Mansouri, Abdol-Reza

Selk, Zachary

English

LSU Scholarly Repository (Louisiana State Univ.)

Journal of Stochastic Analysis Volume 5 Number 2 Article 3 June 2024 STOCHASTIC SOLUTIONS FOR HYPERBOLIC PDE Abdol-Reza Mansouri Queen's University - Kingston, Ontario, mansouri@queensu.ca Zachary Selk Queen's University - Kingston, Ontario, zachary.selk@queensu.ca Follow this and additional works at: https://repository.lsu.edu/josa  Part of the Analysis Commons, and the Other Mathematics Commons Recommended Citation Mansouri, Abdol-Reza and Selk, Zachary (2024) "STOCHASTIC SOLUTIONS FOR HYPERBOLIC PDE," Journal of Stochastic Analysis: Vol. 5: No. 2, Article 3. DOI: 10.31390/josa.5.2.03 Available at: https://repository.lsu.edu/josa/vol5/iss2/3 STOCHASTIC SOLUTIONS FOR HYPERBOLIC PDEABDOL-REZA MANSOURI AND ZACHARY SELK*Abstract. The theory of stochastic representations of solutions to ellipticand parabolic PDE has been extensive. However, the theory for hyperbolicPDE is notably lacking. In this short note we give a stochastic representationfor solutions of hyperbolic PDE.1. IntroductionThere has been extensive work done on representing solutions of parabolic andelliptic PDE stochastically. For example, one simplified version of the celebratedFeynman-Kac theorem states that the heat equation{∂u∂t (t, x) =12∆u(t, x) on (0,∞)× Rd,u(0, x) = f(x) on {0} × Rd,with f continuous and imposing sufficient decay, has solution E[f(x+B(t))], whereB is a standard d-dimensional Brownian motion.Additionally, for domain D ⊂ Rd Kakutani (see [5, 4]) showed that the Dirichletproblem {∆u = 0 in Du∣∣∂D= fwhere f is a continuous function defined on the boundary ∂D has solution E[f(x+B(τ))] where τ = inf{t : x+B(t) ∈ ∂D} is the exit time of Brownian motion fromthe domain D.However, for hyperbolic PDE such as the wave equation, the literature has beenlacking. In the preprint [1], the author presents a way of generating solutions tohyperbolic PDE by making joint use of the exit time analysis for elliptic PDE andthe specific distribution of Cauchy random variables, thereby converting solutionsof elliptic PDE to solutions of hyperbolic PDE. In particular, the author showsthe following theorem.Theorem 1.1. Let d ≥ 1 and let D be a bounded open connected subset of Rd withboundary ∂D. Let f : R× ∂D → R be any bounded measurable function. Let B bea d-dimensional standard Brownian motion, let Z be a standard Normal randomReceived 2024-1-16; Accepted 2024-6-26; Communicated by the editors.2020 Mathematics Subject Classification. Primary 50H30; Secondary 32W50.Key words and phrases. Feynman-Kac, stochastic analysis, complex analysis, PDE.* Corresponding author.1digitalcommons.lsu.edu/josaJournal of Stochastic Analysis Vol. 5, No. 2 (2024) Article 3 (5 pages) DOI: 10.31390/josa.5.2.032 ABDOL-REZA MANSOURI AND ZACHARY SELKvariable and let X be a standard Cauchy random variable. Denote by τ the exittime of B from set D. Thenu(t, x) = E[f(tX +√τZ, x+B(τ))]is twice continuously differentiable and satisfies the wave equation ∂2u∂t2 = ∆u.The idea behind this representation is that the function v given by v(t, x) :=E[f(t+√τZ,B(t)+x)] solves the d+1 dimensional space-time Laplace equation.The Cauchy random variable then changes the sign of the second time derivative.Although innovative and generating a broad class of solutions stochastically, theauthor comments that this representation has several drawbacks. In particular thelimitations are:(1) This only generates bounded solutions.(2) Even among bounded solutions, this only generates solutions for whichlimt→∞ ∂tu(t, x) = 0.(3) The boundary data is “time evolution on the boundary” which might beseen as unnatural, and further the relationship with this boundary dataand more common forms is unclear.(4) The Cauchy random variable introduces some randomness that might beseen as unnatural.(5) This method provides a representation for solutions only to the wave equa-tion.We also refer to [2, 3, 7] for related approaches. In this note, we use complexanalysis and a version of “Wick rotation” (see [8]) to give a new representation.We assume the following hypotheses.Hypothesis 1.2. Let d ≥ 1 where d ∈ N. Let b : Rd → Rd, σ : Rd → Rd×d beLipschitz, thus implying that the stochastic differential equationdXx0(t) = b(Xx0(t))dt+ σ(Xx0(t))dB(t)with Xx0(0) = x0 has a unique strong solution, and its infinitesimal generator isgiven byL∣∣x=d∑i=1bi(x)∂∂xi+d∑i,j=1ai,j(x)∂2∂xi∂xj,where a(x) = 12σ(x)σT (x), with x ∈ Rd.Hypothesis 1.3. Let D ⊂ Rd be a bounded open connected set with regular bound-ary - that is, the exit time τD := inf{s > 0 : Xx0(s) ∈ ∂D} is 0 with probability 1if x ∈ ∂D.One sufficient condition for Hypothesis 1.3 to hold is that the boundary ∂D isC1. In what follows we denote by C2(C × D̄,C) the set of continuous functionsh : C × D̄ → C that are complex differentiable on C in the first component andtwice real differentiable on D in the second component.Hypothesis 1.4. Let Z be a standard normal random variable independent of Xand let B be a standard Brownian motion independent of X. We define the space-time domain D′ = (0,∞) ×D with boundary ∂D′ = ([0,∞) × ∂D) ∪ ({0} ×D).STOCHASTIC SOLUTIONS 3For each (z, x0) ∈ (C \ {0}) × D̄ we define τ = inf{s > 0 : (z + B(s), Xx0(s)) ∈∂((C \ {0})×D)}, i.e. τ is the first exit time of the process (z+B,Xx0) from theset (C \ {0})×D. Let f ∈ C2(C× D̄,C) be so thatE[|f(z + i√τZ,Xx0(τ))|] < ∞for all (z, x0) ∈ C×D̄. Suppose for all compact sets G ⊂ C there is a gG : C×D̄ →[0,∞), with E[gG(i√τZ,Xx0(τ))] < ∞, such that|∂tf(z + i√τZ,Xx0(τ))| ≤ gG(iτZ,Xx0(τ))for all z ∈ G.Remark 1.5. Note that τ is defined as the exit time of the process (z+B,Xx0) fromthe “partially complexified” space-time domain (C \ {0}) × D in anticipation ofthe complexification of the first independent variable. Note also that the stoppingtime τ is defined as the exit time from (C \ {0}) × D and not from C × D; thisallows us to recover the initial conditions in addition to the boundary conditionsfor the hyperbolic problem.Our theorem is the following.Theorem 1.6. Assume Hypotheses 1.2, 1.3, 1.4 hold. Then one solution to theequation 12∂2u∂t2 = Lu,u∣∣∂D′= f,∂tu∣∣{0}×D = ∂tf(0, x),(1.1)is given byu(t, x) = E[f(t+ i√τZ,Xx(τ))] (1.2)for (t, x) ∈ [0,∞)× D̄ where τ is the stopping time associated to (z, x0) = (it, x) ∈C× D̄.The idea behind this representation is to apply a “Wick rotation”. Wick rota-tion consists in “rotating” time by π/2 by sending t 7→ it. If v(t, x) solves a d+ 1dimensional space-time elliptic PDE then (t, x) 7→ u(t, x) := v(it, x) solves a dspace and 1 time dimensional hyperbolic PDE. The approach in [1] uses a Cauchyrandom variable to accomplish this, but using the imaginary i instead allows usto relax the requirements on f .Our result overcomes some of the limitations of the approach in proposed in[1]. We have no boundedness constraints on the function f , nor do we require thatlimt→∞ ∂tu(t, x) = 0. Furthermore, our representation allows for more naturalinitial and boundary conditions. Our representation additionally allows for moregeneral hyperbolic PDE than the classical wave equation. The only limitation ofour approach is that it does not cover all initial and boundary conditions - justones that can be “unified” in a single function f .Remark 1.7. Given any f : C × D̄ → C and ϕ : D̄ → R with ϕ(x) = 0 if x ∈ ∂Dwe can consider f1(z, x) := f(z, x) + zϕ(x). In this case, the initial velocity is∂tf(0, x) + ϕ(x) and the initial position is still f(0, x). Therefore the the initialvelocity is not actually constrained by the initial position, contrary to what thestatement of Theorem 1.6 might suggest.4 ABDOL-REZA MANSOURI AND ZACHARY SELKOne strength of stochastic representations of solutions to PDE is they readilyyield numerical Monte-Carlo methods. However, as in the example given in [1], itis still possible to obtain explicit solutions in some cases.Example 1.8. Let D = (0, 1) ⊂ R and let L = 12∂2∂x2 . Consider the hyperbolicPDE 12∂2u∂t2 =12∂2u∂x2u(t, 0) = etu(t, 1) = et+1∂tu(0, x) = ex(1.3)Letting f(z, x) = ex+z agrees with the boundary and initial conditions. ThenE[f(t+ i√τZ,Xx(τ))] = E[f(t+ i√τZ, x+B(τ))]= E[E[f(t+ i√τZ, x+B(τ))|σ(B(τ))]]= E[E[ex+B(τ)et+i√τZ |σ(B(τ))]]= E[ex+B(τ)E[et+i√τZ |σ(B(τ))]]= ex+tE[eB(τ)−τ/2].As (eB(t)−t/2)t≥0 is a martingale and∣∣eB(t∧τ)−(t∧τ)/2∣∣ ≤ e1 a.s. for all t ≥ 0,Doob’s Optional Stopping Theorem yields E[eB(τ)−τ/2] = 1. Therefore the solu-tion u is given by u(t, x) = ex+t for all (t, x) ∈ [0,∞)× [0, 1]. We emphasize thatthis solution is unbounded, in contrast to the solutions obtainable in [1].2. Proof of Theorem 1.6Proof. The proposed solution u satisfiesu(t, x) = E[f(t+ i√τZ,Xx(τ))]= E[f((it−√τZ)/i,Xx(τ))]:= E[f̂(it−√τZ,Xx(τ))],where f̂(z, x) := f(z/i, x). Note that f̂ ∈ C2(C×D̄,C) as well. Additionally, f̂ en-joys the same integrability conditions as outlined in Hypothesis 1.4. Furthermore,as Z is standard normal and independent of X, u can be written asu(t, x) = E[f̂(it+B(τ), Xx(τ))],where B is a standard Brownian motion independent of X.Define for z ∈ (C\{0}) and x ∈ D the function v(z, x) = E[f̂(z+B(τ), Xx(τ))]where we recall that τ = inf{s > 0 : (z + B(s), Xx(s)) ∈ ∂((C \ {0}) ×D)}. Wewill denote this byv(z, x) := E(z,x)[f̂(X̃(τ))],where X̃ is the d+ 1 dimensional space-time process (B,X). The notation E(z,x)denotes the expectation of X̃ started at the space-time point (z, x) ∈ C ×D. AsX is a Feller process, so is X̃ with generator 12∂2∂z2 + L, where L acts on the xvariable only. As f̂ ∈ C2(C× D̄,C), Lemma 9.2.3 in [6] along with holomorphicitySTOCHASTIC SOLUTIONS 5(recall that if h : C → C is holomorphic, with h(ξ+ iη) = h1(ξ, η)+ ih2(ξ, η) whereξ, η ∈ R, and z = ξ+ iη, then ddzh(z) =∂∂ξh1(ξ, η)+ i∂∂ξh2(ξ, η)) yield that v solvesthe equation12∂2v∂z2+ Lv = 0. (2.1)As u(t, x) = v(it, x), the chain rule and holomorphicity in the first variable thenimply−12∂2u∂t2+ Lu = 0. (2.2)The only thing to check now are the boundary conditions. First, if (t, x) ∈ ∂D′then Hypothesis 1.3 implies that τ = 0 and u(t, x) = f(t, x).Second, by Hypothesis 1.4 we may apply the dominated convergence theoremto interchange differentiation and expectation to get∂tu(t, x) = E[∂tf(t+ i√τZ,Xx(τ))]. (2.3)If t = 0 and x ∈ D then τ = 0 and∂tu(0, x) = E[∂tf(0, x)] = ∂tf(0, x). (2.4)Acknowledgments. The work of Abdol-Reza Mansouri was supported in partby the Natural Sciences and Engineering Research Council of Canada. The au-thors gratefully acknowledge the anonymous referee whose comments have greatlyimproved the presentation of this paper.References1. Chaterjee, S.: Stochastic solutions to the wave equation, arXiv: 1306.2382 (2013).2. Dalang, R. and Mueller, C. and Tribe, R.: A Feynman-Kac-type formula for the deterministicand stochastic wave equations and other P.D.E.’s, Phys. Rev. (2) 360 (2008), 4681–4703.3. Goldstein, S.: On diffusion by discontinuous movements, and on the telegraph equation,Quart. J. Mech. Appl. Math. 4 (1951), 129–156.4. Kakutani, S.: Two-dimensional Brownian motion and harmonic functions, Proc. Imp. Acad.Tokyo 20 (1944), 706–714.5. Kakutani, S.: On Brownian motions in n-space, Proc. Imp. Acad. Tokyo 20 (1944), 648–652.6. Øksendal, B.: Stochastic differential equations: An introduction with applications, Springer-Verlag, Berlin, 2003.7. Wang, Kaizheng: Stochastic representations for the wave equation on graphs and their scalinglimits, J. Math. Anal. Appl. 4 (2017), 808–828.8. Wick, G.: Properties of Bethe-Salpeter wave functions, Phys. Rev. (2) 96 (1954), 1124–1134.Abdol-Reza Mansouri: Department of Mathematics and Statistics, Queen’s Univer-sity, Kingston, ON K7L 3N6, CanadaEmail address: mansouri@queensu.caZachary Selk: Department of Mathematics and Statistics, Queen’s University, King-ston, ON K7L 3N6, CanadaEmail address: zachary.selk@queensu.ca

STOCHASTIC SOLUTIONS FOR HYPERBOLIC PDE

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STOCHASTIC SOLUTIONS FOR HYPERBOLIC PDE

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