance, the second part of (54 ) holds true. For that reason, in the case of general operators (35 ) our results apply to equations with bounded continuous \(\varphi\) and bounded \(f(\cdot,\cdot,0)\).

In what follows \(I_k\) denotes the identity matrix of size \(k\).

6.1 Lévy operators↩︎

The following sufficient condition for the Hartman–Wintner conditon and (70 ) is often easy to check. Suppose that there is a characteristic exponent \(\tilde{\psi}:{\mathbb{R}}^d\rightarrow{\mathbb{R}}\) satisfying the Hartman–Wintner condition (64 ) such that for some \(N\ge1\) we have \[\label{eq6461} \mathrm{Re}\,\psi_n(\xi)\ge\tilde{\psi}(\xi),\quad n\ge N,\tag{71}\] for all sufficiently large \(|\xi|\). Clearly, for each \(n\ge N\), the function \(\psi_n\) satisfies the Hartman–Wintner condition. Furthermore, since \(\tilde{\psi}\) satisfies (64 ), there is \(M\ge1\) such that if \(|\xi|\ge M\), then \[\exp(-(T-s)\tilde{\psi}(\xi)) =\exp\Big(-(T-s)\ln|\xi|\frac{\tilde{\psi}(\xi)}{\ln|\xi|}\Big) \le\exp(-\kappa\ln|\xi|)\] for some constant \(\kappa>d\). Therefore from (71 ) it follows that for a sufficiently large \(M\) we have \(\exp(-(T-s)\mathrm{Re}\psi_n(\xi))\le|\xi|^{-\kappa}\) if \(|\xi|\ge M\) and \(n\ge N\). Hence \[\int_{|\xi|\ge M}e^{-(T-s)\mathrm{Re}\psi_n(\xi)}\,d\xi \le\int_{\{|\xi|\ge M\}}|\xi|^{-\kappa}\,d\xi =C(M,d)\int^{\infty}_Mr^{-\kappa+d-1}\,dr<\infty,\] for \(n\ge N\), which implies (70 ).

Below we provide examples of operators for which the corresponding characteristic exponents \(\psi,\psi_n\) satisfy (48 ), (70 ).

  1.  $\psi(\xi)=(|\xi|^2+m^{2/\alpha})^{\alpha/2}-m$,
      $\psi_n(\xi)=(|\xi|^2+m^{2/\alpha_n})^{\alpha_n/2}-m$,
      $\xi\in{\mathbb{R}}^d$, for some $m>0$ and
      $\alpha,\alpha_n\in(0,2]$. If $\alpha_n\rightarrow\alpha$,
      then (@eq:eq4464 ), (@eq:eq6461 ) are satisfied. Note that
      $L=\Delta$ if $\alpha=2$.

  2.  $\psi(\xi)=|\xi|^{\alpha}$,
      $\psi_n(\xi)=(|\xi|^2+m_n^{2/\alpha})^{\alpha/2}-m_n$,
      $\xi\in{\mathbb{R}}^d$, for some $\alpha\in(0,2)$ and $m_n>0$.
      If $m_n\rightarrow0$, then (@eq:eq4464 ) and (@eq:eq6461 ) are
      satisfied. Note that $L=\Delta^{\alpha/2}$.

6.2 Lévy-type operators↩︎

Below we give examples of operators for which (47 ), (48 ) are satisfied and consequently Theorem 11 is applicable.

References↩︎