The Method of Chernoff Approximation
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Abstract
This survey describes the method of approximation of operator semigroups, based on the Chernoff theorem. We outline recent results in this domain as well as clarify relations between constructed approximations, stochastic processes, numerical schemes for PDEs and SDEs, path integrals. We discuss Chernoff approximations for operator semigroups and Schrödinger groups. In particular, we consider Feller semigroups in \({{\mathbb R}^d}\), (semi)groups obtained from some original (semi)groups by different procedures: additive perturbations of generators, multiplicative perturbations of generators (which sometimes corresponds to a random timechange of related stochastic processes), subordination of semigroups/processes, imposing boundary/external conditions (e.g., Dirichlet or Robin conditions), averaging of generators, “rotation” of semigroups. The developed techniques can be combined to approximate (semi)groups obtained via several iterative procedures listed above. Moreover, this method can be implemented to obtain approximations for solutions of some timefractional evolution equations, although these solutions do not possess the semigroup property.
Keywords
Chernoff approximation Feynman formula Approximation of operator semigroups Approximation of transition probabilities Approximation of solutions of evolution equations Feynman–Kac formulae Euler–Maruyama schemes Feller semigroups Additive perturbations Operator splitting Multiplicative perturbations Dirichlet boundary/external conditions Robin boundary conditions Subordinate semigroups Timefractional evolution equations Schrödinger type equations1 Introduction
Let \((X,\Vert \cdot \Vert _X)\) be a Banach space. A family \((T_t)_{t\ge 0}\) of bounded linear operators on X is called a strongly continuous semigroup (denoted as \(C_0\)semigroup) if \(T_0=\mathop {\mathrm {Id}}\nolimits \), \(T_t\circ T_s=T_{t+s}\) for all \(t,s\ge 0\), and \(\lim _{t\rightarrow 0}\Vert T_t\varphi \varphi \Vert _X=0\) for all \(\varphi \in X\). The generator of the semigroup \((T_t)_{t\ge 0}\) is an operator \((L,\mathop {\mathrm {Dom}}\nolimits (L))\) in X which is given by \(L\varphi :=\lim _{t\rightarrow 0}t^{1}(T_t\varphi \varphi )\), \(\mathop {\mathrm {Dom}}\nolimits (L):=\left\{ \varphi \in X\,:\, \lim _{t\rightarrow 0}t^{1}(T_t\varphi \varphi )\,\text { exists in }\, X \right\} \). In the sequel, we denote the semigroup with a given generator L both as \((T_t)_{t\ge 0}\) and as \((e^{tL})_{t\ge 0}\). The following fundamental result of the theory of operator semigroups connects \(C_0\)semigroups and evolution equations: Let \((L,\mathop {\mathrm {Dom}}\nolimits (L))\) be a densely defined linear operator in X with a nonempty resolvent set. The Cauchy problem \(\frac{\partial f}{\partial t}=Lf\), \(f(0)=f_0\) in X for every \(f_0\in \mathop {\mathrm {Dom}}\nolimits (L)\) has a unique solution f(t) which is continuously differentiable on \([0,+\infty )\) if and only if \((L,\mathop {\mathrm {Dom}}\nolimits (L))\) is the generator of a \(C_0\)semigroup \((T_t)_{t\ge 0}\) on X. And the solution is given by \(f(t):=T_tf_0\).
 (1)
to construct the \(C_0\)semigroup \((T_t)_{t\ge 0}\) with a given generator \((L,\mathop {\mathrm {Dom}}\nolimits (L))\) on a given Banach space X;
 (2)
to solve the Cauchy problem \(\frac{\partial f}{\partial t}=Lf\), \(f(0)=f_0\) in X;
 (3)
to determine the transition kernel P(t, x, dy) of an underlying Markov process \((\xi _t)_{t\ge 0}\).
Theorem 1.1
 (i)
\(F(0)=\mathop {\mathrm {Id}}\nolimits \),
 (ii)
\(\Vert F(t)\Vert \le e^{wt} \) for some \(w\in \mathbb R\) and all \(t\ge 0\),
 (iii)
the limit \(L\varphi :=\lim \limits _{t\rightarrow 0}\frac{F(t)\varphi \varphi }{t}\) exists for all \(\varphi \in D\), where D is a dense subspace in X such that (L, D) is closable and the closure \((L,\mathop {\mathrm {Dom}}\nolimits (L))\) of (L, D) generates a \(C_0\)semigroup \((T_t)_{t\ge 0}\).
This approach allows to create a kind of a LEGOconstructor: we start with a \(C_0\)semigroup which is already known^{2} or Chernoff approximated^{3}; then, applying different operations on its generator, we consider more and more complicated \(C_0\)semigroups and construct their Chernoff approximations.

Operator splitting; additive perturbations of a generator (Sect. 2.1, [20, 21, 27]);

Multiplicative perturbations of a generator/random time change of a process via an additive functional (Sect. 2.3, [20, 21, 27]);

killing of a process upon leaving a given domain/imposing Dirichlet boundary (or external) conditions (Sect. 2.4, [21, 22, 24]);
Moreover, Chernoff approximations have been obtained for some stochastic Schrödinger type equations in [37, 54, 55, 56]; for evolution equations with the Vladimirov operator (this operator is a padic analogue of the Laplace operator) in [65, 66, 67, 68, 69]; for evolution equations containing Lévy Laplacians in [1, 2]; for some nonlinear equations in [58].
If all operators F(t) are integral operators with elementary kernels or pseudodifferential operators with elementary symbols, the identity (3) leads to representation of a given semigroup by nfolds iterated integrals of elementary functions when n tends to infinity. This gives rise to Feynman formulae. A Feynman formula is a representation of a solution of an initial (or initialboundary) value problem for an evolution equation (or, equivalently, a representation of the semigroup solving the problem) by a limit of nfold iterated integrals of some functions as \(n\rightarrow \infty \). One should not confuse the notions of Chernoff approximation and Feynman formula. On the one hand, not all Chernoff approximations can be directly interpreted as Feynman formulae since, generally, the operators \((F(t))_{t\ge 0}\) do not have to be neither integral operators, nor pseudodifferential operators. On the other hand, representations of solutions of evolution equations in the form of Feynman formulae can be obtained by different methods, not necessarily via the Chernoff Theorem. And such Feynman formulae may have no relations to any Chernoff approximation, or their relations may be quite indirect. Richard Feynman was the first who considered representations of solutions of evolution equations by limits of iterated integrals [33, 34]. He has, namely, introduced a construction of a path integral (known nowadays as Feynman path integral) for solving the Schrödinger equation. And this path integral was defined exactly as a limit of iterated finite dimensional integrals. Feynman path integrals can be also understood as integrals with respect to Feynman type pseudomeasures. Analogously, one can sometimes obtain representations of a solution of an initial (or initialboundary) value problem for an evolution equation (or, equivalently, a representation of an operator semigroup resolving the problem) by functional (or, path) integrals with respect to probability measures. Such representations are usually called Feynman–Kac formulae. It is a usual situation that limits in Feynman formulae coincide with (or in some cases define) certain path integrals with respect to probability measures or Feynman type pseudomeasures on a set of paths of a physical system. Hence the iterated integrals in Feynman formulae for some problem give approximations to path integrals representing the solution of the same problem. Therefore, representations of evolution semigroups by Feynman formulae, on the one hand, allow to establish new pathintegralrepresentations and, on the other hand, provide an additional tool to calculate path integrals numerically. Note that different Feynman formulae for the same semigroup allow to establish relations between different path integrals (see, e.g., [21]).
The result of Chernoff has diverse generalizations. Versions, using arbitrary partitions of the time interval [0, t] instead of the equipartition \((t_k)_{k=0}^n\) with \(t_kt_{k1}=t/n\), are presented, e.g., in [60, 71]. Versions, providing stronger type of convergence, can be found in [78]. The analogue of the Chernoff theorem for multivalued generators can be found, e.g., in [32]. Analogues of Chernoff’s result for semigroups, which are continuous in a weaker sense, are obtained, e.g., in [3, 43]. For analogues of the Chernoff theorem in the case of nonlinear semigroups, see, e.g., [5, 16, 17]. The Chernoff Theorem for twoparameter families of operators can be found in [56, 61].
2 Chernoff Approximations for Operator Semigroups and Further Applications
2.1 Chernoff Approximations for the Procedure of Operator Splitting
Theorem 2.1
Note that we do not require from summands \(L_k\) to be generators of \(C_0\)semigroups. For example, \(L_1\) can be a leading term (which generates a \(C_0\)semigroup) and \(L_2,\ldots ,L_m\) can be \(L_1\)bounded additive perturbations such that \(L:=L_1+L_2+\cdots +L_m\) again generates a strongly continuous semigroup. Or L may even be a sum of operators \(L_k\), none of which generates a strongly continuous semigroup itself.
Proof
Remark 2.1
2.2 Chernoff Approximations for Feller Semigroups
Theorem 2.2
Example 2.1
Remark 2.2
Example 2.2
2.3 Chernoff Approximations for Multiplicative Perturbations of a Generator
Assumption 2.1
We assume that Open image in new window generates a strongly continuous semigroup (which is denoted by Open image in new window ) on the Banach space X.
Some conditions assuring the existence and strong continuity of the semigroup Open image in new window can be found, e.g., in [31, 45]. The operator Open image in new window is called a multiplicative perturbation of the generator L and the semigroup Open image in new window , generated by Open image in new window , is called a semigroup with the multiplicatively perturbed with the function a generator. The following result has been shown in [21] (cf. [20, 27]).
Theorem 2.3
Remark 2.3
(i) The statement of Theorem 2.3 remains true for the following Banach spaces (cf. [21]):
(a) \(X=C_\infty (Q):=\left\{ \varphi \in C_b(Q)\,:\, \lim _{\rho (q,q_0)\rightarrow \infty }\varphi (q)=0 \right\} ,\) where \(q_0\) is an arbitrary fixed point of Q and the metric space Q is unbounded with respect to its metric \(\rho \);
(b) \( X=C_0(Q):=\big \{ \varphi \in C_b(Q)\,:\, \forall \,\varepsilon >0\,\,\exists \,\text { a compact }\, K^\varepsilon _\varphi \subset Q\,\text { such that }\, \varphi (q)<\varepsilon \,\) \(\text { for all }\, q\notin K^\varepsilon _\varphi \big \}\), where the metric space Q is assumed to be locally compact.
(ii) As it follows from the proof of Theorem 2.3, if \(\lim _{t\rightarrow 0}\big \Vert \frac{F(t)\varphi \varphi }{t}L\varphi \big \Vert _X=0\) for all \(\varphi \in D\) then also Open image in new window for all \(\varphi \in D\).
Corollary 2.1
Remark 2.4
A multiplicative perturbation of the generator of a Markov process is equivalent to some randome time change of the process (see [32, 76, 77]). Note that Open image in new window is not a transition probability any more. Nevertheless, if the transition probability P(t, q, dy) of the original process is known, formula (20) allows to approximate the unknown transition probability of the modified process.
2.4 Chernoff Approximations for Semigroups Generated by Processes in a Domain with Prescribed Behaviour at the Boundary of/Outside the Domain
Case 1: \(X=C_\infty ({{\mathbb R}^d})\), \((\xi _t)_{t\ge 0}\) is a Feller process whose generator L is given by (6) with A, B, C of the class \(C^{2,\alpha }\), A satisfies (11), and either \(N\equiv 0\) or \(N\ne 0\) and the nonlocal term of L is a relatively bounded perturbation of the local part of L with some extra assumption on jumps of the process (see details in [22, 24]). The family \((F(t))_{t\ge 0}\) is given by (10) (see also (17), or (12) in the corresponding particular cases) and \(D=C^{2,\alpha }_c({{\mathbb R}^d})\). Further, \(\Omega \) is a bounded \(C^{4,\alpha }\)smooth domain, \(Y=C_0(\Omega )\), BC are the homogeneous Dirichlet boundary/external conditions corresponding to killing of the process upon leaving the domain \(\Omega \). A proper extension \(E^*\) has been constructed in [6], and it maps \(\mathop {\mathrm {Dom}}\nolimits (L^*)\cap C^{2,\alpha }(\overline{\Omega })\) into D.
This result can be further generalized for the case of diffusions, using the techniques of Sects. 2.1 and 2.3. This will be demonstrated in Example 2.3. Note, however, that the extension \(E^*\) of [52] maps \(D^*\) into the set of functions which do not belong to \(C^2({{\mathbb R}^d})\). Hence it is not possible to use the family (12) (and \(D=C^{2,\alpha }_c({{\mathbb R}^d})\)) in a straightforward manner for approximation of diffusions with Robin BC.
Example 2.3
2.5 Chernoff Approximations for Subordinate Semigroups
Theorem 2.4
Let \(m:(0,\infty )\rightarrow \mathbb {N}_0\) be a monotone function^{9} such that \(m(t)\rightarrow +\infty \) as \(t\rightarrow 0\). Let the mapping \([F(\cdot /m(t))]^{m(t)}\varphi \,:\,[0,\infty )\rightarrow X \) be Bochner measurable as the mapping from \(([0,\infty ),\mathcal {B}([0,\infty )),\eta ^0_t)\) to \((X,\mathcal {B}(X))\) for each \(t>0\) and each \(\varphi \in X\).
The constructed families \((\mathcal {F}(t))_{t\ge 0}\) and \((\mathcal {F}_\mu (t))_{t\ge 0}\) can be used (in combination with the techniques of Sects. 2.1, 2.3, 2.4 and results of [42, 72]), e.g., to approximate semigroups generated by subordinate Feller diffusions on star graphs and Riemannian manifolds. Note that the family (24) can be used when the convolution semigroup \((\eta ^0_t)_{t\ge 0}\) is known explicitly. This is the case of inverse Gaussian (including 1/2stable) subordinator, Gamma subordinator and some others (see, e.g., [11, 18, 30] for examples).
2.6 Approximation of Solutions of TimeFractional Evolution Equations
Theorem 2.5
Example 2.4
2.7 Chernoff Approximations for Schrödinger Groups
Example 2.5
Theorem 2.6
Footnotes
 1.
We denote the space of continuous functions on \({{\mathbb R}^d}\) vanishing at infinity by \(C_\infty ({{\mathbb R}^d})\) and the Schwartz space by \(S({{\mathbb R}^d})\).
 2.
 3.
 4.
\(C^m_\infty ({{\mathbb R}^d}):=\{\varphi \in C^m({{\mathbb R}^d})\,:\,\partial ^\alpha \varphi \in C_\infty ({{\mathbb R}^d}),\,\alpha \le m \}\).
 5.
The weak convergence of this Euler–Maruyama scheme is, of course, a classical result, cf. [41].
 6.
The family \((F(t))_{t\ge 0}\) of bounded linear operators on a Banach space X is called strongly continuous if \(\lim _{t\rightarrow t_0}\Vert F(t)\varphi F(t_0)\varphi \Vert _X=0\) for all \(t,t_0\ge 0\) and all \(\varphi \in X\).
 7.
Here we consider the Robin BC (23) given in a weaker form via the first Green’s formula; \(d\sigma \) is the surface measure on \(\partial \Omega \).
 8.
A family \((\eta _t)_{t\ge 0}\) of bounded Borel measures on \(\mathbb R\) is called a convolution semigroup if \(\eta _t(\mathbb R)\le 1\) for all \(t\ge 0\), \(\eta _t*\eta _s=\eta _{t+s}\) for all \(t,s\ge 0\), \(\eta _0=\delta _0\), and \(\eta _t\rightarrow \delta _0\) vaguely as \(t\rightarrow 0\), i.e. \(\lim _{t\rightarrow 0}\int _{\mathbb R}\varphi (x)\eta _t(dx)=\int _{\mathbb R}\varphi (x)\delta _0(dx)\equiv \varphi (0)\) for all \(\varphi \in C_c(\mathbb R)\). A convolution semigroup \((\eta _t)_{t\ge 0}\) is supported by \([0,\infty )\) if \(\mathop {\mathrm {supp}}\nolimits \eta _t\subset [0,\infty )\) for all \(t\ge 0\).
 9.
One can take, e.g., \(m(t):=\left\lfloor 1/t \right\rfloor =\) the largest integer \(n\le 1/t\). Recall that \(\mathbb {N}_0:=\mathbb {N}\cup \{0\}.\)
 10.
The semigroup \((S_t)_{t\ge 0}\) is strong Feller iff all the measures \(\eta ^0_t\) admit densities of the class \(L^1([0,\infty ))\) with respect to the Lebesgue measure (cf. Example 4.8.21 of [40]).
 11.
For any bounded operator B, its zero degree \(B^0\) is considered to be the identity operator. For each \(t>0\), a nonnegative integer m(t) and a bounded Bochner measurable mapping \([F(\cdot /m(t))]^{m(t)}\varphi \,:\,[0,\infty )\rightarrow X\), the integral in the right hand side of formula (25) is well defined.
 12.
Hence \((D^\mu _t)_{t\ge 0}\) is a subordinator corresponding to the Bernstein function \(f^\mu (s):=\int _0^1 s^\beta \mu (d\beta )\), \(s>0\), and \(E^\mu _t:=\inf \left\{ \tau \ge 0\,:\,D^\mu _\tau >t \right\} \).
 13.
Actually, \((F(t))_{t\ge 0}\) does not need to fulfill the condition (ii) of the Chernoff Theorem 1.1 in this construction.
 14.
See, e.g., Example 4.7.28 in [40].
Notes
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