Generalized Nonlinear Feynman–Kac Formula

• The Generalized Nonlinear Feynman–Kac formula is a family of probabilistic representation methods that extends the classical formula to include nonlinear, non-Markovian, and path-dependent settings.
• It employs advanced techniques like BSDEs, FBSDEs, G-Brownian motion, and stochastic fixed-point equations to model complex PDEs, PPDEs, and SPDEs.
• Its applications span stochastic control, finance, and regularity theory, offering both analytical insights and numerical strategies for solving nonlinear evolution equations.

The generalized nonlinear Feynman–Kac formula is a family of probabilistic representation principles that extend the classical linear Feynman–Kac identity from linear parabolic PDEs to semilinear, fully nonlinear, path-dependent, nonlocal, infinite-dimensional, and rough-coefficient settings. In the literature considered here, the extension is realized through BSDEs, FBSDEs, 2BSDE-related constructions, G-Brownian motion, stochastic fixed-point equations, backward stochastic variational inequalities, and backward stochastic Volterra integral equations. The common theme is that an analytic object—typically a viscosity, mild, weak, or classical solution of a PDE, PPDE, PIDE, or SPDE—is identified with a stochastic value process, often of the form $u(t,x)=Y_t^{t,x}$, or with a nonlinear expectation or supremum over controlled diffusions (Pham, 2014).

1. From the classical formula to semilinear representations

The classical linear Feynman–Kac formula represents the solution of a linear parabolic PDE by an expectation over a diffusion with generator

$$\mathcal{L}u(t,x)=\tfrac12 \operatorname{Tr}\!\big(a(t,x)D_x^2u(t,x)\big)+b(t,x)\cdot \nabla_x u(t,x),\qquad a=\sigma\sigma^\top.$$

In the Markovian nonlinear extension, a forward SDE

$dX_s=b(s,X_s)\,ds+\sigma(s,X_s)\,dW_s$

is coupled with a BSDE

$$Y_s=g(X_T)+\int_s^T f(r,X_r,Y_r,Z_r)\,dr-\int_s^T Z_r\,dW_r,$$

and the function $u(t,x):=Y_t^{t,x}$ solves the quasilinear PDE

$$\partial_t u+\mathcal{L}u+f\big(t,x,u,\sigma^\top\nabla_x u\big)=0,\qquad u(T,x)=g(x).$$

Peng and Wang present this correspondence as the classical nonlinear Feynman–Kac formula of Pardoux–Peng and then generalize it to path space (Peng et al., 2011). Pham’s review uses the same structure to emphasize that BSDEs provide a nonlinear Feynman–Kac formula for semilinear PDEs, with $Y_s^{t,x}=v(s,X_s^{t,x})$ and $Z_s^{t,x}=\sigma^\top(X_s^{t,x})D_xv(s,X_s^{t,x})$ when smoothness is available (Pham, 2014).

A distinct semilinear extension replaces the backward equation by a stochastic fixed-point equation. For semilinear parabolic PDEs with gradient-dependent nonlinearities,

$$\partial_t u+\langle \mu,\nabla_x u\rangle+\tfrac12\operatorname{Tr}\!\big(\sigma\sigma^*\,\text{Hess}_x u\big)+f(t,x,u,\nabla_x u)=0,$$

one obtains a representation for the pair $(v,\nabla_x v)$,

$$\mathcal{L}u(t,x)=\tfrac12 \operatorname{Tr}\!\big(a(t,x)D_x^2u(t,x)\big)+b(t,x)\cdot \nabla_x u(t,x),\qquad a=\sigma\sigma^\top.$$0

where $$\mathcal{L}u(t,x)=\tfrac12 \operatorname{Tr}\!\big(a(t,x)D_x^2u(t,x)\big)+b(t,x)\cdot \nabla_x u(t,x),\qquad a=\sigma\sigma^\top.$$1 is a Bismut–Elworthy–Li type weight process. This yields a generalized nonlinear Feynman–Kac formula in terms of a stochastic fixed-point equation rather than a backward SDE (Hutzenthaler et al., 2023).

These semilinear formulations preserve the essential classical principle—analytic evolution represented by stochastic dynamics—while allowing the reaction term to depend on $$\mathcal{L}u(t,x)=\tfrac12 \operatorname{Tr}\!\big(a(t,x)D_x^2u(t,x)\big)+b(t,x)\cdot \nabla_x u(t,x),\qquad a=\sigma\sigma^\top.$$2 and gradient variables. This suggests that the first major axis of generalization is not the replacement of probability by an entirely different object, but the replacement of linear conditional expectation by nonlinear stochastic recursion.

2. Fully nonlinear second-order structure: sup-envelopes, $$\mathcal{L}u(t,x)=\tfrac12 \operatorname{Tr}\!\big(a(t,x)D_x^2u(t,x)\big)+b(t,x)\cdot \nabla_x u(t,x),\qquad a=\sigma\sigma^\top.$$3-expectation, and robust diffusion families

A central fully nonlinear development replaces a fixed generator by a supremum envelope of linear or semilinear operators. In one formulation, the PDE is

$$\mathcal{L}u(t,x)=\tfrac12 \operatorname{Tr}\!\big(a(t,x)D_x^2u(t,x)\big)+b(t,x)\cdot \nabla_x u(t,x),\qquad a=\sigma\sigma^\top.$$4

where $$\mathcal{L}u(t,x)=\tfrac12 \operatorname{Tr}\!\big(a(t,x)D_x^2u(t,x)\big)+b(t,x)\cdot \nabla_x u(t,x),\qquad a=\sigma\sigma^\top.$$5 is sublinear in $$\mathcal{L}u(t,x)=\tfrac12 \operatorname{Tr}\!\big(a(t,x)D_x^2u(t,x)\big)+b(t,x)\cdot \nabla_x u(t,x),\qquad a=\sigma\sigma^\top.$$6 and uniformly elliptic. The structural result is that

$$\mathcal{L}u(t,x)=\tfrac12 \operatorname{Tr}\!\big(a(t,x)D_x^2u(t,x)\big)+b(t,x)\cdot \nabla_x u(t,x),\qquad a=\sigma\sigma^\top.$$7

so $$\mathcal{L}u(t,x)=\tfrac12 \operatorname{Tr}\!\big(a(t,x)D_x^2u(t,x)\big)+b(t,x)\cdot \nabla_x u(t,x),\qquad a=\sigma\sigma^\top.$$8 is the support function of a family of linear diffusion generators. With the stochastic control family $$\mathcal{L}u(t,x)=\tfrac12 \operatorname{Tr}\!\big(a(t,x)D_x^2u(t,x)\big)+b(t,x)\cdot \nabla_x u(t,x),\qquad a=\sigma\sigma^\top.$$9, the solution is represented by

$dX_s=b(s,X_s)\,ds+\sigma(s,X_s)\,dW_s$0

where $dX_s=b(s,X_s)\,ds+\sigma(s,X_s)\,dW_s$1 solves a controlled FBSDE. The dynamic programming principle identifies this function as the unique viscosity solution with polynomial growth (Pozza, 2019).

A related formulation defines

$dX_s=b(s,X_s)\,ds+\sigma(s,X_s)\,dW_s$2

and again the value function

$dX_s=b(s,X_s)\,ds+\sigma(s,X_s)\,dW_s$3

is proved to be the unique viscosity solution of the corresponding nonlinear parabolic PDE. The proof is organized through an abstract dynamic programming principle specialized to the BSDE terminal value, and then through viscosity sub- and supersolution arguments plus comparison (Pozza, 2021).

Under volatility uncertainty, Peng’s $dX_s=b(s,X_s)\,ds+\sigma(s,X_s)\,dW_s$4-framework produces a different but closely related nonlinearity. For G-BSDEs driven by G-Brownian motion, if $dX_s=b(s,X_s)\,ds+\sigma(s,X_s)\,dW_s$5, then $dX_s=b(s,X_s)\,ds+\sigma(s,X_s)\,dW_s$6 is the unique viscosity solution of a fully nonlinear PDE of the form

$dX_s=b(s,X_s)\,ds+\sigma(s,X_s)\,dW_s$7

where $dX_s=b(s,X_s)\,ds+\sigma(s,X_s)\,dW_s$8 encodes the volatility-uncertainty set $dX_s=b(s,X_s)\,ds+\sigma(s,X_s)\,dW_s$9 and the quadratic variation enters through $$Y_s=g(X_T)+\int_s^T f(r,X_r,Y_r,Z_r)\,dr-\int_s^T Z_r\,dW_r,$$0 terms in the BSDE (Hu et al., 2012). In a discounted setting, the $$Y_s=g(X_T)+\int_s^T f(r,X_r,Y_r,Z_r)\,dr-\int_s^T Z_r\,dW_r,$$1-conditional expectation of

$$Y_s=g(X_T)+\int_s^T f(r,X_r,Y_r,Z_r)\,dr-\int_s^T Z_r\,dW_r,$$2

is shown to be a viscosity solution of a nonlinear PDE with a linear discount term $$Y_s=g(X_T)+\int_s^T f(r,X_r,Y_r,Z_r)\,dr-\int_s^T Z_r\,dW_r,$$3, again under volatility uncertainty (Akhtari et al., 2020).

Pham’s review places these constructions in a larger control-theoretic context. For HJB equations,

$$Y_s=g(X_T)+\int_s^T f(r,X_r,Y_r,Z_r)\,dr-\int_s^T Z_r\,dW_r,$$4

a generalized nonlinear Feynman–Kac formula is obtained via a randomized control, a BSDE with jumps, and then a constrained BSDE with nonpositive jumps whose minimal supersolution identifies the viscosity solution of the fully nonlinear PDE (Pham, 2014).

Taken together, these results show that full nonlinearity is often encoded as either a supremum over diffusions, a sublinear expectation, or a constrained backward equation. A plausible implication is that “nonlinear expectation” and “robust diffusion family” are two mathematically proximate realizations of the same representation principle.

3. Path dependence, delay, and nonlocal state variables

A second major line of generalization replaces the Markovian state variable $$Y_s=g(X_T)+\int_s^T f(r,X_r,Y_r,Z_r)\,dr-\int_s^T Z_r\,dW_r,$$5 by an entire past trajectory. On path space

$$Y_s=g(X_T)+\int_s^T f(r,X_r,Y_r,Z_r)\,dr-\int_s^T Z_r\,dW_r,$$6

Peng and Wang introduce the path-dependent BSDE

$$Y_s=g(X_T)+\int_s^T f(r,X_r,Y_r,Z_r)\,dr-\int_s^T Z_r\,dW_r,$$7

and the associated PPDE

$$Y_s=g(X_T)+\int_s^T f(r,X_r,Y_r,Z_r)\,dr-\int_s^T Z_r\,dW_r,$$8

They prove both directions of the correspondence: if $$Y_s=g(X_T)+\int_s^T f(r,X_r,Y_r,Z_r)\,dr-\int_s^T Z_r\,dW_r,$$9 solves the PPDE, then $u(t,x):=Y_t^{t,x}$0 solves the BSDE; conversely, the map $u(t,x):=Y_t^{t,x}$1 belongs to $u(t,x):=Y_t^{t,x}$2 and is the unique classical PPDE solution (Peng et al., 2011).

Delay and jumps can be incorporated simultaneously. For a path-dependent forward jump-diffusion

$u(t,x):=Y_t^{t,x}$3

the backward component may depend on the delayed segment

$u(t,x):=Y_t^{t,x}$4

and on the jump integral $u(t,x):=Y_t^{t,x}$5. Under a smallness condition relating $u(t,x):=Y_t^{t,x}$6, the delay Lipschitz constant $u(t,x):=Y_t^{t,x}$7, and the instantaneous Lipschitz constant $u(t,x):=Y_t^{t,x}$8, the resulting BSDE with jumps and delayed generator is well posed, and the representation

$u(t,x):=Y_t^{t,x}$9

identifies a mild solution of a path-dependent nonlinear Kolmogorov equation with both delay and jumps (Persio et al., 2022).

A different nonlocal extension uses extended backward stochastic Volterra integral equations. For the Markovian EBSVIE,

$$\partial_t u+\mathcal{L}u+f\big(t,x,u,\sigma^\top\nabla_x u\big)=0,\qquad u(T,x)=g(x).$$0

the associated nonlocal PDE is

$$\partial_t u+\mathcal{L}u+f\big(t,x,u,\sigma^\top\nabla_x u\big)=0,\qquad u(T,x)=g(x).$$1

with terminal condition $$\partial_t u+\mathcal{L}u+f\big(t,x,u,\sigma^\top\nabla_x u\big)=0,\qquad u(T,x)=g(x).$$2. The generalized nonlinear Feynman–Kac formula is

$$\partial_t u+\mathcal{L}u+f\big(t,x,u,\sigma^\top\nabla_x u\big)=0,\qquad u(T,x)=g(x).$$3

and reduces to the Pardoux–Peng formula when the Volterra structure disappears (Wang, 2019).

Infinite-dimensional analogues also fit this pattern. For semilinear PDEs on a separable Hilbert space $$\partial_t u+\mathcal{L}u+f\big(t,x,u,\sigma^\top\nabla_x u\big)=0,\qquad u(T,x)=g(x).$$4, with unbounded linear operator $$\partial_t u+\mathcal{L}u+f\big(t,x,u,\sigma^\top\nabla_x u\big)=0,\qquad u(T,x)=g(x).$$5 and the $$\partial_t u+\mathcal{L}u+f\big(t,x,u,\sigma^\top\nabla_x u\big)=0,\qquad u(T,x)=g(x).$$6-continuous viscosity framework, the solution of the infinite-dimensional BSDE

$$\partial_t u+\mathcal{L}u+f\big(t,x,u,\sigma^\top\nabla_x u\big)=0,\qquad u(T,x)=g(x).$$7

yields

$$\partial_t u+\mathcal{L}u+f\big(t,x,u,\sigma^\top\nabla_x u\big)=0,\qquad u(T,x)=g(x).$$8

and $$\partial_t u+\mathcal{L}u+f\big(t,x,u,\sigma^\top\nabla_x u\big)=0,\qquad u(T,x)=g(x).$$9 is a $Y_s^{t,x}=v(s,X_s^{t,x})$0-continuous viscosity solution of the corresponding semilinear PDE (Wessels, 2023).

These developments show that the generalized nonlinear Feynman–Kac formula is not confined to finite-dimensional Markov diffusions. It extends to path space, delayed systems, Volterra memory, jump noise, and Hilbert-space state variables, with the analytic equation changing from a PDE to a PPDE, PIDE, or nonlocal evolution equation accordingly.

4. Boundary conditions, singular coefficients, and rough or nonstandard driving mechanisms

Another strand of generalization concerns equations whose coefficients, boundary conditions, or driving noises fall outside the regular semimartingale Markov setting. Pardoux–Răşcanu treat multivalued parabolic equations with nonlinear Neumann or Robin boundary conditions by coupling a reflected SDE

$Y_s^{t,x}=v(s,X_s^{t,x})$1

with a backward stochastic variational inequality

$Y_s^{t,x}=v(s,X_s^{t,x})$2

The function $Y_s^{t,x}=v(s,X_s^{t,x})$3 is deterministic, continuous, and provides a generalized nonlinear Feynman–Kac formula for parabolic variational inequalities with multivalued bulk and boundary terms. The paper also states that an earlier proof of continuity in Pardoux–Zhang was not correct and establishes the required continuity in a more general framework (Pardoux et al., 2016).

Distributional coefficients lead to a different generalization. For the formal BSDE

$Y_s^{t,x}=v(s,X_s^{t,x})$4

with $Y_s^{t,x}=v(s,X_s^{t,x})$5, the term $Y_s^{t,x}=v(s,X_s^{t,x})$6 is undefined pointwise. Issoglio–Russo replace it by an integral operator $Y_s^{t,x}=v(s,X_s^{t,x})$7 acting on distributions and show that $Y_s^{t,x}=v(s,X_s^{t,x})$8 is generally a weak Dirichlet process rather than a semimartingale. The associated PDE

$Y_s^{t,x}=v(s,X_s^{t,x})$9

admits a unique mild solution $Z_s^{t,x}=\sigma^\top(X_s^{t,x})D_xv(s,X_s^{t,x})$0, and $Z_s^{t,x}=\sigma^\top(X_s^{t,x})D_xv(s,X_s^{t,x})$1 yields the generalized nonlinear Feynman–Kac formula (Issoglio et al., 2018).

For SPDEs, the stochastic heat equation with semimartingale multiplicative potential

$Z_s^{t,x}=\sigma^\top(X_s^{t,x})D_xv(s,X_s^{t,x})$2

has the representation

$Z_s^{t,x}=\sigma^\top(X_s^{t,x})D_xv(s,X_s^{t,x})$3

where $Z_s^{t,x}=\sigma^\top(X_s^{t,x})D_xv(s,X_s^{t,x})$4 is a continuous semimartingale with local characteristic $Z_s^{t,x}=\sigma^\top(X_s^{t,x})D_xv(s,X_s^{t,x})$5. This formula is then applied to linearized equations for Malliavin derivatives of a nonlinear stochastic heat equation driven by nonhomogeneous Gaussian noise, leading to density smoothness and Hölder regularity results (Hu et al., 2011). For the multiplicative fractional-noise heat equation with Hurst parameter $Z_s^{t,x}=\sigma^\top(X_s^{t,x})D_xv(s,X_s^{t,x})$6,

$Z_s^{t,x}=\sigma^\top(X_s^{t,x})D_xv(s,X_s^{t,x})$7

the generalized Feynman–Kac representation becomes

$Z_s^{t,x}=\sigma^\top(X_s^{t,x})D_xv(s,X_s^{t,x})$8

where the stochastic integral is defined by symmetric regularization and Malliavin-calculus arguments; the resulting field is a weak Stratonovich solution (Hu et al., 2010).

Anomalous diffusions yield nonlocal-in-time equations rather than standard parabolic PDEs. If $Z_s^{t,x}=\sigma^\top(X_s^{t,x})D_xv(s,X_s^{t,x})$9 is subordinated by the inverse of a subordinator with Laplace exponent $$\partial_t u+\langle \mu,\nabla_x u\rangle+\tfrac12\operatorname{Tr}\!\big(\sigma\sigma^*\,\text{Hess}_x u\big)+f(t,x,u,\nabla_x u)=0,$$0, and

$$\partial_t u+\langle \mu,\nabla_x u\rangle+\tfrac12\operatorname{Tr}\!\big(\sigma\sigma^*\,\text{Hess}_x u\big)+f(t,x,u,\nabla_x u)=0,$$1

then the joint characteristic function $$\partial_t u+\langle \mu,\nabla_x u\rangle+\tfrac12\operatorname{Tr}\!\big(\sigma\sigma^*\,\text{Hess}_x u\big)+f(t,x,u,\nabla_x u)=0,$$2 satisfies a generalized Feynman–Kac equation with memory kernel $$\partial_t u+\langle \mu,\nabla_x u\rangle+\tfrac12\operatorname{Tr}\!\big(\sigma\sigma^*\,\text{Hess}_x u\big)+f(t,x,u,\nabla_x u)=0,$$3 determined by

$$\partial_t u+\langle \mu,\nabla_x u\rangle+\tfrac12\operatorname{Tr}\!\big(\sigma\sigma^*\,\text{Hess}_x u\big)+f(t,x,u,\nabla_x u)=0,$$4

For time-independent $$\partial_t u+\langle \mu,\nabla_x u\rangle+\tfrac12\operatorname{Tr}\!\big(\sigma\sigma^*\,\text{Hess}_x u\big)+f(t,x,u,\nabla_x u)=0,$$5,

$$\partial_t u+\langle \mu,\nabla_x u\rangle+\tfrac12\operatorname{Tr}\!\big(\sigma\sigma^*\,\text{Hess}_x u\big)+f(t,x,u,\nabla_x u)=0,$$6

and explicit time dependence in $$\partial_t u+\langle \mu,\nabla_x u\rangle+\tfrac12\operatorname{Tr}\!\big(\sigma\sigma^*\,\text{Hess}_x u\big)+f(t,x,u,\nabla_x u)=0,$$7 leads to a coupled nonlocal system (Cairoli et al., 2017).

Finally, some papers use “generalized Feynman–Kac” in a broader, not necessarily nonlinear, sense. Under Hörmander’s condition, one can represent smooth solutions of certain boundary-value problems for degenerate diffusions with boundary stopping, obtaining a generalized Feynman–Kac formula that is linear in the unknown but extends classical elliptic theory to hypoelliptic and boundary-stopped regimes (Karrila et al., 15 Jan 2026). This makes the terminology broader than “BSDE representation of semilinear PDE”.

5. Proof mechanisms and analytic solution concepts

Across these formulations, dynamic programming is a primary mechanism in fully nonlinear Markovian settings. In the sublinear operator framework, the value function

$$\partial_t u+\langle \mu,\nabla_x u\rangle+\tfrac12\operatorname{Tr}\!\big(\sigma\sigma^*\,\text{Hess}_x u\big)+f(t,x,u,\nabla_x u)=0,$$8

satisfies a stopping-time consistency property: restarting the BSDE at $$\partial_t u+\langle \mu,\nabla_x u\rangle+\tfrac12\operatorname{Tr}\!\big(\sigma\sigma^*\,\text{Hess}_x u\big)+f(t,x,u,\nabla_x u)=0,$$9 with terminal value $(v,\nabla_x v)$0 reproduces $(v,\nabla_x v)$1. This dynamic programming principle is then converted into viscosity sub- and supersolution inequalities and coupled with comparison to obtain uniqueness (Pozza, 2019). An analogous abstract DPP underlies the sup-envelope framework on $(v,\nabla_x v)$2 and $(v,\nabla_x v)$3 (Pozza, 2021).

In path-dependent settings, the key analytic language is functional calculus. Peng and Wang define vertical and horizontal derivatives $(v,\nabla_x v)$4, $(v,\nabla_x v)$5, and $(v,\nabla_x v)$6 on path space and then apply the functional Itô formula

$(v,\nabla_x v)$7

to identify BSDE solutions with classical PPDE solutions (Peng et al., 2011). In semimartingale-field SPDEs, Kunita’s generalized Itô formula plays the analogous role (Hu et al., 2011).

Regularity identification often relies on Malliavin techniques. In the stochastic fixed-point approach for semilinear PDEs with gradient dependence, a Bismut–Elworthy–Li formula produces the gradient component: $(v,\nabla_x v)$8 which is then embedded into the stochastic fixed-point equation for $(v,\nabla_x v)$9 (Hutzenthaler et al., 2023). In the SPDE setting of nonhomogeneous Gaussian noise, Malliavin representations of $$\mathcal{L}u(t,x)=\tfrac12 \operatorname{Tr}\!\big(a(t,x)D_x^2u(t,x)\big)+b(t,x)\cdot \nabla_x u(t,x),\qquad a=\sigma\sigma^\top.$$00 are the central tool for proving absolute continuity and $$\mathcal{L}u(t,x)=\tfrac12 \operatorname{Tr}\!\big(a(t,x)D_x^2u(t,x)\big)+b(t,x)\cdot \nabla_x u(t,x),\qquad a=\sigma\sigma^\top.$$01 density of the law (Hu et al., 2011).

The notion of solution varies with the regime. The sublinear and $$\mathcal{L}u(t,x)=\tfrac12 \operatorname{Tr}\!\big(a(t,x)D_x^2u(t,x)\big)+b(t,x)\cdot \nabla_x u(t,x),\qquad a=\sigma\sigma^\top.$$02-expectation papers use viscosity solutions (Hu et al., 2012). Path-space work by Peng and Wang proves existence and uniqueness of a $$\mathcal{L}u(t,x)=\tfrac12 \operatorname{Tr}\!\big(a(t,x)D_x^2u(t,x)\big)+b(t,x)\cdot \nabla_x u(t,x),\qquad a=\sigma\sigma^\top.$$03-solution (Peng et al., 2011). The delayed jump framework uses mild solutions on the lifted Hilbert space $$\mathcal{L}u(t,x)=\tfrac12 \operatorname{Tr}\!\big(a(t,x)D_x^2u(t,x)\big)+b(t,x)\cdot \nabla_x u(t,x),\qquad a=\sigma\sigma^\top.$$04 (Persio et al., 2022). Infinite-dimensional semilinear equations use $$\mathcal{L}u(t,x)=\tfrac12 \operatorname{Tr}\!\big(a(t,x)D_x^2u(t,x)\big)+b(t,x)\cdot \nabla_x u(t,x),\qquad a=\sigma\sigma^\top.$$05-continuous viscosity solutions (Wessels, 2023). Distributional-drift BSDEs require weak Dirichlet processes and mild PDE solutions (Issoglio et al., 2018). Fractional-noise SPDEs use weak Stratonovich solutions (Hu et al., 2010). This diversity is not incidental: each notion is matched to the regularity actually available in the corresponding stochastic representation.

A recurrent structural requirement is comparison. The comparison theorem for viscosity solutions is explicitly central in the sublinear parabolic PDEs with polynomial growth (Pozza, 2019) and in the sup-envelope framework (Pozza, 2021). In the $$\mathcal{L}u(t,x)=\tfrac12 \operatorname{Tr}\!\big(a(t,x)D_x^2u(t,x)\big)+b(t,x)\cdot \nabla_x u(t,x),\qquad a=\sigma\sigma^\top.$$06-Brownian setting, comparison for G-BSDEs underlies the viscosity characterization of the associated fully nonlinear PDE (Hu et al., 2012). In the infinite-dimensional $$\mathcal{L}u(t,x)=\tfrac12 \operatorname{Tr}\!\big(a(t,x)D_x^2u(t,x)\big)+b(t,x)\cdot \nabla_x u(t,x),\qquad a=\sigma\sigma^\top.$$07-continuous framework, uniqueness requires additional monotonicity and continuity assumptions, after which a comparison theorem identifies the BSDE-defined map $$\mathcal{L}u(t,x)=\tfrac12 \operatorname{Tr}\!\big(a(t,x)D_x^2u(t,x)\big)+b(t,x)\cdot \nabla_x u(t,x),\qquad a=\sigma\sigma^\top.$$08 as the unique viscosity solution (Wessels, 2023).

6. Applications, scope, and terminological boundaries

The most immediate applications lie in stochastic control, finance, and robust valuation. Pham’s review emphasizes HJB equations, stochastic control problems, and model uncertainty in finance as natural targets of fully nonlinear Feynman–Kac representations (Pham, 2014). Under volatility uncertainty, the $$\mathcal{L}u(t,x)=\tfrac12 \operatorname{Tr}\!\big(a(t,x)D_x^2u(t,x)\big)+b(t,x)\cdot \nabla_x u(t,x),\qquad a=\sigma\sigma^\top.$$09-conditional expectation of a discounted payoff is shown to solve a nonlinear PDE, which the paper presents as useful for computationally efficient calculation of the corresponding sublinear expectation (Akhtari et al., 2020). In the path-dependent delayed-jump framework, the generalized Feynman–Kac formula is applied to a large investor problem in which the stock price follows a jump-diffusion and depends on the investor’s own wealth path (Persio et al., 2022).

Regularity theory is another major application domain. The semimartingale-driven stochastic heat representation is used to derive explicit formulas for Malliavin derivatives, smoothness of densities, and Hölder continuity of solutions (Hu et al., 2011). The fractional-noise heat equation with $$\mathcal{L}u(t,x)=\tfrac12 \operatorname{Tr}\!\big(a(t,x)D_x^2u(t,x)\big)+b(t,x)\cdot \nabla_x u(t,x),\qquad a=\sigma\sigma^\top.$$10 uses the generalized Feynman–Kac formula to identify a weak Stratonovich solution and to prove temporal Hölder regularity (Hu et al., 2010). In the degenerate Hörmander setting, generalized Feynman–Kac formulas provide smooth solutions to certain PDE boundary-value problems and make Girsanov-transform martingales accessible via Itô calculus (Karrila et al., 15 Jan 2026).

The representation principle also underlies modern numerical work. The stochastic fixed-point formulation for semilinear PDEs with gradient-dependent nonlinearities is explicitly designed to support full-history recursive multilevel Picard methods and to avoid solving a backward SDE directly (Hutzenthaler et al., 2023). A recent extension to nonlinear time-dependent Schrödinger equations states that a novel Feynman–Kac formula integrates both the Fisk-Stratonovich and Itô integrals within a BSDE framework and is then used for learning-based numerical approximations in low- and high-dimensional settings (Cheung et al., 2024).

The literature does not use the label “generalized nonlinear Feynman–Kac formula” uniformly. In some papers it denotes a BSDE or G-BSDE representation of semilinear or fully nonlinear PDEs (Hu et al., 2012). In others it refers to path-dependent, delayed, or nonlocal analytic objects (Wang, 2019). In still others, “generalized” primarily signifies distributional coefficients, rough noise, degenerate diffusions, or boundary stopping, even when the resulting equation is linear in the unknown (Karrila et al., 15 Jan 2026). This suggests that the term names a research program rather than a single canonical formula.

A common misconception is that every nonlinear Feynman–Kac formula is merely a restatement of the semilinear Pardoux–Peng theory. The surveyed work shows otherwise. Full nonlinearity may arise through supremum envelopes or $$\mathcal{L}u(t,x)=\tfrac12 \operatorname{Tr}\!\big(a(t,x)D_x^2u(t,x)\big)+b(t,x)\cdot \nabla_x u(t,x),\qquad a=\sigma\sigma^\top.$$11-operators; non-Markovianity may require path-space calculus or Hilbert-space lifting; singular drivers may force replacement of semimartingales by weak Dirichlet processes; and some representations no longer yield a single PDE but a PPDE, PIDE, or nonlocal evolution equation. Another common misconception is that the stochastic representation automatically yields continuity in all generalized settings; the multivalued Neumann framework explicitly identifies continuity of the Feynman–Kac map as a delicate issue requiring separate proof (Pardoux et al., 2016).

In this broader sense, the generalized nonlinear Feynman–Kac formula is best understood as a unifying probabilistic paradigm: analytic evolution equations with nontrivial dependence on uncertainty, memory, geometry, or singular structure are represented by stochastic systems whose law, backward recursion, or nonlinear expectation matches the analytic operator. The exact stochastic object changes across regimes, but the identifying principle remains the same.