May 29, 2026
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We consider the following time-fractional Cauchy problem \[\label{eq:cauchy} \partial_t^\beta u(t,x) = \frac{1}{\Gamma(1-\beta)}\bigl(Lu(t,x) - c\,u(t,x)\bigr), \qquad u(0,x) = f(x),\tag{1}\] where \(c \ge 0\), and \[\partial_t^\beta g(t) = D_{0+ \star}^{\beta} g(t) := \frac{1}{\Gamma(1-\beta)} \int_0^t (t-s)^{-\beta} g'(s)\,ds\] is the Caputo–Dzherbashian fractional derivative acting in the time variable \(t\), whereas \(L\) is the generator of a diffusion process acting in the space variable \(x\).
The solution to 1 admits the following probabilistic representation (see, e.g.the monographs [1], [2]) \[\label{eq:probsol} u(t,x) = {\mathbb{E}}_{x}\bigl[e^{-c \hat{\sigma}_t} f(X_{\hat{\sigma}_t})\bigr],\tag{2}\] where \(X\) is a diffusion process with generator \(L\), and \(\hat{\sigma}_t\) is the inverse of a \(\beta\)-stable subordinator. The representation 2 leads to numerical schemes based on continuous-time random walks (CTRWs). The main question of interest is to obtain quantitative convergence rates of these CTRW-based approximations to the true solution \(u\). The equation itself is meaningful for \(c\ge0\); the fractional convergence theorem proved below is stated in the killed regimes where the discount parameter \(c\) dominates at least the base-space growth rate, with a logarithmic rate before it dominates the stronger smooth-space growth rate.
For bounded diffusion coefficients, weak convergence and convergence rates of CTRW approximations were obtained in [3]. In this paper we extend these results to the case of unbounded diffusion coefficients. This setting includes affine coefficients. These examples are covered in Theorem 5 and Corollary 3.
We work with continuous-time random walk (CTRW) approximations. CTRW processes, originally introduced in physics [4], are known to have scaling limits described by Markov processes with a random time change given by inverse stable subordinators [5]. Numerical methods for time-fractional PDEs have been developed both in the deterministic and probabilistic settings; here we use the probabilistic CTRW approximation framework of [3].
Our analysis relies on the Feller semigroup techniques of [1], [2] and on CTRW approximations of stable subordinators proposed in [3]. To obtain convergence rates for our CTRW approximations, we analyse high-order sensitivities of the diffusion semigroup, \[\mathrm D^k F_t f(x),\] and establish a quasi-contraction property for the diffusion Feller semigroup \(F_t\) on spaces of smooth functions with bounded (weighted) derivatives. We use Kunita’s stochastic flow theory [6]–[8], which allows for pathwise analysis of derivatives. Under mild conditions higher-order derivatives can be organised into tensor fields. We employ a recent chain rule for tensor fields [9], originating from the Faà di Bruno formula [10], [11].
Our main contributions are as follows:
We obtain convergence rate estimates for CTRW schemes approximating the probabilistic solution 2 of the killed time-fractional Cauchy problem 1 with unbounded diffusion coefficients, including a logarithmic fractional rate when the killing dominates the base-space growth but not the stronger smooth-space growth.
We prove uniform convergence rates for random-walk approximations of diffusions with unbounded coefficients, at the level of Feller semigroups.
We develop a methodology for investigating sensitivities of diffusion processes \[\mathrm D^m X_t(x,\omega)\] with respect to the initial condition \(x\) as random tensor fields, based on Kunita’s stochastic flow theory and the tensorial chain rule, and derive corresponding growth estimates.
The structure of the paper is as follows. Section 2 contains the precise setting and the formulation of the main results. Section 3 introduces the stochastic tensor field notation, the tensorial chain rule, and the basic elements of Kunita’s theory used in this work. In Section 4 we study the sensitivities of the diffusion semigroup with respect to the initial point and establish the quasi-contraction property. Section 5 records the weighted Feller facts used for unbounded observables. Section 6 contains the proofs of the main results, including the weak convergence rates for the time-fractional diffusion.
We collect here the conventions used in the statements of the main results. Throughout, \({\mathbb{R}}_+=[0,\infty)\) and \({\mathbb{N}}_0={\mathbb{N}}\cup\{0\}\). If \(r\in{\mathbb{R}}\), then \([r]\) denotes its integer part. The state space is \({\cal X}={\mathbb{R}}^k\), equipped with the Euclidean norm \(|\cdot|\). Constants denoted by \(C\), possibly with subscripts, are positive and may change from line to line; their dependence is recorded when it is relevant for an estimate.
For a diffusion started from \(x\), expectations are denoted by \({\mathbb{E}}_x\). The diffusion semigroup is \[F_t f(x):={\mathbb{E}}_x f(X_t),\] and \(L\) denotes its generator. The killed semigroup is written \(e^{-ct}F_t\). The one-step Markov approximation with time step \(h\) is denoted by \(U_h\); thus \(U_h^n\) is the \(n\)-step transition operator, and in continuous-time estimates we use the shorthand \(U_h^{\lfloor s/h\rfloor}\) for the approximation at operational time \(s\).
The fractional clock is denoted as follows. The process \(\hat{\Sigma}\) is a \(\beta\)-stable subordinator and \(\hat{\sigma}_T:=\sup\{s:\hat{\Sigma}_s\le T\}\) is its inverse. Its discrete counterpart is generated by the heavy-tailed walk \(S_n^h=h^{1/\beta}\sum_{i=1}^n\tau_i\); the first passage index and the corresponding operational time are \[n_T^h:=\min\{n\ge0:S_n^h>T\}, \qquad N_T^h:=h\,n_T^h.\]
Derivatives with respect to the initial point are written in tensor form: \(\mathrm D^j f(x)\) is the \(j\)-linear derivative of \(f\) at \(x\), with the convention \(\mathrm D^0f=f\). Tensor arguments are placed in square brackets, for example \(\mathrm D^j f(x)[v_1,\dots,v_j]\), and their norms are the operator tensor norms introduced in 4 . Symmetric tensor products are denoted by \(\vee\).
We use two regularity scales. For a weight \(G\ge1\), \(C_G({\cal X})\) and \(C_{G,\infty}({\cal X})\) are the weighted continuous spaces with norm \(\|f\|_G=\sup_x |f(x)|/G(x)\). The bounded-derivative scale uses \[V(x)=1+|x|^2,\qquad W(x)=1+|x|^4,\qquad D_l\subset C_{W,\infty}({\cal X}),\] with norm \(\|\cdot\|_l\) defined below from bounded derivatives and one base point value. The weighted-regular scale uses \(w_\alpha(x)=(1+|x|^2)^{\alpha/2}\) and the spaces \({\cal C}^r_{\alpha,\infty}({\cal X})\) with norm \(\|\cdot\|_{\alpha,r}\). In the abstract fractional theorem, \(B\) denotes the base Banach space in which the error is measured, \(D\subset B\) denotes the smoother space controlling the local approximation error, \(\bar\mu_B\) is the base-space growth rate, and \(\mu_D\) is the corresponding growth rate in the smoother norm.
Let \((\Omega,{\cal F},({\cal F}_t)_{t\in{\mathbb{R}}_+},P,W)\) be a stochastic basis such that
\({\cal F}\) is complete;
\({\cal F}_0\) contains all \(P\)-null sets \({\cal N}\) from \({\cal F}\);
\(W_t=(W_t^1,\dots,W_t^d)\) is a standard \(d\)-dimensional Brownian motion;
\({\cal F}_t=\sigma\{{\cal N},W_s:0\le s\le t\}\).
Thus the filtration satisfies the usual conditions. We also fix, independently of \(W\), a \(\beta\)-stable subordinator \(\hat{\Sigma}\) with generator \[\hat{L}_\beta g(t) := \int_t^\infty \frac{g(s) - g(t)}{(s-t)^{1 + \beta}} \, ds.\]
Put \({\cal X}:={\mathbb{R}}^k\), let \(\sigma_j:{\cal X}\to{\cal X}\) for \(j=0,\dots,d\), and write \(b:=\sigma_0\). We consider the Itô SDE \[\label{eq:SDE-flow} dX_t = b(X_t)\,dt + \sum_{j=1}^d \sigma_j(X_t)\,dW_t^j, \qquad X_0 = x \in {\cal X}.\tag{3}\]
For \(m\in{\mathbb{N}}\) and \(\theta\in(0,1)\), let \(C^{m,\theta}_g({\cal X};{\cal X})\) denote the maps whose derivatives up to order \(m\) are bounded and continuous and whose \(m\)-th derivative is globally \(\theta\)-Hölder continuous.
Assumption 1. The coefficients of 3 satisfy the following:
for all \(x\in{\cal X}\), \[|\sigma_j(x)| \leq K (1 + |x|), \qquad j = 0, \dots, d,\] for some \(K>0\);
for all \(x,x'\in{\cal X}\), \[|\sigma_j(x) - \sigma_j(x')| \leq K |x - x'|, \qquad j = 0, \dots, d;\]
for some integer \(m\ge4\) and some \(\theta\in(0,1)\), the coefficients \(\sigma_j\) belong to \(C^{m,\theta}_g({\cal X};{\cal X})\) and \[\|\mathrm D^i \sigma_j(x)\| \le K \quad\text{for all } x\in{\cal X},\;i=1,\dots,m,\;j=0,\dots,d,\] in the tensor norm 4 .
Under Assumption 1, 3 has a unique strong solution \(X_t(x)\) for every \(x\in{\cal X}\), and \(X_t(x)\in L^p\) for all \(p\ge2\).
We introduce weight functions \[V(x) := 1 + |x|^2, \qquad W(x) := 1 + |x|^4.\] We use two related scales. First, for a weight \(G\ge1\), let \(C_G({\cal X})\) be the space of continuous functions with \[\|f\|_G := \sup_{x \in {\cal X}} \frac{|f(x)|}{G(x)} < \infty.\] The subspace \(C_{G,\infty}({\cal X})\) consists of those \(f\in C_G({\cal X})\) for which \(f/G\) vanishes at infinity. Fix a base point \(x_*\in{\cal X}\); for the ambient state space \({\mathbb{R}}^k\) we take \(x_*=0\). For \(1 \le l \le m\), we introduce the subspace \[D_l := \Bigl\{ f \in C^l({\cal X}) : f/W\in C_\infty({\cal X})\text{ and } \|\mathrm D^j f(\cdot)\| \in C_\infty({\cal X}) \text{ for all } j=1,\dots,l \Bigr\}.\] The norm on \(D_l\) is \[\| f \|_{l} := |f(x_*)| + \sum_{j=1}^l \sup_{x \in {\cal X}} \| \mathrm D^j f (x) \|,\] where derivative norms are the tensor norms from 4 . Since \(f\in D_l\) has bounded first derivative, \(|f(x)|\le C_f(1+|x|)\) and therefore \(D_l\subset C_{W,\infty}({\cal X})\) continuously. Thus the bounded-derivative theorem below gives a coarse \(W\)-norm estimate for bounded-derivative observables; the weighted-regular theorem is the sharper scale for genuinely weighted observables. The vanishing-at-infinity condition on each derivative is the natural analogue of the \({\cal C}^r_{\alpha,\infty}\)-condition used in the weighted-regular scale; it is what makes the semigroup \((F_t)_{t\ge0}\) strongly continuous on \(D_l\) in the norm \(\|\cdot\|_l\) under linear-growth coefficients (Lemma 10 below).
Second, for the weighted regular estimates put \[w_\alpha(x):=(1+|x|^2)^{\alpha/2}.\] For \(k\in{\mathbb{N}}_0\) define \[\|f\|_{\alpha,k} := \max_{0\le j\le k} \sup_{x\in{\cal X}} \frac{\|\mathrm D^j f(x)\|}{w_{\alpha-j}(x)}.\] Let \({\cal C}^k_\alpha({\cal X})\) be the space of all \(f\in C^k({\cal X})\) with \(\|f\|_{\alpha,k}<\infty\), and by \({\cal C}^k_{\alpha,\infty}({\cal X})\) the closed subspace of functions satisfying \[\frac{\|\mathrm D^j f(\cdot)\|}{w_{\alpha-j}(\cdot)}\in C_\infty({\cal X}), \qquad 0\le j\le k.\] These spaces carry the norm \(\|\cdot\|_{\alpha,k}\).
Assumption 2. For fixed \(\alpha\in{\mathbb{R}}\) and \(r\in{\mathbb{N}}_0\), the coefficients satisfy \[\sigma_j\in {\cal C}^{r+4}_1({\cal X};{\cal X}), \qquad j=0,\dots,d,\] and their \({\cal C}^{r+4}_1\)-norms are bounded by a common constant \(K_{\alpha,r}\). Thus, for every \(q=0,\dots,r+4\), \[\|\mathrm D^q\sigma_j(x)\| \le K_{\alpha,r} w_{1-q}(x), \qquad x\in{\cal X},\quad j=0,\dots,d.\] In addition, \(\mathrm D^{r+4}\sigma_j\) is globally \(\theta\)-Hölder continuous for some \(\theta\in(0,1)\).
Remark 1. Affine coefficients, including Black–Scholes and Langevin-type coefficients, satisfy this condition.
The one-step random-walk approximation of 3 is given by the transition operator \(U_h\).
Assumption 3. The random–walk transition operator is \[(U_h f)(x) := {\mathbb{E}}\Bigl[ f\bigl(x + b(x)h + \sum_{j=1}^d \sigma_j(x)\,\xi_j\sqrt{h}\bigr) \Bigr],\] where \(\xi\) is an \({\mathbb{R}}^d\)-valued random vector with \[{\mathbb{E}}[\xi]=0,\qquad {\mathbb{E}}[\xi\wedge\xi]=I_d,\qquad M_4 := {\mathbb{E}}\|\xi\|^4 < \infty,\] and vanishing third moments: \[{\mathbb{E}}[\xi_{i_1}\xi_{i_2}\xi_{i_3}] = 0, \qquad i_1,i_2,i_3 = 1,\dots,d.\]
Remark 2. The random walk replaces the Brownian increment. The choice \(\xi=h^{-1/2}W_h\) gives the Euler–Maruyama scheme. The odd moments vanish, for example, when \({\cal L}(\xi)={\cal L}(-\xi)\).
Assumption 4. The random vector \(\xi\) from Assumption 3 is bounded: \[\|\xi\|\le R_\xi \qquad\text{a.s.}\]
Assumption 5. For given \(\alpha\in{\mathbb{R}}\) and \(r\in{\mathbb{N}}_0\), there are \(q_{\alpha,r}\ge0\) and \(h_0>0\) such that \[\|U_h g\|_{\alpha,r} \le e^{q_{\alpha,r}h}\|g\|_{\alpha,r}, \qquad g\in{\cal C}^r_{\alpha,\infty}({\cal X}),\quad 0<h\le h_0.\]
Remark 3. For \(r=0\) this follows from the usual weighted Lyapunov estimate for \(x+\eta_h(x)\), exactly as in Lemma 15 for the weight \(W\). For \(r\ge1\) it is a genuinely stronger, derivative-level stability condition: it is the one-step discrete analogue of the weighted jet bounds in Lemma 19. We impose it as a hypothesis rather than derive it, since its verification is scheme-dependent; establishing it for concrete walks under Assumption 2 is left open.
The first convergence result is the bounded-derivative version of the \(C_W\) argument.
Theorem 1. Let \(F_t\) be the Feller semigroup of 3 , and let Assumptions 1 and 3 hold. Then, for any \(f\in D_4\), \[\sup_{s \leq t} \| U_h^{\lfloor s/h\rfloor} f - F_s f \|_W \leq C_{\mathrm{RW}} e^{\mu t} h \|f \|_4,\] where \(\mu:=\mu_4\) is the renormed \(D_4\)-growth rate of Corollary 4, and \(C_{\mathrm{RW}}\) depends only on the constants in Assumptions 1 and 3. For the killed semigroups, \[\sup_{s \leq t} \| e^{-c\lfloor s/h\rfloor h} U_h^{\lfloor s/h\rfloor} f - e^{-c s} F_s f \|_W \leq C_{\mathrm{RW}}' e^{(\mu - c) t} h \|f \|_4,\] with \(C_{\mathrm{RW}}'\) depending on the same data and on \(c\).
The companion estimate uses the weighted regular scale \({\cal C}^r_{\alpha,\infty}({\cal X})\) and the bounded-increment assumption.
Theorem 2. Fix \(\alpha\in{\mathbb{R}}\) and \(r\in{\mathbb{N}}_0\). Let Assumptions 1, 2, 3, and 4 hold, and let the walk satisfy Assumption 5. Suppose also that Assumption 1 is valid for an order \(m\ge r+5\). Let \(F_t\) be the semigroup of 3 . Then, for every \(f\in {\cal C}^{r+4}_{\alpha,\infty}({\cal X})\), \[\label{eq:uniform-weighted-regular} \sup_{s\le t} \|U_h^{\lfloor s/h\rfloor}f-F_s f\|_{\alpha,r} \le C_{\alpha,r} e^{\mu_{\alpha,r}t}h \|f\|_{\alpha,r+4}.\qquad{(1)}\] For the killed semigroups, if \(c\ge0\), then \[\label{eq:uniform-weighted-regular-killed} \sup_{s\le t} \|e^{-c\lfloor s/h\rfloor h}U_h^{\lfloor s/h\rfloor}f-e^{-cs}F_s f\|_{\alpha,r} \le C'_{\alpha,r}e^{(\mu_{\alpha,r}-c)t}h \|f\|_{\alpha,r+4}.\qquad{(2)}\] The estimate holds for all sufficiently small \(h>0\). The constants depend on \(\alpha,r,K_{\alpha,r},d,R_\xi\), and the moment constants in Assumption 3.
The inverse stable subordinator is \[\hat{\sigma}_T := \sup \{s \colon \hat{\Sigma}_s \leq T \}.\] Recall from [1], [3] that the Caputo–Dzherbashian derivative also has the representation \[D_{0+ \star}^{\beta} g(t) = \frac{1}{\Gamma(-\beta)} \int_0^t \frac{g(s) - g(t)}{s^{1+\beta}} ds + \frac{g(t) - g(0)}{\Gamma(1 - \beta) t^\beta} = - \frac{1}{\Gamma(1 - \beta)} \tilde{L}_\beta g(t),\] where \(\tilde{L}_\beta\) is the generator of a decreasing \(\beta\)-stable subordinator started at \(t\) and stopped at zero.
We approximate the stable subordinator by the heavy-tailed random walk \[S_n^h:=h^{1/\beta}\sum_{i=1}^n\tau_i, \qquad \Phi^h_s := S_{\lfloor s/h\rfloor}^h,\] where \(\tau_i\) are i.i.d. positive random variables. For a terminal time \(T>0\) we define the discrete inverse clock by \[n_T^h:=\min\{n\ge0:S_n^h>T\}, \qquad N_T^h:=h\,n_T^h.\]
Assumption 6. There is a probability density \(p(x)\) of \(\tau_i\) such that \[p(x) = x^{-1-\beta}, \, x > B > 0,\] and \(p\) is otherwise arbitrary on \((0,B)\). Moreover, \(p(0)=0\) and \(p\) is continuously differentiable on \({\mathbb{R}}_+\).
Under this assumption, \(\beta B^\beta > 1\).
Remark 4. More general heavy-tail assumptions are possible; we use this form to keep the clock estimates explicit.
Let \(T^\beta_t\) denote the Feller semigroup of the stable subordinator \(\hat{\Sigma}_t\). We recall two estimates for the stable clock and its discrete approximation. We recall [3].
Theorem 3. Under Assumption 6, the following estimate holds \[\int_0^\infty | P(\Phi^h_t > a) - P(\hat{\Sigma}_t > a)| dt \leq (1 + a + a^{-1}) C(\beta) h^{\chi(\beta)},\] where \(a>0\), and \(C(\beta) > 0, \, \chi(\beta) \in (0, 1)\) can be found explicitly.
We also use [3].
Proposition 4. Let Assumption 6 hold. Then, for any \(a > 1\), \(h < 1\) and \(t > t_0\), where \(t_0\) can be calculated explicitly, we have \[P(\Phi^h_t < a) \leq 2^{1 + 2 / \beta} a t^{-1/\beta}.\]
The forward estimate of Theorem 3 transfers to the inverse clocks \(\hat{\sigma}_T\) and \(N_T^h\):
Lemma 1 (inverse-clock rate). Under Assumption 6, for every terminal time \(T>0\) and all \(h>0\), \[\label{eq:inverse-clock-rate} \int_0^\infty \bigl| P(\hat{\sigma}_T>s)-P(N_T^h>s) \bigr|\,ds \le C_{\mathrm{clock}}(T)\,h^{\chi(\beta)},\qquad{(3)}\] with \(C_{\mathrm{clock}}(T):=(1+T+T^{-1})C(\beta)\), where \(C(\beta)\) and \(\chi(\beta)\) are the constants of Theorem 3.
Proof. By the duality between a non-decreasing process and its inverse, \(\{\hat{\sigma}_T>s\}=\{\hat{\Sigma}_s\le T\}\) pathwise. For the discrete clock, since \(n_T^h=\min\{n\ge0:S_n^h>T\}\) and \(N_T^h=hn_T^h\), \[\{N_T^h>s\} =\{n_T^h>\lfloor s/h\rfloor\} =\{S^h_{\lfloor s/h\rfloor}\le T\} =\{\Phi_s^h\le T\},\] because \(S^h\) is non-decreasing. Therefore \[|P(\hat{\sigma}_T>s)-P(N_T^h>s)| =|P(\hat{\Sigma}_s\le T)-P(\Phi_s^h\le T)| =|P(\Phi_s^h>T)-P(\hat{\Sigma}_s>T)|.\] Integrating in \(s\) and applying Theorem 3 with \(a=T\) gives ?? . ◻
We approximate the representation 2 by a continuous-time random walk. In the simplest case, a CTRW is a sum \[\sum_{i=1}^{N^h_t} X^h_i,\] where \((X_i^h)\) are i.i.d. increments. Here the spatial motion is the discrete Markov chain with transition operator \(U_h\), sampled at the inverse clock \(N_t^h\).
We state the fractional estimate in an abstract form covering both spatial scales. Let \((B,\|\cdot\|_B)\) and \((D,\|\cdot\|_D)\) be Banach spaces of functions on \({\cal X}\), with \(D\subset B\). For \(f\in D\) put \[u(T,\cdot):={\mathbb{E}}\,e^{-c\hat{\sigma}_T}F_{\hat{\sigma}_T}f, \qquad u_h(T,\cdot):={\mathbb{E}}\,e^{-c h n_T^h}U_h^{n_T^h}f.\]
Theorem 5. Let Assumption 6 hold. Assume that \((F_t)_{t\ge0}\) is a strongly continuous semigroup on \(B\) with generator \(L_B\), \(D\subset\mathrm{Dom}(L_B)\), and the restriction of \(L_B\) to \(D\) coincides with \(L\). Assume further that the map \(s\mapsto \hat{F}_s f := e^{-cs}F_s f\) belongs to \(C^1([0,\infty);B)\) with \(\tfrac{d}{ds}\hat{F}_s f=\hat{F}_s(L-c)f\). Finally, assume that there exist constants \(\bar\mu_B\le \mu_D\) and \(C_B,C_{\mathrm{RW}},C_L>0\) such that \[\|e^{-cs}F_s f\|_B + \|e^{-c\lfloor s/h\rfloor h}U_h^{\lfloor s/h\rfloor}f\|_B \le C_B e^{(\bar\mu_B-c)s}\|f\|_D, \qquad s\ge0,\] \[\|e^{-cs}F_s f-e^{-c\lfloor s/h\rfloor h}U_h^{\lfloor s/h\rfloor}f\|_B \le C_{\mathrm{RW}}e^{(\mu_D-c)s}h\|f\|_D, \qquad s\ge0,\] and \[\|e^{-cs}F_s(L-c)f\|_B \le C_L e^{(\bar\mu_B-c)s}\|f\|_D, \qquad s\ge0.\] If \(c\ge\mu_D\ge\bar\mu_B\), then for \(T>t_0\) and all sufficiently small \(h\), \[\label{eq:frac-abstract} \|u_h(T,\cdot)-u(T,\cdot)\|_B \le C_{\mathrm{frac}}\bigl(h+h^{\chi(\beta)}\bigr)\|f\|_D.\qquad{(4)}\] If \(\bar\mu_B\le c<\mu_D\), then for \(T>t_0\) and all sufficiently small \(h\), \[\label{eq:frac-abstract-log} \|u_h(T,\cdot)-u(T,\cdot)\|_B \le C_{\mathrm{frac}}^{\log} \Bigl((\log(1/h))^{-1/\beta}+h^{\chi(\beta)}\Bigr)\|f\|_D.\qquad{(5)}\] The constants depend on the clock parameters, on \(T\), on \(C_{\mathrm{clock}}(T)\), and on the displayed bounds. In the logarithmic case the admissible upper bound on \(h\) may also depend on \(\mu_D-c\).
Corollary 1. Let Assumptions 1, 3, and 6 hold. Let \(\bar\mu_W,\mu_4\ge0\) be admissible growth bounds on \(C_{W,\infty}({\cal X})\) and \(D_4\), respectively, with the equivalent-norm renorming used in the proof of Theorem 1. If \(c\ge\mu_4\ge\bar\mu_W\), then for every \(f\in D_4\), \[\label{eq:frac95main} \|u_h(T,\cdot)-u(T,\cdot)\|_W \le C_{\mathrm{frac}}\bigl(h+h^{\chi(\beta)}\bigr)\|f\|_4.\qquad{(6)}\] If \(\bar\mu_W\le c<\mu_4\), then \[\label{eq:frac95main95log} \|u_h(T,\cdot)-u(T,\cdot)\|_W \le C_{\mathrm{frac}}^{\log} \Bigl((\log(1/h))^{-1/\beta}+h^{\chi(\beta)}\Bigr)\|f\|_4.\qquad{(7)}\]
Corollary 2 (Black–Scholes coefficients). Consider the componentwise geometric Brownian motion \[dX_t^i=\mu_iX_t^i\,dt+\sigma_iX_t^i\,dW_t^i, \qquad i=1,\dots,k,\] started from \(x\in(0,\infty)^k\). The natural state space is the invariant domain \((0,\infty)^k\), although the coefficients extend linearly to \({\mathbb{R}}^k\). Put \[K_{\mathrm{GBM}} := \max_{1\le i\le k}\{|\mu_i|,|\sigma_i|\}.\] Then Assumption 1 holds with \(K=K_{\mathrm{GBM}}\) and \(d=k\): first derivatives are bounded by \(K_{\mathrm{GBM}}\) and all derivatives of order at least two vanish. For the quartic weight \(W=1+|x|^4\), Lemma 4 gives the explicit Lyapunov coefficient \[C_2=(8K_{\mathrm{GBM}}+24K_{\mathrm{GBM}}^2)d.\] If the random-walk and clock assumptions in Corollary 1 also hold for the chosen discretization, then the bounded-derivative fractional estimate ?? applies to \(f\in D_4\) whenever \(c\ge\mu_4\ge\bar\mu_W\), where \(\mu_4\) is the renormed \(D_4\) growth bound from Corollary 4 and \(\bar\mu_W\) may be taken from the above Lyapunov coefficient. In the scalar case \(\mu=0.05\), \(\sigma=0.2\), \(K_{\mathrm{GBM}}=0.2\), and \(d=1\), this gives \(C_2=8(0.2)+24(0.2)^2=2.56\); the minimum killing condition in this theorem remains \(c\ge\mu_4\). The estimate is stated for the linear extension of the coefficients to \({\mathbb{R}}^k\); the one-step increment \(x+\eta_h(x)\) may leave the positivity domain \((0,\infty)^k\), so positivity-preserving discretizations are a separate matter not addressed here.
Corollary 3. Fix \(\alpha\in{\mathbb{R}}\) and \(r\in{\mathbb{N}}_0\). Let Assumptions 1, 2, 3, 4, 5, and 6 hold. Let \(\bar\mu_{\alpha,r}\) and \(\mu_{\alpha,r+4}\) denote admissible growth bounds for \(F_t\) and \(U_h^{\lfloor s/h\rfloor}\) on \({\cal C}^r_{\alpha,\infty}({\cal X})\) and for \(F_t\) on \({\cal C}^{r+4}_{\alpha,\infty}({\cal X})\), respectively. If \(c\ge\mu_{\alpha,r+4}\ge\bar\mu_{\alpha,r}\), then for every \(f\in{\cal C}^{r+4}_{\alpha,\infty}({\cal X})\), \[\label{eq:frac-weighted-regular} \|u_h(T,\cdot)-u(T,\cdot)\|_{\alpha,r} \le C_{\alpha,r,\mathrm{frac}}\bigl(h+h^{\chi(\beta)}\bigr) \|f\|_{\alpha,r+4}.\qquad{(8)}\] If \(\bar\mu_{\alpha,r}\le c<\mu_{\alpha,r+4}\), then \[\label{eq:frac-weighted-regular-log} \|u_h(T,\cdot)-u(T,\cdot)\|_{\alpha,r} \le C_{\alpha,r,\mathrm{frac}}^{\log} \Bigl((\log(1/h))^{-1/\beta}+h^{\chi(\beta)}\Bigr) \|f\|_{\alpha,r+4}.\qquad{(9)}\]
Remark 5. The killing coefficient is needed because the operational time \(\hat{\sigma}_T\) has only polynomial tails. Thus the map \(s\mapsto e^{-cs}F_s f\) must be integrable against the law of \(\hat{\sigma}_T\). Theorem 5 separates the supercritical regime \(c\ge\mu_D\), where the deterministic approximation error remains of order \(h\) at random operational times, from the logarithmic regime \(\bar\mu_B\le c<\mu_D\), where the same term must be truncated in operational time. We state no theorem below \(c=\bar\mu_B\).
The proofs are given in Section 6.
Let \(X = {\mathbb{R}}^k\) and \(Y = {\mathbb{R}}^\ell\) be finite-dimensional Euclidean spaces. For integers \(m \ge 0\), denote by \[\mathfrak{L}_m(X,Y)\] the space of continuous \(m\)-linear mappings \[A : X^m \to Y.\] In coordinates (for simplicity, when \(Y={\mathbb{R}}\)) an element \(A \in \mathfrak{L}_m(X,{\mathbb{R}})\) is represented by \[\bigl(a_{i_1,\dots,i_m} \bigr), \qquad 1 \le i_p \le k,\] and acts via \[A[x_1,\dots,x_m] = \sum_{i_1,\dots,i_m=1}^k a_{i_1,\dots,i_m} \, x_1^{i_1} \cdots x_m^{i_m}.\] To avoid ambiguity, we put the arguments of a tensor inside square brackets.
For \(A\in\mathfrak{L}_m(X,Y)\), define the operator (spectral) norm \[\label{eq:tensor95norm} \|A\| := \sup_{\substack{x_1,\dots,x_m\in X \\ x_i\neq 0}} \frac{\|A(x_1,\dots,x_m)\|}{\|x_1\|\cdots\|x_m\|}.\tag{4}\] Because all norms in finite dimensions are equivalent, the operator norm is comparable to the Frobenius norm \[\|A\|_F^2 := \sum_{i_1,\dots,i_m} \bigl(a_{i_1,\dots,i_m}\bigr)^2.\]
For elements \(x_1, \dots, x_m \in X\), we denote their tensor product by \(x_1 \otimes \dots \otimes x_m\) or \(\bigotimes_{j=1}^m x_j\). The tensor product of normed linear spaces \(X_1, \dots, X_n\) is denoted by \(X_1 \otimes \dots \otimes X_n\) or \(\bigotimes_{i=1}^n X_i\). The \(m\)-th tensor power of \(X\) is denoted by \(X^{\otimes m}\).
For an element \(u \in X_1 \otimes X_2\), define the projective tensor norm \(\|\cdot\|_\pi\) by \[\|u\|_\pi := \inf\Bigl\{ \sum_{i=1}^N \|x_{1,i}\| \,\|x_{2,i}\| \;\Big|\; u = \sum_{i=1}^N x_{1,i} \otimes x_{2,i},\; x_{1,i} \in X_1,\; x_{2,i} \in X_2,\; N \in \mathbb{N} \Bigr\}.\] In the finite-dimensional case, \(X_1 \otimes X_2\) equipped with the projective tensor norm \(\| \cdot \|_{\pi}\) is complete.
Any tensor \(A \in \mathfrak{L}_m(X,Y)\) can be viewed as an element of \(\mathcal{L}(X^{\otimes m}, Y)\). In fact, the tensor norm on the former coincides with the operator norm on the latter (up to equivalence of norms in finite dimensions).
For a tensor \(A \in \mathfrak{L}_m(X,Y)\), define its symmetrisation as \[\operatorname{Sym}(A)[x_1,\dots,x_m] := \frac{1}{m!} \sum_{\sigma \in \mathrm{Perm}(m)} A[x_{\sigma(1)}, \dots, x_{\sigma(m)}].\] We define the symmetric product \(x_1 \vee x_2\) as \(\operatorname{Sym}(x_1 \otimes x_2)\), and in general \[x_1 \vee \dots \vee x_m := \operatorname{Sym}\bigl(x_1 \otimes \dots \otimes x_m\bigr).\] When all arguments coincide, we also write \(v^{\vee m} := v \vee \dots \vee v\).
Fix \(m \in \mathbb{N}\). For a function \(f \in C^m (X)\) (scalar-valued), the tensor \(\nabla f (x) \in \mathfrak{L}_1(X,{\mathbb{R}})\) is defined as the first-order directional derivative \[\nabla f (x)[u] := \lim_{\varepsilon \to 0} \frac{f(x + \varepsilon u) - f(x)}{\varepsilon},\] and the tensor \(\mathrm D^m f (x) \in \mathfrak{L}_m(X,{\mathbb{R}})\) is the iterated \(m\)-th order directional derivative. In coordinates, \[\mathrm D^m f (x)[u_1, \dots, u_m] = \sum_{i_1, \dots, i_m = 1}^k \frac{\partial^m f(x)}{\partial x_{i_1} \dots \partial x_{i_m}}\, u^{i_1}_1 \dots u^{i_m}_m.\] The tensor \(\mathrm D^m f(x)\) is symmetric, so we also write \[\mathrm D^m f(x)\bigl[ x_1 \vee \dots \vee x_m \bigr]\] instead of \(\mathrm D^m f(x)[x_1,\dots,x_m]\).
We view the derivatives \(\mathrm D^jX_t(x,\omega)\) with respect to the initial point as random tensor fields and recall the needed parts of Kunita’s stochastic flow theory [6] in tensor notation.
We recall [6] in the notation used here.
Theorem 6. Consider the SDE 3 . Let Assumption 1 (items (1)–(2)) hold. Assume that \(\nabla \sigma_j(x)\) is well-defined and (globally) \(\alpha\)-Hölder continuous for some \(\alpha \in (0,1)\) and all \(j = 0, \dots, d\). Then the process \(\nabla X_t(x)\) is well-defined and is locally \(\beta\)-Hölder continuous in \(x\), almost surely, for any \(\beta < \alpha\). Furthermore, it is the unique strong solution of the SDE \[\label{eq:jet} \nabla X_t(x) = e + \int_0^t \bigl(\nabla b\bigl(X_s(x)\bigr)\bigr)\,\nabla X_s(x)\,ds + \sum_{l=1}^d \int_0^t \bigl(\nabla \sigma_l\bigl(X_s(x)\bigr)\bigr)\,\nabla X_s(x)\,dW_s^l,\qquad{(10)}\] where \(e[u] = u\) is the identity mapping on \(X\). Moreover, for every \(u \in X\) and \(p \ge 2\) we have \(\nabla X_t(x)[u] \in L^p\).
For higher-order derivatives we use a tensorial Faà di Bruno formula. For a finite set \(A\), denote by \(\mathcal{P}(A,k)\) the set of partitions of \(A\) into \(k\) disjoint non-empty subsets: \[\mathcal{P}(A,k) := \left\{ P = \{P_1, \dots, P_k\} : A = \bigcup_{i=1}^k P_i,\; P_i \cap P_j = \varnothing,\;i \ne j,\; P_i \neq \varnothing,\;i=1,\dots,k \right\}.\] For brevity, write \(\mathcal{P}(m,k) := \mathcal{P}(\{1,\dots,m\},k)\).
We use the following tensor form of the Faà di Bruno formula; cf.[9].
Proposition 7. Let \(X, Y, Z\) be Banach spaces and let \(f \colon X \to Y\), \(g \colon Y \to Z\). Let \(x \in X\) and set \(y = f(x)\). Assume that \(f\) and \(g\) have continuous derivatives up to order \(m\) in a neighbourhood of \(x\) and \(y\), respectively. Then, for \(v_1,\dots,v_m\in X\), \[\begin{align} \mathrm D^m (g \circ f)(x)[ v_1, \dots, v_m] &= \sum_{k=1}^{m} \;\sum_{P \in \mathcal{P}(m,k)} \mathrm D^k g\bigl(f(x)\bigr) \Bigl[ \mathrm D^{|P_1|} f(x)\bigl[ \bigvee_{i \in P_1} v_i \bigr], \dots, \mathrm D^{|P_k|} f(x)\bigl[ \bigvee_{i \in P_k} v_i \bigr] \Bigr]. \end{align}\]
This is the tensor form of the Faà di Bruno formula. The classical and multivariate versions are discussed in [10], [12]; see also [11] for probabilistic applications.
For \(m\in\mathbb{N}\) and \(\alpha\in(0,1)\), let \(C^{m,\alpha}\) and \(C^{m,\alpha}_g\) denote the classes of maps \(f:X\to X\) whose derivatives up to order \(m\) are continuous and whose \(m\)-th derivatives are locally, respectively globally, \(\alpha\)-Hölder continuous.
We also use [6].
Theorem 8. Consider the SDE 3 . Let \(m \ge 2\) and \(\alpha \in (0,1)\), and suppose \(b, \sigma_j \in C^{m,\alpha}_g\) for all \(j=0,\dots,d\). Assume that all derivatives of \(b\) and \(\sigma_j\) up to order \(m\) are bounded. Then, for each \(t \ge 0\), the mapping \(x \mapsto \mathrm D^m X_t(x)\) is of class \(C^{m,\beta}\) almost surely, for any \(\beta < \alpha\). Furthermore, \(\mathrm D^m X_t(x)\) satisfies an SDE of the form \[\label{eq:higher-jet} \mathrm D^m X_t(x) = \int_0^t \bigl(\mathrm D^m b\bigl(X_s(x)\bigr)\bigr) \bigl[\cdot\bigr]\,ds + \sum_{l=1}^d \int_0^t \bigl(\mathrm D^m \sigma_l\bigl(X_s(x)\bigr)\bigr) \bigl[\cdot\bigr]\,dW_s^l,\qquad{(11)}\] where the dots indicate multilinear dependence on \(\mathrm D^k X_s(x)\), \(k=1,\dots,m\), given explicitly via Proposition 7. Moreover, for any \(u_1,\dots,u_m \in X\) and \(p \ge 2\), \[\mathrm D^m X_t(x)[u_1,\dots,u_m] \in L^p.\]
Equation ?? involves the lower-order derivatives \(\mathrm D^k X_t(x)\), \(k = 1,\dots, m-1\). In particular, by Proposition 7, the coefficients can be written in closed form in terms of the tensors \(\mathrm D^k b\) and \(\mathrm D^k \sigma_j\).
This section estimates the moments of the flow derivatives \(\mathrm D^mX_t(x)\) and the corresponding derivatives of the semigroup \(F_t\).
Lemma 2. Consider the SDE 3 under Assumption 1. Let \(p \ge 2\) and \(v \in {\cal X}\) with \(|v|=1\). Then there exists a constant \(\mu_{1,p} = \mu_{1,p}(K,d,p) > 0\) such that \[\label{eq:first95order95estim} {\mathbb{E}}\bigl| \nabla X_t(x)[v] \bigr|^p \;\le\; e^{\mu_{1,p} t}, \qquad t \ge 0.\qquad{(12)}\]
Proof. For \(p \ge 2\), the map \(y \mapsto |y|^p\) on \({\cal X}\) is twice continuously differentiable. Set \[Y_t := \nabla X_t(x)[v].\] By Theorem 6, \(Y_t\) satisfies the linear SDE \[dY_t = \bigl(\nabla b(X_t(x))\,Y_t\bigr)\,dt + \sum_{l=1}^d \bigl(\nabla \sigma_l(X_t(x))\,Y_t\bigr)\,dW_t^l, \qquad Y_0 = v.\] Applying Itô’s formula to \(t \mapsto |Y_t|^p\) gives \[d |Y_t|^p = p |Y_t|^{p-2} \langle Y_t, dY_t \rangle + \frac{p(p-1)}{2} |Y_t|^{p-2} d\langle Y \rangle_t,\] where \(\langle Y \rangle_t\) is the quadratic variation of \(Y\). Taking expectations and using the boundedness of \(\nabla b\) and \(\nabla\sigma_l\) from Assumption 1, we obtain \[\begin{align} \frac{d}{dt} {\mathbb{E}}|Y_t|^p &= p\,{\mathbb{E}}\Bigl[ |Y_t|^{p-2} \big\langle Y_t, \nabla b(X_t) Y_t\big\rangle \Bigr] + \frac{p(p-1)}{2} \sum_{l=1}^d {\mathbb{E}}\Bigl[ |Y_t|^{p-2}\,\big|\nabla \sigma_l(X_t) Y_t\big|^2 \Bigr] \\ &\le C_{1,p}\,{\mathbb{E}}|Y_t|^p, \end{align}\] where \(C_{1,p}=C_{1,p}(K,d,p)\) is finite and can be computed explicitly by Young and Cauchy–Schwarz inequalities. Since \(Y_0 = v\) and \(|v|=1\), Grönwall’s lemma yields \[{\mathbb{E}}|Y_t|^p \le e^{C_{1,p} t}, \qquad t\ge0.\] Setting \(\mu_{1,p} := C_{1,p}\) gives ?? . ◻
We derive inductive bounds for higher-order derivatives of the flow.
Proposition 9. Consider the SDE 3 under Assumption 1. Fix \(m \in \mathbb{N}\) and \(p \ge 2\). Then there exist functions \(A_m(\cdot,p) : [0,\infty) \to [0,\infty)\) such that, for all \(v_1,\dots,v_m \in {\cal X}\), \[{\mathbb{E}}\bigl\| \mathrm D^m X_t(x)[v_1,\dots,v_m] \bigr\|^p \;\le\; A_m(t,p) \prod_{j=1}^m \|v_j\|^p, \qquad t \ge 0.\] Moreover, there exist constants \(\mu_{m,p}>0\) and \(\lambda_{m,p}>0\) depending only on \(K,d,m,p\) such that \[\begin{align} A_1(t,p) &\le e^{\mu_{1,p} t}, \\ A_m(t,p) &\le \lambda_{m,p} \bigl( e^{\mu_{m,p} t } - 1 \bigr), \qquad m \ge 2. \end{align}\]
Proof. We proceed by induction on \(m\). The case \(m=1\) is given by Lemma 2, which yields \(A_1(t,p) \le e^{\mu_{1,p}t}\).
Let \(m \ge 2\) and suppose the claim holds for all orders \(1,\dots,m-1\). For brevity, set \[Z_t^{(m)} := \mathrm D^m X_t(x)[v_1,\dots,v_m].\] By Kunita’s theory and the tensorial Faà di Bruno formula (Proposition 7), \(Z_t^{(m)}\) satisfies a linear SDE whose drift and diffusion coefficients at time \(s\) are finite linear combinations of terms of the form \[\mathrm D^k b(X_s(x)) \Bigl[ \mathrm D^{|P_1|} X_s(x)[\vee_{i\in P_1} v_i],\dots, \mathrm D^{|P_k|} X_s(x)[\vee_{i\in P_k} v_i] \Bigr],\] and similarly with \(\sigma_l\) instead of \(b\), where \(1 \le k \le m\) and \(P=\{P_1,\dots,P_k\}\) runs over partitions \(P \in \mathcal{P}(m,k)\).
Using the boundedness of the derivatives of \(b\) and \(\sigma_l\) in Assumption 1, we have for each such term \[\Bigl\| \mathrm D^k \sigma_l(X_s(x)) \bigl[ \mathrm D^{|P_1|} X_s(x)[\vee_{i\in P_1} v_i],\dots, \mathrm D^{|P_k|} X_s(x)[\vee_{i\in P_k} v_i] \bigr] \Bigr\| \le K \prod_{j=1}^k \Bigl\| \mathrm D^{|P_j|} X_s(x)[\vee_{i\in P_j} v_i] \Bigr\|.\] In this expansion the single-block partition \(k=1\) produces the term \(\mathrm Db(X_s(x))\,Z_s^{(m)}\), and likewise \(\mathrm D\sigma_l(X_s(x))\,Z_s^{(m)}\) in the diffusion part; these are the only terms linear in \(Z_s^{(m)}\), they are bounded by Assumption 1, and we keep them on the homogeneous side. Every remaining partition has \(k\ge2\) blocks, and since \(\sum_j|P_j|=m\) each such block has size at most \(m-1\). Applying Itô’s formula to \(t \mapsto \|Z_t^{(m)}\|^p\), taking expectations and dropping martingale terms, we obtain an integral inequality of the form \[\frac{d}{dt} {\mathbb{E}}\bigl\|Z_t^{(m)}\bigr\|^p \le C_{m,p}\, {\mathbb{E}}\bigl\|Z_t^{(m)}\bigr\|^p + \sum_{k=2}^{m} \sum_{P \in \mathcal{P}(m,k)} {\mathbb{E}}\Bigl[ \prod_{j=1}^k \bigl\| \mathrm D^{|P_j|} X_t(x)[\vee_{i\in P_j} v_i] \bigr\|^p \Bigr],\] where \(C_{m,p}\) depends only on \(K,d,m,p\).
Fix a partition \(P=\{P_1,\dots,P_k\} \in \mathcal{P}(m,k)\) with \(2\le k \le m\). Applying Hölder’s inequality with exponents \((q_1,\dots,q_k)\) given by \(q_j := k\) to the product of \(k\) factors, \[{\mathbb{E}}\Bigl[ \prod_{j=1}^k \bigl\|\mathrm D^{|P_j|} X_t(x)[\vee_{i\in P_j}v_i]\bigr\|^p \Bigr] \le \prod_{j=1}^k \Bigl( {\mathbb{E}}\bigl\|\mathrm D^{|P_j|}X_t(x)[\vee_{i\in P_j}v_i]\bigr\|^{kp} \Bigr)^{1/k}.\] Since \(|P_j| \le m-1\) for every block, the induction hypothesis applied with exponent \(kp \ge 2\) gives \({\mathbb{E}}\|\mathrm D^{|P_j|}X_t(x)[\vee v_i]\|^{kp} \le A_{|P_j|}(t,kp)\prod_{i\in P_j}|v_i|^{kp}\), and therefore \[\frac{d}{dt} A_m(t,p) \le C_{m,p}\,A_m(t,p) + \tilde{C}_{m,p} \sum_{k=2}^{m}\sum_{P\in{\cal P}(m,k)} \prod_{j=1}^k A_{|P_j|}(t,kp)^{1/k},\] where \(\tilde{C}_{m,p}\) depends only on \(K,d,m,p\). Each \(A_l(t,kp)\) with \(1\le l\le m-1\) is, by the induction hypothesis, bounded by \(\lambda_{l,kp}(e^{\mu_{l,kp}t}-1)\) for \(l\ge2\) and by \(e^{\mu_{1,kp}t}\) for \(l=1\). Substituting these bounds and applying Grönwall’s lemma to the resulting linear inequality for \(A_m(t,p)\) yields \[A_m(t,p) \le \lambda_{m,p}\bigl(e^{\mu_{m,p} t}-1\bigr),\] for suitable constants \(\lambda_{m,p},\mu_{m,p} > 0\). This completes the induction. ◻
Lemma 3. Let \(m\in\mathbb{N}\), \(f \in D_m\), and let \(F_t\) be the Markov semigroup corresponding to the solution \(X_t(x)\) of 3 . Let Assumption 1 hold. Then there exist constants \(C_m,\bar\mu_m>0\), depending only on \(K,d,m\), such that \[\label{eq:semigroup-deriv-growth} \big\| \mathrm D^m F_t f(x) \big\| \;\le\; C_m e^{\bar \mu_m t} \, \| f \|_{m}, \qquad t \ge 0,\qquad{(13)}\] uniformly in \(x \in {\cal X}\). Strong continuity of \((F_t)_{t\ge0}\) in the norm \(\|\cdot\|_m\) on the space \(D_m\) defined in Section 2 is proved as Lemma 10 below.
Proof. Without loss of generality we may assume \(|v_j|=1\) for all \(j=1,\dots,m\). By definition and dominated convergence, \[\mathrm D^m F_t f(x)[v_1,\dots,v_m] = \mathrm D^m {\mathbb{E}}\bigl[f(X_t(x))\bigr][v_1,\dots,v_m].\] Applying the tensorial Faà di Bruno formula (Proposition 7) with \(g=f\) and \(f(\cdot)=X_t(\cdot)\) yields \[\begin{gather} \bigl|\mathrm D^m F_t f(x)[v_1,\dots,v_m]\bigr| \le \sum_{k=1}^m \sum_{P \in \mathcal{P}(m,k)} {\mathbb{E}}\Bigl[ \bigl\|\mathrm D^k f(X_t(x))\bigr\| \prod_{j=1}^k \bigl\| \mathrm D^{|P_j|} X_t(x)[\vee_{l\in P_j} v_l] \bigr\| \Bigr] \\ \le \|f\|_m \sum_{k=1}^m \sum_{P \in \mathcal{P}(m,k)} \prod_{j=1}^k \Bigl( {\mathbb{E}}\bigl\| \mathrm D^{|P_j|} X_t(x)[\vee_{l\in P_j} v_l] \bigr\|^{k} \Bigr)^{1/k}. \end{gather}\] Using Proposition 9 with \(p=k\) and \(|v_i|=1\), we obtain \[{\mathbb{E}}\bigl\| \mathrm D^{|P_j|} X_t(x)[\vee_{l\in P_j} v_l] \bigr\|^{k} \le A_{|P_j|}(t,k),\] so that \[\bigl|\mathrm D^m F_t f(x)[v_1,\dots,v_m]\bigr| \le \|f\|_m \sum_{k=1}^m \sum_{P \in \mathcal{P}(m,k)} \prod_{j=1}^k A_{|P_j|}(t,k)^{1/k}.\] By Proposition 9, \(A_1(t,k)\le e^{\mu_{1,k} t}\) and \(A_{l}(t,k)\le \lambda_{l,k}(e^{\mu_{l,k} t}-1)\) for \(l\ge2\). Each partition \(P\in\mathcal{P}(m,k)\) with \(k<m\) contains at least one block of size \(\ge2\), so we obtain an upper bound of the form \[\bigl|\mathrm D^m F_t f(x)[v_1,\dots,v_m]\bigr| \le C_m e^{\bar\mu_m t} \|f\|_m,\] for suitable constants \(C_m,\bar\mu_m>0\) depending only on \(K,d,m\). Taking the supremum over all unit vectors \(v_1,\dots,v_m\) and \(x\in{\cal X}\) yields ?? . ◻
The previous lemma gives growth bounds, but not a quasi-contraction in the natural norm. We use the equivalent renorming below.
Proposition 10. Let \((P_t)_{t\ge 0}\) be a strongly continuous semigroup on a Banach space \((X,\|\cdot\|)\). Assume there exist constants \(M\ge 1\) and \(\omega\in\mathbb{R}\) such that \[\label{eq:semigroup-bound} \|P_t\|_{X\to X} \;\le\; M e^{\omega t}, \qquad t\ge 0.\qquad{(14)}\] Then, for every \(\varepsilon>0\), there exists an equivalent norm \(\|\cdot\|_{*}\) on \(X\) such that \[\label{eq:quasi-contract} \|P_t x\|_{*} \;\le\; e^{(\omega+\varepsilon)t}\, \|x\|_{*}, \qquad x\in X,\; t\ge 0.\qquad{(15)}\] In particular, with respect to \(\|\cdot\|_{*}\), the semigroup \((P_t)_{t\ge0}\) is quasi-contractive.
Proof. Fix \(\varepsilon>0\) and define \[\|x\|_{*} := \sup_{t\ge 0} e^{-(\omega+\varepsilon)t}\, \|P_t x\|.\] \(\|\cdot\|_{*}\) is a norm. Moreover, \[\|x\|_{*} \;\ge\; \|P_0 x\| = \|x\|,\] and by ?? , \[\|x\|_{*} \le \sup_{t\ge0} e^{-(\omega+\varepsilon)t} M e^{\omega t} \|x\| = M \|x\|.\] Thus \(\|\cdot\|_{*}\) is equivalent to \(\|\cdot\|\).
For \(s\ge0\), \[\begin{align} \|P_s x\|_{*} &= \sup_{t\ge 0} e^{-(\omega+\varepsilon)t}\, \|P_t(P_s x)\| = \sup_{t\ge 0} e^{-(\omega+\varepsilon)t}\, \|P_{t+s} x\| \\ &= \sup_{r\ge s} e^{-(\omega+\varepsilon)(r-s)}\, e^{-(\omega+\varepsilon)s} \|P_r x\| \le e^{(\omega+\varepsilon)s}\, \sup_{r\ge 0} e^{-(\omega+\varepsilon)r}\|P_r x\| \\ &= e^{(\omega+\varepsilon)s}\, \|x\|_{*}, \end{align}\] which gives ?? . ◻
We need the following polynomial Lyapunov bound. For an integer \(q \ge 1\) define \[V_q(x) := 1 + |x|^{2q}, \qquad x \in {\mathbb{R}}^k,\] and the weighted norm \[\|f\|_{V_q} := \sup_{x \in {\mathbb{R}}^k} \frac{|f(x)|}{V_q(x)}.\]
Lemma 4. Let \(X_t^x\) solve 3 under Assumption 1, and let \(V_q(x)=1+|x|^{2q}\) with \(q\in\mathbb{N}\). Then there are constants \(C_q,D_q>0\), depending only on \(q,K,d\), such that \[\label{eq:LVk-bound} L V_q(x) \;\le\; C_q \, V_q(x), \qquad x \in {\cal X},\qquad{(16)}\] and \[\label{eq:L2Vk-bound} \bigl|L^2 V_q(x)\bigr| \;\le\; D_q \, V_q(x), \qquad x \in {\cal X}.\qquad{(17)}\] Consequently, for every measurable \(f\) with \(\|f\|_{V_q}<\infty\), \[\|F_t f\|_{V_q} \;\le\; e^{C_q t} \, \|f\|_{V_q}, \qquad t \ge 0.\] One can take \[C_q := (4qK + 6q^2 K^2)\, d, \qquad D_q := C_* \, q^4 K^4 \, d^2\] for a universal constant \(C_* > 0\) independent of \(q,K,d\), and \(k\).
Proof. We first estimate \(L V_q\). For \[V_q(x)=1+|x|^{2q},\] one has \[\nabla V_q(x)=2q\,|x|^{2q-2}x,\] and \[\mathrm D^2 V_q(x)[u,v] = 2q\,|x|^{2q-2}\langle u,v\rangle +4q(q-1)\,|x|^{2q-4}\langle x,u\rangle\langle x,v\rangle.\] Hence \[|\nabla V_q(x)| \le C_q(1+|x|^{2q-1}), \qquad \|\mathrm D^2 V_q(x)\| \le C_q(1+|x|^{2q-2}).\] Using the linear growth of \(b,\sigma\) gives \[L V_q(x) = \nabla V_q(x)[b(x)] + \frac{1}{2} \sum_{j=1}^d \mathrm D^2 V_q(x)[\sigma_j(x),\sigma_j(x)] \le C_q V_q(x),\] which is ?? . By Dynkin’s formula and Grönwall’s lemma, \[{\mathbb{E}}[V_q(X_t^x)] \le e^{C_q t}V_q(x),\] and therefore \[\|F_t f\|_{V_q} \le e^{C_q t}\|f\|_{V_q}.\]
To estimate \(L^2V_q\), write \[g(x):=LV_q(x).\] For every multiindex \(\alpha\) with \(|\alpha|\le 4\), \[|\partial^\alpha V_q(x)| \le C(q)\bigl(1+|x|^{2q-|\alpha|}\bigr).\] Because \(b,\sigma_j\) have bounded derivatives up to order \(m\ge4\) and at most linear growth, every derivative of \(g\) of order at most two is a finite sum of products involving derivatives of \(V_q\) of order at most four, bounded derivatives of \(b,\sigma_j\), and at most one undifferentiated factor of \(b\) or two undifferentiated factors of \(\sigma_j\). Consequently, differentiating lowers the polynomial degree exactly as expected, and there exist constants \(C_0,C_1,C_2>0\) such that \[\label{eq:g-derivative-bounds} |g(x)| \le C_0(1+|x|^{2q}), \qquad |\nabla g(x)| \le C_1(1+|x|^{2q-1}), \qquad \|\mathrm D^2 g(x)\| \le C_2(1+|x|^{2q-2}).\tag{5}\] Applying \(L\) once more gives \[L^2V_q(x) = \nabla g(x)[b(x)] + \frac{1}{2} \sum_{j=1}^d \mathrm D^2 g(x)[\sigma_j(x),\sigma_j(x)].\] Using 5 and the linear growth of \(b,\sigma\), \[\begin{align} |L^2V_q(x)| &\le |b(x)|\,|\nabla g(x)| + \frac{1}{2} \|\mathrm D^2 g(x)\| \sum_{j=1}^d |\sigma_j(x)|^2 \\ &\le K(1+|x|)\,C_1(1+|x|^{2q-1}) + \frac{1}{2} dK^2(1+|x|)^2\,C_2(1+|x|^{2q-2}) \\ &\le D_q(1+|x|^{2q}) = D_q V_q(x), \end{align}\] which proves ?? . Enlarging the universal constant \(C_*\) if needed, one may take \[D_q \le C_* q^4 K^4 d^2.\] ◻
Combining Lemma 3, Lemma 4, Lemma 10, and Proposition 10, we obtain the following.
Corollary 4. For each \(m\in\mathbb{N}\) there exists an equivalent norm \(\|\cdot\|_m^{*}\) on \(D_m\) such that \[\|F_t f\|_m^{*} \;\le\; e^{\mu_m t} \|f\|_m^{*}, \qquad t\ge0,\] where \(\mu_m \ge 0\) depends only on \(K,d,m\).
We record the weighted semigroup facts needed in Section 6. The point is to work on Banach spaces that are invariant under \(F_t\) and on which \(F_tf\to f\) strongly as \(t\downarrow0\). Under these conditions the generator is well defined on a dense domain; see [2].
For a weight \(V\ge1\) with \(V(x)\to\infty\) as \(|x|\to\infty\), put \[C_{V,\infty}({\cal X}) := \Bigl\{ f\in C({\cal X}): \|f\|_V:=\sup_{x\in{\cal X}}\frac{|f(x)|}{V(x)}<\infty,\quad f/V\in C_\infty({\cal X}) \Bigr\}.\]
We use the following weighted Feller consequence of [2].
Lemma 5. Let \((X_t^x)_{t\ge0}\) satisfy Assumption 1, and suppose that a twice continuously differentiable weight \(V\ge1\) satisfies \[V(x)\xrightarrow[|x|\to\infty]{}\infty, \qquad LV\le cV\] for some \(c\ge0\). Then \((F_t)_{t\ge0}\) is a bounded strongly continuous semigroup on \(C_{V,\infty}({\cal X})\) and \[\|F_t f\|_V\le e^{ct}\|f\|_V,\qquad \|F_t f-f\|_V\xrightarrow[t\downarrow0]{}0.\]
Proof. The Lyapunov bound and Dynkin’s formula give \({\mathbb{E}}V(X_t^x)\le e^{ct}V(x)\). The cited weighted Feller theorem then gives invariance of \(C_{V,\infty}({\cal X})\) and strong continuity in the weighted norm. ◻
The next lemma records the escape-to-infinity property of the diffusion that underlies the vanishing-at-infinity bookkeeping in Section 6. It is the only place where we use that the process, started far away, does not return to a fixed compact set within a bounded time.
Lemma 6. Let Assumption 1 hold. Then, for every compact \(K\subset{\cal X}\) and every \(T>0\), \[\sup_{0\le t\le T}P_x(X_t\in K)\xrightarrow[\;|x|\to\infty\;]{}0 .\]
Proof. Fix \(p>0\) and consider the bounded weight \(w_{-p}(x)=(1+|x|^2)^{-p/2}\), which vanishes at infinity. Writing \(\nabla w_{-p}(x)=-p(1+|x|^2)^{-p/2-1}x\) and \(\mathrm D^2 w_{-p}(x)=-p(1+|x|^2)^{-p/2-1}I+p(p+2)(1+|x|^2)^{-p/2-2}\,x\otimes x\), the linear growth \(|b(x)|,|\sigma_\ell(x)|\le K(1+|x|)\) together with \(|x|(1+|x|)\le C(1+|x|^2)\) gives, after collecting terms, a constant \(c_p=c_p(K,d,p)\) with \[L w_{-p}(x)\le c_p\,w_{-p}(x),\qquad x\in{\cal X}.\] Since \(w_{-p}\) is bounded with bounded derivatives, Dynkin’s formula and Grönwall’s lemma yield \({\mathbb{E}}w_{-p}(X_t(x))\le e^{c_pt}w_{-p}(x)\). On the compact set \(K\) the weight is bounded below, so \(\mathbf{1}_K\le(\inf_K w_{-p})^{-1}w_{-p}\), and therefore \[\sup_{0\le t\le T}P_x(X_t\in K) \le\frac{e^{c_pT}}{\inf_K w_{-p}}\,w_{-p}(x) \xrightarrow[\;|x|\to\infty\;]{}0 . \qedhere\] ◻
We use the following variant of [3].
Proposition 11. Let \(F_t = e^{tL}\) be a strongly continuous bounded semigroup on a Banach space \((B,\|\cdot\|_B)\): \[\max_{s \in [0,t]} \|F_s\|_{B \to B} \leq e^{\bar m t}.\]
Let \(L\) be the generator of \(F_t\), and let \(D \subset \operatorname{dom} L\) be another Banach space with \(\|\cdot\|_D \geq \|\cdot\|_B\) such that \[\|Lf\|_B \leq l \|f\|_D \, \forall f \in D.\] Assume also that \(F_t\) is bounded in \(D\): \[\max_{s \in [0,t]} \|F_s\|_{D \to D} \leq e^{m t},\] for some \(m \geq 0\).
Let \(U_h\) be a family of linear bounded operators in \(B\) with \[\| U_h \|_{B \to B} \leq e^{q h}, \qquad q\ge0,\] and let \[\begin{align} \label{eq:prop1aneq} \left\| \left( \frac{U_h - 1}{h} - L \right) f \right\|_B &\leq \epsilon_h \|f\|_D, \\ \label{eq:prop1aneq2} \left\| \left( \frac{F_h - 1}{h} - L \right) f \right\|_B &\leq \chi_h \|f\|_D, \end{align}\] {#eq: sublabel=eq:eq:prop1aneq,eq:eq:prop1aneq2} where \(\epsilon_h,\chi_h>0\) are monotone and tend to zero as \(h\to0\).
Then \[\label{eq:prop1461} \begin{align} \sup_{s \leq t} \left\| (U_h)^{\lfloor s/h\rfloor} f - F_s f \right\|_B &\leq (\chi_h + \epsilon_h) \|f\|_D \int_0^t e^{ms + q (t-s)} \, ds \\ &\quad + e^{\bar m t} h (l + \epsilon_h) \|f\|_D . \end{align}\qquad{(18)}\]
For the killed semigroups with rate \(c>0\), \[\label{eq:prop1462} \begin{align} \sup_{s \leq t} \left\| e^{-c\lfloor s/h\rfloor h } (U_h)^{\lfloor s/h\rfloor} f - e^{-cs} F_s f \right\|_B &\leq (\chi_h + \epsilon_h) \|f\|_D \int_0^t e^{(m-c)s + (q-c) (t-s)} \, ds \\ &\quad + e^{(\bar m - c) t} h (l + \epsilon_h + c) \|f\|_D . \end{align}\qquad{(19)}\]
Remark 6. Contrary to [3], we do not require \(D\) to be dense in \(B\).
Proof. The estimate ?? is equivalent to \[\left\| \frac{1}{h} \int_0^h (F_s - 1)L f\, ds \right\|_B \leq \chi_h \|f\|_D,\] because \[F_t f - f = t L f + \int_0^t (F_s - 1) L f\, ds.\]
By ?? and \[U_h - F_h = (U_h - 1) - \int_0^h F_s L\, ds = (U_h - 1) - h L - \int_0^h (F_s - 1) L\, ds,\] we have \[\| U_h f - F_h f \|_B \leq h (\epsilon_h + \chi_h) \|f\|_D.\]
Since \[U^{k}_h f - F_{kh} f = U^{k}_h f - F_h^k f,\] the telescoping expansion gives \[U^{k}_h f - F_h^k f = U^{k}_h f - U^{(k-1)}_h F_h f + U^{(k-1)}_h F_h f - U^{(k-2)}_h F_h^2 f + \dots + U_h F_h^{k-1} f - F_h^k f,\] and, since for \(a \geq 1\) and \(b \geq 0\) \[\begin{align} \| U^a_h F^b_h f - U^{a-1}_h F^{b+1}_h f \|_B &\leq \|U^{a-1}_h \|_{B \to B} \| (U_h - F_h) F^{b}_h f \|_B \\ &\leq \|U_h \|_{B \to B}^{a-1} h (\epsilon_h + \chi_h) \| F_h \|_{D \to D}^{b} \|f\|_D . \end{align}\] hence \[\| U^{k}_h f - F_h^k f \|_B \leq h(\epsilon_h + \chi_h) \|f\|_D (e^{(k-1)qh} + e^{(k-2) qh} e^{mh} + \dots + e^{(k-1)mh}),\] implying that \[\label{eq:prop1d} \max_{k \leq [t/h]} \| U^{k}_h f - F_h^k f \|_B \leq (\epsilon_h + \chi_h) \|f\|_D \int_0^t e^{ms + q(t-s)}ds.\tag{6}\]
For arbitrary times, \[\| U^{[t/h]}_h f - F_t f \|_B \leq \| U^{[t/h]}_h f - F_h^{[t/h]} f\|_B + \| F_{h[t/h]} f - F_t f \|_B.\] For \(\delta \leq h\), we have \[\label{eq:prop1c} \begin{align} \| F_\delta f - f \|_B &= \left\| Lf \delta + \int_0^\delta (F_s- 1) L f ds \right\|_B \\ &\leq \delta ( l + \chi_h) \|f\|_D . \end{align}\tag{7}\] With \(\delta = t - h[t/h]\), \[\begin{gather} \| F_{h[t/h]} f - F_t f \|_B = \| F_{h[t/h]} (f - F_{\delta} f)\|_B \leq \|F_{h[t/h]}\|_{B\to B}\, \|F_{\delta} f - f\|_B \leq e^{\bar m t} \delta ( l + \chi_h) \|f\|_D \\ \leq e^{\bar m t} h (l + \chi_h) \|f\|_D. \end{gather}\] This proves ?? . For ?? , set \[\hat{F}_t := e^{-c t} F_t, \qquad \hat{U}_h := e^{-c h} U_h.\] The semigroup \(\hat{F}_t\) has generator \(\hat{L}:=L-c\), and \[\| \hat{F}_t \|_{B \to B} = e^{-ct} \| F_t \|_{B \to B} \leq e^{(\bar m - c)t},\] and \[\| \hat{F}_t \|_{D \to D} = e^{-ct} \| F_t \|_{D \to D} \leq e^{( m - c)t}.\] Also, \[\| \hat{L} f \|_B \leq \| Lf\|_B + \|cf\|_B \leq (l + c) \|f\|_D.\]
For \(\delta \leq h\), we have \[\hat{F}_\delta f - f = e^{-c \delta} F_\delta f - f = e^{-c\delta} (F_\delta f - f) - (1 - e^{-c\delta}) f.\] By 7 and Bernoulli’s inequality, \[\|\hat{F}_\delta f - f \| \leq \delta ( l + c + \chi_h) \|f\|_D.\] Taking supremum for \(\delta \leq h\), we get \[\sup_{0 \leq \delta \leq h} \|\hat{F}_\delta f - f \| \leq h ( l + c + \chi_h) \|f\|_D.\] Combining this bound with 6 gives ?? . ◻
We apply Proposition 11 with \(V(x):=V_1(x)\), noting that \(V\le2W\). The preceding Lyapunov estimate gives an admissible base growth rate \(\bar m=4K+K^2\). We next estimate \(L\) and \(L^2\) in the weighted norms used below.
Lemma 7. Let Assumption 1 hold. Let \(L\) be the generator of the diffusion corresponding to 3 , acting on \(f\in D_2\) by \[Lf(x) = \nabla f(x)[b(x)] + \frac{1}{2} \sum_{l=1}^d \mathrm D^2 f(x)\bigl[\sigma_l(x),\sigma_l(x)\bigr].\] Then for a constant \(l = 2 K + d K^2 > 0\) and for all \(f \in D_2\), \[\label{eq:L-weighted-estimate} \|Lf\|_V \;\le\; l \,\|f\|_2.\qquad{(20)}\] Furthermore, if \(f \in D_4\), then there exists a constant \(l' = l'(K,d) >0\) such that \[\label{eq:L2-weighted-estimate} \|L^2 f\|_W \;\le\; l' \,\|f\|_4.\qquad{(21)}\]
Proof. The first bound follows directly from the definition of \(L\): \[|Lf(x)| \le \|\nabla f(x)\|\,|b(x)| + \frac{1}{2} \|\mathrm D^2 f(x)\| \sum_{l=1}^d |\sigma_l(x)|^2.\] Using \(|b(x)|\le K(1+|x|)\), \(|\sigma_l(x)|\le K(1+|x|)\), and \[1+|x| \le 1+|x|^2 = V(x),\qquad (1+|x|)^2 \le 2V(x),\] we obtain \[|Lf(x)| \le \Bigl(K + dK^2\Bigr)V(x)\|f\|_2 \le \bigl(2K+dK^2\bigr)V(x)\|f\|_2,\] which proves ?? .
For the second estimate, introduce the diffusion tensor \[a(x):=\sum_{l=1}^d \sigma_l(x)\vee \sigma_l(x).\] Then \[Lf = \nabla f[b] + \frac{1}{2} \mathrm D^2 f[a].\] By differentiating once and twice, we obtain for \(u,v\in{\cal X}\) \[\begin{align} \nabla(Lf)[u] &= \mathrm D^2 f[u,b] + \nabla f[\nabla b[u]] + \frac{1}{2} \mathrm D^3 f[u,a] + \frac{1}{2} \mathrm D^2 f[\nabla a[u]], \end{align}\] and \[\begin{align} \mathrm D^2(Lf)[u,v] &= \mathrm D^3 f[u,v,b] + \mathrm D^2 f[u,\nabla b[v]] + \mathrm D^2 f[v,\nabla b[u]] + \nabla f[\mathrm D^2 b[u,v]] \\ &\quad + \frac{1}{2} \mathrm D^4 f[u,v,a] + \frac{1}{2} \mathrm D^3 f[u,\nabla a[v]] + \frac{1}{2} \mathrm D^3 f[v,\nabla a[u]] + \frac{1}{2} \mathrm D^2 f[\mathrm D^2 a[u,v]]. \end{align}\] Hence \[L^2 f = \nabla(Lf)[b] + \frac{1}{2} \mathrm D^2(Lf)[a]\] is a finite sum of terms involving \(\mathrm D^r f\) for \(r=1,2,3,4\) applied to tensors built from \[b,\;a,\;\nabla b,\;\mathrm D^2 b,\;\nabla a,\;\mathrm D^2 a.\] Under Assumption 1, \[|b(x)|\le K(1+|x|),\qquad \|a(x)\|\le dK^2(1+|x|)^2,\] while \(\|\nabla b(x)\|+\|\mathrm D^2 b(x)\|\le K\) and \[\|\nabla a(x)\| \le 2dK^2(1+|x|),\qquad \|\mathrm D^2 a(x)\| \le 4dK^2(1+|x|).\] Therefore every term in \(L^2f(x)\) is bounded by \[C(K,d)\,(1+|x|)^m\,\|f\|_4, \qquad m\le 4.\] Summing all contributions and using \((1+|x|)^m\le C_m W(x)\) for \(m\le4\), we get \[|L^2f(x)| \le l'(K,d)\,W(x)\,\|f\|_4,\] which is ?? . ◻
Lemma 8. For every integer \(l \ge 0\), the space \(C_c^\infty({\cal X})\) is dense in \(D_l\) with respect to \(\|\cdot\|_l\).
Proof. Let \(f \in D_l\). Choose \(\chi \in C_c^\infty({\cal X})\) with \(0\le\chi\le1\), \(\chi=1\) on the unit ball and \(\chi=0\) outside the ball of radius \(2\), and set \(\chi_R(x) := \chi(x/R)\), \(f_R := \chi_R f\). By Leibniz’ rule, for \(0 \le j \le l\), \[\mathrm D^j(f - f_R) = (1-\chi_R)\mathrm D^j f + \sum_{\ell=1}^{j} C_{j,\ell}\,\mathrm D^\ell(1-\chi_R)\,\mathrm D^{j-\ell}f .\] The first term tends to zero in sup norm because \(\|\mathrm D^j f\| \in C_\infty({\cal X})\). For \(1 \le \ell \le j\), \(\mathrm D^\ell(1-\chi_R)\) is supported on \(\{R \le |x| \le 2R\}\) with \(\|\mathrm D^\ell\chi_R\|_\infty \le C_\ell R^{-\ell}\). When \(j - \ell \ge 1\), the cross term is bounded by \(C_\ell R^{-\ell}\sup_{|x|\ge R}\|\mathrm D^{j-\ell}f\| \to 0\) because \(\|\mathrm D^{j-\ell}f\| \in C_\infty({\cal X})\). When \(j = \ell \ge 1\), write \(x = \rho e\) with \(|e|=1\); by the fundamental theorem of calculus, \[\frac{|f(\rho e)|}{\rho} \le \frac{|f(0)|}{\rho} + \frac{1}{\rho}\int_0^\rho \|\nabla f(s e)\|\,ds.\] Since \(\|\nabla f\| \in C_\infty({\cal X})\), the right-hand side tends to zero uniformly in the direction \(e\), and therefore \(R^{-1}\sup_{|x|\ge R}|f(x)| \to 0\). For \(j = \ell \ge 2\) a similar estimate using \(f/W \in C_\infty\) and \(W(x) = 1+|x|^4\) gives \(R^{-j}\sup_{|x|\ge R}|f(x)| \to 0\). Hence \(f_R \to f\) in \(\|\cdot\|_l\).
For fixed \(R\), mollify \(f_R\) by a standard mollifier \(\rho_\varepsilon\). The functions \(f_{R,\varepsilon} := \rho_\varepsilon * f_R\) belong to \(C_c^\infty({\cal X})\) and converge to \(f_R\) in the usual \(C^l\) norm on \(\overline{B_{2R}}\), hence in \(\|\cdot\|_l\). This proves density. ◻
Lemma 9. Let Assumption 1 hold with \(m \ge l\), and let \(g \in C_c^\infty({\cal X})\). Then for every compact \(K \subset {\cal X}\) and every \(0 \le r \le l\), \[\sup_{x \in K}\bigl\|\mathrm D^r F_t g(x) - \mathrm D^r g(x)\bigr\| \to 0, \qquad t \downarrow 0.\]
Proof. The case \(r = 0\) follows from continuity of \(g\) and the uniform-on-compacts convergence \(X_t(x) \to x\) in probability for SDEs with linear-growth Lipschitz coefficients, combined with the moment bound \(\sup_{x \in K}{\mathbb{E}}|X_t(x)|^p < \infty\) (\(p\ge1\)) and Vitali’s theorem. Fix \(1 \le r \le l\), write \(v = (v_1,\dots,v_r)\) with \(|v_i| \le 1\), and decompose the Faà di Bruno representation \[\mathrm D^r F_t g(x)[v] = S_t(x)[v] + R_t(x)[v],\] with \[S_t(x)[v] := {\mathbb{E}}_x\bigl[\mathrm D^r g(X_t(x))[\nabla X_t(x)v_1,\dots,\nabla X_t(x)v_r]\bigr]\] the singleton-partition part and \(R_t(x)[v]\) the sum over partitions \(P \in {\cal P}(r,q)\), \(q < r\), each containing at least one block of size \(|P_i| \ge 2\).
For \(R_t\), every summand contains a higher jet \(\mathrm D^{|P_i|}X_t(x)\) with \(|P_i| \ge 2\). Hölder’s inequality with \(q+1\) factors combined with Proposition 9 (the \(A_{|P_i|}(t,\cdot)\) bounds vanish as \(t\downarrow 0\) with rate \((e^{ct}-1)^{1/p}\)) gives \[\sup_{x \in K} \|R_t(x)\| \le C_{g,K}\bigl(e^{ct}-1\bigr)^{1/p}\,e^{ct} \to 0, \qquad t \downarrow 0,\] for some \(p > 1\) depending on \(r\).
For the singleton part, expand \(\bigotimes_i(I + (\nabla X_t(x) - I))v_i\) and split \[S_t(x)[v] - \mathrm D^r g(x)[v] = \bigl({\mathbb{E}}_x[\mathrm D^r g(X_t(x))[v]] - \mathrm D^r g(x)[v]\bigr) + B_t(x)[v],\] where \(B_t(x)[v]\) collects the \(2^r - 1\) non-identity multilinear terms. The first difference tends to zero uniformly for \(x \in K\) by continuity of \(\mathrm D^r g\) and Vitali’s theorem as above. For \(B_t\), multilinearity and boundedness of \(\mathrm D^r g\) give \[|B_t(x)[v]| \le C_g\,{\mathbb{E}}_x\bigl[(1 + \|\nabla X_t(x)\|^{r-1})\,\|\nabla X_t(x) - I\|\bigr].\] The SDE ?? for \(\nabla X_t(x) - I\) starts from zero. The Burkholder–Davis–Gundy inequality, together with the linear-growth moment bound from Lemma 2, gives \(\sup_{x \in K}{\mathbb{E}}_x\|\nabla X_t(x) - I\|^p \le C_K t^{p/2}\) for \(0 < t \le 1\). Hence \(\sup_{x \in K}|B_t(x)[v]| \le C_{g,K} t^{1/2} \to 0\). ◻
Lemma 10. Let Assumption 1 hold with \(m \ge l\). Then \(F_t D_l \subset D_l\) for every \(t \ge 0\), and \((F_t)_{t\ge0}\) is strongly continuous on \(D_l\) in the norm \(\|\cdot\|_l\).
Proof. Step 1 (\(F_tD_l \subset D_l\)). Let \(f \in D_l\). Lemma 3 gives \(\sup_x\|\mathrm D^jF_t f(x)\| \le C_{j,t}\|f\|_j\) for \(1 \le j \le l\), so \(F_t f \in C^l({\cal X})\) with bounded derivatives. Since \(D_l \subset C_{W,\infty}({\cal X})\), the weighted Feller property (Lemma 5) gives \(F_t f / W \in C_\infty({\cal X})\).
To prove \(\|\mathrm D^j F_t f\| \in C_\infty({\cal X})\), choose \(g_n \in C_c^\infty({\cal X})\) with \(g_n \to f\) in \(\|\cdot\|_l\) (Lemma 8). By Lemma 3, \(\|\mathrm D^j F_t (g_n - f)\|_\infty \le C_{l,t}\|g_n - f\|_l \to 0\), so \(\mathrm D^j F_t f\) is a uniform limit of bounded continuous functions. For each \(g_n\), the Faà di Bruno representation \[\mathrm D^j F_t g_n(x) = {\mathbb{E}}\sum_{q=1}^j\sum_{P\in{\cal P}(j,q)} \mathrm D^q g_n(X_t(x)) \bigl[\mathrm D^{|P_1|}X_t(x),\dots,\mathrm D^{|P_q|}X_t(x)\bigr]\] is supported, via \(\mathrm D^q g_n(X_t(x))\), on \(\{X_t(x) \in \mathrm{supp}\,g_n\}\). By Lemma 6, for any compact \(K \subset {\cal X}\), \[\sup_{0\le t\le1}P_x(X_t \in K) \to 0 \qquad \text{as } |x|\to\infty .\] Applying Hölder’s inequality and combining with the moment estimates of Proposition 9 yields \(\|\mathrm D^j F_t g_n(x)\| \to 0\) as \(|x| \to \infty\). Hence \(\|\mathrm D^j F_t g_n\| \in C_\infty({\cal X})\), and the uniform limit \(\|\mathrm D^j F_t f\|\) also belongs to \(C_\infty({\cal X})\). Therefore \(F_t f \in D_l\).
Step 2 (strong continuity on \(C_c^\infty\)). Fix \(g \in C_c^\infty({\cal X})\). By Lemma 9, for every compact \(K \subset {\cal X}\) and every \(0 \le j \le l\), \[\sup_{x \in K}\|\mathrm D^j F_t g(x) - \mathrm D^j g(x)\| \to 0 \quad\text{as }t \downarrow 0.\] For tail control, fix \(\varepsilon > 0\), choose \(K = \overline{B_M}\) with \(M\) large enough that \(\|\mathrm D^j g\| < \varepsilon\) outside \(K\), and apply the Faà di Bruno representation above: each summand of \(\mathrm D^j F_t g(x)\) is bounded by \(\|g\|_{C^l} \cdot {\mathbb{E}}_x\bigl[\mathbf{1}_{\{X_t \in \mathrm{supp}\,g\}} \cdot \prod_i \|\mathrm D^{|P_i|}X_t(x)\|\bigr]\). Hölder’s inequality and the escape estimate of Lemma 6, \(\sup_{t\in[0,1]}P_x(X_t \in \mathrm{supp}\,g) \to 0\) as \(|x|\to\infty\), give \(\sup_{|x|\ge M_1}\|\mathrm D^j F_t g(x)\| < \varepsilon\) for \(M_1\) sufficiently large, uniformly in \(t \in [0,1]\). Combining the compact and tail estimates yields \(\|\mathrm D^j(F_t g - g)\|_\infty \to 0\). The base-point term \(|F_t g(x_*) - g(x_*)|\to 0\) by continuity at \(x_*\) and bounded convergence. Hence \(\|F_t g - g\|_l \to 0\).
Step 3 (strong continuity on \(D_l\)). Let \(f \in D_l\) and choose \(g_n \in C_c^\infty({\cal X})\) with \(g_n \to f\) in \(\|\cdot\|_l\). By the triangle inequality, \[\|F_t f - f\|_l \le \|F_t(f - g_n)\|_l + \|F_t g_n - g_n\|_l + \|g_n - f\|_l .\] Lemma 3 bounds the first summand by \(C_{l}(1 + e^{\bar\mu_l t})\|f - g_n\|_l\), uniformly for \(t \in [0,1]\). Step 2 sends the middle summand to zero as \(t\downarrow 0\) for each fixed \(n\). Taking \(n\) large first and then \(t \downarrow 0\) proves \(\|F_t f - f\|_l \to 0\). ◻
Lemma 11. Let Assumption 1 hold with \(m\ge4\). Then \((F_t)_{t\ge0}\) is a bounded strongly continuous semigroup on \(C_{W,\infty}({\cal X})\), with \[\|F_t g\|_W\le e^{\bar\mu_Wt}\|g\|_W.\] Moreover \(F_tD_4\subset D_4\), and \((F_t)_{t\ge0}\) is strongly continuous on \(D_4\) in the norm \(\|\cdot\|_4\).
Proof. The Lyapunov estimate for \(W=1+|x|^4\) follows from Lemma 4 with \(q=2\), and the weighted Feller argument recalled in Lemma 5 gives bounded strong continuity on \(C_{W,\infty}({\cal X})\). The \(D_4\)-invariance and strong continuity in \(\|\cdot\|_4\) are Lemma 10 with \(l = 4\). ◻
Lemma 12. Let Assumption 1 hold with \(m\ge4\). For every \(f\in D_4\), \[f\in\operatorname{Dom}_{C_{W,\infty}}(L), \qquad Lf\in\operatorname{Dom}_{C_{W,\infty}}(L),\] and \[\|Lf\|_W+\|L^2f\|_W\le C(K,d)\|f\|_4.\] Consequently, \[F_t f-f=\int_0^tF_sLf\,ds, \qquad F_tLf-Lf=\int_0^tF_sL^2f\,ds\] as identities in \(C_{W,\infty}({\cal X})\).
Proof. Lemma 7 gives \(Lf,L^2f\in C_{W,\infty}({\cal X})\) and the displayed norm estimate. To identify the generator in \(C_{W,\infty}\), stop the diffusion at \(\tau_R=\inf\{t:|X_t|\ge R\}\). For the stopped diffusion, Itô’s formula gives \[{\mathbb{E}}f(X_{t\wedge\tau_R})-f(x) = {\mathbb{E}}\int_0^{t\wedge\tau_R}Lf(X_s)\,ds.\] The estimate ?? , the Lyapunov bound for \(W\), and localization let \(R\to\infty\) and yield \[F_t f-f=\int_0^tF_sLf\,ds \quad\text{in }C_{W,\infty}({\cal X}).\] Hence \(f\in\operatorname{Dom}_{C_{W,\infty}}(L)\). Repeating the same argument with \(Lf\), whose pointwise generator is \(L^2f\), gives \(Lf\in\operatorname{Dom}_{C_{W,\infty}}(L)\) and \[F_tLf-Lf=\int_0^tF_sL^2f\,ds.\] The displayed bound on \(\|Lf\|_W+\|L^2f\|_W\) is precisely Lemma 7. ◻
In the following lemma, we prove an estimate for the continuous semigroup.
Lemma 13. Let Assumption 1 hold. Then for all \(0<h \le h_0\), \[\left\| Lf - \frac{F_h f - f}{h} \right\|_W \;\le\; C\, h\, \|f\|_{4},\] where \(C\) depends only on the constants in Assumption 1.
Proof. Lemma 12 gives \(f,Lf\in \operatorname{Dom}_{C_{W,\infty}}(L)\) and \(\|L^2f\|_W\le C\|f\|_4\). We use the Dynkin formula in the Banach space \(C_{W,\infty}({\cal X})\): \[F_h f - f = \int_0^h L F_s f ds.\] Applying this formula twice, we obtain \[F_h f - f = hLf + \int_0^h\!\!\int_0^s F_r L^2 f \, dr\, ds.\] Dividing by \(h\), \[\frac{F_h f - f}{h} - Lf = \frac{1}{h} \int_0^h\!\!\int_0^s F_r L^2 f\,dr\,ds = \int_0^h \Bigl(1-\frac{r}{h}\Bigr) F_r L^2 f\, dr.\]
Since \(F_r\) is quasi-contractive in \(\|\cdot\|_W\), thanks to Lemma 4, \[\|F_r g\|_W \le e^{\bar \mu r}\|g\|_W \le e\,\|g\|_W,\qquad 0\le r\le \bar \mu^{-1}.\] Thus, thanks to ?? , for all \(h \leq h_0 = \bar \mu^{-1}\), \[\left\| \frac{F_h f - f}{h} - Lf \right\|_W \le \int_0^h e\,\|L^2 f\|_W\, dr = e\,h\,\|L^2 f\|_W \leq e\,h\,l' \, \|f\|_4,\] which completes the proof. ◻
The next lemma computes the error of a random walk scheme.
Lemma 14. Let Assumption 1 hold. Consider a random walk with a one-step transition operator \(U_h\) satisfying Assumption 3.
Then there exists a constant \(C = C(K,d,M_4) > 0\) such that for all \(f\in D_4\) and all \(0<h\le 1\), \[\left\| \left( \frac{U_h - I}{h} - L \right) f \right\|_W \;\le\; C\, h\, \|f\|_4,\] The constant depends only on \(K,d\) and \(M_4\).
Proof. Fix \(x\in{\cal X}\) and abbreviate \[B := b(x), \qquad S_l := \sigma_l(x),\qquad \eta := \eta_h(x).\] We write \[\eta = hB + \sqrt{h}\,Y, \qquad Y := \sum_{j=1}^d \sigma_j(x) \xi_j.\] We have \({\mathbb{E}}[Y]=0\), and by the third–moment assumption \[{\mathbb{E}}[Y^{\wedge 3}] = 0.\]
The tensor Taylor expansion with remainder gives \[\begin{align} f(x+\eta) &= f(x) + \nabla f(x)[\eta] + \frac{1}{2} \mathrm D^2 f(x)[\eta,\eta] + \frac{1}{6}\mathrm D^3 f(x)[\eta,\eta,\eta] \\ &\qquad + \frac{1}{24}\mathrm D^4 f(x+\theta\eta)[\eta,\eta,\eta,\eta], \end{align}\] for some (random) \(\theta\in(0,1)\). Let us take expectations of these values from the Taylor extension. Since \({\mathbb{E}}[\xi]=0\) and \({\mathbb{E}}[\xi_l\xi_m]=\delta_{lm}\), \[{\mathbb{E}}[\eta] = hB, \qquad {\mathbb{E}}[\eta\wedge\eta] = h\sum_{l=1}^d S_l\wedge S_l + h^2 B\wedge B.\] Hence \[\begin{align} {\mathbb{E}}[f(x+\eta)] &= f(x) + h \nabla f(x)[B] + \frac{h}{2} \sum_{l=1}^d \mathrm D^2 f(x)[S_l,S_l] \\ &\qquad + \frac{h^2}{2}\,\mathrm D^2 f(x)[B,B] + R_3(x) + R_4(x), \end{align}\] where \[R_3(x) := \frac{1}{6}{\mathbb{E}}[\mathrm D^3 f(x)[\eta,\eta,\eta]],\qquad R_4(x) := \frac{1}{24}{\mathbb{E}}[\mathrm D^4 f(x+\theta\eta)[\eta,\eta,\eta,\eta]].\] The generator is \[Lf(x) = \nabla f(x)[B] + \frac{1}{2}\sum_{l=1}^d \mathrm D^2 f(x)[S_l,S_l].\] Thus \[(U_h f)(x) - f(x) - hLf(x) = \frac{h^2}{2}\mathrm D^2 f(x)[B,B] + R_3(x) + R_4(x).\] Using \(\|B\|\le K(1+\|x\|)\), \[\left|\frac{h^2}{2}\mathrm D^2 f(x)[B,B]\right| \le h^2 \|f\|_4\, K^2(1+\|x\|)^2.\] Dividing by \(hW(x)\) and using \((1+\|x\|)^2 \le 2(1+\|x\|^4)=2W(x)\), \[\frac{1}{W(x)} \left| \frac{1}{h}\cdot \frac{h^2}{2}\mathrm D^2f(x)[B,B] \right| \le 2K^2 h\,\|f\|_4.\] Since \(\mathrm D^3 f(x)\) is a symmetric tensor, \[R_3(x) = \frac{1}{6}\mathrm D^3 f(x)\big[{\mathbb{E}}(\eta^{\wedge 3})\big].\] We compute \({\mathbb{E}}(\eta^{\wedge 3})\) explicitly. With \(\eta = hB + \sqrt{h}Y\), \[\eta^{\wedge 3} = (hB + \sqrt{h}Y)^{\wedge 3} = h^3 B^{\wedge 3} + 3h^2\sqrt{h}\, B^{\wedge 2}\wedge Y + 3h(\sqrt{h})^2 B\wedge Y^{\wedge 2} + (\sqrt{h})^3 Y^{\wedge 3}.\] Taking expectations and using \({\mathbb{E}}[Y]=0\) and \({\mathbb{E}}[Y^{\wedge 3}]=0\), \[{\mathbb{E}}(\eta^{\wedge 3}) = h^3 B^{\wedge 3} + 3h^2\, B\wedge {\mathbb{E}}(Y^{\wedge 2}).\] Since \({\mathbb{E}}[\xi_i\xi_j]=\delta_{ij}\), \({\mathbb{E}}(Y^{\wedge 2}) = \sum_{l=1}^d \sigma_l(x)\vee\sigma_l(x)\) is deterministic, with norm bounded by \[\|{\mathbb{E}}(Y^{\wedge 2})\| \le \sum_{l=1}^d \|\sigma_l(x)\|^2 \le d K^2(1+\|x\|)^2,\] and \(\|B\|\le K(1+\|x\|)\), so \[\|{\mathbb{E}}(\eta^{\wedge 3})\| \le (1 + 3d) K^3 h^2 (1+\|x\|)^3.\] Since \((1+\|x\|)^3 \le 8(1+\|x\|^4)=8 W(x)\), we get \[\|{\mathbb{E}}(\eta^{\wedge 3})\| \le 8 (1 + 3d) K^3 h^2 W(x).\] Thus, \[\frac{|R_3(x)|}{hW(x)} \le \frac{1}{6}\|\mathrm D^3 f(x)\|\,\|{\mathbb{E}}(\eta^{\wedge 3})\| \le \frac{4}{3} (1 + 3d) K^3 h\,\|f\|_4.\] The fourth-order term \[|R_4(x)| \le \frac{1}{24}\|f\|_4\, {\mathbb{E}}\|\eta\|^4.\] Using the vector inequality \(\|u+v\|^4 \le 8(\|u\|^4 + \|v\|^4)\), \[{\mathbb{E}}\|\eta\|^4 \le 8h^4\|B\|^4 + 8h^2\,{\mathbb{E}}\|Y\|^4.\]
For the drift part, we obtain \[8h^4\|B\|^4 \le 8h^4 K^4(1+\|x\|)^4 \le 64K^4 W(x) h^4.\]
Now we proceed with the diffusion part. Thanks to the Cauchy–Schwarz inequality, \[\left( \sum_{j=1}^d a_j \right)^2 \leq d \sum_{j=1}^d a_j^2.\] Therefore, \[{\mathbb{E}}\|Y\|^4 \le d^3 \sum_{j=1}^d |\sigma_j(x)|^4 |\xi_j|^4 \le 2 d^4 M_4 K^4 (1+|x|^4).\] Combining the two contributions gives (recall that \(h \leq 1\)) \[{\mathbb{E}}\|\eta\|^4 \le C K^4\,(1 + d^4 M_4)\, W(x)\, h^2,\]
Collecting the three contributions, we obtain for all \(x\in{\cal X}\) and \(0<h\le1\), \[\begin{align} \frac{1}{W(x)} \left| \left( \frac{U_h-I}{h}-L \right)f(x) \right| &\le \Big( 2K^2 + \frac{4}{3}(1 + 3d) K^3 + C K^4(1+d^4M_4) \Big) h\,\|f\|_4 \\ &\le C(K,d,M_4)\, h\,\|f\|_4, \end{align}\] which finishes the proof. ◻
The same one-step computation gives the base-space stability of \(U_h\) on \(C_W({\cal X})\), which is the second hypothesis of Proposition 11.
Lemma 15. Let Assumptions 1 and 3 hold. Then there is a constant \(q_W=q_W(K,d,M_4)>0\) such that, for all \(0<h\le1\), \[{\mathbb{E}}\,W\bigl(x+\eta_h(x)\bigr)\le e^{q_Wh}\,W(x), \qquad x\in{\cal X},\] and consequently \(\|U_hg\|_W\le e^{q_Wh}\|g\|_W\) for every \(g\in C_W({\cal X})\).
Proof. Write \(z:=x+hb(x)\) and \(Y:=\sum_{\ell=1}^d\sigma_\ell(x)\xi_\ell\), so that \(x+\eta_h(x)=z+\sqrt h\,Y\) with \({\mathbb{E}}Y=0\). Expanding \(|z+\sqrt h\,Y|^4\) and using \({\mathbb{E}}Y=0\), the term of order \(h^{1/2}\) vanishes, while the terms of order \(h^{3/2}\) and higher are bounded, for \(0<h\le1\), by \(C\,h\,W(x)\), through \({\mathbb{E}}|Y|^2\le dK^2(1+|x|)^2\) and \({\mathbb{E}}|Y|^4\le C(d,M_4)K^4(1+|x|)^4\) together with \(|z|\le(1+hK)(1+|x|)\). Hence \({\mathbb{E}}|x+\eta_h(x)|^4\le|x+hb(x)|^4+C\,h\,W(x)\), and the linear growth of \(b\) gives \(|x+hb(x)|^4\le|x|^4+C\,h\,W(x)\). Therefore \({\mathbb{E}}W(x+\eta_h(x))\le W(x)+C\,h\,W(x)\le e^{q_Wh}W(x)\) with \(q_W:=C\). The bound on \(\|U_hg\|_W\) follows from \(|U_hg(x)|\le\|g\|_W\,{\mathbb{E}}W(x+\eta_h(x))\). ◻
We next record the estimates in the weighted regular spaces \({\cal C}^r_{\alpha,\infty}({\cal X})\).
Lemma 16. Let \(\alpha,\beta\in{\mathbb{R}}\) and \(k\in{\mathbb{N}}_0\).
If \(u\in{\cal C}^k_{\alpha,\infty}({\cal X})\) and \(v\in{\cal C}^k_{\beta,\infty}({\cal X})\), then \(uv\in{\cal C}^k_{\alpha+\beta,\infty}({\cal X})\) and \[\|uv\|_{\alpha+\beta,k} \le C_{\alpha,\beta,k}\|u\|_{\alpha,k}\|v\|_{\beta,k}.\] The same estimate holds for contractions of tensor-valued functions.
Let \[\eta_h(x)=h b(x)+\sqrt h\sum_{\ell=1}^d\sigma_\ell(x)\xi_\ell\] and let Assumptions 2 and 4 hold. Then, for \(0<h\le h_0\) and \(0\le\theta\le1\), \[C^{-1}w_\gamma(x)\le w_\gamma(x+\theta\eta_h(x)) \le Cw_\gamma(x), \qquad \gamma\in{\mathbb{R}},\] and, for \(1\le j\le k+4\), \[\|\mathrm D^j\eta_h(x)\| \le C\sqrt h\,w_{1-j}(x).\] Moreover, if \(\varphi_h^\theta(x):=x+\theta\eta_h(x)\), then \[\|\mathrm D\varphi_h^\theta(x)\|\le C, \qquad \|\mathrm D^j\varphi_h^\theta(x)\| \le C\sqrt h\,w_{1-j}(x), \quad j\ge2.\]
If \(g\in{\cal C}^k_{\alpha,\infty}({\cal X})\), then \(x\mapsto {\mathbb{E}}g(x+\eta_h(x))\) belongs to \({\cal C}^k_{\alpha,\infty}({\cal X})\).
Proof. The product estimate follows from Leibniz’ rule. For \(0\le j\le k\), every term of \(\mathrm D^j(uv)\) is a contraction of \(\mathrm D^\ell u\) with \(\mathrm D^{j-\ell}v\), and \[w_{\alpha-\ell}(x)\,w_{\beta-(j-\ell)}(x) =w_{\alpha+\beta-j}(x).\] Dividing by \(w_{\alpha+\beta-j}\) and taking the supremum gives the estimate. The same calculation applies to any fixed tensor contraction because the contraction norm is bounded by the product of the participating tensor norms. The vanishing-at-infinity condition is preserved since each weighted derivative ratio is a product of bounded ratios with at least one factor which vanishes at infinity.
For the comparability estimate, boundedness of \(\xi\) and the linear growth of the coefficients give \[|\eta_h(x)| \le C\sqrt h\,(1+|x|), \qquad 0<h\le1.\] Choose \(h_0\) so that \(C\sqrt h_0\le1/2\). Then \(1+|x+\theta\eta_h(x)|\) and \(1+|x|\) are comparable uniformly in \(\theta\in[0,1]\) and in the bounded random variable \(\xi\). Raising the resulting two-sided estimate to the power \(\gamma\) gives the displayed weight comparison for positive and negative \(\gamma\). Differentiating \(\eta_h\) gives \[\mathrm D^j\eta_h(x) = h\,\mathrm D^j b(x) +\sqrt h\sum_{\ell=1}^d \mathrm D^j\sigma_\ell(x)\xi_\ell ,\] and Assumption 2 yields \(\|\mathrm D^j\eta_h(x)\|\le C\sqrt h\,w_{1-j}(x)\) for \(h\le1\). The estimates for \(\varphi_h^\theta\) follow because \(\mathrm D\varphi_h^\theta=I+\theta\mathrm D\eta_h\) and \(\mathrm D^j\varphi_h^\theta=\theta\mathrm D^j\eta_h\) for \(j\ge2\).
Finally put \(\varphi_h(x)=x+\eta_h(x)\). By Proposition 7, for \(0\le j\le k\), \(\mathrm D^j(g\circ\varphi_h)(x)\) is a finite sum over partitions \(P=\{P_1,\dots,P_q\}\in{\cal P}(j,q)\) of contractions \[\mathrm D^q g(\varphi_h(x)) \big[ \mathrm D^{|P_1|}\varphi_h(x),\dots, \mathrm D^{|P_q|}\varphi_h(x) \big].\] The \(g\)-factor is bounded by \(\|g\|_{\alpha,k}w_{\alpha-q}(\varphi_h(x))\), hence by \(C\|g\|_{\alpha,k}w_{\alpha-q}(x)\). Each singleton block contributes a bounded first derivative of \(\varphi_h\), and each block of size \(m\ge2\) contributes \(C\sqrt h\,w_{1-m}(x)\). Since \(\sum_i|P_i|=j\), the product of the weights is \(w_{\alpha-q}(x)\prod_iw_{1-|P_i|}(x)=w_{\alpha-j}(x)\), with singleton factors equal to \(w_0=1\). Thus the weighted derivative ratios are uniformly bounded. Taking expectations preserves the same bounds.
To prove vanishing at infinity, fix \(j\) and a partition term. The ratio of the corresponding term to \(w_{\alpha-j}(x)\) is bounded by a constant times \[\frac{\|\mathrm D^q g(\varphi_h(x))\|}{w_{\alpha-q}(\varphi_h(x))}\] multiplied by bounded coefficient ratios. Since \(|\varphi_h(x)|\to\infty\) uniformly in the bounded increment variable as \(|x|\to\infty\), and the displayed ratio for \(g\) belongs to \(C_\infty({\cal X})\), every partition term vanishes at infinity. The expectation is over a bounded random variable and therefore preserves this limit by dominated convergence. ◻
Lemma 17. For every \(\alpha\in{\mathbb{R}}\) and \(r\in{\mathbb{N}}_0\), \(C_c^\infty({\cal X})\) is dense in \({\cal C}^r_{\alpha,\infty}({\cal X})\) with respect to \(\|\cdot\|_{\alpha,r}\). In particular, \({\cal C}^m_{\alpha,\infty}({\cal X})\) is dense in \({\cal C}^r_{\alpha,\infty}({\cal X})\) whenever \(m\ge r\).
Proof. Let \(f\in{\cal C}^r_{\alpha,\infty}({\cal X})\) and choose \(\chi\in C_c^\infty({\cal X})\) with \(0\le\chi\le1\), \(\chi=1\) on \(B_1\), and \(\chi=0\) outside \(B_2\). Put \(\chi_R(x)=\chi(x/R)\) and \(f_R=\chi_Rf\). For \(0\le j\le r\), Leibniz’ rule gives \[\mathrm D^j(f-f_R) = (1-\chi_R)\mathrm D^j f + \sum_{\ell=1}^j C_{j,\ell}\,\mathrm D^\ell(1-\chi_R)\,\mathrm D^{j-\ell}f .\] The first term tends to zero in the weighted norm because \(\mathrm D^j f/w_{\alpha-j}\) vanishes at infinity. The remaining terms are supported in \(\{R\le |x|\le2R\}\) and satisfy \(\|\mathrm D^\ell\chi_R(x)\|\le C_\ell R^{-\ell}\le C_\ell w_{-\ell}(x)\) there. Since \(w_{-\ell}w_{\alpha-(j-\ell)}=w_{\alpha-j}\), these terms also tend to zero uniformly after division by \(w_{\alpha-j}\). Hence \(\|f_R-f\|_{\alpha,r}\to0\).
For fixed \(R\), mollify \(f_R\) by a standard mollifier \(\rho_\varepsilon\). The functions \(f_{R,\varepsilon}:=\rho_\varepsilon*f_R\) belong to \(C_c^\infty({\cal X})\) and converge to \(f_R\) in the ordinary \(C^r\) norm on a fixed compact set. On that compact set the weighted norm is equivalent to the usual \(C^r\) norm, so \(\|f_{R,\varepsilon}-f_R\|_{\alpha,r}\to0\). This proves density. ◻
Lemma 18. Fix \(\alpha\in{\mathbb{R}}\) and \(r\in{\mathbb{N}}_0\), and assume Assumption 2. Then there exists \(C_{\alpha,r}>0\) such that, for every \(f\in{\cal C}^{r+4}_{\alpha,\infty}({\cal X})\), \[\|Lf\|_{\alpha,r+2} + \|L^2f\|_{\alpha,r} \le C_{\alpha,r}\|f\|_{\alpha,r+4}.\] In particular, \[L:{\cal C}^{r+2}_{\alpha,\infty}({\cal X}) \to {\cal C}^r_{\alpha,\infty}({\cal X})\] is a bounded operator.
Proof. We write \[Lf=\mathrm Df[b] +\frac{1}{2}\sum_{\ell=1}^d\mathrm D^2f[\sigma_\ell,\sigma_\ell].\] Assumption 2 gives \(\|\mathrm D^j b(x)\|\le Cw_{1-j}(x)\). By Leibniz’ rule, the derivatives of the tensor field \(a:=\sum_{\ell=1}^d\sigma_\ell\otimes\sigma_\ell\) satisfy \[\|\mathrm D^j a(x)\|\le C w_{2-j}(x), \qquad 0\le j\le r+4.\] For \(0\le q\le r+2\), every term in \(\mathrm D^q(\mathrm Df[b])\) is bounded by a product of the form \[\|\mathrm D^{q-\ell+1}f(x)\|\, \|\mathrm D^\ell b(x)\| \le C\|f\|_{\alpha,r+4} w_{\alpha-(q-\ell+1)}(x)w_{1-\ell}(x) = C\|f\|_{\alpha,r+4}w_{\alpha-q}(x).\] The diffusion term is identical, using \(w_{\alpha-(q-\ell+2)}w_{2-\ell}=w_{\alpha-q}\). This proves \(\|Lf\|_{\alpha,r+2}\le C\|f\|_{\alpha,r+4}\). The same derivative expansion proves the vanishing-at-infinity condition: after division by \(w_{\alpha-q}\) each term contains a weighted derivative ratio of \(f\), evaluated at \(x\), multiplied by bounded coefficient ratios. Thus \(Lf\in{\cal C}^{r+2}_{\alpha,\infty}\). Applying the already proved bound with \(Lf\) in place of \(f\) gives \(L^2f\in{\cal C}^r_{\alpha,\infty}\) and \(\|L^2f\|_{\alpha,r}\le C\|Lf\|_{\alpha,r+2}\le C\|f\|_{\alpha,r+4}\). ◻
Lemma 19. Fix \(\alpha\in{\mathbb{R}}\) and \(r\in{\mathbb{N}}_0\), and assume Assumption 2. The diffusion semigroup \((F_t)_{t\ge0}\) leaves \({\cal C}^r_{\alpha,\infty}({\cal X})\) invariant and is strongly continuous on this space. If \(A_{\alpha,r}\) denotes its generator on \({\cal C}^r_{\alpha,\infty}({\cal X})\), then \[{\cal C}^{r+2}_{\alpha,\infty}({\cal X}) \subset \operatorname{Dom}(A_{\alpha,r}), \qquad A_{\alpha,r}f=Lf.\] The space \({\cal C}^{r+2}_{\alpha,\infty}({\cal X})\) is dense in \({\cal C}^r_{\alpha,\infty}({\cal X})\) and is an invariant core for \(A_{\alpha,r}\). Moreover, for each \(T>0\) there is \(M_{\alpha,r,T}\) such that \[\sup_{0\le t\le T}\|F_tg\|_{\alpha,r} \le M_{\alpha,r,T}\|g\|_{\alpha,r}.\]
Proof. We first record the weighted flow estimates used below. The Lyapunov calculation for the weights \(w_\gamma\) gives, for every \(\gamma\in{\mathbb{R}}\), \(p\ge1\), and \(T>0\), \[\sup_{0\le t\le T}{\mathbb{E}}w_\gamma(X_t(x))^p \le C_{\gamma,p,T}w_\gamma(x)^p .\] Kunita’s equations for the flow jets, combined with \(\|\mathrm D^q\sigma_\ell(y)\|\le Cw_{1-q}(y)\), yield by induction \[\sup_{0\le t\le T}{\mathbb{E}}\|\mathrm DX_t(x)\|^p\le C_{p,T}, \qquad \sup_{0\le t\le T}{\mathbb{E}}\|\mathrm D^jX_t(x)\|^p \le C_{j,p,T}w_{1-j}(x)^p,\quad j\ge2.\] Indeed, in the \(j\)th jet equation each forcing term associated with a partition \(P=\{P_1,\dots,P_q\}\) has weight \[w_{1-q}(X_s)\prod_{i=1}^q w_{1-|P_i|}(X_s),\] and after Hölder’s inequality, the Lyapunov estimate, and the induction hypothesis this becomes \(w_{1-j}(x)\) because \(\sum_i|P_i|=j\). Grönwall’s lemma then gives the displayed bound.
Let \(g\in{\cal C}^r_{\alpha,\infty}({\cal X})\). The derivative representation for \(F_tg\) follows from Kunita’s flow theorem and Proposition 7. For \(1\le j\le r\), \[\mathrm D^jF_tg(x) = {\mathbb{E}}\sum_{q=1}^{j}\sum_{P\in{\cal P}(j,q)} \mathrm D^q g(X_t(x)) \big[ \mathrm D^{|P_1|}X_t(x),\dots, \mathrm D^{|P_q|}X_t(x) \big],\] while \(F_tg(x)={\mathbb{E}}g(X_t(x))\) for \(j=0\). Hölder’s inequality and the estimates above give \[\|\mathrm D^jF_tg(x)\| \le C_{\alpha,r,T}\|g\|_{\alpha,r}w_{\alpha-j}(x), \qquad 0\le j\le r,\quad 0\le t\le T.\] This proves the boundedness estimate on \({\cal C}^r_\alpha\).
We next prove the vanishing condition and strong continuity. First let \(g\in C_c^\infty({\cal X})\). The same derivative representation and the continuity of the stochastic flow jets at \(t=0\) give local uniform convergence \(\mathrm D^jF_tg\to\mathrm D^jg\) for \(0\le j\le r\). To control the tail, choose \(\beta<\alpha\). Since \(g\) is compactly supported, \(g\in{\cal C}^r_{\beta,\infty}({\cal X})\), and the estimate already proved gives, for \(0\le t\le1\), \[\frac{\|\mathrm D^jF_tg(x)\|}{w_{\alpha-j}(x)} \le C\|g\|_{\beta,r}w_{\beta-\alpha}(x), \qquad |x|\;\text{large}.\] The right-hand side tends to zero as \(|x|\to\infty\), uniformly in \(t\in[0,1]\). Thus \(F_tg\in{\cal C}^r_{\alpha,\infty}\) and \(\|F_tg-g\|_{\alpha,r}\to0\) for compactly supported smooth \(g\). Lemma 17 and the boundedness estimate extend both statements to arbitrary \(g\in{\cal C}^r_{\alpha,\infty}({\cal X})\).
If \(f\in{\cal C}^{r+2}_{\alpha,\infty}({\cal X})\), then Lemma 18 gives \(Lf\in{\cal C}^r_{\alpha,\infty}({\cal X})\). Applying Itô’s formula to the stopped diffusion and then removing the stopping by the weighted Lyapunov estimates gives \[F_t f-f=\int_0^tF_sLf\,ds \quad\text{in }{\cal C}^r_{\alpha,\infty}({\cal X}),\] and therefore \[\frac{F_tf-f}{t}-Lf = \frac{1}{t}\int_0^t(F_sLf-Lf)\,ds \to0\] in \({\cal C}^r_{\alpha,\infty}\) by strong continuity. This identifies the generator. Density follows from Lemma 17. Invariance of \({\cal C}^{r+2}_{\alpha,\infty}\) follows from the same argument applied at derivative order \(r+2\). Hence \({\cal C}^{r+2}_{\alpha,\infty}\) is a dense invariant subspace contained in \(\operatorname{Dom}(A_{\alpha,r})\), and the semigroup core criterion makes it an invariant core. ◻
Lemma 20. Fix \(\alpha\in{\mathbb{R}}\) and \(r\in{\mathbb{N}}_0\), and assume Assumption 2. Then, for every \(f\in{\cal C}^{r+4}_{\alpha,\infty}({\cal X})\) and all sufficiently small \(h>0\), \[\left\| \frac{F_h-I}{h}f-Lf \right\|_{\alpha,r} \le C_{\alpha,r}h\|f\|_{\alpha,r+4}.\]
Proof. By Lemma 18, \(Lf\in{\cal C}^{r+2}_{\alpha,\infty}({\cal X})\) and \(L^2f\in{\cal C}^{r}_{\alpha,\infty}({\cal X})\). Lemma 19 therefore gives \(f\in\operatorname{Dom}(A_{\alpha,r})\) and \(Lf\in\operatorname{Dom}(A_{\alpha,r})\), where \(A_{\alpha,r}\) is the generator of \((F_t)\) on \({\cal C}^r_{\alpha,\infty}({\cal X})\). Dynkin’s formula in this Banach space, applied twice, gives \[\frac{F_hf-f}{h}-Lf = \int_0^h\left(1-\frac{s}{h}\right)F_sL^2f\,ds.\] The regularity estimate from Section 4 gives boundedness of \(F_s\) on \({\cal C}^r_{\alpha,\infty}({\cal X})\) for \(0\le s\le1\). Combining this bound with Lemma 18 yields the claim. ◻
Lemma 21. Fix \(\alpha\in{\mathbb{R}}\) and \(r\in{\mathbb{N}}_0\). Let Assumptions 2, 3, 4, and 5 hold. Then there are constants \(C_{\alpha,r},q_{\alpha,r}>0\) and \(h_0>0\) such that, for all \(0<h\le h_0\), \[\|U_hg\|_{\alpha,r} \le e^{q_{\alpha,r}h}\|g\|_{\alpha,r}, \qquad g\in{\cal C}^r_{\alpha,\infty}({\cal X}),\] and \[\left\| \left(\frac{U_h-I}{h}-L\right)f \right\|_{\alpha,r} \le C_{\alpha,r}h\|f\|_{\alpha,r+4}, \qquad f\in{\cal C}^{r+4}_{\alpha,\infty}({\cal X}).\]
Proof. Put \[\eta_h(x):=h b(x)+\sqrt h\sum_{\ell=1}^d\sigma_\ell(x)\xi_\ell.\] Lemma 16 gives the weight comparability, the derivative bounds for \(\eta_h\), and the preservation of \({\cal C}^r_{\alpha,\infty}\) by \(U_h\). The displayed stability estimate is precisely Assumption 5. It remains to prove the local consistency estimate.
For consistency, use Taylor’s formula with integral fourth-order remainder: \[\begin{align} f(x+\eta_h) &=f(x)+\mathrm Df(x)[\eta_h] +\frac{1}{2}\mathrm D^2f(x)[\eta_h,\eta_h] +\frac{1}{6}\mathrm D^3f(x)[\eta_h,\eta_h,\eta_h]\\ &\quad +\frac{1}{6}\int_0^1(1-\theta)^3 \mathrm D^4f(x+\theta\eta_h) [\eta_h,\eta_h,\eta_h,\eta_h]\,d\theta . \end{align}\] The coefficient \(1/6\) is the integral form of the fourth-order remainder, since \(\int_0^1(1-\theta)^3\,d\theta=1/4\). Let \[a(x):=\sum_{\ell=1}^d\sigma_\ell(x)\otimes\sigma_\ell(x).\] The moment assumptions give \[{\mathbb{E}}\eta_h=hb,\qquad {\mathbb{E}}(\eta_h^{\otimes2})=h\,a+h^2 b^{\otimes2},\] and, because all third moments of \(\xi\) vanish, \[{\mathbb{E}}(\eta_h^{\otimes3}) = h^2\,\mathcal{S}(b\otimes a)+h^3 b^{\otimes3},\] where \(\mathcal{S}(b\otimes a)\) denotes the finite symmetrised sum of the three tensors obtained by placing \(b\) in one of the slots. Hence \[\begin{align} (U_h-I-hL)f(x) &= \frac{h^2}{2}\mathrm D^2f(x)[b,b] +\frac{h^2}{6}\mathrm D^3f(x)[\mathcal{S}(b\otimes a)]\\ &\quad +\frac{h^3}{6}\mathrm D^3f(x)[b,b,b] +R_{4,h}f(x), \end{align}\] where \[R_{4,h}f(x) := \frac{1}{6}\int_0^1(1-\theta)^3 {\mathbb{E}}\,\mathrm D^4f(x+\theta\eta_h) [\eta_h,\eta_h,\eta_h,\eta_h]\,d\theta .\]
Differentiate the four displayed terms. For the first term, \(\mathrm D^j\{\mathrm D^2f[b,b]\}\) is a finite sum of contractions of \(\mathrm D^{j_0+2}f\) with derivatives \(\mathrm D^{j_1}b\) and \(\mathrm D^{j_2}b\), where \(j_0+j_1+j_2=j\). After division by \(w_{\alpha-j}\), each product is bounded by \[\|f\|_{\alpha,r+4} \frac{ w_{\alpha-(j_0+2)}w_{1-j_1}w_{1-j_2} }{w_{\alpha-j}} = \|f\|_{\alpha,r+4}.\] The same calculation applies to \(\mathrm D^3f[\mathcal{S}(b\otimes a)]\) because \(\|\mathrm D^\ell a(x)\|\le Cw_{2-\ell}(x)\), and to \(\mathrm D^3f[b,b,b]\) because the coefficient weight is \(w_{1-j_1}w_{1-j_2}w_{1-j_3}\). Thus these three deterministic terms contribute at most \(C h^2\|f\|_{\alpha,r+4}w_{\alpha-j}(x)\) for \(0\le j\le r\).
For the integral remainder, apply Proposition 7 and Leibniz’ rule to \[\mathrm D^4f(x+\theta\eta_h) [\eta_h,\eta_h,\eta_h,\eta_h].\] Every \(j\)th derivative is a finite sum of contractions containing one derivative \(\mathrm D^{4+q}f(x+\theta\eta_h(x))\), with \(q\le j\), several derivatives of \(\varphi_h^\theta(x)=x+\theta\eta_h(x)\), and four factors obtained by differentiating the four copies of \(\eta_h\). Lemma 16 gives the weight comparison \(w_{\alpha-(4+q)}(x+\theta\eta_h(x))\le Cw_{\alpha-(4+q)}(x)\) and the bounds \(\|\mathrm D^\ell\eta_h(x)\|\le C\sqrt h\,w_{1-\ell}(x)\). Since four copies of \(\eta_h\) are present before differentiation, every term contains the factor \(h^2\) and its coefficient weights multiply exactly to \(w_{\alpha-j}(x)\). Boundedness of \(\xi\) permits taking expectation inside the same estimate. Therefore \[\|\mathrm D^jR_{4,h}f(x)\| \le C_{\alpha,r}h^2 \|f\|_{\alpha,r+4}w_{\alpha-j}(x).\] The same product estimates also show that each weighted derivative ratio vanishes at infinity, because one of the factors is a weighted derivative ratio of \(f\) evaluated at \(x\) or at \(x+\theta\eta_h(x)\) and the latter tends to infinity uniformly in the bounded increment variable. Combining the deterministic and remainder estimates gives \[\|\mathrm D^j[(U_h-I-hL)f](x)\| \le C_{\alpha,r}h^2\|f\|_{\alpha,r+4}w_{\alpha-j}(x), \qquad 0\le j\le r.\] Dividing by \(h\) and taking the maximum over \(j\) proves the local error estimate. ◻
Completion of the proof of Theorem 1. Apply Proposition 11 with \[B = C_{W,\infty}({\cal X}), \qquad D = D_4.\]
The quasi-contraction property \(\|F_t f\|_W \leq e^{\bar \mu t} \|f\|_W\) and strong continuity on \(B\) follow from Lemma 11.
The random-walk operator satisfies \(\|U_hg\|_W\le e^{q_Wh}\|g\|_W\) for \(g\in C_W({\cal X})\) and \(0<h\le1\), by Lemma 15.
Corollary 4 and Proposition 10 give an equivalent norm \(\|\cdot\|_4^*\) on \(D_4\) such that \(\|\cdot\|_4\le\|\cdot\|_4^*\le C_4\|\cdot\|_4\) and \(\|F_t f\|_4^*\le e^{mt}\|f\|_4^*\). Since \(D_4\hookrightarrow C_{W,\infty}({\cal X})\) continuously, we multiply this equivalent norm by a fixed constant, if necessary, so that \(\|f\|_W\le\|f\|_4^*\).
The inequality \[\left\| \frac{F_h - I}{h}f - Lf \right\|_W \leq \chi_h \|f\|_4 \leq \chi_h \|f\|^*_4\] follows from Lemma 13.
The inequality \[\left\| \frac{U_h - I}{h}f - Lf \right\|_W \leq \epsilon_h \|f\|_4 \leq \epsilon_h \|f\|_4^*\] follows from Lemma 14.
Proposition 11 gives the estimate in the equivalent norm, and \(\|f\|_4^*\le C_4\|f\|_4\) returns it to the stated \(D_4\) norm.
We apply Proposition 11 with \[B={\cal C}^r_{\alpha,\infty}({\cal X}), \qquad D={\cal C}^{r+4}_{\alpha,\infty}({\cal X}).\] The boundedness and strong continuity of \(F_t\) on \(B\) and \(D\) follow from Lemma 19; after applying Proposition 10 to the \(D\)-norm, we may write \(\|F_t f\|_D\le e^{\mu_{\alpha,r}t}\|f\|_D\). The stability of \(U_h\) on \(B\) and the local error estimate for \(U_h\) are contained in Lemma 21; the local error estimate for \(F_h\) is Lemma 20. Proposition 11 then gives ?? ; applying the killed part of the same proposition gives ?? .
Define \[\hat{F}_t := e^{-ct}F_t, \qquad \hat{U}_h := e^{-ch}U_h,\] and decompose \[\left\| {\mathbb{E}}\hat{F}_{\hat{\sigma}_T} f - {\mathbb{E}}\hat{U}_h^{n_T^h} f \right\|_B \le I + II,\] where \[I := \left\| {\mathbb{E}}\hat{F}_{N_T^h} f - {\mathbb{E}}\hat{U}_h^{n_T^h} f \right\|_B, \qquad II := \left\| {\mathbb{E}}\hat{F}_{\hat{\sigma}_T} f - {\mathbb{E}}\hat{F}_{N_T^h} f \right\|_B.\] Here the norm is the norm of the abstract space \(B\). We start with \(I\). For every deterministic \(s\ge0\), the assumed killed deterministic estimate yields \[\|\hat{F}_s f - \hat{U}_h^{\lfloor s/h\rfloor}f\|_B \le C_{\mathrm{RW}} h e^{(\mu_D-c)s}\|f\|_D\] If \(c\ge\mu_D\), this gives immediately \[I \le {\mathbb{E}}\bigl[\|\hat{F}_{N_T^h}f - \hat{U}_h^{n_T^h}f\|_B\bigr] \le C_{\mathrm{RW}} h \|f\|_D.\] If \(\bar\mu_B\le c<\mu_D\), put \[z_h:=\frac{\log(1/h)}{2(\mu_D-c)}.\] For \(h\) sufficiently small we have \(z_h>t_0\). Splitting the expectation according to \(\{N_T^h\le z_h\}\) and \(\{N_T^h>z_h\}\) gives \[\begin{align} I &\le C_{\mathrm{RW}}h e^{(\mu_D-c)z_h}\|f\|_D +2C_B\|f\|_D\,P(N_T^h>z_h) \\ &\le C h^{1/2}\|f\|_D +2C_B\|f\|_D\,P(\Phi^h_{z_h}\le T). \end{align}\] By Proposition 4, enlarged in the constant to cover \(T\le1\) and the replacement of \(<\) by \(\le\), we have \[P(\Phi^h_{z_h}\le T)\le C_T z_h^{-1/\beta}.\] Hence \[I \le C_T(\log(1/h))^{-1/\beta}\|f\|_D,\] after absorbing the stronger term \(h^{1/2}\) into the logarithmic bound for small \(h\).
Now consider \(II\). Writing \(\mu_{\hat{\sigma}_T}\) and \(\mu_{N_T^h}\) for the laws of \(\hat{\sigma}_T\) and \(N_T^h=h n_T^h\), we have \[II = \left\| \int_{[0,\infty)} \hat{F}_s f \, d(\mu_{\hat{\sigma}_T}-\mu_{N_T^h})(s) \right\|_B.\] Since \(f\in D\), the map \(s\mapsto \hat{F}_s f\) is continuously differentiable in \(B\) and \[\frac{d}{ds}\hat{F}_s f = \hat{F}_s (L-c)f.\] Hence, using \(c\ge\bar\mu_B\) and the assumed derivative estimate, \[\left\|\frac{d}{ds}\hat{F}_s f\right\|_B \le C_L e^{(\bar\mu_B-c)s}\|f\|_D \le C_L\|f\|_D, \qquad s\ge0.\] Write \(G_h(s):=P(\hat{\sigma}_T>s)-P(N_T^h>s) =(\mu_{\hat{\sigma}_T}-\mu_{N_T^h})\bigl((s,\infty)\bigr)\), so that \(d(\mu_{\hat{\sigma}_T}-\mu_{N_T^h})(s)=-\,dG_h(s)\) on \((0,\infty)\). The Banach-valued Lebesgue–Stieltjes integration-by-parts identity for the bounded variation function \(s\mapsto G_h(s)\) on \([0,\infty)\) then reads \[\int_{[0,\infty)}\hat{F}_s f\,d(\mu_{\hat{\sigma}_T}-\mu_{N_T^h})(s) = -\bigl[\hat{F}_s f\cdot G_h(s)\bigr]_0^\infty +\int_0^\infty\Bigl(\tfrac{d}{ds}\hat{F}_s f\Bigr)\,G_h(s)\,ds.\] At \(s=0\), both inverse clocks satisfy \(\hat{\sigma}_T,N_T^h>0\) almost surely, so \(G_h(0)=0\). As \(s\to\infty\), the tail estimate Proposition 4 and the analogous tail bound for \(\hat{\sigma}_T\) (a polynomial of order \(-1/\beta\) in \(s\)) give \(P(\hat{\sigma}_T>s),P(N_T^h>s)\to0\), hence \(G_h(s)\to0\); combined with the uniform bound \(\|\hat{F}_sf\|_B\le C_B\|f\|_D\) from the standing hypothesis, the boundary term at infinity vanishes. Therefore \[II \le C_L \|f\|_D \int_0^\infty \bigl| P(\hat{\sigma}_T > s) - P(N_T^h > s) \bigr|\,ds.\] The continuous identity is \(\{\hat{\sigma}_T>s\}=\{\hat{\Sigma}_s\le T\}\). For the discrete clock, since \(n_T^h=\min\{n:S_n^h>T\}\), \[\{N_T^h>s\} = \{h n_T^h>s\} = \{n_T^h>\lfloor s/h\rfloor\} = \{S_{\lfloor s/h\rfloor}^h\le T\} = \{\Phi_s^h\le T\}.\] These identities also underlie Lemma 1, which converts the forward subordinator estimate of Theorem 3 into the inverse-clock bound. Applying that lemma, \[II \le C_L C_{\mathrm{clock}}(T) h^{\chi(\beta)} \|f\|_D.\]
Combining the bounds for \(I\) and \(II\) gives, in the case \(c\ge\mu_D\), \[\|u_h(T,\cdot)-u(T,\cdot)\|_B \le C_{\mathrm{frac}} \bigl(h+h^{\chi(\beta)}\bigr)\|f\|_D,\] which is ?? . In the case \(\bar\mu_B\le c<\mu_D\), the same clock bound together with the logarithmic estimate for \(I\) gives ?? .
For Corollary 1, apply Theorem 5 with \[B=C_{W,\infty}({\cal X}),\qquad D=D_4.\] Strong continuity of \((F_t)_{t\ge0}\) on \(B\) and the inclusion \(D\subset\mathrm{Dom}(L_B)\) with \(L_B|_D=L\) are supplied by Lemmas 11 and 12, which also give \(F_s f-f=\int_0^s F_rLf\,dr\) in \(B\); combining with the semigroup identity yields \(s\mapsto\hat{F}_s f\in C^1([0,\infty);B)\) with \(\tfrac{d}{ds}\hat{F}_s f=\hat{F}_s(L-c)f\). The deterministic killed estimate is the killed part of Theorem 1. The base-space killed growth bound for \(F_s\) follows from the weighted Lyapunov estimate for \(W\) (Lemma 4), and the one for \(U_h^{\lfloor s/h\rfloor}\) from the one-step bound Lemma 15. The bound on \(e^{-cs}F_s(L-c)f\) follows from Lemma 7 and the \(C_{W,\infty}\)-growth estimate for \(F_s\).
For Corollary 3, apply Theorem 5 with \[B={\cal C}^r_{\alpha,\infty}({\cal X}), \qquad D={\cal C}^{r+4}_{\alpha,\infty}({\cal X}).\] Strong continuity of \((F_t)_{t\ge0}\) on \(B\), the inclusion \(D\subset\mathrm{Dom}(A_{\alpha,r})\) with \(A_{\alpha,r}|_D=L\), and the resulting \(C^1\)-differentiability of \(s\mapsto\hat{F}_s f\) in \(B\) are all contained in Lemma 19. The deterministic killed estimate is ?? . The base-space killed growth bound is supplied by Lemma 19 for \(F_s\) and by Assumption 5 for \(U_h^{\lfloor s/h\rfloor}\). The bound on \(e^{-cs}F_s(L-c)f\) follows from Lemma 18 and the weighted regularity estimate for \(F_t\) on \({\cal C}^r_{\alpha,\infty}({\cal X})\).
This project was supported by Vega Institute Foundation.
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